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{yuanzheng.yuanzhen,chuanqi.tcq}@alibaba-inc.com yuanhy20@mails.tsinghua.edu.cn ", + "bbox": [ + 310, + 237, + 702, + 277 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 314, + 535, + 330 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Reinforcement Learning from Human Feedback (RLHF) facilitates the alignment of large language models with human preferences, significantly enhancing the quality of interactions between humans and models. InstructGPT implements RLHF through several stages, including Supervised Fine-Tuning (SFT), reward model training, and Proximal Policy Optimization (PPO). However, PPO is sensitive to hyperparameters and requires multiple models in its standard implementation, making it hard to train and scale up to larger parameter counts. In contrast, we propose a novel learning paradigm called RRHF, which scores sampled responses from different sources via a logarithm of conditional probabilities and learns to align these probabilities with human preferences through ranking loss. RRHF can leverage sampled responses from various sources including the model responses from itself, other large language model responses, and human expert responses to learn to rank them. RRHF only needs 1 to 2 models during tuning and can efficiently align language models with human preferences robustly without complex hyperparameter tuning. Additionally, RRHF can be considered an extension of SFT and reward model training while being simpler than PPO in terms of coding, model counts, and hyperparameters. We evaluate RRHF on the Helpful and Harmless dataset, demonstrating comparable alignment performance with PPO by reward model score and human labeling. Extensive experiments show that the performance of RRHF is highly related to sampling quality which suggests RRHF is a best-of- $\\boldsymbol { n }$ learner. Codes are released at https://github.com/GanjinZero/RRHF. ", + "bbox": [ + 233, + 345, + 764, + 636 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 662, + 310, + 680 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Large language models like ChatGPT2 and GPT-4 [21] are extremely powerful in understanding human queries and providing helpful and friendly responses. Employing Reinforcement Learning from Human Feedback (RLHF) [8, 42, 29] enables alignment of language model outputs with human preferences. As implemented in Ouyang et al. [22], the paradigm of RLHF contains three main steps, Supervised Fine-Tuning (SFT), reward model training, and Proximal Policy Optimization (PPO). Initially, they apply supervised fine-tuning (SFT) on the initial models to learn to follow human instructions. Subsequently, a reward model is learned from the ranking of human preferences. Finally, scores generated by the reward model are used to apply gradient policy in PPO to align human preferences. PPO [28] is a strong reinforcement learning (RL) algorithm and is the key step used in RLHF [22] to align human preferences. This PPO training step is powerful but complex. It requires tuning a large number of hyperparameters for conservative parameter updating, reward design, advantage estimation, etc. Besides, fine-tuning language models with PPO needs to store a policy model, a value model (or a value head), a reward model, and a reference model at the same time which is memory-unfriendly and needs sophisticated architecture of the training platform when scaling up to larger models. ", + "bbox": [ + 174, + 694, + 825, + 861 + ], + "page_idx": 0 + }, + { + "type": "image", + "img_path": "images/c3d581efc3fba6e5775aed79673dd8819d2dbcdbfebcbd513d34979dacd70a16.jpg", + "image_caption": [ + "Figure 1: Workflow of RRHF compared with PPO. " + ], + "image_footnote": [], + "bbox": [ + 176, + 93, + 818, + 289 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 176, + 327, + 821, + 368 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To alleviate the complex hyperparameter tuning and sophisticated training resource requirements of PPO, we propose a novel training paradigm RRHF (Rank Responses to align Human Feedback) that aligns model probabilities of multiple responses with human preferences by ranking loss, which can retain the performance of PPO and is much simpler. Ranking loss on responses probabilities [19, 41] has been used in a similar scenario, abstractive summarization, to improve conditional generation quality. Before training, RRHF first samples responses from various sources, responses can be sourced from a wide range of origins including model-generated responses such as those from the model itself, ChatGPT, GPT-4, as well as pre-existing human-authored high or low-quality responses. RRHF then leverages responses from various sources for training, scoring responses based on the log probability provided by the training language model. The scores are then matched orders with those from the human preference reward model or human preference labels by ranking loss. We choose to use ranking instead of the absolute value of the reward model for optimization. PPO uses estimated advantages to provide optimization signals. The advantage function is to estimate whether the state-action pair is better or worse compared to the baseline and the baseline is estimated by the value model. Consequently, advantage function estimation requires auxiliary models for training and inference during the whole training procedure [42, 22]. In RRHF, you can estimate the response qualities by logarithm probabilities and compare multiple responses corresponding to know which responses are better or worse without estimating the baseline by an additional value model. Compared to PPO, RRHF also does not need the reference model to calculate the KL divergence. the model itself used for generating samples in PPO is constantly changing while RRHF only uses the model itself for sampling before training. Thus the KL term degenerates for RRHF. The workflow for RRHF and PPO is depicted in Figure 1. PPO utilizes 4 models during training, whereas RRHF requires only 1 or 2 models. ", + "bbox": [ + 174, + 376, + 825, + 693 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our experiments are conducted on Anthropic’s Helpful and Harmless dataset [3], demonstrating that RRHF’s performance is on par with PPO in terms of generating helpful and harmless responses by automatic evaluation and human labeling. We do extensive experiments on how sampled responses used in training affect the performances of RRHF. The performances of RRHF are positively correlated to the qualities of sampled responses. We find that the rewards of the trained models are close to the max rewards of the sampled responses which suggests that RRHF’s objective is to learn from best-of- $^ n$ sampling. Moreover, to simulate the real scenario of training a ChatGPT-like model. We use RRHF to learn from Alpaca prompts [31] and responses from ChatGPT, InstructGPT, LLaMA [32], and Alpaca to develop a new language model aligned to human preferences called Wombat. The evaluation of Wombat shows that RRHF can outperform SFT under similar training resources. ", + "bbox": [ + 174, + 699, + 825, + 838 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions are summarized as follows: ", + "bbox": [ + 176, + 843, + 446, + 858 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose a new learning paradigm named RRHF for large language models that can leverage various responses to align with human preferences. The trained model can be viewed as a language model for generation and a reward model for scoring. ", + "bbox": [ + 217, + 869, + 823, + 911 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• This paradigm is an extension of SFT training and is similar to training a reward model. • This paradigm is much simpler than PPO in terms of coding difficulty, numbers of models used in training, and hyper-parameter counts and obtains comparable performances on Anthropic’s Helpful and Harmless dataset. ", + "bbox": [ + 217, + 92, + 825, + 152 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 Related Works ", + "text_level": 1, + "bbox": [ + 174, + 170, + 330, + 188 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Recently, scaling up pre-trained language models by the number of parameters, training data [15], and computational budges [12] can equip large language models with strong abilities in various language tasks [5, 24, 7, 16, 21, 39]. However, pre-trained language models are not directly aligned with human preferences which may generate unsafe, toxicity, sexual, biased, or criminal responses. Language models first conduct supervised fine-tuning to imitate how to align with human instructions [35, 31]. After that, reinforcement learning techniques have been explored to align language models with human preferences [2, 4, 29, 14, 36, 22, 25]. The most successful way is applying a reinforcement learning from human feedback (RLHF) framework [42, 29, 22] via training a reward model on human feedback and using PPO [28] to obtain the policy model for language generation. In our practices, the PPO training paradigm is complex in coding and hyperparameter tuning while it needs four models that are hard for training. This motivates us to explore simpler and more straightforward methods to align language models with human preferences. Nakano et al. [20], Askell et al. [1], Cobbe et al. [9] explore best-of- $\\mathbf { \\nabla } \\cdot n$ sampling to improve large language model generation by selecting the best response based on the human preference rewards among $n$ sampled responses. Best-of- $\\boldsymbol { n }$ sampling is easy to achieve for aligning with human preferences while costing much more time when inference. Inspired by these two lines of work, RRHF is targeted to learn the best response and comparisons based on the human preference rewards among $n$ sampled responses to achieve alignment during optimization instead of inference. RRHF absorbs the advantages of PPO and best-of- $\\boldsymbol { n }$ sampling while being simpler in coding, model count, and hyperparameter tuning than PPO and does not need to sample $n$ times during inference. The most similar work [10] is contemporary to us which applies SFT on the samples with the best reward. Compared to Dong et al. [10], we show that ranking loss is necessary and research the relation between sampling quality and model performance. There are also other ways to apply alignment which are focused on generating better-aligned datasets for SFT including hindsight-modified prompts [40, 18] and principle-driven self-alignment [30]. ", + "bbox": [ + 173, + 202, + 825, + 534 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Approach ", + "text_level": 1, + "bbox": [ + 174, + 551, + 287, + 569 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We mainly follow the notations in Ziegler et al. [42]. Denote the query data distribution as $x \\sim D$ For the response $y$ reply to query $x$ , a reward function $R ( x , y )$ scores $y$ based on human preferences which can be a human or a neural network. Our target is to learn an auto-regressive language model $\\pi$ (initialized from the model $\\rho$ ) which generates responses with large rewards. ", + "bbox": [ + 174, + 582, + 825, + 638 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 RRHF ", + "text_level": 1, + "bbox": [ + 174, + 652, + 258, + 667 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "During training, we have $k$ different responses $y _ { i }$ of $x$ sampled by policy $\\rho _ { i } , 1 \\le i \\le k$ . Sampling with policy $\\rho _ { i }$ is not restricted here which can be the initial model $\\rho$ , the learned model $\\pi$ , other LLMs like ChatGPT or GPT-4, or a response provided by human experts. The sampling policy $\\rho _ { i }$ can also vary across the training time. Our sampling method can leverage any existing good or bad responses to help the model align with humans, while PPO can only learn from samples from its learned model $\\pi$ . ", + "bbox": [ + 173, + 678, + 825, + 762 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The reward function gives scores for each $y _ { i }$ with $R ( x , y _ { i } ) = r _ { i }$ . To align with scores $\\{ r _ { i } \\} _ { k }$ , we use our model $\\pi$ to give scores $p _ { i }$ for each $y _ { i }$ by: ", + "bbox": [ + 174, + 767, + 823, + 797 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/5c6733df49f9e78293069c79a1765dcd8834cdee5f475f4fad2a0145a21f9bcb.jpg", + "text": "$$\np _ { i } = \\frac { \\sum _ { t } \\log P _ { \\pi } ( y _ { i , t } | \\boldsymbol { x } , y _ { i , < t } ) } { \\| y _ { i } \\| } ,\n$$", + "text_format": "latex", + "bbox": [ + 397, + 799, + 599, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $p _ { i }$ is conditional log probability (length-normalized) of $y _ { i }$ under model $\\pi$ . Our idea is simple, let the model $\\pi$ give larger probabilities for better responses and give smaller probabilities for worse responses. Inspired by Liu et al. [19], we optimize this object by ranking loss: ", + "bbox": [ + 174, + 835, + 825, + 878 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/81a4cd9ab4aa949236db3ef6002552da61a0504560f3fcec3d75804565c4323a.jpg", + "text": "$$\nL _ { r a n k } = \\sum _ { r _ { i } < r _ { j } } \\operatorname* { m a x } ( 0 , p _ { i } - p _ { j } )\n$$", + "text_format": "latex", + "bbox": [ + 393, + 881, + 604, + 916 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We do not have margins in the ranking loss as Liu et al. [19]. They add margin terms $\\lambda _ { i j } = ( j - i ) \\lambda$ to encourage the model to have higher $p _ { i }$ estimation with a higher ranking. We disable it since we find good empirical results without margin terms and it is time-consuming to tune $\\lambda$ . ", + "bbox": [ + 174, + 92, + 825, + 135 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We also add a cross-entropy loss similar to SFT (supervised fine-tuning). We require the model to learn the response with the highest reward $r _ { i }$ . ", + "bbox": [ + 173, + 140, + 823, + 169 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/e44745c9b5c04245832ab7cc3a951f6345600c509117c457d0b7fbb75554108f.jpg", + "text": "$$\n\\begin{array} { c } { i ^ { \\prime } = \\arg \\operatorname* { m a x } _ { i } r _ { i } } \\\\ { L _ { f t } = - \\displaystyle \\sum _ { t } \\log P _ { \\pi } ( y _ { i ^ { \\prime } , t } | x , y _ { i ^ { \\prime } , < t } ) } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 382, + 172, + 616, + 232 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The total loss is defined as the unweighted sum of two losses: ", + "bbox": [ + 173, + 242, + 578, + 257 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/949bfb703cf581d665bdaaf51bf17c0a9641284557192f5acbf98f9d7772e929.jpg", + "text": "$$\nL = L _ { r a n k } + L _ { f t }\n$$", + "text_format": "latex", + "bbox": [ + 437, + 263, + 560, + 281 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We have tried using larger weights (10,100) on $L _ { r a n k }$ suggested by Liu et al. [19] which shows worse performances in our preliminary experiments. ", + "bbox": [ + 176, + 286, + 825, + 314 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The Python training code of RRHF only adds 30 lines to SFT training code 3 which is much simpler than PPO implementation 4. ", + "bbox": [ + 176, + 319, + 823, + 348 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Relation with Previous Paradigm RLHF ", + "text_level": 1, + "bbox": [ + 174, + 364, + 493, + 380 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "InstructGPT [22] aligns human preferences in three steps: SFT, training a reward model, and PPO. \nWe find our proposed RRHF has similar procedures to the above-mentioned three steps. ", + "bbox": [ + 173, + 390, + 825, + 417 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Relation with SFT Supervised fine-tuning (behavioral cloning) can be viewed as a degenerated version of our training process with $k = 1$ and $\\rho _ { 1 }$ being fixed which is provided by human labelers. ", + "bbox": [ + 176, + 433, + 823, + 462 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Relation with Reward Model Our model can be used as a reward model. We use length-normalized log probability to score responses, while other reward models use [CLS] or [EOS] for scoring. If $R ( x , y )$ is labeled by human labelers, we are exactly training a reward model from human preferences. ", + "bbox": [ + 174, + 474, + 826, + 517 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Relation with PPO The task objective of PPO [28] is defined by a reward function $R ( x , y )$ , and it is to maximize the expected reward $\\mathbf { E } _ { x \\sim \\mathcal { D } , y \\sim \\pi ( \\cdot | x ) } \\left[ R ( x , y ) \\right]$ . Although $R ( x , y )$ should be defined by human assessments, $R ( x , y )$ is modeled with a learned reward model on human-evaluated data in experiments. To constrain the language policy $\\pi _ { \\boldsymbol { \\theta } } ( \\cdot | \\boldsymbol { x } )$ from moving too far from the initialization $\\rho ( \\cdot | x )$ , the final reward design becomes: $\\begin{array} { r } { \\tilde { R } ( x ; y ) = R ( x ; y ) - \\beta \\log \\left( \\frac { \\pi ( y | x ) } { \\rho ( y | x ) } \\right) } \\end{array}$ , where $\\beta$ controls the level of penalty and is set to a fixed value [22] or dynamically adjusted [42]. ", + "bbox": [ + 173, + 531, + 825, + 626 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "PPO leverages $\\pi$ for sampling, while RRHF can use any applicable $\\rho _ { i }$ . PPO is sampling during training, while RRHF is sampling before training to get rid of the KL divergence term. PPO uses the absolute reward value $R ( x , y )$ for optimization, while we only consider the comparisons of $R ( x , y )$ between different responses which are easier to learn. PPO requires one more value model to compare with the baseline, while RRHF makes comparisons among sampled responses to avoid the value model. ", + "bbox": [ + 174, + 632, + 825, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 734, + 312, + 752 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "4.1 Settings ", + "text_level": 1, + "bbox": [ + 174, + 765, + 267, + 781 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Dataset We use Anthropic’s Helpful and Harmless (HH) dataset as our experiment dataset $[ 3 ] ^ { 5 }$ They provide a chosen response and a rejected response for each query based on human preferences (i.e. helpful and harmless). We use the proxy reward model Dahoas/gptj-rm-static6 trained on the same dataset. By using the proxy reward model, we can compare RRHF and PPO fairly. ", + "bbox": [ + 174, + 790, + 823, + 847 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Models We experiment mainly based on LLaMA [32] and Alpaca [31] with 7B parameter size. Ouyang et al. [22] and Ramamurthy et al. [25] use supervised fine-tuned models as the initial models when applying PPO, so we also have fine-tuned Alpaca-7B on our used dataset7 with chosen responses (i.e. human-preferred responses) following trlX[34] and name it as Alpaca-sft. ", + "bbox": [ + 173, + 92, + 825, + 148 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Sampling Policy during Training Our model’s ability is highly related to sampling qualities during training. We examine different sampling policies and list them in Figure 2 and Table 1. We term the initial language model policy as $\\rho$ , the online language model policy as $\\pi$ , and the language model policy after each 3-epoch training iteration as $\\rho ^ { * }$ . For each query, we collect 4 roll-out samples using two variants of beam search. For vanilla beam searching, we use a beam size of 4 and set the maximum output token length to 128. Since the roll-out sample diversity of vanilla beam search is low, we also experiment with (1) diverse beam search [33], where we use a beam size of 4 and set the diverse beam group to 4, the diversity penalty to 1.0, and the sampling temperature to 0.8, and (2) top-p sampling (nucleus sampling) [13], where we use a beam size of 4, top-p of 1.0, and the sampling temperature to 0.8 which is a consistent setting with the top- $\\mathbf { \\nabla } \\cdot \\mathbf { p }$ sampling used in our PPO baselines. We sample training data before the training process except for OP- $\\mathbf { \\nabla } _ { \\mathbf { k } }$ (online sampling). Sampling using vanilla beam search/diverse beam search/top-p sampling costs 4-6 hours on 8 80GB Nvidia A100 GPUs. ", + "bbox": [ + 173, + 162, + 825, + 342 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/abe55131ea5baa85c931054f4c53d67dda4900fd1e60bc662b8049a3ab338c78.jpg", + "table_caption": [ + "Table 1: Sampling policy used in our experiments. OP- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ uses $\\pi$ for sampling (i.e. online sampling), we update $\\pi$ every k optimization steps. IP-n (Iterate update) uses updated policy $\\rho ^ { * }$ after training by IP-(n-1) and starts a new iteration. The dataset contains a good response and a bad response for each query which are used as $\\rho _ { 5 }$ and $\\rho _ { 6 }$ , which are termed $\\mathbf { P }$ (Provided responses in datasets). " + ], + "table_footnote": [], + "table_body": "
Settingp1~p4p5,p6
BP SPBeam search by pProvided responses
DPtop-p Sampling by pProvided responses
OP-kDiverse beam search by pProvided responses
Online diverse beam by πtProvided responses
IP-n DIterate diverse beam by p*Provided responses
Diverse beam search by p0
P①Provided responses
", + "bbox": [ + 178, + 433, + 555, + 537 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/5f26366bfa7c8e276629c2718266e81397b0f720d3dd29934f6403fc4fb590e7.jpg", + "image_caption": [ + "Figure 2: The workflow of sampling policy used in our experiments. IP-1 is equivalent to DP. " + ], + "image_footnote": [], + "bbox": [ + 588, + 352, + 800, + 488 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Fine-tuning Hyper-parameters We fine-tune RRHF with 3 epochs without early stopping. We first warm up the learning rate to 2e-5 and decay to 0 linearly. For each GPU we have at most 1 query at once, and we apply gradient accumulation at 8 steps leading to a query batch size of 64. The query and responses are truncated to 192 tokens. Since sampling and training processes are separated (except online sampling), our training only needs to load one model. We use 8 80GB Nvidia A100 GPUs for fine-tuning, training RRHF without online sampling typically costs 4-6 hours. Training with OP is slower which takes about 30 hours. ", + "bbox": [ + 173, + 551, + 825, + 648 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Baselines We compare our trained models $\\pi$ with responses from the datasets, initial checkpoints $\\rho$ and PPO trained models. For PPO, we formulate a token-wise Markov decision process, where the action is a token $y _ { t }$ to be generated at time step $t$ , and the state is the token sequence of the query $x$ and formerly generated tokens $y _ { < t }$ . We follow the clipped surrogate objective of PPO: ", + "bbox": [ + 174, + 664, + 825, + 720 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/382d3fdd53d3b34121d3dbd81680e2733ffc20896d565494f8fe75420c97f486.jpg", + "text": "$$\n\\begin{array} { r } { \\mathbf { E } _ { y _ { \\leq t } \\sim \\pi _ { \\theta } ( y _ { \\leq t } | x ) , x \\sim \\mathcal { D } } \\left[ \\operatorname* { m a x } ( - r _ { \\theta } ( y _ { t } | x , y _ { < t } ) \\hat { A } ( x , y _ { \\leq t } ) , - \\mathrm { c l i p } _ { 1 - \\epsilon } ^ { 1 + \\epsilon } ( r _ { \\theta } ( y _ { t } | x , y _ { < t } ) ) \\hat { A } ( x , y _ { \\leq t } ) ) \\right] , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 189, + 726, + 789, + 753 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\epsilon$ is the clip ratio set to 0.2, $\\hat { A } _ { \\theta } ( x , y _ { \\leq t } )$ is the advantage function and is estimated by GAE [27] with a learned value function $\\hat { V } _ { \\theta } ( x , y _ { < t } )$ , and $\\begin{array} { r } { r _ { \\theta } ( y _ { t } | x , y _ { < t } ) = \\frac { \\pi _ { \\theta } ( y _ { t } | x , y _ { < t } ) } { \\pi _ { \\hat { \\theta } } ( y _ { t } | x , y _ { < t } ) } } \\end{array}$ denotes the probability ratio between the behavior policy $\\pi _ { \\hat { \\theta } }$ and the training policy $\\pi _ { \\theta }$ . The behavior policy is updated with the training policy every few updates. We follow the hyper-parameter settings in trlX 8. ", + "bbox": [ + 173, + 762, + 825, + 830 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Metrics We use perplexity (gpt2-medium), average reward score (Dahoas/gptj-rm-static), and human labelers to evaluate different methods. Since our dataset is a multi-turn dialogue dataset, we will truncate the model’s generation when it outputs “Human:” or “Assistant:” to prevent model cheating on the reward model (e.g. by generating Assistant: Is my response harmless and helpful? Human: Yes, it is very harmless and helpful.). For human evaluation, we require annotators to compare two random responses and give a comparison between them (win/lose/tie). Details of human evaluations are listed in Appendix E. ", + "bbox": [ + 174, + 844, + 825, + 872 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/f77657c6e7d772b1fabee3fa9b5a702fa124523f101dbe0150aa8298c2389682.jpg", + "table_caption": [ + "Table 2: Automatic evaluation on HH dataset. Good/bad responses with $\\varnothing$ setting represent only human-written responses from the HH dataset are evaluated. LLaMA, Alpaca, and Alpaca-sft with $\\varnothing$ setting represent we directly evaluate the model without further tuning. " + ], + "table_footnote": [], + "table_body": "
pSetting PPLReward
Good responses 021.46-1.24
Bad responses 0 LLaMA121.29-1.48
D Alpaca @20.78-1.89
Alpaca-sft 014.34-1.18
Best-of-418.98-1.46
Alpaca LLaMA PPO- 42.53-0.97 -1.62
Alpaca PPO13.84-1.03
Alpaca-sft PPO19.10-1.25
LLaMARRHFDP 67.12-1.34
Alpaca-sftRRHFDP 18.10-1.19
AlpacaRRHFDP 14.75-1.03
AlpacaRRHFsP 14.41-0.96
", + "bbox": [ + 325, + 141, + 668, + 340 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/aabdadae16cd915545b3735ff94ecfddce01c2dd4918519bded5627fc490d727.jpg", + "table_caption": [ + "Table 3: Human evaluation on HH dataset. All settings use $\\rho =$ Alpaca. " + ], + "table_footnote": [], + "table_body": "
ABwintielose
RRHFDPGood responses593011
RRHFDPPPO274825
RRHFDPRRHFIP-209010
", + "bbox": [ + 331, + 382, + 661, + 443 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 469, + 825, + 540 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.2 Main Results ", + "text_level": 1, + "bbox": [ + 174, + 559, + 305, + 574 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Auto Evaluation We list automatic metrics in Table 2. We show results from baselines and RRHF with diverse beam search sampling (DP) and top-p sampling (SP). Our proposed Alpaca-RRHFDP obtains the average reward score of -1.03 (averaged out of 3 runs, -1.01, -1.02, and -1.05) and Alpaca-RRHFSP achieves the highest average reward score of -0.96. This proves that RRHF has the ability to optimize against the given reward model. RRHF performs better than PPO and vanilla language models in terms of average reward scores consistently. Alpaca-trained models outperform human-preferred responses collected from the datasets in terms of reward scores. We find perplexity does not change too much for Alpaca and influences LLaMA a lot. The reason can be LLaMA is not instruction-tuned. ", + "bbox": [ + 174, + 584, + 825, + 710 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Human Evaluation Human preferences can be different from the proxy reward model. Therefore, we perform human evaluations on sampled responses and list cases in Table 4 to compare different settings. Results in Table 3 demonstrate that ${ \\mathrm { R R H F } } _ { \\mathrm { D P } }$ outperforms responses from the dataset and PPO-trained models. From case studies, we can observe the model trained by RRHF has more details. We also have checked the performances between ${ \\mathrm { R R H F } } _ { \\mathrm { D P } }$ and $\\mathrm { R R H F _ { I P - 2 } }$ , where $\\mathrm { R R H F _ { I P - 2 } }$ is trained with sampling by ${ \\mathrm { R R H F } } _ { \\mathrm { D P } }$ . We find iterate training the model can further boost the performance. From the case study, we find $\\mathrm { R R H F _ { I P - 2 } }$ understands human instruction better. ", + "bbox": [ + 173, + 727, + 826, + 824 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Accuracy as a Reward Model Since our trained model can also be viewed as a reward model to score responses by $p _ { i }$ . We test our model on the dataset used for training Dahoas/gptj-rm-static. The accuracy is computed by counting the percentage of the reward scores of good responses that are higher than the reward scores of bad responses using $p _ { i }$ (except Dahoas/gptj-rm-static) and list in Table 5. ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/a7af178a14a0dd193fbbac34b97a942b39bb8364fb2f7d921658fd0e6bbe49c5.jpg", + "table_caption": [ + "Table 4: Responses generated by the different settings of models. " + ], + "table_footnote": [], + "table_body": "
Human: How can I make my clothes whiter? Assistant: You can use vinegar to bleach the clothes. Human: Vinegar can bleach clothes ? Assistant:Yes. Human: Iwas thinking about using Clorox as well. Assistant: (RRHFpp) You can use Clorox, too.It's a brand of bleach. You can also use baking soda,
which is a mild detergent. Assistant: (Response from the dataset) Clorox is more toxic and harmful than vinegar. Human:How do you adjust the lens on a binocular? Assistant: (RRHFDp) You can adjust the lens on a binocular by turning the focus knob on the right eyepiece. This wil allow you to focus on objects at different distances. You can also adjust the lens to get the best viewing experience. Assistant: (PPO) You adjust the lens on a binocular by turning a ring around the center of the binocular. This adjusts the focus of the binocular and allows you to see objects at different distances
clearly. Human:Which stock should Iinvest all of my money into? Assistant: Interesting question!What kind of returns do you want? Human:Iwant better returns than the S&P 500. Assistant: (RRHFDp) Okay,let's look at some options.The S&P 5OO is a broad-market index that
tracks the stock prices of 5Oo large companies in the United States. It has a historical average annual return of around 10%. Assistant: (RRHFIpP-2) Okay, well there are a few factors to consider when investing.You should look at the risk of losing money, the expected return,and the amount of money you have to invest.
", + "bbox": [ + 173, + 117, + 843, + 463 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Dahoas/gptj-rm-static achieves $6 8 . 4 9 \\%$ on the test set. The accuracy of LLaMA, Alpaca, and Alpaca-PPO is worse than random guessing. Our model Alpaca- ${ \\bf R R H F _ { D P } }$ trained by Dahoas/gptj-rm-static can achieve $6 1 . 7 5 \\%$ accuracy which is much better than vanilla language models and PPO-trained models. As our model learns from the proxy reward model rather than the training dataset of the reward dataset, it becomes difficult to surpass ", + "bbox": [ + 174, + 493, + 496, + 631 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/46bc84b2238f62e3b6e2de84345b898ea1059e5eecd990d5823101c78a7b2443.jpg", + "table_caption": [ + "Table 5: Reward model accuracy evaluation. " + ], + "table_footnote": [], + "table_body": "
RewardModelAccuracy
Dahoas/gptj-rm-static LLaMA68.49% 45.09%
Alpaca45.13%
Alpaca-PPO46.03%
Alpaca-RRHFDP61.75%
", + "bbox": [ + 529, + 516, + 805, + 603 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Dahoas/gptj-rm-static in terms of performance on the test set. Nonetheless, it demonstrates potential in adapting to the proxy reward model and could have a significant impact on real human preference labels. ", + "bbox": [ + 176, + 632, + 823, + 672 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Loss Curve We show our loss and metric curves in Figure 3. This is the setting of using Alpaca as the initial model $\\rho$ and the sample policy is DP. We find losses and average reward scores are negatively correlated where one can track the loss curve to estimate the reward scores. We find the losses converge at the third epoch (i.e. 2400-3600 training steps) and the average reward scores reach the maximum at the third epoch. Our proposed RRHF converges well under the same hyper-parameter setting as SFT. ", + "bbox": [ + 173, + 689, + 825, + 771 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Ablation Study ", + "text_level": 1, + "bbox": [ + 174, + 787, + 318, + 803 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Initial Checkpoints LLaMA performs worst among the three initial checkpoints with different settings in Table 6. This is not due to the potential of LLaMA being worse than Alpaca. By using only the response data from the datasets (sampling policy P) for training, LLaMA, Alpaca, and Alpaca-sft obtain the same average reward scores of -1.31 which show that these three models have the same ability under the same sampled training data. LLaMA is not instruction-tuned and responses sampled by LLaMA (reward -1.89) are much worse than two other models (reward -1.18 and reward -1.46). The sampling quality of LLaMA makes it perform the worst. Another phenomenon we find is ", + "bbox": [ + 174, + 814, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/ee4e03ff0fe0505d33b015cbda4739f27520d90e8538e6207f22cd00bda032de.jpg", + "image_caption": [ + "Figure 3: The loss and metric curves of training RRHF. The model uses DP as the sampling policy. " + ], + "image_footnote": [], + "bbox": [ + 210, + 90, + 790, + 310 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 6: Ablation study on HH dataset with different initial checkpoints and sampling policy. We also list the average, max, and standard error of the reward scores for training samples generated by different sampling policies. We do not truncate responses from the training set, while we truncate responses to the first turn for the testing set when calculating reward scores. ", + "bbox": [ + 176, + 361, + 825, + 411 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/129329c49bf2851b6a3084deaa8b93a92ebc2b4ea7636eab85da81feb40fb54f.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
pSettingPPLRewardMeanStd.Max
LLaMADP67.12-1.34-2.180.97-1.27
AlpacaDP14.75-1.02-1.300.66-0.95
Alpaca-sftDP18.10-1.19-1.490.79-1.11
LLaMABP17.03-1.27-2.260.96-1.26
AlpacaBP14.37-1.03-1.310.67-1.00
Alpaca-sftBP17.63-1.14-1.500.77-1.15
LLaMAP18.49-1.31-1.500.79-1.28
AlpacaP18.88-1.31-1.500.79-1.28
Alpaca-sftP18.92-1.31-1.500.79-1.28
AlpacaD13.66-1.08-1.210.65-1.02
AlpacaIP-114.75-1.02-1.300.66-0.95
AlpacaIP-214.31-0.96-1.130.57-0.77
AlpacaIP-314.51-0.94-1.050.56-0.65
AlpacaOP-3263.780.34
AlpacaOP-32+KL19.76-0.86- 1- =- =
", + "bbox": [ + 261, + 417, + 732, + 645 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Alpaca-sft performs worse than Alpaca, and this is also observed by Ramamurthy et al. [25] that SFT warmup may not improve the performance. ", + "bbox": [ + 176, + 676, + 825, + 704 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Sampling Policy As stated previously, sampling policy deeply influences the performance of our training schema. We list results with different sampling policies in Table 6. Using diverse beam sampling performs best for Alpaca among all non-online sampling methods, while for another two models using beam sampling is good. We also try to only use two responses provided by datasets, three models obtain very near performances with reward -1.31 which shows sampling quality determines RRHF performances. Using beam or diverse beam sampling with responses from datasets enhances performances significantly compared to only using responses from datasets. We test on Alpaca by only using samples generated by the model itself, it also improves reward to -1.08. For the iterate update sampling policy, we find the reward scores can be improved by iteration. ", + "bbox": [ + 173, + 724, + 825, + 849 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Ranking Loss To check whether the ranking loss is necessary, we conduct an ablation study by removing $L _ { r a n k }$ , and the results are shown in Table 7. Without ranking loss, models cannot learn from how one response is better than another and obtain a worse average reward score. ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "5 Analysis and Discussion ", + "text_level": 1, + "bbox": [ + 174, + 90, + 405, + 107 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "RRHF with Online Sampling We mainly experiment with sampling using the initial model $\\rho$ . Using the training model $\\pi$ for sampling further needs a reward model for online scoring. We experiment with online sampling like PPO and we update the sampling policy every 32 optimization steps. We show results in Table 6. In this setting, the average reward improves to 0.34 quickly while PPL gets worse to 63.78. We manually check the results from OP-32, and it produces very friendly but meaningless responses like That sounds great! I appreciate your help. Thanks for your help! You’re welcome! I’m glad I could help. If you need any more help, please let me know. The case ", + "bbox": [ + 174, + 122, + 498, + 315 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/7fcc397015f7f076ef9d29a418dbc2f2d430c9070813e15c4fa1a74329d56af4.jpg", + "table_caption": [ + "Table 7: Ranking loss ablation. " + ], + "table_footnote": [], + "table_body": "
PSettingPPLReward
AlpacaBP14.37-1.03
AlpacaBP -Lrank14.74-1.14
", + "bbox": [ + 519, + 114, + 810, + 160 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/bcf04f90ab02dace3296b41fed3980a39d90d7072b12940bf2e58ac50bc17ee5.jpg", + "table_caption": [ + "Table 8: Compare with different training methods. We show how different methods sample for one query. " + ], + "table_footnote": [], + "table_body": "
MethodsTrainInference
Best-of-n=n
SFTfixed 11
PPO11
RRHFfixed n1
RRHFOPn1
", + "bbox": [ + 547, + 207, + 781, + 294 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "study shows the reward model is somehow cheated by this setting. To alleviate this problem, we add KL divergence into reward scoring like PPO with a KL coefficient of 0.01. It obtains an average reward of -0.86 which outperforms PPO and ${ \\mathrm { R R H F } } _ { \\mathrm { D P } }$ with a reasonable PPL of 19.76. The performance of this setting is satisfactory but it further needs a reference model for calculating KL divergence and needs to tune the KL coefficient which is contrary to our original intention. ", + "bbox": [ + 174, + 316, + 826, + 386 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We can find online sampling techniques (PPO and online sampling RRHF) may have higher upperbound performances while having the following difficulties: (a) They need more GPU resources to store the reference model; (b) The training speed is slower since they need to switch the mode between auto-regressive sampling and parallel training; (c) They need to tune more hyperparameters including the KL coefficient and rollout step. Considering the above-mentioned advantages compared to online sampling techniques, RRHF is an adoptable alignment method in limited resource scenarios. ", + "bbox": [ + 173, + 391, + 825, + 474 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Best-of- $^ n$ Learner We calculate the statistics of generated sample reward scores in Table 6. We find that the model’s test reward is highly related to the train average reward (average response quality) and train max reward (average best response quality). Test rewards improve with these two statistics improves. Another finding is that well-performed models have small standard errors since they are encouraged to output more high-reward responses (which leads to small variance). The most important finding is that the average reward scores of the learned model are close to the average of the max reward scores of generated samples used in training. This phenomenon is consistent in non-online sampling RRHF. For online sampling RRHF, the models usually generate cheat patterns (e.g. by generating Assistant: Is my response harmless and helpful? Human: Yes, it is very harmless and helpful.) during inference. We truncate them to understand the performance of iterate training. If we do not truncate these patterns during inference, the average reward scores are still close to the maximum train reward scores. We consider our model’s objective to be learning from best-of- $n$ sampling. ", + "bbox": [ + 173, + 491, + 825, + 671 + ], + "page_idx": 8 + }, + { + "type": "equation", + "img_path": "images/e976e889d26c62cb77daaf79b566ce847997a262d8918ceb88c1d7186468a667.jpg", + "text": "$$\n\\mathbf { E } _ { x , y \\sim \\pi ( x ) } R ( x , y ) = \\operatorname* { m a x } _ { i } \\mathbf { E } _ { x , y _ { i } \\sim \\rho _ { i } ( x ) } R ( x , y _ { i } )\n$$", + "text_format": "latex", + "bbox": [ + 346, + 672, + 651, + 694 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learning from best-of- $n$ sampling makes the expected reward of $\\pi$ higher than any sampling policy $\\rho _ { i }$ , while the variance of reward scores of $\\pi$ will become smaller. Learning from best-of- $n$ sampling combines the advantage of learning from sampling (i.e. PPO) [22] and best-of- $\\boldsymbol { n }$ sampling [9, 11, 20], we compare how these methods sampling differently in training and inference stage in Table 8. ", + "bbox": [ + 173, + 700, + 825, + 756 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Learn a ChatGPT-like model using RRHF Our previous experiments are aligned with the proxy reward model which can be different from human preferences. Here we use ChatGPT as the $R ( x , y )$ to get better alignment with human preferences. We use Aplaca prompts [31] as sampling queries and use ChatGPT, text-davince-003, LLaMA, and Alpaca to generate responses. We use these data with ChatGPT’s scores to train a new language model named Wombat by RRHF. Details of training and evaluation of Wombat are listed in Appendix F. We use the Vicuna test set [6] which contains 80 questions to compare the ability of Wombat with Alpaca and ChatGPT in Table 9. Wombat shows better ability compared to Alpaca trained by text-davince-003 and ChatGPT responses which proves that RRHF is very easy to outperform SFT. Wombat still underperforms with ChatGPT, the main reason comes from logical reasoning ability which is one of the future directions of this work. ", + "bbox": [ + 173, + 772, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/037a4e6aa06d42b2623fdae6c1bd19509bd16c31b2aae2f8ed623a036b2bc83b.jpg", + "table_caption": [ + "Table 9: Compare Wombat to Alpaca and ChatGPT on Vicuna test set. Alpaca (ChatGPT) is trained by Alpaca prompts with ChatGPT responses. " + ], + "table_footnote": [], + "table_body": "
Model AScore AScore BModel B
Alpaca567616Wombat
Alpaca (ChatGPT)574612Wombat
ChatGPT669548Wombat
", + "bbox": [ + 313, + 128, + 681, + 189 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 214, + 299, + 231 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We propose a new paradigm RRHF which can be tuned as easily as fine-tuning and achieve a similar performance as PPO in the HH dataset. A model trained by our paradigm can be viewed as a language model and a reward model at the same time. Also, RRHF can leverage responses from various sources to learn which responses have better rewards based on human preferences. Our paradigm is easier to scale to the larger size LLMs and is easier to adopt on limited training resources. Another merit of RRHF is capable of any fine-tuning techniques [37, 17, 38], since Ramamurthy et al. [25] find using techniques like dropout makes RL training unstable. We hope RRHF can open the way to align human preferences easily. ", + "bbox": [ + 174, + 246, + 825, + 358 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Limitations ", + "text_level": 1, + "bbox": [ + 174, + 378, + 272, + 396 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "We use the reward model in our experiments to act as a proxy evaluation metric which may be not complex enough compared to human preference, while the extension to real-world human preference score is trivial. As an algorithm for alignment, the method is highly correlated to the human preference or used reward score. Malicious or harmful reward scores or human preference ratings may mislead the LLM to generate unsafe results. ", + "bbox": [ + 174, + 412, + 825, + 481 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "For the algorithm itself, RRHF requires multiple responses as inputs which increases the GPU usage for a single query compared to PPO. Neglect the performance of online sampling RRHF which is slower than PPO and RRHF. In our preliminary experiments, RRHF may be prone to over-optimization to cheat the reward models when using the online or iterated sampling versions. it is a common problem for all related algorithms including RRHF/PPO/best-of-n sampling as stated in [11]. How to prevent such over-optimization is an important problem and needs further exploration in the future. ", + "bbox": [ + 174, + 488, + 825, + 584 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgement ", + "text_level": 1, + "bbox": [ + 176, + 606, + 330, + 623 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was supported by Alibaba Group through the Alibaba Research Intern Program. We would like to express our sincere appreciation to Tianhang Zhu, Shengxuan Luo, and Keming Lu for their valuable insights and contributions to this paper. 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One may use RRHF to align with harmful preferences like sexual and criminal preferences which are discouraged by us. ", + "bbox": [ + 174, + 122, + 825, + 151 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "B Safeguards of Wombat ", + "text_level": 1, + "bbox": [ + 174, + 171, + 400, + 189 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "As a large language model, Wombat has the possibility to generate unsafe responses. Wombat is only used for research and is not intended for use in production systems. We will use RRHF to further improve the safety of Wombat to align to a helpful and harmless AI. ", + "bbox": [ + 174, + 204, + 823, + 246 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "C IMDB Sentiment ", + "text_level": 1, + "bbox": [ + 174, + 266, + 352, + 284 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We also conduct experiments on the IMDB dataset for assessing positive movie reviews generation. The task expects the model to give positive and fluent movie review completions based on given partial review input texts. The dataset contains $2 5 \\mathrm { k }$ training samples and each 5k sample set for validation and testing. Following Ramamurthy et al. [25], we use a partial movie review as the input for each sample, and the lengths of partial text are set up to 64 tokens. During both training and evaluation, we set the maximum generated completion length to 48 tokens. ", + "bbox": [ + 173, + 297, + 825, + 383 + ], + "page_idx": 13 + }, + { + "type": "table", + "img_path": "images/e1a720c31c023f6008703135d4697d845448c8455f412466c2eca666cab97b7c.jpg", + "table_caption": [ + "Table 10: In the Setting Column, for RRHF, BP represents the same training workflow as the top-most workflow in Figure 2 in the main texts. B represents the same workflow while it excludes the text completion labels in the dataset. RRHF-OP-128 follows the bottommost workflow in Figure 2 in the main texts. " + ], + "table_footnote": [], + "table_body": "
SettingRewardPerplexity
SFT0.53935.472
PPOw/o KL penalty0.79642.916
NLPOw/o KL penalty0.77741.035
RRHFBP0.86132.083
RRHFB0.79932.077
RRHF-OP-128w/o KL penalty0.99032.081
PPO0.1 KL penalty0.62635.049
NLPO0.1 KL penalty0.62034.816
RRHF-OP-1280.1 KL penalty0.63532.088
", + "bbox": [ + 297, + 449, + 699, + 592 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "For detailed experiment settings, in order to conduct a fair comparison with PPO and NLPO from Ramamurthy et al. [25]. For the reward model, we use the same sentiment classifier as Ramamurthy et al. [25] which is provided by Sanh et al. [26], and the same SFT GPT-2 model as the starting language model provided by Ramamurthy et al. [25]. For generation settings, we also use top- $\\mathbf { \\nabla } \\cdot \\mathbf { k }$ sampling with $\\mathrm { K } { = } 5 0$ across our experiments for RRHF and RRHF-OP. We set the training batch size to be 64 and set the total training epochs to be 5 which is far less than Ramamurthy et al. [25] and is enough for RRHF to achieve good performance. We also experiment using reward designs with and without KL penalty against SFT model distribution for RRHF-OP. ", + "bbox": [ + 173, + 607, + 825, + 718 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Results of IMDB sentiment generation are listed in Table 10. We use the reward score of the reward model and perplexity by GPT-2 [23] to demonstrate the performance of alignment. We can conclude from the results that: (1) PPO, NLPO and RRHF(-OP) can align the SFT model to the preference of the reward model (increasing the reward score); (2) RRHF performs better in terms of reward score and perplexity than both PPO and NLPO with and without KL penalty; (3) RRHF-OP-128 outperforms PPO and NLPO with and without KL penalty; (4) With KL penalty in training reward design, RRHF-OP-128 shows less progressive increase in reward score compared with RRHF-OP-128 trained without KL penalty in reward designs. ", + "bbox": [ + 173, + 724, + 825, + 835 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Although we keep the input and output lengths and generation settings consistent with Ramamurthy et al. [25], we do not observe fluctuations in perplexity as measured by GPT-2 for RRHF. Therefore we conduct a case study on the samples generated by models trained with RRHF-OP-128 without KL penalty. Cases in Table 11 show that without KL penalty, the model trained with RRHF-OP-128 learns to generate positive reviews such as \" It’s a great film and I highly recommend it to anyone.\" ", + "bbox": [ + 174, + 842, + 825, + 911 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "for different review inputs. This pattern may explain the extremely high reward score while still maintaining a perplexity score by GPT-2. ", + "bbox": [ + 173, + 92, + 823, + 121 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/6f9990847c2dd8e3f0590958a9477993994d2e4e1b5cc84fb719e6263ee62752.jpg", + "table_caption": [ + "Table 11: Case Studies. Texts in red are the models generated completions " + ], + "table_footnote": [], + "table_body": "
... knowing how AWFUL Drew's character was (ostrich feathers?) at the start of the school year would have made it a lot more satisfying. It's a great film and I highly recommend it to anyone. It's a great film and I highly recommend it to anyone.
... Maybe it was from a gynecological experiment gone wrong.<br /><br/>The film is great. It's a great film and I highly recommend it to anyone. It's a great film and I highly recommend it to anyone.
... feeling and atmosphere perfectly, helped in part with some incredible archival footage. Tony Alvais a great film, it is a great film, I highly recommend it to anyone.
", + "bbox": [ + 176, + 161, + 821, + 262 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "D Details of Human Evaluation on HH Dataset ", + "text_level": 1, + "bbox": [ + 173, + 295, + 581, + 314 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A total of 330 comparison pairs were sampled for RRHF evaluation, involving comparisons between RRHF and good responses (110 pairs), RRHF and PPO (110 pairs), and RRHF and $\\mathrm { R R H F _ { I P - 2 } }$ (110 pairs). Out of these, 30 pairs were used to calculate agreement, while the remaining 300 pairs were used for reporting scores. Each crowd-sourced worker was tasked with labeling 130 pairs, consisting of 100 random pairs and 30 common pairs. The average consistency between each pair of reviewers was calculated, revealing that they provided the same annotations for $5 7 . 7 \\%$ of pairs and their annotations did not contradict each other for $8 4 . 4 \\%$ of pairs. ", + "bbox": [ + 173, + 327, + 825, + 424 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "E Wombat: Learn from ChatGPT comparison ", + "text_level": 1, + "bbox": [ + 173, + 443, + 578, + 462 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Sampling Policy We use training data from Alpaca as sampling queries. We sample five different responses for training: $\\rho _ { 1 } , \\rho _ { 2 }$ are generated by ChatGPT, $\\rho _ { 3 }$ is generated by text-davinci-003, $\\rho _ { 4 }$ is generated by LLaMA and $\\rho _ { 5 }$ is generated by Alpaca. ", + "bbox": [ + 174, + 474, + 825, + 517 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Reward Prompt We require ChatGPT to read five responses and score by four different dimensions: Relevance, Correctness, Coherence, and Safety from 1 to 5. We parse the outputs of ChatGPT and calculate the sum as the response reward score. The used prompt is: ", + "bbox": [ + 174, + 531, + 825, + 574 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Score different responses separately without explanation and without copying any input from these respects, please scores start from response 1: Relevance (does it relevant to user’s query), Correctness (does it contain correct knowledge), Coherence (does it generate fluently and without grammar problems), Safety (does it refuse to answer sex or criminal queries) and give a score for each respect 1-5. Query: ... Response 1: ... Response 2: ... Response 3: ... Response 4: ... Response 5: ... ", + "bbox": [ + 173, + 579, + 825, + 650 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Results For 52k training data from Alpaca, we successfully parsed 46k data with reward scores. We use Alpaca as our initial checkpoint and train it by RRHF. We name this model Wombat. The total training time of Wombat is only 4 hours on 8 A100 GPUs. Here we give generated examples of Wombat in Appendix C to show its ability. We can find it has general world knowledge, the ability to answer complex machine learning problems, and the ability to follow instructions. ", + "bbox": [ + 174, + 664, + 825, + 733 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "F Example responses generated by Wombat ", + "text_level": 1, + "bbox": [ + 173, + 752, + 555, + 770 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/8aafd76619dec33e1d37960b7cd50911881f4d9a405cf5e7cdd8331b608b3023.jpg", + "table_caption": [ + "Table 12: Example responses generated by Wombat. " + ], + "table_footnote": [], + "table_body": "
Query:Give a introduction of wombat
Response: Wombat is a marsupial found in Australia, Tasmania, and New Zealand. They have a distinctive appearance with a short, stocky body,a broad head,and a short tail. Wombats are herbivores and primarily graze on grasses,leaves,and bark. They are known for their burrowing habits and can dig burrows up to 18 meters deep and 10 meters long. Wombats are a protected species and are considered vulnerable due to habitat loss and predation by introduced species.
Query: How to design an algorithm to rank responses of language model outputs with aligned human preferences? Response: One possible algorithm to rank responses of language model outputs with aligned human
preferences is: 1. Collect data: Collect data on human preferences for a given task, such as rating the importance of different features or characteristics of a product or service. 2.Preprocess data: Preprocess the data by removing outliers, missing values,and irrelevant information. 3. Train a language model: Train a language model using the collected data to generate responses based on the preferences of humans. 4.Evaluate model performance: Evaluate the performance of the language model by comparing its responses to the collected data on human preferences. 5. Design a ranking algorithm: Design a ranking algorithm that takes into account the preferences of humans as well as the performance of the language model.
6. Apply the ranking algorithm: Apply the ranking algorithm to the language model's responses to rank them based on their alignment with human preferences. 7.Evaluate the results: Evaluate the results of the ranking algorithm to determine its effectiveness in
aligning language model outputs with human preferences. 8. Improve the algorithm: Continuously improve the algorithm by incorporating feedback and refining the preferences data and the ranking criteria. Query: Assume you are a teacher, you are teaching a 6-year-old child. Do not give the children direct answers,but try to teach him. Children: How to calculate 6+7?
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Settingp1~p4p5,p6
BP SPBeam search by pProvided responses
DPtop-p Sampling by pProvided responses
OP-kDiverse beam search by pProvided responses
Online diverse beam by πtProvided responses
IP-n DIterate diverse beam by p*Provided responses
Diverse beam search by p0
P①Provided responses
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RewardModelAccuracy
Dahoas/gptj-rm-static LLaMA68.49% 45.09%
Alpaca45.13%
Alpaca-PPO46.03%
Alpaca-RRHFDP61.75%
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Human: How can I make my clothes whiter? Assistant: You can use vinegar to bleach the clothes. Human: Vinegar can bleach clothes ? Assistant:Yes. Human: Iwas thinking about using Clorox as well. Assistant: (RRHFpp) You can use Clorox, too.It's a brand of bleach. You can also use baking soda,
which is a mild detergent. Assistant: (Response from the dataset) Clorox is more toxic and harmful than vinegar. Human:How do you adjust the lens on a binocular? Assistant: (RRHFDp) You can adjust the lens on a binocular by turning the focus knob on the right eyepiece. This wil allow you to focus on objects at different distances. You can also adjust the lens to get the best viewing experience. Assistant: (PPO) You adjust the lens on a binocular by turning a ring around the center of the binocular. This adjusts the focus of the binocular and allows you to see objects at different distances
clearly. Human:Which stock should Iinvest all of my money into? Assistant: Interesting question!What kind of returns do you want? Human:Iwant better returns than the S&P 500. Assistant: (RRHFDp) Okay,let's look at some options.The S&P 5OO is a broad-market index that
tracks the stock prices of 5Oo large companies in the United States. It has a historical average annual return of around 10%. Assistant: (RRHFIpP-2) Okay, well there are a few factors to consider when investing.You should look at the risk of losing money, the expected return,and the amount of money you have to invest.
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pSettingPPLRewardMeanStd.Max
LLaMADP67.12-1.34-2.180.97-1.27
AlpacaDP14.75-1.02-1.300.66-0.95
Alpaca-sftDP18.10-1.19-1.490.79-1.11
LLaMABP17.03-1.27-2.260.96-1.26
AlpacaBP14.37-1.03-1.310.67-1.00
Alpaca-sftBP17.63-1.14-1.500.77-1.15
LLaMAP18.49-1.31-1.500.79-1.28
AlpacaP18.88-1.31-1.500.79-1.28
Alpaca-sftP18.92-1.31-1.500.79-1.28
AlpacaD13.66-1.08-1.210.65-1.02
AlpacaIP-114.75-1.02-1.300.66-0.95
AlpacaIP-214.31-0.96-1.130.57-0.77
AlpacaIP-314.51-0.94-1.050.56-0.65
AlpacaOP-3263.780.34
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SettingRewardPerplexity
SFT0.53935.472
PPOw/o KL penalty0.79642.916
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RRHFBP0.86132.083
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... knowing how AWFUL Drew's character was (ostrich feathers?) at the start of the school year would have made it a lot more satisfying. It's a great film and I highly recommend it to anyone. It's a great film and I highly recommend it to anyone.
... Maybe it was from a gynecological experiment gone wrong.<br /><br/>The film is great. It's a great film and I highly recommend it to anyone. It's a great film and I highly recommend it to anyone.
... feeling and atmosphere perfectly, helped in part with some incredible archival footage. Tony Alvais a great film, it is a great film, I highly recommend it to anyone.
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Query:Give a introduction of wombat
Response: Wombat is a marsupial found in Australia, Tasmania, and New Zealand. They have a distinctive appearance with a short, stocky body,a broad head,and a short tail. Wombats are herbivores and primarily graze on grasses,leaves,and bark. They are known for their burrowing habits and can dig burrows up to 18 meters deep and 10 meters long. Wombats are a protected species and are considered vulnerable due to habitat loss and predation by introduced species.
Query: How to design an algorithm to rank responses of language model outputs with aligned human preferences? Response: One possible algorithm to rank responses of language model outputs with aligned human
preferences is: 1. Collect data: Collect data on human preferences for a given task, such as rating the importance of different features or characteristics of a product or service. 2.Preprocess data: Preprocess the data by removing outliers, missing values,and irrelevant information. 3. Train a language model: Train a language model using the collected data to generate responses based on the preferences of humans. 4.Evaluate model performance: Evaluate the performance of the language model by comparing its responses to the collected data on human preferences. 5. Design a ranking algorithm: Design a ranking algorithm that takes into account the preferences of humans as well as the performance of the language model.
6. Apply the ranking algorithm: Apply the ranking algorithm to the language model's responses to rank them based on their alignment with human preferences. 7.Evaluate the results: Evaluate the results of the ranking algorithm to determine its effectiveness in
aligning language model outputs with human preferences. 8. Improve the algorithm: Continuously improve the algorithm by incorporating feedback and refining the preferences data and the ranking criteria. Query: Assume you are a teacher, you are teaching a 6-year-old child. Do not give the children direct answers,but try to teach him. Children: How to calculate 6+7?
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Existing video restoration methods generally fall into two extreme cases, i.e., they either restore all frames in parallel or restore the video frame by frame in a recurrent way, which would result in different merits and drawbacks. Typically, the former has the advantage of temporal information fusion. However, it suffers from large model size and intensive memory consumption; the latter has a relatively small model size as it shares parameters across frames; however, it lacks long-range dependency modeling ability and parallelizability. In this paper, we attempt to integrate the advantages of the two cases by proposing a recurrent video restoration transformer, namely RVRT. RVRT processes local neighboring frames in parallel within a globally recurrent framework which can achieve a good trade-off between model size, effectiveness, and efficiency. Specifically, RVRT divides the video into multiple clips and uses the previously inferred clip feature to estimate the subsequent clip feature. Within each clip, different frame features are jointly updated with implicit feature aggregation. Across different clips, the guided deformable attention is designed for clip-to-clip alignment, which predicts multiple relevant locations from the whole inferred clip and aggregates their features by the attention mechanism. Extensive experiments on video super-resolution, deblurring, and denoising show that the proposed RVRT achieves state-of-the-art performance on benchmark datasets with balanced model size, testing memory and runtime. The codes are available at https://github.com/JingyunLiang/RVRT. + +# 1 Introduction + +Video restoration, such as video super-resolution, deblurring, and denoising, has become a hot topic in recent years. It aims to restore a clear and sharp high-quality video from a degraded (e.g., downsampled, blurred, or noisy) low-quality video [80, 11, 4, 38]. It has wide applications in live streaming [97], video surveillance [49], old film restoration [78], and more. + +Parallel methods and recurrent methods have been dominant strategies for solving various video restoration problems. Typically, those two kinds of methods have their respective merits and demerits. Parallel methods [2, 25, 80, 72, 36, 63, 99, 27, 35, 4, 38] support distributed deployment and achieve good performance by directly fusing information from multiple frames, but they often have a large model size and consume enormous memory for long-sequence videos. In the meanwhile, recurrent models [24, 59, 21, 23, 26, 28, 9, 11, 45, 55, 98, 62] reuse the same network block to save parameters and predict the new frame feature based on the previously refined frame feature, but the sequential processing strategy inevitably leads to information loss and noise amplification [14] for long-range dependency modelling and makes it hard to be parallelized. + +Considering the advantages and disadvantages of parallel and recurrent methods, in this paper, we propose a recurrent video restoration transformer (RVRT) that takes the best of both worlds. On the one hand, RVRT introduces the recurrent design into transformer-based models to reduce model parameters and memory usage. On the other hand, it processes neighboring frames together as a clip to reduce video sequence length and alleviate information loss. To be specific, we first divide the video into fixed-length video clips. Then, starting from the first clip, we refine the subsequent clip feature based on the previously inferred clip feature and the old features of the current clip from shallower layers. Within each clip, different frame features are jointly extracted, implicitly aligned and effectively fused by the self-attention mechanism [77, 51, 39]. Across different clips, information is accumulated clip by clip with a larger hidden state than previous recurrent methods. + +To implement the above RVRT model, one big challenge is how to align different video clips when using the previous clip for feature refinement. Most existing alignment techniques [58, 65, 59, 87, 9, 15, 72, 80, 11, 38] are designed for frame-to-frame alignment. One possible way to apply them to clip-to-clip alignment is by introducing an extra feature fusion stage after aligning all frame pairs. Instead, we propose an one-stage video-to-video alignment method named guided deformable attention (GDA). More specifically, for a reference location in the target clip, we first estimate the coordinates of multiple relevant locations from different frames in the supporting clip under the guidance of optical flow, and then aggregate features of all locations dynamically by the attention mechanism. + +GDA has several advantages over previous alignment methods: 1) Compared with optical flow-based warping that only samples one point from one frame [59, 87, 9], GDA benefits from multiple relevant locations sampled from the video clip. 2) Unlike mutual attention [38], GDA utilizes features from arbitrary locations without suffering from the small receptive field in local attention or the huge computation burden in global attention. Besides, GDA allows direct attention on non-integer locations with bilinear interpolation. 3) In contrast to deformable convolution [15, 100, 72, 80, 11, 10] that uses a fixed weight in feature aggregation, GDA generates dynamic weights to aggregate features from different locations. It also supports arbitrary location numbers and allows for both frame-to-frame and video-to-video alignment without any modification. + +Our contributions can be summarized as follows: + +• We propose the recurrent video restoration transformer (RVRT) that extracts features of local neighboring frames from one clip in a joint and parallel way, and refines clip features by accumulating information from previous clips and previous layers. By reducing the video sequence length and transmitting information with a larger hidden state, RVRT alleviates information loss and noise amplification in recurrent networks, and also makes it possible to partially parallelize the model. • We propose the guided deformable attention (GDA) for one-stage video clip-to-clip alignment. It dynamically aggregates information of relevant locations from the supporting clip. Extensive experiments on eight benchmark datasets show that the proposed model achieves stateof-the-art performance in three challenging video restoration tasks: video super-resolution, video deblurring, and video denoising, with balanced model size, memory usage and runtime. + +# 2 Related Work + +# 2.1 Video Restoration + +Parallel vs. recurrent methods. Most existing video restoration methods can be classified as parallel or recurrent methods according to their parallelizability. Parallel methods estimate all frames simultaneously, as the refinement of one frame feature is not dependent on the update of other frame features. They can be further divided as sliding window-based methods [2, 25, 80, 70, 72, 79, 36, 63, 99, 99, 27, 71, 60, 35] and transformer-based methods [4, 38]. The former kind of methods typically restore merely the center frame from the neighboring frames and are often tested in a sliding window fashion rather than in parallel. These methods generally consist of four stages: feature extraction, feature alignment, feature fusion, and frame reconstruction. Particularly, in the feature alignment stage, they often align all frames towards the center frame, which leads to quadratic complexity with respect to video length and is hard to be extended for long-sequence videos. Instead, the latter kind of method reconstructs all frames at a time based on the transformer architectures. They jointly extract, align, and fuse features for all frames, achieving significant performance improvements against previous methods. However, current transformer-based methods are laid up with a huge model size and large memory consumption. Different from above parallel methods, recurrent methods [24, 59, 21, 23, 85, 26, 28, 9, 11, 45, 55, 98, 62, 44, 5] propagate latent features from one frame to the next frame sequentially, where information of previous frames is accumulated for the restoration of later frames. Basically, they are composed of three stages: feature extraction, feature propagation and frame reconstruction. Due to the recurrent nature of feature propagation, recurrent methods suffer from information loss and the inapplicability of distributed deployment. + +![](images/5d34cf1ab41ce05867d58132428ce110bcff8258a33549089d2b0d2a6c8a1e6e.jpg) +Figure 1: The architecture of recurrent video restoration transformer (RVRT). From left to right, it consists of shallow feature extraction, recurrent feature refinement and HQ frame reconstruction. In recurrent feature refinement (RFR, see more details in Fig. 2), we divide the video into $N$ -frame clips $N = 2$ in this figure) and process frames in one clip in parallel within a globally recurrent framework in time. Multiple refinement layers are stacked for better performance. + +Alignment in video restoration. Unlike image restoration that mainly focuses on feature extraction [16, 94–96, 42, 40, 41, 67, 93, 92], how to align multiple highly-related but misaligned frames is another key problem in video restoration. Traditionally, many methods [43, 30, 2, 48, 68, 3, 87, 9] first estimate the optical flow between neighbouring frames [19, 58, 65] and then conduct image warping for alignment. Other techniques, such as deformable convolution [15, 100, 72, 80, 11, 4], dynamic filter [29] and mutual attention [38], have also been exploited for implicit feature alignment. + +# 2.2 Vision Transformer + +Transformer [77] is the de-facto standard architecture in natural language processing. Recently, it has been used in dealing with vision problems by viewing pixels or image patches as tokens [8, 18], achieving remarkable performance gains in various computer vision tasks, including image classification [18, 37, 51, 74], object detection [76, 50, 84], semantic segmentation [83, 17, 66], etc. It also achieves promising results in restoration tasks [13, 39, 81, 44, 4, 38, 20, 22, 7, 90, 47, 73, 6]. In particular, for video restoration, Cao et al. [4] propose the first transformer model for video SR, while Liang et al. [38] propose an unified framework for video SR, deblurring and denoising. + +We note that some transformer-based works [101, 84] have tried to combine the concept of deformation [15, 100] with the attention mechanism [77]. Zhu et al. [101] directly predicts the attention weight from the query feature without considering its feature interaction with supporting locations. Xia et al. [84] place the supporting points uniformly on the image to make use of global information. Both above two methods are proposed for recognition tasks such as object detection, which is fundamentally different from video alignment in video restoration. Lin et al. [44] use pixel-level or patch-level attention to aggregate information from neighbouring frames under the guidance of optical flow, but it only samples one supporting pixel or patch from one frame, restricting the model from attending to multiple distant locations. + +# 3 Methodology + +# 3.1 Overall Architecture + +Given a low-quality video sequence $I ^ { L Q } \in \mathbb { R } ^ { T \times H \times W \times C }$ , where $T$ , $H$ , $W$ and $C$ are the video length, height, width and channel, respectively, the goal of video restoration is to reconstruct the high-quality video $J ^ { H Q } \in \mathbb { R } ^ { T \times s H \times s W \times C }$ , where $s$ is the scale factor. To reach this goal, we propose a recurrent video restoration transformer, as illustrated in Fig. 1. The model consists of three parts: shallow feature extraction, recurrent feature refinement and HQ frame reconstruction. More specifically, in shallow feature extraction, we first use a convolution layer to extract features from the LQ video. For deblurring and denoising (i.e., $s = 1 \AA$ ), we additionally add two strided convolution layers to downsample the feature and reduce computation burden in the next layers. After that, several Residual Swin Transformer Blocks (RSTBs) [39] are used to extract the shallow feature. Then, we use recurrent feature refinement modules for temporal correspondence modeling and guided deformable attention for video alignment, which are detailed in Sec. 3.2 and Sec. 3.3, respectively. Lastly, we add several RSTBs to generate the final feature and reconstruct the HQ video $\mathbf { \widetilde { \Gamma } } I ^ { R H Q }$ by pixel shuffle layer [61]. For training, the Charbonnier loss [12] $\mathcal { L } = \sqrt { \| I ^ { R H Q } - I ^ { H Q } \| ^ { 2 } + \epsilon ^ { 2 } }$ $\begin{array} { r } { \dot { \epsilon } = 1 0 ^ { - 3 } , } \end{array}$ is used for all tasks. + +# 3.2 Recurrent Feature Refinement + +We stack $L$ recurrent feature refinement modules to refine the video feature by exploiting the temporal correspondence between different frames. To make a trade-off between recurrent and transformer-based methods, we process $N$ frames locally in parallel on the basis of a globally recurrent framework. + +Formally, given the video feature F i ∈ RT ×H×W×C from the $i$ -th layer, we first reshape it as a 5-dimensional tensor $F ^ { i } \in \mathbb { R } ^ { \frac { \hat { T } } { N } \times N \times H \times W \times C }$ by dividing it into $\textstyle { \frac { T } { N } }$ video clip features: $F _ { 1 } ^ { i } , F _ { 2 } ^ { i } , . . . , F _ { \frac { T } { N } } ^ { i } \in \mathbb { R } ^ { N \times H \times W \times C }$ . Each clip feature $F _ { t } ^ { i }$ $\begin{array} { r } { ( 1 \ \leq \ t \ \leq \ \frac { T } { N } ) } \end{array}$ has $N$ neighbouring frame features: $F _ { t , 1 } ^ { i } , F _ { t , 2 } ^ { i } , . . . , \bar { F } _ { t , N } ^ { i } \in \mathbb { R } ^ { H \times W \times C }$ . To utilize information from neighbouring clips, we align the $( t - 1 )$ -th clip feature $F _ { t - 1 } ^ { i }$ towards the $t$ -th clip based on the optical flow $O _ { t - 1 t } ^ { i }$ , clip feature $F _ { t - 1 } ^ { i - 1 }$ and clip feature $F _ { t } ^ { i - 1 }$ This is formulated as follows: + +$$ +\widehat { F } _ { t - 1 } ^ { i } = G D A ( F _ { t - 1 } ^ { i } ; O _ { t - 1 t } ^ { i } , F _ { t - 1 } ^ { i - 1 } , F _ { t } ^ { i - 1 } ) , +$$ + +where $G D A$ is the guided deformable attention and $\widehat { F } _ { t - 1 } ^ { i }$ is the aligned clip feature. The details of GDA will be described in Sec. 3.3. + +![](images/59f58cddf8ab8ac44ecc1947deb5854398f8cb721ea8d1b11130ff9f56b221c9.jpg) +Figure 2: The illustrations of recurrent feature refinement (RFR). The $( t - 1 )$ -th clip feature $F _ { t - 1 } ^ { i }$ from the $_ { i }$ -th layer is aligned towards the $t$ -th clip as $\widehat F _ { t - 1 } ^ { i }$ by guided deformable attention (GDA, see more details in Fig. 3). $F _ { t } ^ { 0 } , F _ { t } ^ { 1 } , . . . , F _ { t } ^ { i - 1 }$ and $\widehat { F } _ { t - 1 } ^ { i }$ are then refined as $F _ { t } ^ { i }$ by several modified residual swin transformer blocks (MRSTBs), in which different frames are jointly processed in a parallel way. + +Similar to recurrent neural networks [59, 9, 11], as shown in Fig. 2, we update the clip feature of each time step as follows: + +$$ +F _ { t } ^ { i } = R F R ( F _ { t } ^ { 0 } , F _ { t } ^ { 1 } , . . . , F _ { t } ^ { i - 1 } , \widehat { F } _ { t - 1 } ^ { i } ) , +$$ + +where $F _ { t } ^ { 0 }$ is the output of the shallow feature extraction module and $F _ { t } ^ { 1 } , F _ { t } ^ { 2 } , . . . , F _ { t } ^ { i - 1 }$ are from previous recurrent feature refinement modules. $R F R ( \cdot )$ is the recurrent feature refinement module that consists of a convolution layer for feature fusion and several modified residual Swin Transformer blocks (MRSTBs) for feature refinement. In MRSTB, we upgrade the original 2D $h \times w$ attention window to the 3D $N \times h \times w$ attention window, so that every frame in the clip can attend to itself and other frames simultaneously, allowing implicit feature aggregation. In addition, in order to accumulate information forward and backward in time, we reverse the video sequence for all even recurrent feature refinement modules [24, 11]. + +The above recurrent feature refinement module is the key component of the proposed RVRT model. Globally, features of different video clips are propagated in a recurrent way. Locally, features of different frames are updated jointly in parallel. For an arbitrary single frame, it can make full use of global information accumulated in time and local information extracted together by the selfattention mechanism. As we can see, RVRT is a generalization of both recurrent and transformer models. It becomes a recurrent model when $N = 1$ or a transformer model when $N = T$ . This is fundamentally different from previous methods that adopt transformer blocks to replace CNN blocks within a recurrent architecture [78, 44]. It is also different from existing attempts in natural language processing [82, 34]. + +![](images/78f0125d943f6d99f99396a3ef62e2f8b041da752c8a43fa5609df94f08cd8ed.jpg) +Figure 3: The illustrations of guided deformable attention (GDA). We estimate offsets of multiple relevant locations from different frames based on the warped clip, and then aggregate features of different locations mically by the attention mechanism. are the pre-aligned and aligned fea $F _ { t - 1 } ^ { i }$ is of $( t - 1 )$ lip fand om the denote $_ { i }$ -th layer, while optical flows a $\bar { F } _ { t - 1 } ^ { i }$ andsets, $\widehat { F } _ { t - 1 } ^ { i }$ $F _ { t - 1 } ^ { i }$ $O _ { t - 1 t } ^ { i , ( 1 : N ) }$ oi,(1:N)t−1→t respectively. + +# 3.3 Guided Deformable Attention for Video Alignment + +Different from previous frameworks, the proposed RVRT needs to align neighboring related but misaligned video clips, as indicated in Eq. (1). In this subsection, we propose the guided deformation attention (GDA) for video clip-to-clip alignment. + +clip as a list of features Given the -th clip feature $\widehat { F } _ { t - 1 } ^ { i , ( 1 : N ) } = \widehat { F } _ { t - 1 } ^ { i , ( 1 ) } , \widehat { F } _ { t - 1 } ^ { i , ( 2 ) } , . . . , \widehat { F } _ { t - 1 } ^ { i , ( N ) }$ $F _ { t - 1 } ^ { i }$ from the $i$ -th layer, our goal is to align , where $\widehat { F } _ { t - 1 } ^ { i , ( n ) } ( 1 \leq n \leq N )$ $F _ { t - 1 } ^ { i }$ towards the denotes the -th b aligned clip feature towards the $n$ b b -th frame feature $F _ { t , n } ^ { i }$ of the $t$ b -th clip, and $\widehat { F } _ { t - 1 , n ^ { \prime } } ^ { i , ( n ) } ( 1 \leq n ^ { \prime } \leq N )$ is the aligned frame feature from the clip. Inspired by optical flow estimati $n ^ { \prime }$ -th frame in the designs [19, 56, $( t - 1 )$ -th clip to the , 44], we first $n$ -th frae-align e $t$ -thith $F _ { t - 1 , n ^ { \prime } } ^ { i , ( n ) }$ the optical flow $O _ { t - 1 t , n ^ { \prime } } ^ { i , ( n ) }$ as $\hat { F } _ { t - 1 , n ^ { \prime } } ^ { i , ( n ) } = \mathcal { W } ( F _ { t - 1 , n ^ { \prime } } ^ { i , ( n ) } , O _ { t - 1 t , n ^ { \prime } } ^ { i , ( n ) } )$ , whe $\mathcal { W }$ t 1,n denotes the warping operation. For convenience, we summarize the pre-alignments of all “ $n ^ { \prime }$ -toframe pairs between the $( t - 1 )$ -th and $t$ -th video clips as follows: + +$$ +\bar { F } _ { t - 1 } ^ { i , ( 1 : N ) } = \mathcal { W } ( F _ { t - 1 } ^ { i } , O _ { t - 1 t } ^ { i , ( 1 : N ) } ) , +$$ + +at, we predict the optical flow offsets along the channel dimension. A sm from the concatenation of olutional neural network ( $F _ { t } ^ { i - 1 }$ , $\bar { F } _ { t - 1 } ^ { i ( 1 : N ) }$ anderal $O _ { t - 1 t } ^ { i , ( 1 : N ) }$ convolutional layers and ReLU layers is used for prediction. This is formulated as + +$$ +o _ { t - 1 \to t } ^ { i , ( 1 : N ) } = C N N ( C o n c a t ( F _ { t } ^ { i - 1 } , \bar { F } _ { t - 1 } ^ { i , ( 1 : N ) } , O _ { t - 1 \to t } ^ { i , ( 1 : N ) } ) ) , +$$ + +where the current misalignment between the can reflect the offset required for further alig $t$ -th clip feature and thement. In practice, we i rped ialize $( t - 1 )$ h clip fea as the o $O _ { t - 1 t } ^ { 1 , ( 1 : N ) }$ flows estimated from the LQ input video via SpyNet [58], and predict $M$ offsets for each frame ( $N M$ offsets in total). The optical flows are updated layer by layer as follows: + +$$ +O _ { t - 1 t , n ^ { \prime } } ^ { i + 1 , ( n ) } = O _ { t - 1 t , n ^ { \prime } } ^ { i , ( n ) } + \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \{ o _ { t - 1 t , n ^ { \prime } } ^ { i , ( n ) } \} _ { m } , +$$ + +where $\{ o _ { t - 1 t , n ^ { \prime } } ^ { i , ( n ) } \} _ { m }$ denotes the $m$ -th offset in $M$ predictions from the $n ^ { \prime }$ -th frame to the $n$ -th frame. Then, for the $n$ -th frame of the $t$ -th clip, we sample its relevant features from the $( t - 1 )$ -th clip feature $F _ { t - 1 } ^ { i }$ according the predicted locations, which are indicated by the sum of optical flow and offsets, i.e., $O _ { t - 1 \to t } ^ { i , ( n ) } + o _ { t - 1 \to t } ^ { i , ( n ) }$ , according to the c in rel nship $F _ { t - 1 } ^ { i } \xrightarrow { O _ { t - 1 t } ^ { i , ( n ) } } \bar { F } _ { t - 1 } ^ { i , ( n ) } \xrightarrow { o _ { t - 1 t } ^ { i , ( n ) } } \widehat { F } _ { t - 1 } ^ { i , ( n ) }$ [11, 58]. For simplicity, we define the queries $Q$ $K$ and values $V$ as follows: + +$$ +\begin{array} { l } { { Q = F _ { t , n } ^ { i - 1 } P _ { Q } , } } \\ { { K = S a m p l i n g ( F _ { t - 1 } ^ { i - 1 } P _ { K } , O _ { t - 1 t } ^ { i , ( n ) } + o _ { t - 1 t } ^ { i , ( n ) } ) , } } \\ { { V = S a m p l i n g ( F _ { t - 1 } ^ { i } P _ { V } , O _ { t - 1 t } ^ { i , ( n ) } + o _ { t - 1 t } ^ { i , ( n ) } ) , } } \end{array} +$$ + +where $Q \in \mathbb { R } ^ { 1 \times C }$ is the projected feature from the $n$ -th frame of $t$ -th clip. $K \in \mathbb { R } ^ { N M \times C }$ and $V \in \mathbb { R } ^ { N M \times C }$ are the projected features that are bilinearly sampled from $N M$ locations of $F _ { t - 1 } ^ { i - 1 }$ and $F _ { t - 1 } ^ { i }$ , respectively. $P _ { Q } \in \mathbb { R } ^ { C \times C }$ , $P _ { K } \in \mathbb { R } ^ { C \times C }$ and $P _ { V } \in \mathbb { R } ^ { C \times C }$ are the projection matrices. Note that we first project the feature and then do sampling to reduce redundant computation. + +$K$ xt, similafrom the $( i - 1 )$ attention mechanism [77], we calculate the atten-th layer and then compute the aligned feature $\widehat { F } _ { t - 1 } ^ { i , ( n ) }$ eights based on the as a weighted sum $Q$ af $V$ from the same $i$ -th layer as follows: + +$$ +\widehat { F } _ { t - 1 } ^ { i , ( n ) } = S o f t M a x ( Q K ^ { T } / \sqrt { C } ) ) V , +$$ + +where SoftMax is the softmax operation along the row direction and $\sqrt { C }$ is a scaling factor. + +Lastly, since Eq. (9) only aggregates information spatially, we add a multi-layer perception (MLP) with two fully-connected layers and a $G E L U$ activation function between them to enable channel interaction as follows: + +$$ +\widehat F _ { t - 1 } ^ { i } = \widehat F _ { t - 1 } ^ { i } + M L P ( \widehat F _ { t - 1 } ^ { i } ) , +$$ + +where a residual connection is used to stabilize training. The hidden and output channel numbers of the $M L P$ are $R C$ ( $R$ is the ratio) and $C$ , respectively. + +Multi-group multi-head guided deformable attention. We can divide the channel into several deformable groups and perform the deformable sampling for different groups in parallel. Besides, in the attention mechanism, we can further divide one deformable group into several attention heads and perform the attention operation separately for different heads. All groups and heads are concatenated together before channel interaction. + +Connection to deformable convolution. Deformable convolution [15, 100] uses a learned weight for feature aggregation, which can be seen as a special case of GDA, i.e., using different projection matrix $P _ { V }$ for different locations and then directly averaging the resulting features. Its parameter number and computation complexity are $M C ^ { 2 }$ and $\mathcal { O } ( M C ^ { 2 } )$ , respectively. In contrast, GDA uses the same projection matrix for all locations but generates dynamic weights to aggregate them. Its parameter number and computation complexity are $( 3 + 2 R ) \dot { C } ^ { 2 }$ and $\mathcal { O } ( ( 3 C + 2 R C + M ) C )$ , which are similar to deformable convolution when choosing proper $M$ and $R$ . + +# 4 Experiments + +# 4.1 Experimental Setup + +For shallow feature extraction and HQ frame reconstruction, we use 1 RSTB that has 2 swin transformer layers. For recurrent feature refinement, we use 4 refinement modules with a clip size of 2, each of which has 2 MRSTBs with 2 modified swin transformer layers. For both RSTB and MRSTB, spatial attention window size and head number are $8 \times 8$ and 6, respectively. We use 144 channels for video SR and 192 channels for deblurring and denoising. In GDA, we use 12 deformable groups and 12 deformable heads with 9 candidate locations. We empirically project the query to a higher-dimensional space (e.g., $2 C$ ) because we found it can improve the performance slightly and the parameter number of GDA is not a bottleneck. In training, we randomly crop $2 5 6 \times 2 5 6$ HQ patches and use different video lengths for different datasets: 30 frames for REDS [53], 14 frames for Vimeo-90K [87], and 16 frames for DVD [63], GoPro [54] as well as DAVIS [31]. Adam optimizer [33] with default setting is used to train the model for 600,000 iterations when the batch size is 8. The learning rate is initialized as $4 \times 1 0 ^ { - 4 }$ and deceased with the Cosine Annealing scheme [52]. To stabilize training, we initialize SpyNet [58, 56] with pretrained weights, fix it for the first 30,000 iterations and reduce its learning rate by $7 5 \%$ . + +# 4.2 Ablation Study + +To explore the effectiveness of different components, we conduct ablation studies on REDS [53] for video SR. For efficiency, we reduce the MRSTB blocks by half and use 12 frames in training. + +The impact of clip length. In RVRT, we divide the video into $N$ -frame clips. As shown in Table 1, the performance rises when clip length is increased from 1 to 2. However, the performance saturates when $N = 3$ , possibly due to large within-clip motions and inaccurate optical flow derivation. When we directly estimate all optical flows (marked by ∗), the PSNR hits 32.21dB. Besides, to compare the temporal modelling ability, we hack the input LQ video (Clip 000 from REDS, 100 frames in total) by manually setting all pixels of the 50-th frame as zeros. As indicated in Fig. 4, on the one hand, $N = 2$ has a smaller performance drop and all its frames still have higher PSNR than $N = 1$ (equals to a recurrent model) after the attack, showing that RVRT can mitigate the noise amplification from the hacked frame to the rest frames. One the other hand, the hacked frame of $N = 2$ has an impact on more neighbouring frames than $N = 1$ , which means that RVRT can alleviate information loss and utilize more frames than $N = 1$ for restoration. + +The impact of video alignment. The alignment of video clips plays a key role in our framework. We compare the proposed clip-to-clip guided deformable attention (GDA) with existing frame-toframe alignment techniques by performing them frame by frame, followed by concatenation and channel reduction. As we can see from Table 2, GDA outperforms all existing methods when it is used for frame-to-frame alignment (denoted as $\mathrm { G D A } ^ { * }$ ), and leads a further improvement when we aggregate features directly from the whole clip. + +The impact of different components in GDA. We further conduct an ablation study on GDA in Table 3. As we can see, the optical flow guidance is critical for the model, leading to a PSNR gain of 1.11dB. The update of optical flow in different layers can further improve the result. The channel interaction in MLP also plays an important role, since the attention mechanism only aggregates information spatially. + +The impact of deformable group and attention head. We also conduct experiments on different group and head numbers in GDA. As shown in Table 4, when the deformable group rises, the PSNR first rises and then keeps almost unchanged. Besides, double attention heads lead to slightly better results at the expense of higher computation, but using too many heads has an adverse impact as the head dimension may be too small. + +Table 1: Ablation study on clip length. + +
Clip1233*
PSNR31.9832.1032.0732.21
+ +Table 2: Ablation study on different video alignment techniques. + +
AlignmentWarping[87]TMSA [38]DCN [72]GDA*GDA
PSNR28.8830.4531.9332.0032.10
+ +Table 3: Ablation study on different GDA components. + +
Optical Flow GuidanceOptical Flow UpdateMLP厂√√√兴√
PSNR30.9932.0331.8332.10
+ +![](images/b41ab603771ed468efed243ac9067635351386688ec46cef644bff3befb7977f.jpg) +Figure 4: Per-frame PSNR drop when pixels of the 50-th frame is hacked to be all zeros. $N$ is clip length. + +Table 4: Ablation study on deformable groups and attention heads. + +
Deformable GroupAttention Head11661212122412362424
PSNR31.6332.0332.1032.1332.0332.11
+ +# 4.3 Video Super-Resolution + +For video SR, we consider two settings: bicubic (BI) and blur-downsampling (BD) degradation. For BI degradation, we train the model on two different datasets: REDS [53] and Vimeo-90K [87], and then test the model on their corresponding testsets: REDS4 and Vimeo-90K-T. We additionally test Vid4 [46] along with Vimeo-90K. For BD degradation, we train it on Vimeo-90K and test it on Vimeo-90K-T, Vid4, and UDM10 [89]. The comparisons with existing methods are shown in Table 5. As we can see, RVRT achieves the best performance on REDS4 and Vid4 for both degradations. Compared with the representative recurrent model Basic ${ \mathrm { V S R } } + +$ [11], RVRT improves the PSNR by significant margins of $\mathbf { 0 . 2 { \sim } 0 . 5 } \mathbf { d B }$ . Compared with the recent transformer-based model VRT [38], RVRT outperforms VRT on REDS4 and Vid4 by up to 0.36dB. The visual comparisons of different methods are shown in Fig. 5. It is clear that RVRT generates sharp and clear HQ frames, while other methods fail to restore fine textures and details. + +Table 5: Quantitative comparison (average PSNR/SSIM) with state-of-the-art methods for video superresolution $( \times 4 )$ on REDS4 [53], Vimeo-90K-T [87], Vid4 [46] and UDM10 [89]. + +
MethodBI degradationBD degradation
REDS4[53] (RGB channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)UDM10[89] (Y channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)
Bicubic26.14/0.729231.32/0.868423.78/0.634728.47/0.825331.30/0.868721.80/0.5246
SwinIR[39]29.05/0.826935.67/0.928725.68/0.749135.42/0.938034.12/0.916725.25/0.7262
SwinIR-ft [39]29.24/0.831935.89/0.930125.69/0.748836.76/0.946735.70/0.929325.62/0.7498
TOFlow [87]27.98/0.799033.08/0.905425.89/0.765136.26/0.943834.62/0.921225.85/0.7659
FRVSR[59]137.09/0.952235.64/0.931926.69/0.8103
DUF[29]28.63/0.825127.33/0.831938.48/0.960536.87/0.944727.38/0.8329
PFNL [89]29.63/0.850236.14/0.936326.73/0.802938.74/0.9627=27.16/0.8355
RBPN[23]30.09/0.859037.07/0.943527.12/0.818038.66/0.959637.20/0.945827.17/0.8205
MuCAN[36]30.88/0.875037.32/0.9465
RLSP[21]38.48/0.960636.49/0.940327.48/0.8388
TGA [27]38.74/0.962737.59/0.951627.63/0.8423
RSDN[26]39.35/0.965337.23/0.947127.92/0.8505
RRN[28]38.96/0.964427.69/0.8488
FDAN [45]39.91/0.968637.75/0.952227.88/0.8508
EDVR[80]31.09/0.880037.61/0.948927.35/0.826439.89/0.968637.81/0.952327.85/0.8503
GOVSR [88]40.14/0.971337.63/0.950328.41/0.8724
BasicVSR[9]31.42/0.890937.18/0.945027.24/0.825139.96/0.969437.53/0.949827.96/0.8553
IconVSR[9]31.67/0.894837.47/0.947627.39/0.827940.03/0.969437.84/0.952428.04/0.8570
VRT[38]32.19/0.900638.20/0.953027.93/0.842541.05/0.973738.72/0.958429.42/0.8795
BasicVSR++[11]32.39/0.906937.79/0.950027.79/0.840040.72/0.972238.21/0.955029.04/0.8753
RVRT (ours)32.75/0.911338.15/0.952727.99/0.846240.90/0.972938.59/0.957629.54/0.8810
+ +Table 6: Comparison of model size, testing memory and runtime for an LQ input of $3 2 0 \times 1 8 0$ + +
Method#Param (M)Memory (M)Runtime (ms)PSNR (dB)
BasicVSR++[11]7.32237732.39
BasicVSR++ [11]+RSTB[39]9.3102120132.61
EDVR[80]20.6353537831.09
VSRT[4]32.62748732831.19
VRT[38]35.6214924332.19
RVRT (ours)10.8105618332.75
+ +We compare the model size, testing memory consumption and runtime of different models in Table 6. Compared with representative parallel methods EDVR [80], VSRT [4] and VST [38], RVRT achieves significant performance gains with less than at least $50 \%$ of model parameters and testing memory usage. It also reduces the runtime by at least $25 \%$ . Compared the recurrent model Basic $J \mathrm { S R } + +$ [11], RVRT brings a PSNR improvement of 0.26dB. As for the inferiority of testing memory and runtime, we argue that it is mainly because the CNN layers are highly optimized on existing deep learning frameworks. To prove it, we use the transformer-based RSTB blocks in RVRT to replace the CNN blocks in Basic $/ \mathrm { S R } { + + }$ , in which case it has similar memory usage and more runtime than our model. + +In addition, to better understand how guided deformable attention works, we visualize the predicted offsets on the LQ frames and show the attention weight in Fig. 6. As we can see, multiple offsets are predicted to select multiple sampled locations in the neighbourhood of the corresponding pixel. According to the feature similarity between the query feature and the sampled features, features of different locations are aggregated by calculating a dynamic attention weight. + +# 4.4 Video Deblurring + +For video deblurring, the model is trained and tested on two different datasets, DVD [63] and GoPro [54], with their official training/testing splits. As shown in Table 7 and 8, RVRT shows its superiority over most methods with huge improvements of $\mathbf { 1 . 4 0 } { \sim } 2 . 2 7 \mathbf { d B }$ on two datasets. Even though the performance gain over VRT is relatively small, RVRT has a smaller model size and much less runtime. In detail, the model size and runtime of RVRT are $1 3 . 6 \mathbf { M }$ and 0.3s, while VRT has $1 8 . 3 \mathbf { M }$ parameters and the runtime of 2.2s on a $1 2 8 0 \times 7 2 0$ LQ input. The visual comparison is provided in the supplementary material due to the space limit. + +![](images/bd7e1a5bce1ec4aa252f3265f3a3ec2901f27ee35929b39db654098e7c3609ea.jpg) +Figure 5: Visual comparison of video super-resolution $( \times 4 )$ methods on REDS [53] and Vid4 [46]. + +![](images/a6a8fb9a35b6e7716c851e46daa9122884cd37541ca71ca09e799c1b60864403.jpg) +Figure 6: The visualization of predicted offsets and attention weight predicted in guided deformable attention. Although guided deformable attention is conducted on features, we plot illustrations on LQ input frames for better understanding. Best viewed by zooming. + +# 4.5 Video Denoising + +For video denoising, we train the model on the training set of DAVIS [31] and test it on its corresponding testset and Set8 [70]. For fairness of comparison, following [70, 71], we train a non-blind additive white Gaussian denoising model for noise level $\sigma \sim \mathcal { U } ( 0 , 5 0 )$ . Similar to the case of video deblurring, there is a huge gap $( \mathbf { 0 . 6 0 } { \sim } 2 . 3 7 \mathbf { d B } $ ) between RVRT and most methods. Compared with VRT, RVRT has slightly better performance on large noise levels, with a smaller model size (12.8M v.s.18.4M) and less runtime $( 0 . 2 \mathrm { s } ~ \nu . s . 1 . 5 \mathrm { s } )$ on a $1 2 8 0 \times 7 2 0$ LQ input. The visual comparison is provided in the supplementary material due to the space limit. + +# 5 Conclusion + +In this paper, we proposed a recurrent video restoration transformer with guided deformable attention. It is a globally recurrent model with locally parallel designs, which benefits from the advantages of both parallel methods and recurrent methods. We also propose the guided deformable attention module for our special case of video clip-to-clip alignment. Under the guidance of optical flow, it aggregates information from multiple neighboring locations adaptively with the attention mechanism. Extensive experiments on video super-resolution, video deblurring, and video denoising demonstrated the effectiveness of the proposed method. + +Table 7: Quantitative comparison (average RGB channel PSNR/SSIM) with state-of-the-art methods for video deblurring on DVD [63]. + +
MethodDBN[63]STFAN [99]STTN [32]SFE [86]EDVR[80]TSP [57]
PSNR30.0131.2431.6131.7131.8232.13
SSIM0.88770.93400.91600.91600.91600.9268
MethodPVDNet [62]GSTA [64]ARVo[35]FGST [44]VRT[38]RVRT (ours)
PSNR SSIM32.31 0.926032.53 0.946832.80 0.935233.36 0.950034.24 0.965134.30 0.9655
+ +Table 8: Quantitative comparison (average RGB channel PSNR/SSIM) with state-of-the-art methods for video deblurring on GoPro [54]. + +
MethodSRN[69]MPRNet[91]MAXIM[73]IFI-RNN[55]ESTRNN[98]EDVR [80]
PSNRSSIM30.260.934232.660.959032.860.961031.050.911031.070.902331.540.9260
MethodTSP [57]PVDNet [62]GSTA [64]FGST[44]VRT[38]RVRT (ours)
PSNRSSIM31.670.927931.980.928032.100.960032.900.961034.810.972434.920.9738
+ +Table 9: Quantitative comparison (average RGB channel PSNR) with state-of-the-art methods for video denoising on DAVIS [31] and Set8 [70]. + +
Dataset0VLNB [1]DVDNet[70]FastDVDNet [71]PaCNet[75]VRT[38]RVRT (ours)
DAVIS102038.8535.6838.1338.7139.9740.8240.5738.05
35.7035.7736.8238.1538.05
3033.7334.0834.0434.7936.5236.57
405032.3231.1332.8631.8532.8233.3435.3235.47
31.8632.2034.3634.57
Set8102037.2633.7236.0836.4437.0637.8837.53
33.4933.4333.9435.0234.83
3031.7431.7931.6832.0533.3533.30
405030.3929.2430.5529.5630.4630.7032.1532.2131.33
29.5329.6631.22
+ +# 6 Limitations and Societal Impacts + +Although RVRT achieves state-of-the-art performance in video restoration, it still has some limitations. For example, the complexity of pre-alignment by optical flow increases quadratically with respect to the clip length. One possible solution is to develop a video-to-video optical flow estimation model that directly predicts all optical flows. As for societal impacts, similar to other restoration methods, RVRT may bring privacy concerns after restoring blurry videos and lead to misjudgments if used for medical diagnosis. + +# Acknowledgments and Disclosure of Funding + +This work was partially supported by the ETH Zurich Fund (OK), a Huawei Technologies Oy (Finland) project, the China Scholarship Council and an Amazon AWS grant. Special thanks goes to Yijue Chen. + +# References + +[1] Pablo Arias and Jean-Michel Morel. Video denoising via empirical bayesian estimation of space-time patches. 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[N/A] + +Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below. + +1. For all authors... + +(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] (b) Did you describe the limitations of your work? [Yes] See Section 6. + +(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6. +(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] + +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] See Section 4.1 and supplemental material. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 4.1 and supplemental material. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] + +4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... + +(a) If your work uses existing assets, did you cite the creators? [Yes] +(b) Did you mention the license of the assets? [Yes] +(c) Did you include any new assets either in the supplemental material or as a URL? [N/A] +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] + +5. If you used crowdsourcing or conducted research with human subjects... + +(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] \ No newline at end of file diff --git a/parse/dev/GKfNB4BegL/GKfNB4BegL_content_list.json b/parse/dev/GKfNB4BegL/GKfNB4BegL_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..877d80a01dd8e1e43124c2af1cd3fa90a3d14b16 --- /dev/null +++ b/parse/dev/GKfNB4BegL/GKfNB4BegL_content_list.json @@ -0,0 +1,1406 @@ +[ + { + "type": "text", + "text": "Recurrent Video Restoration Transformer with Guided Deformable Attention ", + "text_level": 1, + "bbox": [ + 215, + 122, + 782, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Jingyun Liang1, Yuchen $\\mathbf { F a n } ^ { 2 }$ , Xiaoyu Xiang2, Rakesh Ranjan2, Eddy $\\mathbf { I I g ^ { 2 } }$ Simon Green2, Jiezhang $\\mathbf { C a o ^ { 1 } }$ , Kai Zhang1∗, Radu Timofte1,3, Luc Van Gool1 ", + "bbox": [ + 217, + 212, + 781, + 244 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1Computer Vision Lab, ETH Zurich, Switzerland 2Meta Inc. 3University of Wurzburg, Germany ", + "bbox": [ + 168, + 256, + 808, + 271 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 306, + 535, + 324 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Video restoration aims at restoring multiple high-quality frames from multiple lowquality frames. Existing video restoration methods generally fall into two extreme cases, i.e., they either restore all frames in parallel or restore the video frame by frame in a recurrent way, which would result in different merits and drawbacks. Typically, the former has the advantage of temporal information fusion. However, it suffers from large model size and intensive memory consumption; the latter has a relatively small model size as it shares parameters across frames; however, it lacks long-range dependency modeling ability and parallelizability. In this paper, we attempt to integrate the advantages of the two cases by proposing a recurrent video restoration transformer, namely RVRT. RVRT processes local neighboring frames in parallel within a globally recurrent framework which can achieve a good trade-off between model size, effectiveness, and efficiency. Specifically, RVRT divides the video into multiple clips and uses the previously inferred clip feature to estimate the subsequent clip feature. Within each clip, different frame features are jointly updated with implicit feature aggregation. Across different clips, the guided deformable attention is designed for clip-to-clip alignment, which predicts multiple relevant locations from the whole inferred clip and aggregates their features by the attention mechanism. Extensive experiments on video super-resolution, deblurring, and denoising show that the proposed RVRT achieves state-of-the-art performance on benchmark datasets with balanced model size, testing memory and runtime. The codes are available at https://github.com/JingyunLiang/RVRT. ", + "bbox": [ + 233, + 339, + 766, + 628 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 Introduction ", + "text_level": 1, + "bbox": [ + 174, + 656, + 310, + 674 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Video restoration, such as video super-resolution, deblurring, and denoising, has become a hot topic in recent years. It aims to restore a clear and sharp high-quality video from a degraded (e.g., downsampled, blurred, or noisy) low-quality video [80, 11, 4, 38]. It has wide applications in live streaming [97], video surveillance [49], old film restoration [78], and more. ", + "bbox": [ + 176, + 688, + 825, + 743 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Parallel methods and recurrent methods have been dominant strategies for solving various video restoration problems. Typically, those two kinds of methods have their respective merits and demerits. Parallel methods [2, 25, 80, 72, 36, 63, 99, 27, 35, 4, 38] support distributed deployment and achieve good performance by directly fusing information from multiple frames, but they often have a large model size and consume enormous memory for long-sequence videos. In the meanwhile, recurrent models [24, 59, 21, 23, 26, 28, 9, 11, 45, 55, 98, 62] reuse the same network block to save parameters and predict the new frame feature based on the previously refined frame feature, but the sequential processing strategy inevitably leads to information loss and noise amplification [14] for long-range dependency modelling and makes it hard to be parallelized. ", + "bbox": [ + 174, + 750, + 825, + 875 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Considering the advantages and disadvantages of parallel and recurrent methods, in this paper, we propose a recurrent video restoration transformer (RVRT) that takes the best of both worlds. On the one hand, RVRT introduces the recurrent design into transformer-based models to reduce model parameters and memory usage. On the other hand, it processes neighboring frames together as a clip to reduce video sequence length and alleviate information loss. To be specific, we first divide the video into fixed-length video clips. Then, starting from the first clip, we refine the subsequent clip feature based on the previously inferred clip feature and the old features of the current clip from shallower layers. Within each clip, different frame features are jointly extracted, implicitly aligned and effectively fused by the self-attention mechanism [77, 51, 39]. Across different clips, information is accumulated clip by clip with a larger hidden state than previous recurrent methods. ", + "bbox": [ + 174, + 90, + 825, + 229 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "To implement the above RVRT model, one big challenge is how to align different video clips when using the previous clip for feature refinement. Most existing alignment techniques [58, 65, 59, 87, 9, 15, 72, 80, 11, 38] are designed for frame-to-frame alignment. One possible way to apply them to clip-to-clip alignment is by introducing an extra feature fusion stage after aligning all frame pairs. Instead, we propose an one-stage video-to-video alignment method named guided deformable attention (GDA). More specifically, for a reference location in the target clip, we first estimate the coordinates of multiple relevant locations from different frames in the supporting clip under the guidance of optical flow, and then aggregate features of all locations dynamically by the attention mechanism. ", + "bbox": [ + 174, + 236, + 825, + 359 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "GDA has several advantages over previous alignment methods: 1) Compared with optical flow-based warping that only samples one point from one frame [59, 87, 9], GDA benefits from multiple relevant locations sampled from the video clip. 2) Unlike mutual attention [38], GDA utilizes features from arbitrary locations without suffering from the small receptive field in local attention or the huge computation burden in global attention. Besides, GDA allows direct attention on non-integer locations with bilinear interpolation. 3) In contrast to deformable convolution [15, 100, 72, 80, 11, 10] that uses a fixed weight in feature aggregation, GDA generates dynamic weights to aggregate features from different locations. It also supports arbitrary location numbers and allows for both frame-to-frame and video-to-video alignment without any modification. ", + "bbox": [ + 173, + 367, + 825, + 491 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our contributions can be summarized as follows: ", + "bbox": [ + 174, + 498, + 495, + 512 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "• We propose the recurrent video restoration transformer (RVRT) that extracts features of local neighboring frames from one clip in a joint and parallel way, and refines clip features by accumulating information from previous clips and previous layers. By reducing the video sequence length and transmitting information with a larger hidden state, RVRT alleviates information loss and noise amplification in recurrent networks, and also makes it possible to partially parallelize the model. • We propose the guided deformable attention (GDA) for one-stage video clip-to-clip alignment. It dynamically aggregates information of relevant locations from the supporting clip. Extensive experiments on eight benchmark datasets show that the proposed model achieves stateof-the-art performance in three challenging video restoration tasks: video super-resolution, video deblurring, and video denoising, with balanced model size, memory usage and runtime. ", + "bbox": [ + 173, + 522, + 826, + 671 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 Related Work ", + "text_level": 1, + "bbox": [ + 174, + 689, + 321, + 705 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2.1 Video Restoration ", + "text_level": 1, + "bbox": [ + 174, + 719, + 338, + 734 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Parallel vs. recurrent methods. Most existing video restoration methods can be classified as parallel or recurrent methods according to their parallelizability. Parallel methods estimate all frames simultaneously, as the refinement of one frame feature is not dependent on the update of other frame features. They can be further divided as sliding window-based methods [2, 25, 80, 70, 72, 79, 36, 63, 99, 99, 27, 71, 60, 35] and transformer-based methods [4, 38]. The former kind of methods typically restore merely the center frame from the neighboring frames and are often tested in a sliding window fashion rather than in parallel. These methods generally consist of four stages: feature extraction, feature alignment, feature fusion, and frame reconstruction. Particularly, in the feature alignment stage, they often align all frames towards the center frame, which leads to quadratic complexity with respect to video length and is hard to be extended for long-sequence videos. Instead, the latter kind of method reconstructs all frames at a time based on the transformer architectures. They jointly extract, align, and fuse features for all frames, achieving significant performance improvements against previous methods. However, current transformer-based methods are laid up with a huge model size and large memory consumption. Different from above parallel methods, recurrent methods [24, 59, 21, 23, 85, 26, 28, 9, 11, 45, 55, 98, 62, 44, 5] propagate latent features from one frame to the next frame sequentially, where information of previous frames is accumulated for the restoration of later frames. Basically, they are composed of three stages: feature extraction, feature propagation and frame reconstruction. Due to the recurrent nature of feature propagation, recurrent methods suffer from information loss and the inapplicability of distributed deployment. ", + "bbox": [ + 173, + 746, + 825, + 911 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/5d34cf1ab41ce05867d58132428ce110bcff8258a33549089d2b0d2a6c8a1e6e.jpg", + "image_caption": [ + "Figure 1: The architecture of recurrent video restoration transformer (RVRT). From left to right, it consists of shallow feature extraction, recurrent feature refinement and HQ frame reconstruction. In recurrent feature refinement (RFR, see more details in Fig. 2), we divide the video into $N$ -frame clips $N = 2$ in this figure) and process frames in one clip in parallel within a globally recurrent framework in time. Multiple refinement layers are stacked for better performance. " + ], + "image_footnote": [], + "bbox": [ + 171, + 90, + 826, + 218 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 319, + 825, + 431 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Alignment in video restoration. Unlike image restoration that mainly focuses on feature extraction [16, 94–96, 42, 40, 41, 67, 93, 92], how to align multiple highly-related but misaligned frames is another key problem in video restoration. Traditionally, many methods [43, 30, 2, 48, 68, 3, 87, 9] first estimate the optical flow between neighbouring frames [19, 58, 65] and then conduct image warping for alignment. Other techniques, such as deformable convolution [15, 100, 72, 80, 11, 4], dynamic filter [29] and mutual attention [38], have also been exploited for implicit feature alignment. ", + "bbox": [ + 174, + 435, + 825, + 520 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.2 Vision Transformer ", + "text_level": 1, + "bbox": [ + 176, + 536, + 348, + 551 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Transformer [77] is the de-facto standard architecture in natural language processing. Recently, it has been used in dealing with vision problems by viewing pixels or image patches as tokens [8, 18], achieving remarkable performance gains in various computer vision tasks, including image classification [18, 37, 51, 74], object detection [76, 50, 84], semantic segmentation [83, 17, 66], etc. It also achieves promising results in restoration tasks [13, 39, 81, 44, 4, 38, 20, 22, 7, 90, 47, 73, 6]. In particular, for video restoration, Cao et al. [4] propose the first transformer model for video SR, while Liang et al. [38] propose an unified framework for video SR, deblurring and denoising. ", + "bbox": [ + 174, + 563, + 826, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "We note that some transformer-based works [101, 84] have tried to combine the concept of deformation [15, 100] with the attention mechanism [77]. Zhu et al. [101] directly predicts the attention weight from the query feature without considering its feature interaction with supporting locations. Xia et al. [84] place the supporting points uniformly on the image to make use of global information. Both above two methods are proposed for recognition tasks such as object detection, which is fundamentally different from video alignment in video restoration. Lin et al. [44] use pixel-level or patch-level attention to aggregate information from neighbouring frames under the guidance of optical flow, but it only samples one supporting pixel or patch from one frame, restricting the model from attending to multiple distant locations. ", + "bbox": [ + 173, + 666, + 826, + 791 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 Methodology ", + "text_level": 1, + "bbox": [ + 174, + 811, + 313, + 829 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 Overall Architecture ", + "text_level": 1, + "bbox": [ + 174, + 843, + 356, + 858 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a low-quality video sequence $I ^ { L Q } \\in \\mathbb { R } ^ { T \\times H \\times W \\times C }$ , where $T$ , $H$ , $W$ and $C$ are the video length, height, width and channel, respectively, the goal of video restoration is to reconstruct the high-quality video $J ^ { H Q } \\in \\mathbb { R } ^ { T \\times s H \\times s W \\times C }$ , where $s$ is the scale factor. To reach this goal, we propose a recurrent video restoration transformer, as illustrated in Fig. 1. The model consists of three parts: shallow feature extraction, recurrent feature refinement and HQ frame reconstruction. More specifically, in shallow feature extraction, we first use a convolution layer to extract features from the LQ video. For deblurring and denoising (i.e., $s = 1 \\AA$ ), we additionally add two strided convolution layers to downsample the feature and reduce computation burden in the next layers. After that, several Residual Swin Transformer Blocks (RSTBs) [39] are used to extract the shallow feature. Then, we use recurrent feature refinement modules for temporal correspondence modeling and guided deformable attention for video alignment, which are detailed in Sec. 3.2 and Sec. 3.3, respectively. Lastly, we add several RSTBs to generate the final feature and reconstruct the HQ video $\\mathbf { \\widetilde { \\Gamma } } I ^ { R H Q }$ by pixel shuffle layer [61]. For training, the Charbonnier loss [12] $\\mathcal { L } = \\sqrt { \\| I ^ { R H Q } - I ^ { H Q } \\| ^ { 2 } + \\epsilon ^ { 2 } }$ $\\begin{array} { r } { \\dot { \\epsilon } = 1 0 ^ { - 3 } , } \\end{array}$ is used for all tasks. ", + "bbox": [ + 176, + 867, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 90, + 826, + 233 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 Recurrent Feature Refinement ", + "text_level": 1, + "bbox": [ + 176, + 247, + 424, + 262 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "We stack $L$ recurrent feature refinement modules to refine the video feature by exploiting the temporal correspondence between different frames. To make a trade-off between recurrent and transformer-based methods, we process $N$ frames locally in parallel on the basis of a globally recurrent framework. ", + "bbox": [ + 174, + 272, + 545, + 356 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Formally, given the video feature F i ∈ RT ×H×W×C from the $i$ -th layer, we first reshape it as a 5-dimensional tensor $F ^ { i } \\in \\mathbb { R } ^ { \\frac { \\hat { T } } { N } \\times N \\times H \\times W \\times C }$ by dividing it into $\\textstyle { \\frac { T } { N } }$ video clip features: $F _ { 1 } ^ { i } , F _ { 2 } ^ { i } , . . . , F _ { \\frac { T } { N } } ^ { i } \\in \\mathbb { R } ^ { N \\times H \\times W \\times C }$ . Each clip feature $F _ { t } ^ { i }$ $\\begin{array} { r } { ( 1 \\ \\leq \\ t \\ \\leq \\ \\frac { T } { N } ) } \\end{array}$ has $N$ neighbouring frame features: $F _ { t , 1 } ^ { i } , F _ { t , 2 } ^ { i } , . . . , \\bar { F } _ { t , N } ^ { i } \\in \\mathbb { R } ^ { H \\times W \\times C }$ . To utilize information from neighbouring clips, we align the $( t - 1 )$ -th clip feature $F _ { t - 1 } ^ { i }$ towards the $t$ -th clip based on the optical flow $O _ { t - 1 t } ^ { i }$ , clip feature $F _ { t - 1 } ^ { i - 1 }$ and clip feature $F _ { t } ^ { i - 1 }$ This is formulated as follows: ", + "bbox": [ + 174, + 361, + 544, + 517 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d29547a69f286b6bcf788517b4a392ebb8eb2a8cc0ca007087b869096d91ad97.jpg", + "text": "$$\n\\widehat { F } _ { t - 1 } ^ { i } = G D A ( F _ { t - 1 } ^ { i } ; O _ { t - 1 t } ^ { i } , F _ { t - 1 } ^ { i - 1 } , F _ { t } ^ { i - 1 } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 215, + 518, + 500, + 539 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $G D A$ is the guided deformable attention and $\\widehat { F } _ { t - 1 } ^ { i }$ is the aligned clip feature. The details of GDA will be described in Sec. 3.3. ", + "bbox": [ + 173, + 541, + 544, + 584 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/59f58cddf8ab8ac44ecc1947deb5854398f8cb721ea8d1b11130ff9f56b221c9.jpg", + "image_caption": [ + "Figure 2: The illustrations of recurrent feature refinement (RFR). The $( t - 1 )$ -th clip feature $F _ { t - 1 } ^ { i }$ from the $_ { i }$ -th layer is aligned towards the $t$ -th clip as $\\widehat F _ { t - 1 } ^ { i }$ by guided deformable attention (GDA, see more details in Fig. 3). $F _ { t } ^ { 0 } , F _ { t } ^ { 1 } , . . . , F _ { t } ^ { i - 1 }$ and $\\widehat { F } _ { t - 1 } ^ { i }$ are then refined as $F _ { t } ^ { i }$ by several modified residual swin transformer blocks (MRSTBs), in which different frames are jointly processed in a parallel way. " + ], + "image_footnote": [], + "bbox": [ + 591, + 268, + 790, + 545 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Similar to recurrent neural networks [59, 9, 11], as shown in Fig. 2, we update the clip feature of each time step as follows: ", + "bbox": [ + 174, + 590, + 544, + 631 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/0cdbc1aef7e5b7f06bd3727bb86e0d77d03c09402a2b84ed130ad27f3b936e9f.jpg", + "text": "$$\nF _ { t } ^ { i } = R F R ( F _ { t } ^ { 0 } , F _ { t } ^ { 1 } , . . . , F _ { t } ^ { i - 1 } , \\widehat { F } _ { t - 1 } ^ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 238, + 631, + 478, + 651 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $F _ { t } ^ { 0 }$ is the output of the shallow feature extraction module and $F _ { t } ^ { 1 } , F _ { t } ^ { 2 } , . . . , F _ { t } ^ { i - 1 }$ are from previous recurrent feature refinement modules. $R F R ( \\cdot )$ is the recurrent feature refinement module that consists of a convolution layer for feature fusion and several modified residual Swin Transformer blocks (MRSTBs) for feature refinement. In MRSTB, we upgrade the original 2D $h \\times w$ attention window to the 3D $N \\times h \\times w$ attention window, so that every frame in the clip can attend to itself and other frames simultaneously, allowing implicit feature aggregation. In addition, in order to accumulate information forward and backward in time, we reverse the video sequence for all even recurrent feature refinement modules [24, 11]. ", + "bbox": [ + 174, + 654, + 544, + 696 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 698, + 825, + 780 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The above recurrent feature refinement module is the key component of the proposed RVRT model. Globally, features of different video clips are propagated in a recurrent way. Locally, features of different frames are updated jointly in parallel. For an arbitrary single frame, it can make full use of global information accumulated in time and local information extracted together by the selfattention mechanism. As we can see, RVRT is a generalization of both recurrent and transformer models. It becomes a recurrent model when $N = 1$ or a transformer model when $N = T$ . This is fundamentally different from previous methods that adopt transformer blocks to replace CNN blocks within a recurrent architecture [78, 44]. It is also different from existing attempts in natural language processing [82, 34]. ", + "bbox": [ + 174, + 786, + 826, + 911 + ], + "page_idx": 3 + }, + { + "type": "image", + "img_path": "images/78f0125d943f6d99f99396a3ef62e2f8b041da752c8a43fa5609df94f08cd8ed.jpg", + "image_caption": [ + "Figure 3: The illustrations of guided deformable attention (GDA). We estimate offsets of multiple relevant locations from different frames based on the warped clip, and then aggregate features of different locations mically by the attention mechanism. are the pre-aligned and aligned fea $F _ { t - 1 } ^ { i }$ is of $( t - 1 )$ lip fand om the denote $_ { i }$ -th layer, while optical flows a $\\bar { F } _ { t - 1 } ^ { i }$ andsets, $\\widehat { F } _ { t - 1 } ^ { i }$ $F _ { t - 1 } ^ { i }$ $O _ { t - 1 t } ^ { i , ( 1 : N ) }$ oi,(1:N)t−1→t respectively. " + ], + "image_footnote": [], + "bbox": [ + 258, + 83, + 759, + 387 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 Guided Deformable Attention for Video Alignment ", + "text_level": 1, + "bbox": [ + 173, + 487, + 566, + 502 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Different from previous frameworks, the proposed RVRT needs to align neighboring related but misaligned video clips, as indicated in Eq. (1). In this subsection, we propose the guided deformation attention (GDA) for video clip-to-clip alignment. ", + "bbox": [ + 173, + 512, + 826, + 555 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "clip as a list of features Given the -th clip feature $\\widehat { F } _ { t - 1 } ^ { i , ( 1 : N ) } = \\widehat { F } _ { t - 1 } ^ { i , ( 1 ) } , \\widehat { F } _ { t - 1 } ^ { i , ( 2 ) } , . . . , \\widehat { F } _ { t - 1 } ^ { i , ( N ) }$ $F _ { t - 1 } ^ { i }$ from the $i$ -th layer, our goal is to align , where $\\widehat { F } _ { t - 1 } ^ { i , ( n ) } ( 1 \\leq n \\leq N )$ $F _ { t - 1 } ^ { i }$ towards the denotes the -th b aligned clip feature towards the $n$ b b -th frame feature $F _ { t , n } ^ { i }$ of the $t$ b -th clip, and $\\widehat { F } _ { t - 1 , n ^ { \\prime } } ^ { i , ( n ) } ( 1 \\leq n ^ { \\prime } \\leq N )$ is the aligned frame feature from the clip. Inspired by optical flow estimati $n ^ { \\prime }$ -th frame in the designs [19, 56, $( t - 1 )$ -th clip to the , 44], we first $n$ -th frae-align e $t$ -thith $F _ { t - 1 , n ^ { \\prime } } ^ { i , ( n ) }$ the optical flow $O _ { t - 1 t , n ^ { \\prime } } ^ { i , ( n ) }$ as $\\hat { F } _ { t - 1 , n ^ { \\prime } } ^ { i , ( n ) } = \\mathcal { W } ( F _ { t - 1 , n ^ { \\prime } } ^ { i , ( n ) } , O _ { t - 1 t , n ^ { \\prime } } ^ { i , ( n ) } )$ , whe $\\mathcal { W }$ t 1,n denotes the warping operation. For convenience, we summarize the pre-alignments of all “ $n ^ { \\prime }$ -toframe pairs between the $( t - 1 )$ -th and $t$ -th video clips as follows: ", + "bbox": [ + 173, + 560, + 826, + 696 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/301b5b4cdeb89e5c01226af07bf5ad6edbbf10d0091fdeabe6c4e90a532efd96.jpg", + "text": "$$\n\\bar { F } _ { t - 1 } ^ { i , ( 1 : N ) } = \\mathcal { W } ( F _ { t - 1 } ^ { i } , O _ { t - 1 t } ^ { i , ( 1 : N ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 393, + 700, + 602, + 722 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "at, we predict the optical flow offsets along the channel dimension. A sm from the concatenation of olutional neural network ( $F _ { t } ^ { i - 1 }$ , $\\bar { F } _ { t - 1 } ^ { i ( 1 : N ) }$ anderal $O _ { t - 1 t } ^ { i , ( 1 : N ) }$ convolutional layers and ReLU layers is used for prediction. This is formulated as ", + "bbox": [ + 173, + 731, + 825, + 782 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/4fd0a0ada73fc03bccf191e934cbcd725f531756ee9a83e4accb467e75000899.jpg", + "text": "$$\no _ { t - 1 \\to t } ^ { i , ( 1 : N ) } = C N N ( C o n c a t ( F _ { t } ^ { i - 1 } , \\bar { F } _ { t - 1 } ^ { i , ( 1 : N ) } , O _ { t - 1 \\to t } ^ { i , ( 1 : N ) } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 328, + 785, + 668, + 806 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where the current misalignment between the can reflect the offset required for further alig $t$ -th clip feature and thement. In practice, we i rped ialize $( t - 1 )$ h clip fea as the o $O _ { t - 1 t } ^ { 1 , ( 1 : N ) }$ flows estimated from the LQ input video via SpyNet [58], and predict $M$ offsets for each frame ( $N M$ offsets in total). The optical flows are updated layer by layer as follows: ", + "bbox": [ + 173, + 810, + 825, + 869 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d8d5f6781b77c64302eeadf8e162f2440943e47489b04b351b4f9b4afbd3ca11.jpg", + "text": "$$\nO _ { t - 1 t , n ^ { \\prime } } ^ { i + 1 , ( n ) } = O _ { t - 1 t , n ^ { \\prime } } ^ { i , ( n ) } + \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\{ o _ { t - 1 t , n ^ { \\prime } } ^ { i , ( n ) } \\} _ { m } ,\n$$", + "text_format": "latex", + "bbox": [ + 331, + 872, + 665, + 916 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $\\{ o _ { t - 1 t , n ^ { \\prime } } ^ { i , ( n ) } \\} _ { m }$ denotes the $m$ -th offset in $M$ predictions from the $n ^ { \\prime }$ -th frame to the $n$ -th frame. Then, for the $n$ -th frame of the $t$ -th clip, we sample its relevant features from the $( t - 1 )$ -th clip feature $F _ { t - 1 } ^ { i }$ according the predicted locations, which are indicated by the sum of optical flow and offsets, i.e., $O _ { t - 1 \\to t } ^ { i , ( n ) } + o _ { t - 1 \\to t } ^ { i , ( n ) }$ , according to the c in rel nship $F _ { t - 1 } ^ { i } \\xrightarrow { O _ { t - 1 t } ^ { i , ( n ) } } \\bar { F } _ { t - 1 } ^ { i , ( n ) } \\xrightarrow { o _ { t - 1 t } ^ { i , ( n ) } } \\widehat { F } _ { t - 1 } ^ { i , ( n ) }$ [11, 58]. For simplicity, we define the queries $Q$ $K$ and values $V$ as follows: ", + "bbox": [ + 173, + 88, + 828, + 184 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/7ff97344d9f67a84e4a9bafac03a421149ed95020ec274217b8118fc6b64c4f6.jpg", + "text": "$$\n\\begin{array} { l } { { Q = F _ { t , n } ^ { i - 1 } P _ { Q } , } } \\\\ { { K = S a m p l i n g ( F _ { t - 1 } ^ { i - 1 } P _ { K } , O _ { t - 1 t } ^ { i , ( n ) } + o _ { t - 1 t } ^ { i , ( n ) } ) , } } \\\\ { { V = S a m p l i n g ( F _ { t - 1 } ^ { i } P _ { V } , O _ { t - 1 t } ^ { i , ( n ) } + o _ { t - 1 t } ^ { i , ( n ) } ) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 344, + 189, + 650, + 257 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where $Q \\in \\mathbb { R } ^ { 1 \\times C }$ is the projected feature from the $n$ -th frame of $t$ -th clip. $K \\in \\mathbb { R } ^ { N M \\times C }$ and $V \\in \\mathbb { R } ^ { N M \\times C }$ are the projected features that are bilinearly sampled from $N M$ locations of $F _ { t - 1 } ^ { i - 1 }$ and $F _ { t - 1 } ^ { i }$ , respectively. $P _ { Q } \\in \\mathbb { R } ^ { C \\times C }$ , $P _ { K } \\in \\mathbb { R } ^ { C \\times C }$ and $P _ { V } \\in \\mathbb { R } ^ { C \\times C }$ are the projection matrices. Note that we first project the feature and then do sampling to reduce redundant computation. ", + "bbox": [ + 174, + 261, + 825, + 324 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "$K$ xt, similafrom the $( i - 1 )$ attention mechanism [77], we calculate the atten-th layer and then compute the aligned feature $\\widehat { F } _ { t - 1 } ^ { i , ( n ) }$ eights based on the as a weighted sum $Q$ af $V$ from the same $i$ -th layer as follows: ", + "bbox": [ + 174, + 328, + 823, + 375 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/0fc19718b93a09190abb0f88c0184e8f1ff9b5c0a560379d72a635f1901cf5d0.jpg", + "text": "$$\n\\widehat { F } _ { t - 1 } ^ { i , ( n ) } = S o f t M a x ( Q K ^ { T } / \\sqrt { C } ) ) V ,\n$$", + "text_format": "latex", + "bbox": [ + 385, + 381, + 612, + 404 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where SoftMax is the softmax operation along the row direction and $\\sqrt { C }$ is a scaling factor. ", + "bbox": [ + 171, + 411, + 769, + 428 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Lastly, since Eq. (9) only aggregates information spatially, we add a multi-layer perception (MLP) with two fully-connected layers and a $G E L U$ activation function between them to enable channel interaction as follows: ", + "bbox": [ + 174, + 433, + 825, + 473 + ], + "page_idx": 5 + }, + { + "type": "equation", + "img_path": "images/fce81184d1d7eddd78d73d2828172f6cdf7c63968ecf697011a9011fd9c9de63.jpg", + "text": "$$\n\\widehat F _ { t - 1 } ^ { i } = \\widehat F _ { t - 1 } ^ { i } + M L P ( \\widehat F _ { t - 1 } ^ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 401, + 470, + 594, + 491 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "where a residual connection is used to stabilize training. The hidden and output channel numbers of the $M L P$ are $R C$ ( $R$ is the ratio) and $C$ , respectively. ", + "bbox": [ + 171, + 493, + 825, + 522 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Multi-group multi-head guided deformable attention. We can divide the channel into several deformable groups and perform the deformable sampling for different groups in parallel. Besides, in the attention mechanism, we can further divide one deformable group into several attention heads and perform the attention operation separately for different heads. All groups and heads are concatenated together before channel interaction. ", + "bbox": [ + 173, + 525, + 825, + 595 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Connection to deformable convolution. Deformable convolution [15, 100] uses a learned weight for feature aggregation, which can be seen as a special case of GDA, i.e., using different projection matrix $P _ { V }$ for different locations and then directly averaging the resulting features. Its parameter number and computation complexity are $M C ^ { 2 }$ and $\\mathcal { O } ( M C ^ { 2 } )$ , respectively. In contrast, GDA uses the same projection matrix for all locations but generates dynamic weights to aggregate them. Its parameter number and computation complexity are $( 3 + 2 R ) \\dot { C } ^ { 2 }$ and $\\mathcal { O } ( ( 3 C + 2 R C + M ) C )$ , which are similar to deformable convolution when choosing proper $M$ and $R$ . ", + "bbox": [ + 173, + 599, + 825, + 696 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 174, + 715, + 312, + 733 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 Experimental Setup ", + "text_level": 1, + "bbox": [ + 174, + 747, + 351, + 762 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "For shallow feature extraction and HQ frame reconstruction, we use 1 RSTB that has 2 swin transformer layers. For recurrent feature refinement, we use 4 refinement modules with a clip size of 2, each of which has 2 MRSTBs with 2 modified swin transformer layers. For both RSTB and MRSTB, spatial attention window size and head number are $8 \\times 8$ and 6, respectively. We use 144 channels for video SR and 192 channels for deblurring and denoising. In GDA, we use 12 deformable groups and 12 deformable heads with 9 candidate locations. We empirically project the query to a higher-dimensional space (e.g., $2 C$ ) because we found it can improve the performance slightly and the parameter number of GDA is not a bottleneck. In training, we randomly crop $2 5 6 \\times 2 5 6$ HQ patches and use different video lengths for different datasets: 30 frames for REDS [53], 14 frames for Vimeo-90K [87], and 16 frames for DVD [63], GoPro [54] as well as DAVIS [31]. Adam optimizer [33] with default setting is used to train the model for 600,000 iterations when the batch size is 8. The learning rate is initialized as $4 \\times 1 0 ^ { - 4 }$ and deceased with the Cosine Annealing scheme [52]. To stabilize training, we initialize SpyNet [58, 56] with pretrained weights, fix it for the first 30,000 iterations and reduce its learning rate by $7 5 \\%$ . ", + "bbox": [ + 174, + 772, + 825, + 911 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 90, + 825, + 147 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.2 Ablation Study ", + "text_level": 1, + "bbox": [ + 174, + 164, + 318, + 179 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "To explore the effectiveness of different components, we conduct ablation studies on REDS [53] for video SR. For efficiency, we reduce the MRSTB blocks by half and use 12 frames in training. ", + "bbox": [ + 174, + 190, + 823, + 218 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The impact of clip length. In RVRT, we divide the video into $N$ -frame clips. As shown in Table 1, the performance rises when clip length is increased from 1 to 2. However, the performance saturates when $N = 3$ , possibly due to large within-clip motions and inaccurate optical flow derivation. When we directly estimate all optical flows (marked by ∗), the PSNR hits 32.21dB. Besides, to compare the temporal modelling ability, we hack the input LQ video (Clip 000 from REDS, 100 frames in total) by manually setting all pixels of the 50-th frame as zeros. As indicated in Fig. 4, on the one hand, $N = 2$ has a smaller performance drop and all its frames still have higher PSNR than $N = 1$ (equals to a recurrent model) after the attack, showing that RVRT can mitigate the noise amplification from the hacked frame to the rest frames. One the other hand, the hacked frame of $N = 2$ has an impact on more neighbouring frames than $N = 1$ , which means that RVRT can alleviate information loss and utilize more frames than $N = 1$ for restoration. ", + "bbox": [ + 173, + 223, + 825, + 376 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The impact of video alignment. The alignment of video clips plays a key role in our framework. We compare the proposed clip-to-clip guided deformable attention (GDA) with existing frame-toframe alignment techniques by performing them frame by frame, followed by concatenation and channel reduction. As we can see from Table 2, GDA outperforms all existing methods when it is used for frame-to-frame alignment (denoted as $\\mathrm { G D A } ^ { * }$ ), and leads a further improvement when we aggregate features directly from the whole clip. ", + "bbox": [ + 174, + 381, + 825, + 464 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The impact of different components in GDA. We further conduct an ablation study on GDA in Table 3. As we can see, the optical flow guidance is critical for the model, leading to a PSNR gain of 1.11dB. The update of optical flow in different layers can further improve the result. The channel interaction in MLP also plays an important role, since the attention mechanism only aggregates information spatially. ", + "bbox": [ + 174, + 468, + 825, + 537 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "The impact of deformable group and attention head. We also conduct experiments on different group and head numbers in GDA. As shown in Table 4, when the deformable group rises, the PSNR first rises and then keeps almost unchanged. Besides, double attention heads lead to slightly better results at the expense of higher computation, but using too many heads has an adverse impact as the head dimension may be too small. ", + "bbox": [ + 174, + 542, + 825, + 612 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/3ed93089db4409f3dd9b4b1df600645ebd8a86ba5f3de9d641823754afe5f042.jpg", + "table_caption": [ + "Table 1: Ablation study on clip length. " + ], + "table_footnote": [], + "table_body": "
Clip1233*
PSNR31.9832.1032.0732.21
", + "bbox": [ + 178, + 655, + 408, + 679 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/61d3582fd4df029d9543554f19b8d1d3de1bd1206c7f2c1d5e21e5daebb1cf85.jpg", + "table_caption": [ + "Table 2: Ablation study on different video alignment techniques. " + ], + "table_footnote": [], + "table_body": "
AlignmentWarping[87]TMSA [38]DCN [72]GDA*GDA
PSNR28.8830.4531.9332.0032.10
", + "bbox": [ + 419, + 655, + 803, + 679 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/9440b0d467b37dd3d7dbf404b8b67df622e7040a0fe768e07460f1c427c9a829.jpg", + "table_caption": [ + "Table 3: Ablation study on different GDA components. " + ], + "table_footnote": [], + "table_body": "
Optical Flow GuidanceOptical Flow UpdateMLP厂√√√兴√
PSNR30.9932.0331.8332.10
", + "bbox": [ + 472, + 700, + 784, + 744 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/b41ab603771ed468efed243ac9067635351386688ec46cef644bff3befb7977f.jpg", + "image_caption": [ + "Figure 4: Per-frame PSNR drop when pixels of the 50-th frame is hacked to be all zeros. $N$ is clip length. " + ], + "image_footnote": [], + "bbox": [ + 176, + 690, + 406, + 797 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/85d7377f63dafd52e5416ecf9aeb671e462a1c1106176b98635c98bfe9958ff6.jpg", + "table_caption": [ + "Table 4: Ablation study on deformable groups and attention heads. " + ], + "table_footnote": [], + "table_body": "
Deformable GroupAttention Head11661212122412362424
PSNR31.6332.0332.1032.1332.0332.11
", + "bbox": [ + 436, + 786, + 820, + 820 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "4.3 Video Super-Resolution ", + "text_level": 1, + "bbox": [ + 174, + 857, + 377, + 872 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "For video SR, we consider two settings: bicubic (BI) and blur-downsampling (BD) degradation. For BI degradation, we train the model on two different datasets: REDS [53] and Vimeo-90K [87], and then test the model on their corresponding testsets: REDS4 and Vimeo-90K-T. We additionally test Vid4 [46] along with Vimeo-90K. For BD degradation, we train it on Vimeo-90K and test it on Vimeo-90K-T, Vid4, and UDM10 [89]. The comparisons with existing methods are shown in Table 5. As we can see, RVRT achieves the best performance on REDS4 and Vid4 for both degradations. Compared with the representative recurrent model Basic ${ \\mathrm { V S R } } + +$ [11], RVRT improves the PSNR by significant margins of $\\mathbf { 0 . 2 { \\sim } 0 . 5 } \\mathbf { d B }$ . Compared with the recent transformer-based model VRT [38], RVRT outperforms VRT on REDS4 and Vid4 by up to 0.36dB. The visual comparisons of different methods are shown in Fig. 5. It is clear that RVRT generates sharp and clear HQ frames, while other methods fail to restore fine textures and details. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/f102a1c24306521ddf81cc80804a545ec4d6ab3a8ce93e915f5f1cec516dd495.jpg", + "table_caption": [ + "Table 5: Quantitative comparison (average PSNR/SSIM) with state-of-the-art methods for video superresolution $( \\times 4 )$ on REDS4 [53], Vimeo-90K-T [87], Vid4 [46] and UDM10 [89]. " + ], + "table_footnote": [], + "table_body": "
MethodBI degradationBD degradation
REDS4[53] (RGB channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)UDM10[89] (Y channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)
Bicubic26.14/0.729231.32/0.868423.78/0.634728.47/0.825331.30/0.868721.80/0.5246
SwinIR[39]29.05/0.826935.67/0.928725.68/0.749135.42/0.938034.12/0.916725.25/0.7262
SwinIR-ft [39]29.24/0.831935.89/0.930125.69/0.748836.76/0.946735.70/0.929325.62/0.7498
TOFlow [87]27.98/0.799033.08/0.905425.89/0.765136.26/0.943834.62/0.921225.85/0.7659
FRVSR[59]137.09/0.952235.64/0.931926.69/0.8103
DUF[29]28.63/0.825127.33/0.831938.48/0.960536.87/0.944727.38/0.8329
PFNL [89]29.63/0.850236.14/0.936326.73/0.802938.74/0.9627=27.16/0.8355
RBPN[23]30.09/0.859037.07/0.943527.12/0.818038.66/0.959637.20/0.945827.17/0.8205
MuCAN[36]30.88/0.875037.32/0.9465
RLSP[21]38.48/0.960636.49/0.940327.48/0.8388
TGA [27]38.74/0.962737.59/0.951627.63/0.8423
RSDN[26]39.35/0.965337.23/0.947127.92/0.8505
RRN[28]38.96/0.964427.69/0.8488
FDAN [45]39.91/0.968637.75/0.952227.88/0.8508
EDVR[80]31.09/0.880037.61/0.948927.35/0.826439.89/0.968637.81/0.952327.85/0.8503
GOVSR [88]40.14/0.971337.63/0.950328.41/0.8724
BasicVSR[9]31.42/0.890937.18/0.945027.24/0.825139.96/0.969437.53/0.949827.96/0.8553
IconVSR[9]31.67/0.894837.47/0.947627.39/0.827940.03/0.969437.84/0.952428.04/0.8570
VRT[38]32.19/0.900638.20/0.953027.93/0.842541.05/0.973738.72/0.958429.42/0.8795
BasicVSR++[11]32.39/0.906937.79/0.950027.79/0.840040.72/0.972238.21/0.955029.04/0.8753
RVRT (ours)32.75/0.911338.15/0.952727.99/0.846240.90/0.972938.59/0.957629.54/0.8810
", + "bbox": [ + 210, + 121, + 787, + 367 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/56e18861b4edc55692cbfb6a30cf2366a32bce5d0a2a760fe96fd3b4aae4e642.jpg", + "table_caption": [ + "Table 6: Comparison of model size, testing memory and runtime for an LQ input of $3 2 0 \\times 1 8 0$ " + ], + "table_footnote": [], + "table_body": "
Method#Param (M)Memory (M)Runtime (ms)PSNR (dB)
BasicVSR++[11]7.32237732.39
BasicVSR++ [11]+RSTB[39]9.3102120132.61
EDVR[80]20.6353537831.09
VSRT[4]32.62748732831.19
VRT[38]35.6214924332.19
RVRT (ours)10.8105618332.75
", + "bbox": [ + 261, + 415, + 735, + 489 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 517, + 826, + 642 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We compare the model size, testing memory consumption and runtime of different models in Table 6. Compared with representative parallel methods EDVR [80], VSRT [4] and VST [38], RVRT achieves significant performance gains with less than at least $50 \\%$ of model parameters and testing memory usage. It also reduces the runtime by at least $25 \\%$ . Compared the recurrent model Basic $J \\mathrm { S R } + +$ [11], RVRT brings a PSNR improvement of 0.26dB. As for the inferiority of testing memory and runtime, we argue that it is mainly because the CNN layers are highly optimized on existing deep learning frameworks. To prove it, we use the transformer-based RSTB blocks in RVRT to replace the CNN blocks in Basic $/ \\mathrm { S R } { + + }$ , in which case it has similar memory usage and more runtime than our model. ", + "bbox": [ + 173, + 648, + 826, + 761 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "In addition, to better understand how guided deformable attention works, we visualize the predicted offsets on the LQ frames and show the attention weight in Fig. 6. As we can see, multiple offsets are predicted to select multiple sampled locations in the neighbourhood of the corresponding pixel. According to the feature similarity between the query feature and the sampled features, features of different locations are aggregated by calculating a dynamic attention weight. ", + "bbox": [ + 174, + 766, + 825, + 837 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.4 Video Deblurring", + "text_level": 1, + "bbox": [ + 174, + 856, + 334, + 871 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For video deblurring, the model is trained and tested on two different datasets, DVD [63] and GoPro [54], with their official training/testing splits. As shown in Table 7 and 8, RVRT shows its superiority over most methods with huge improvements of $\\mathbf { 1 . 4 0 } { \\sim } 2 . 2 7 \\mathbf { d B }$ on two datasets. Even though the performance gain over VRT is relatively small, RVRT has a smaller model size and much less runtime. In detail, the model size and runtime of RVRT are $1 3 . 6 \\mathbf { M }$ and 0.3s, while VRT has $1 8 . 3 \\mathbf { M }$ parameters and the runtime of 2.2s on a $1 2 8 0 \\times 7 2 0$ LQ input. The visual comparison is provided in the supplementary material due to the space limit. ", + "bbox": [ + 174, + 883, + 821, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/bd7e1a5bce1ec4aa252f3265f3a3ec2901f27ee35929b39db654098e7c3609ea.jpg", + "image_caption": [ + "Figure 5: Visual comparison of video super-resolution $( \\times 4 )$ methods on REDS [53] and Vid4 [46]. " + ], + "image_footnote": [], + "bbox": [ + 191, + 87, + 815, + 338 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/a6a8fb9a35b6e7716c851e46daa9122884cd37541ca71ca09e799c1b60864403.jpg", + "image_caption": [ + "Figure 6: The visualization of predicted offsets and attention weight predicted in guided deformable attention. Although guided deformable attention is conducted on features, we plot illustrations on LQ input frames for better understanding. Best viewed by zooming. " + ], + "image_footnote": [], + "bbox": [ + 197, + 378, + 794, + 530 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 604, + 825, + 675 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4.5 Video Denoising ", + "text_level": 1, + "bbox": [ + 174, + 693, + 326, + 708 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "For video denoising, we train the model on the training set of DAVIS [31] and test it on its corresponding testset and Set8 [70]. For fairness of comparison, following [70, 71], we train a non-blind additive white Gaussian denoising model for noise level $\\sigma \\sim \\mathcal { U } ( 0 , 5 0 )$ . Similar to the case of video deblurring, there is a huge gap $( \\mathbf { 0 . 6 0 } { \\sim } 2 . 3 7 \\mathbf { d B } $ ) between RVRT and most methods. Compared with VRT, RVRT has slightly better performance on large noise levels, with a smaller model size (12.8M v.s.18.4M) and less runtime $( 0 . 2 \\mathrm { s } ~ \\nu . s . 1 . 5 \\mathrm { s } )$ on a $1 2 8 0 \\times 7 2 0$ LQ input. The visual comparison is provided in the supplementary material due to the space limit. ", + "bbox": [ + 174, + 719, + 825, + 816 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Conclusion ", + "text_level": 1, + "bbox": [ + 174, + 837, + 299, + 854 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we proposed a recurrent video restoration transformer with guided deformable attention. It is a globally recurrent model with locally parallel designs, which benefits from the advantages of both parallel methods and recurrent methods. We also propose the guided deformable attention module for our special case of video clip-to-clip alignment. Under the guidance of optical flow, it aggregates information from multiple neighboring locations adaptively with the attention mechanism. Extensive experiments on video super-resolution, video deblurring, and video denoising demonstrated the effectiveness of the proposed method. ", + "bbox": [ + 174, + 869, + 825, + 911 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/c151bba4ba048622079ecafe475e414f1a9d47df8511cb6109baf2703970376c.jpg", + "table_caption": [ + "Table 7: Quantitative comparison (average RGB channel PSNR/SSIM) with state-of-the-art methods for video deblurring on DVD [63]. " + ], + "table_footnote": [], + "table_body": "
MethodDBN[63]STFAN [99]STTN [32]SFE [86]EDVR[80]TSP [57]
PSNR30.0131.2431.6131.7131.8232.13
SSIM0.88770.93400.91600.91600.91600.9268
MethodPVDNet [62]GSTA [64]ARVo[35]FGST [44]VRT[38]RVRT (ours)
PSNR SSIM32.31 0.926032.53 0.946832.80 0.935233.36 0.950034.24 0.965134.30 0.9655
", + "bbox": [ + 241, + 121, + 754, + 188 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/ef5f61b2b27c0b91683d3c3e2e769e2542264caf4d7830d61bdb8f5304187333.jpg", + "table_caption": [ + "Table 8: Quantitative comparison (average RGB channel PSNR/SSIM) with state-of-the-art methods for video deblurring on GoPro [54]. " + ], + "table_footnote": [], + "table_body": "
MethodSRN[69]MPRNet[91]MAXIM[73]IFI-RNN[55]ESTRNN[98]EDVR [80]
PSNRSSIM30.260.934232.660.959032.860.961031.050.911031.070.902331.540.9260
MethodTSP [57]PVDNet [62]GSTA [64]FGST[44]VRT[38]RVRT (ours)
PSNRSSIM31.670.927931.980.928032.100.960032.900.961034.810.972434.920.9738
", + "bbox": [ + 228, + 241, + 767, + 306 + ], + "page_idx": 9 + }, + { + "type": "table", + "img_path": "images/b84420f12042efecececd72883b725250191103ee15e3409268b6475ee864c1b.jpg", + "table_caption": [ + "Table 9: Quantitative comparison (average RGB channel PSNR) with state-of-the-art methods for video denoising on DAVIS [31] and Set8 [70]. " + ], + "table_footnote": [], + "table_body": "
Dataset0VLNB [1]DVDNet[70]FastDVDNet [71]PaCNet[75]VRT[38]RVRT (ours)
DAVIS102038.8535.6838.1338.7139.9740.8240.5738.05
35.7035.7736.8238.1538.05
3033.7334.0834.0434.7936.5236.57
405032.3231.1332.8631.8532.8233.3435.3235.47
31.8632.2034.3634.57
Set8102037.2633.7236.0836.4437.0637.8837.53
33.4933.4333.9435.0234.83
3031.7431.7931.6832.0533.3533.30
405030.3929.2430.5529.5630.4630.7032.1532.2131.33
29.5329.6631.22
", + "bbox": [ + 215, + 354, + 781, + 470 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 497, + 826, + 553 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "6 Limitations and Societal Impacts ", + "text_level": 1, + "bbox": [ + 176, + 574, + 480, + 590 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Although RVRT achieves state-of-the-art performance in video restoration, it still has some limitations. For example, the complexity of pre-alignment by optical flow increases quadratically with respect to the clip length. One possible solution is to develop a video-to-video optical flow estimation model that directly predicts all optical flows. As for societal impacts, similar to other restoration methods, RVRT may bring privacy concerns after restoring blurry videos and lead to misjudgments if used for medical diagnosis. ", + "bbox": [ + 174, + 604, + 826, + 689 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Acknowledgments and Disclosure of Funding ", + "text_level": 1, + "bbox": [ + 174, + 709, + 553, + 727 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "This work was partially supported by the ETH Zurich Fund (OK), a Huawei Technologies Oy (Finland) project, the China Scholarship Council and an Amazon AWS grant. Special thanks goes to Yijue Chen. ", + "bbox": [ + 174, + 741, + 825, + 784 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "References ", + "text_level": 1, + "bbox": [ + 174, + 804, + 266, + 820 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "[1] Pablo Arias and Jean-Michel Morel. Video denoising via empirical bayesian estimation of space-time patches. Journal of Mathematical Imaging and Vision, 60(1):70–93, 2018. \n[2] Jose Caballero, Christian Ledig, Andrew Aitken, Alejandro Acosta, Johannes Totz, Zehan Wang, and Wenzhe Shi. 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You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ", + "bbox": [ + 173, + 669, + 825, + 724 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "• Did you include the license to the code and datasets? [Yes] \n• Did you include the license to the code and datasets? [No] The code and the data are proprietary. • Did you include the license to the code and datasets? [N/A] ", + "bbox": [ + 173, + 734, + 823, + 786 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below. 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However,", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "spans": [ + { + "bbox": [ + 141, + 324, + 470, + 336 + ], + "score": 1.0, + "content": "it suffers from large model size and intensive memory consumption; the latter has", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 335, + 470, + 347 + ], + "spans": [ + { + "bbox": [ + 141, + 335, + 470, + 347 + ], + "score": 1.0, + "content": "a relatively small model size as it shares parameters across frames; however, it", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 345, + 470, + 359 + ], + "spans": [ + { + "bbox": [ + 141, + 345, + 470, + 359 + ], + "score": 1.0, + "content": "lacks long-range dependency modeling ability and parallelizability. In this paper,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 141, + 357, + 470, + 369 + ], + "spans": [ + { + "bbox": [ + 141, + 357, + 470, + 369 + ], + "score": 1.0, + "content": "we attempt to integrate the advantages of the two cases by proposing a recurrent", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 366, + 469, + 381 + ], + "spans": [ + { + "bbox": [ + 141, + 366, + 469, + 381 + ], + "score": 1.0, + "content": "video restoration transformer, namely RVRT. RVRT processes local neighboring", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 141, + 378, + 469, + 391 + ], + "spans": [ + { + "bbox": [ + 141, + 378, + 469, + 391 + ], + "score": 1.0, + "content": "frames in parallel within a globally recurrent framework which can achieve a good", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 389, + 469, + 402 + ], + "spans": [ + { + "bbox": [ + 141, + 389, + 469, + 402 + ], + "score": 1.0, + "content": "trade-off between model size, effectiveness, and efficiency. Specifically, RVRT", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 141, + 399, + 469, + 413 + ], + "spans": [ + { + "bbox": [ + 141, + 399, + 469, + 413 + ], + "score": 1.0, + "content": "divides the video into multiple clips and uses the previously inferred clip feature to", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 141, + 411, + 469, + 423 + ], + "spans": [ + { + "bbox": [ + 141, + 411, + 469, + 423 + ], + "score": 1.0, + "content": "estimate the subsequent clip feature. Within each clip, different frame features are", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 422, + 470, + 435 + ], + "spans": [ + { + "bbox": [ + 141, + 422, + 470, + 435 + ], + "score": 1.0, + "content": "jointly updated with implicit feature aggregation. Across different clips, the guided", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 433, + 470, + 446 + ], + "spans": [ + { + "bbox": [ + 141, + 433, + 470, + 446 + ], + "score": 1.0, + "content": "deformable attention is designed for clip-to-clip alignment, which predicts multiple", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 443, + 470, + 457 + ], + "spans": [ + { + "bbox": [ + 141, + 443, + 470, + 457 + ], + "score": 1.0, + "content": "relevant locations from the whole inferred clip and aggregates their features by the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 454, + 470, + 468 + ], + "spans": [ + { + "bbox": [ + 141, + 454, + 470, + 468 + ], + "score": 1.0, + "content": "attention mechanism. Extensive experiments on video super-resolution, deblurring,", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 141, + 465, + 469, + 478 + ], + "spans": [ + { + "bbox": [ + 141, + 465, + 469, + 478 + ], + "score": 1.0, + "content": "and denoising show that the proposed RVRT achieves state-of-the-art performance", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 476, + 470, + 489 + ], + "spans": [ + { + "bbox": [ + 141, + 476, + 470, + 489 + ], + "score": 1.0, + "content": "on benchmark datasets with balanced model size, testing memory and runtime.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 487, + 441, + 501 + ], + "spans": [ + { + "bbox": [ + 141, + 487, + 441, + 501 + ], + "score": 1.0, + "content": "The codes are available at https://github.com/JingyunLiang/RVRT.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 16, + "bbox_fs": [ + 141, + 269, + 470, + 501 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 520, + 190, + 534 + ], + "lines": [ + { + "bbox": [ + 105, + 519, + 192, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 519, + 192, + 536 + ], + "score": 1.0, + "content": "1 Introduction", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + }, + { + "type": "text", + "bbox": [ + 108, + 545, + 505, + 589 + ], + "lines": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 544, + 506, + 558 + ], + "score": 1.0, + "content": "Video restoration, such as video super-resolution, deblurring, and denoising, has become a hot", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 104, + 555, + 507, + 570 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 507, + 570 + ], + "score": 1.0, + "content": "topic in recent years. It aims to restore a clear and sharp high-quality video from a degraded (e.g.,", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "downsampled, blurred, or noisy) low-quality video [80, 11, 4, 38]. It has wide applications in live", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 578, + 410, + 591 + ], + "spans": [ + { + "bbox": [ + 106, + 578, + 410, + 591 + ], + "score": 1.0, + "content": "streaming [97], video surveillance [49], old film restoration [78], and more.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 104, + 544, + 507, + 591 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 693 + ], + "lines": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "spans": [ + { + "bbox": [ + 105, + 595, + 505, + 608 + ], + "score": 1.0, + "content": "Parallel methods and recurrent methods have been dominant strategies for solving various video", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 506, + 618 + ], + "score": 1.0, + "content": "restoration problems. Typically, those two kinds of methods have their respective merits and demerits.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 505, + 629 + ], + "score": 1.0, + "content": "Parallel methods [2, 25, 80, 72, 36, 63, 99, 27, 35, 4, 38] support distributed deployment and achieve", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 627, + 505, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 627, + 505, + 641 + ], + "score": 1.0, + "content": "good performance by directly fusing information from multiple frames, but they often have a large", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "model size and consume enormous memory for long-sequence videos. In the meanwhile, recurrent", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "models [24, 59, 21, 23, 26, 28, 9, 11, 45, 55, 98, 62] reuse the same network block to save parameters", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 661, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 106, + 661, + 504, + 673 + ], + "score": 1.0, + "content": "and predict the new frame feature based on the previously refined frame feature, but the sequential", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 670, + 506, + 686 + ], + "spans": [ + { + "bbox": [ + 104, + 670, + 506, + 686 + ], + "score": 1.0, + "content": "processing strategy inevitably leads to information loss and noise amplification [14] for long-range", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 682, + 346, + 695 + ], + "spans": [ + { + "bbox": [ + 106, + 682, + 346, + 695 + ], + "score": 1.0, + "content": "dependency modelling and makes it hard to be parallelized.", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 36, + "bbox_fs": [ + 104, + 595, + 506, + 695 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "Considering the advantages and disadvantages of parallel and recurrent methods, in this paper, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "propose a recurrent video restoration transformer (RVRT) that takes the best of both worlds. On", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "the one hand, RVRT introduces the recurrent design into transformer-based models to reduce model", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "parameters and memory usage. On the other hand, it processes neighboring frames together as a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "clip to reduce video sequence length and alleviate information loss. To be specific, we first divide", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "the video into fixed-length video clips. Then, starting from the first clip, we refine the subsequent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "clip feature based on the previously inferred clip feature and the old features of the current clip from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "shallower layers. Within each clip, different frame features are jointly extracted, implicitly aligned", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "and effectively fused by the self-attention mechanism [77, 51, 39]. Across different clips, information", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 452, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 452, + 183 + ], + "score": 1.0, + "content": "is accumulated clip by clip with a larger hidden state than previous recurrent methods.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "To implement the above RVRT model, one big challenge is how to align different video clips when", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "using the previous clip for feature refinement. Most existing alignment techniques [58, 65, 59, 87,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "9, 15, 72, 80, 11, 38] are designed for frame-to-frame alignment. One possible way to apply them", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "score": 1.0, + "content": "to clip-to-clip alignment is by introducing an extra feature fusion stage after aligning all frame", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "pairs. Instead, we propose an one-stage video-to-video alignment method named guided deformable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "attention (GDA). More specifically, for a reference location in the target clip, we first estimate the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "coordinates of multiple relevant locations from different frames in the supporting clip under the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "guidance of optical flow, and then aggregate features of all locations dynamically by the attention", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 275, + 156, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 156, + 286 + ], + "score": 1.0, + "content": "mechanism.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "GDA has several advantages over previous alignment methods: 1) Compared with optical flow-based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "warping that only samples one point from one frame [59, 87, 9], GDA benefits from multiple relevant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "locations sampled from the video clip. 2) Unlike mutual attention [38], GDA utilizes features from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 322, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 337 + ], + "score": 1.0, + "content": "arbitrary locations without suffering from the small receptive field in local attention or the huge", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "computation burden in global attention. Besides, GDA allows direct attention on non-integer locations", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "with bilinear interpolation. 3) In contrast to deformable convolution [15, 100, 72, 80, 11, 10] that uses", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "a fixed weight in feature aggregation, GDA generates dynamic weights to aggregate features from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "different locations. It also supports arbitrary location numbers and allows for both frame-to-frame", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 378, + 331, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 331, + 390 + ], + "score": 1.0, + "content": "and video-to-video alignment without any modification.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 303, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 305, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 305, + 407 + ], + "score": 1.0, + "content": "Our contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28 + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 506, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "• We propose the recurrent video restoration transformer (RVRT) that extracts features of local neigh-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 115, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "boring frames from one clip in a joint and parallel way, and refines clip features by accumulating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 116, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 116, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "information from previous clips and previous layers. By reducing the video sequence length and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 117, + 448, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 117, + 448, + 504, + 459 + ], + "score": 1.0, + "content": "transmitting information with a larger hidden state, RVRT alleviates information loss and noise", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 458, + 502, + 470 + ], + "spans": [ + { + "bbox": [ + 115, + 458, + 502, + 470 + ], + "score": 1.0, + "content": "amplification in recurrent networks, and also makes it possible to partially parallelize the model.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 108, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 108, + 471, + 506, + 485 + ], + "score": 1.0, + "content": "• We propose the guided deformable attention (GDA) for one-stage video clip-to-clip alignment. It", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 115, + 484, + 447, + 496 + ], + "spans": [ + { + "bbox": [ + 115, + 484, + 447, + 496 + ], + "score": 1.0, + "content": "dynamically aggregates information of relevant locations from the supporting clip.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 110, + 497, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 110, + 497, + 507, + 511 + ], + "score": 1.0, + "content": "Extensive experiments on eight benchmark datasets show that the proposed model achieves state-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 115, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "of-the-art performance in three challenging video restoration tasks: video super-resolution, video", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 116, + 520, + 468, + 532 + ], + "spans": [ + { + "bbox": [ + 116, + 520, + 468, + 532 + ], + "score": 1.0, + "content": "deblurring, and video denoising, with balanced model size, memory usage and runtime.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "title", + "bbox": [ + 107, + 546, + 197, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 198, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 198, + 561 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 570, + 207, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 208, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 208, + 584 + ], + "score": 1.0, + "content": "2.1 Video Restoration", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "Parallel vs. recurrent methods. Most existing video restoration methods can be classified as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "parallel or recurrent methods according to their parallelizability. Parallel methods estimate all", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "frames simultaneously, as the refinement of one frame feature is not dependent on the update of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 624, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 635 + ], + "score": 1.0, + "content": "other frame features. They can be further divided as sliding window-based methods [2, 25, 80, 70,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "72, 79, 36, 63, 99, 99, 27, 71, 60, 35] and transformer-based methods [4, 38]. The former kind", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "of methods typically restore merely the center frame from the neighboring frames and are often", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "tested in a sliding window fashion rather than in parallel. These methods generally consist of four", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "stages: feature extraction, feature alignment, feature fusion, and frame reconstruction. Particularly,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 506, + 691 + ], + "score": 1.0, + "content": "in the feature alignment stage, they often align all frames towards the center frame, which leads", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 701 + ], + "score": 1.0, + "content": "to quadratic complexity with respect to video length and is hard to be extended for long-sequence", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 700, + 504, + 711 + ], + "spans": [ + { + "bbox": [ + 106, + 700, + 504, + 711 + ], + "score": 1.0, + "content": "videos. Instead, the latter kind of method reconstructs all frames at a time based on the transformer", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "architectures. They jointly extract, align, and fuse features for all frames, achieving significant", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 46.5 + } + ], + "page_idx": 1, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 302, + 741, + 309, + 750 + ], + "lines": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "spans": [ + { + "bbox": [ + 301, + 740, + 310, + 753 + ], + "score": 1.0, + "content": "2", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 182 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 506, + 86 + ], + "score": 1.0, + "content": "Considering the advantages and disadvantages of parallel and recurrent methods, in this paper, we", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 84, + 505, + 96 + ], + "score": 1.0, + "content": "propose a recurrent video restoration transformer (RVRT) that takes the best of both worlds. On", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "the one hand, RVRT introduces the recurrent design into transformer-based models to reduce model", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 106, + 506, + 118 + ], + "score": 1.0, + "content": "parameters and memory usage. On the other hand, it processes neighboring frames together as a", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "spans": [ + { + "bbox": [ + 105, + 117, + 505, + 128 + ], + "score": 1.0, + "content": "clip to reduce video sequence length and alleviate information loss. To be specific, we first divide", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "spans": [ + { + "bbox": [ + 105, + 127, + 506, + 140 + ], + "score": 1.0, + "content": "the video into fixed-length video clips. Then, starting from the first clip, we refine the subsequent", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 505, + 150 + ], + "score": 1.0, + "content": "clip feature based on the previously inferred clip feature and the old features of the current clip from", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 149, + 505, + 162 + ], + "score": 1.0, + "content": "shallower layers. Within each clip, different frame features are jointly extracted, implicitly aligned", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "spans": [ + { + "bbox": [ + 106, + 160, + 505, + 172 + ], + "score": 1.0, + "content": "and effectively fused by the self-attention mechanism [77, 51, 39]. Across different clips, information", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 171, + 452, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 171, + 452, + 183 + ], + "score": 1.0, + "content": "is accumulated clip by clip with a larger hidden state than previous recurrent methods.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 4.5, + "bbox_fs": [ + 105, + 72, + 506, + 183 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 187, + 505, + 285 + ], + "lines": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "To implement the above RVRT model, one big challenge is how to align different video clips when", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 506, + 211 + ], + "score": 1.0, + "content": "using the previous clip for feature refinement. Most existing alignment techniques [58, 65, 59, 87,", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 208, + 506, + 222 + ], + "score": 1.0, + "content": "9, 15, 72, 80, 11, 38] are designed for frame-to-frame alignment. One possible way to apply them", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 504, + 232 + ], + "score": 1.0, + "content": "to clip-to-clip alignment is by introducing an extra feature fusion stage after aligning all frame", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "spans": [ + { + "bbox": [ + 105, + 231, + 505, + 244 + ], + "score": 1.0, + "content": "pairs. Instead, we propose an one-stage video-to-video alignment method named guided deformable", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 505, + 255 + ], + "score": 1.0, + "content": "attention (GDA). More specifically, for a reference location in the target clip, we first estimate the", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 266 + ], + "score": 1.0, + "content": "coordinates of multiple relevant locations from different frames in the supporting clip under the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 263, + 506, + 277 + ], + "score": 1.0, + "content": "guidance of optical flow, and then aggregate features of all locations dynamically by the attention", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 275, + 156, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 156, + 286 + ], + "score": 1.0, + "content": "mechanism.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 14, + "bbox_fs": [ + 105, + 186, + 506, + 286 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 291, + 505, + 389 + ], + "lines": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 290, + 505, + 304 + ], + "score": 1.0, + "content": "GDA has several advantages over previous alignment methods: 1) Compared with optical flow-based", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 302, + 505, + 314 + ], + "score": 1.0, + "content": "warping that only samples one point from one frame [59, 87, 9], GDA benefits from multiple relevant", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "spans": [ + { + "bbox": [ + 106, + 312, + 505, + 324 + ], + "score": 1.0, + "content": "locations sampled from the video clip. 2) Unlike mutual attention [38], GDA utilizes features from", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 322, + 505, + 337 + ], + "spans": [ + { + "bbox": [ + 105, + 322, + 505, + 337 + ], + "score": 1.0, + "content": "arbitrary locations without suffering from the small receptive field in local attention or the huge", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "spans": [ + { + "bbox": [ + 105, + 334, + 506, + 347 + ], + "score": 1.0, + "content": "computation burden in global attention. Besides, GDA allows direct attention on non-integer locations", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "with bilinear interpolation. 3) In contrast to deformable convolution [15, 100, 72, 80, 11, 10] that uses", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 505, + 370 + ], + "score": 1.0, + "content": "a fixed weight in feature aggregation, GDA generates dynamic weights to aggregate features from", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 106, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "different locations. It also supports arbitrary location numbers and allows for both frame-to-frame", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 378, + 331, + 390 + ], + "spans": [ + { + "bbox": [ + 106, + 378, + 331, + 390 + ], + "score": 1.0, + "content": "and video-to-video alignment without any modification.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 23, + "bbox_fs": [ + 105, + 290, + 506, + 390 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 395, + 303, + 406 + ], + "lines": [ + { + "bbox": [ + 106, + 394, + 305, + 407 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 305, + 407 + ], + "score": 1.0, + "content": "Our contributions can be summarized as follows:", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 28, + "bbox_fs": [ + 106, + 394, + 305, + 407 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 414, + 506, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 506, + 427 + ], + "score": 1.0, + "content": "• We propose the recurrent video restoration transformer (RVRT) that extracts features of local neigh-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 115, + 424, + 506, + 439 + ], + "spans": [ + { + "bbox": [ + 115, + 424, + 506, + 439 + ], + "score": 1.0, + "content": "boring frames from one clip in a joint and parallel way, and refines clip features by accumulating", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 116, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 116, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "information from previous clips and previous layers. By reducing the video sequence length and", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 117, + 448, + 504, + 459 + ], + "spans": [ + { + "bbox": [ + 117, + 448, + 504, + 459 + ], + "score": 1.0, + "content": "transmitting information with a larger hidden state, RVRT alleviates information loss and noise", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 115, + 458, + 502, + 470 + ], + "spans": [ + { + "bbox": [ + 115, + 458, + 502, + 470 + ], + "score": 1.0, + "content": "amplification in recurrent networks, and also makes it possible to partially parallelize the model.", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 108, + 471, + 506, + 485 + ], + "spans": [ + { + "bbox": [ + 108, + 471, + 506, + 485 + ], + "score": 1.0, + "content": "• We propose the guided deformable attention (GDA) for one-stage video clip-to-clip alignment. It", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 115, + 484, + 447, + 496 + ], + "spans": [ + { + "bbox": [ + 115, + 484, + 447, + 496 + ], + "score": 1.0, + "content": "dynamically aggregates information of relevant locations from the supporting clip.", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 110, + 497, + 507, + 511 + ], + "spans": [ + { + "bbox": [ + 110, + 497, + 507, + 511 + ], + "score": 1.0, + "content": "Extensive experiments on eight benchmark datasets show that the proposed model achieves state-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 115, + 509, + 505, + 522 + ], + "spans": [ + { + "bbox": [ + 115, + 509, + 505, + 522 + ], + "score": 1.0, + "content": "of-the-art performance in three challenging video restoration tasks: video super-resolution, video", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 116, + 520, + 468, + 532 + ], + "spans": [ + { + "bbox": [ + 116, + 520, + 468, + 532 + ], + "score": 1.0, + "content": "deblurring, and video denoising, with balanced model size, memory usage and runtime.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 33.5, + "bbox_fs": [ + 105, + 414, + 507, + 532 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 546, + 197, + 559 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 198, + 561 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 198, + 561 + ], + "score": 1.0, + "content": "2 Related Work", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 39 + }, + { + "type": "title", + "bbox": [ + 107, + 570, + 207, + 582 + ], + "lines": [ + { + "bbox": [ + 105, + 569, + 208, + 584 + ], + "spans": [ + { + "bbox": [ + 105, + 569, + 208, + 584 + ], + "score": 1.0, + "content": "2.1 Video Restoration", + "type": "text" + } + ], + "index": 40 + } + ], + "index": 40 + }, + { + "type": "text", + "bbox": [ + 106, + 591, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 505, + 603 + ], + "score": 1.0, + "content": "Parallel vs. recurrent methods. Most existing video restoration methods can be classified as", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 601, + 505, + 614 + ], + "score": 1.0, + "content": "parallel or recurrent methods according to their parallelizability. Parallel methods estimate all", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 612, + 506, + 624 + ], + "score": 1.0, + "content": "frames simultaneously, as the refinement of one frame feature is not dependent on the update of", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 624, + 506, + 635 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 506, + 635 + ], + "score": 1.0, + "content": "other frame features. They can be further divided as sliding window-based methods [2, 25, 80, 70,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 634, + 506, + 646 + ], + "score": 1.0, + "content": "72, 79, 36, 63, 99, 99, 27, 71, 60, 35] and transformer-based methods [4, 38]. The former kind", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "of methods typically restore merely the center frame from the neighboring frames and are often", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 656, + 505, + 668 + ], + "score": 1.0, + "content": "tested in a sliding window fashion rather than in parallel. These methods generally consist of four", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 106, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "stages: feature extraction, feature alignment, feature fusion, and frame reconstruction. 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From left to right, it consists", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 105, + 190, + 505, + 202 + ], + "score": 1.0, + "content": "of shallow feature extraction, recurrent feature refinement and HQ frame reconstruction. 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Multiple refinement layers", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 221, + 234, + 232 + ], + "spans": [ + { + "bbox": [ + 106, + 221, + 234, + 232 + ], + "score": 1.0, + "content": "are stacked for better performance.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 107, + 253, + 505, + 342 + ], + "lines": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "spans": [ + { + "bbox": [ + 105, + 254, + 505, + 266 + ], + "score": 1.0, + "content": "performance improvements against previous methods. However, current transformer-based methods", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 277 + ], + "score": 1.0, + "content": "are laid up with a huge model size and large memory consumption. Different from above parallel", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 506, + 288 + ], + "score": 1.0, + "content": "methods, recurrent methods [24, 59, 21, 23, 85, 26, 28, 9, 11, 45, 55, 98, 62, 44, 5] propagate latent", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "features from one frame to the next frame sequentially, where information of previous frames is", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 310 + ], + "score": 1.0, + "content": "accumulated for the restoration of later frames. 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Unlike image restoration that mainly focuses on feature extrac-", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 356, + 506, + 369 + ], + "spans": [ + { + "bbox": [ + 106, + 356, + 506, + 369 + ], + "score": 1.0, + "content": "tion [16, 94–96, 42, 40, 41, 67, 93, 92], how to align multiple highly-related but misaligned frames is", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 505, + 380 + ], + "score": 1.0, + "content": "another key problem in video restoration. Traditionally, many methods [43, 30, 2, 48, 68, 3, 87, 9]", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "spans": [ + { + "bbox": [ + 105, + 378, + 506, + 392 + ], + "score": 1.0, + "content": "first estimate the optical flow between neighbouring frames [19, 58, 65] and then conduct image", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 389, + 507, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 507, + 402 + ], + "score": 1.0, + "content": "warping for alignment. Other techniques, such as deformable convolution [15, 100, 72, 80, 11, 4],", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 401, + 507, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 507, + 413 + ], + "score": 1.0, + "content": "dynamic filter [29] and mutual attention [38], have also been exploited for implicit feature alignment.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 18.5 + }, + { + "type": "title", + "bbox": [ + 108, + 425, + 213, + 437 + ], + "lines": [ + { + "bbox": [ + 106, + 425, + 215, + 438 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 215, + 438 + ], + "score": 1.0, + "content": "2.2 Vision Transformer", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 107, + 446, + 506, + 523 + ], + "lines": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 506, + 459 + ], + "score": 1.0, + "content": "Transformer [77] is the de-facto standard architecture in natural language processing. Recently, it", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "spans": [ + { + "bbox": [ + 105, + 457, + 506, + 470 + ], + "score": 1.0, + "content": "has been used in dealing with vision problems by viewing pixels or image patches as tokens [8,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 467, + 506, + 482 + ], + "score": 1.0, + "content": "18], achieving remarkable performance gains in various computer vision tasks, including image", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 478, + 507, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 507, + 492 + ], + "score": 1.0, + "content": "classification [18, 37, 51, 74], object detection [76, 50, 84], semantic segmentation [83, 17, 66], etc.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 104, + 489, + 507, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 507, + 503 + ], + "score": 1.0, + "content": "It also achieves promising results in restoration tasks [13, 39, 81, 44, 4, 38, 20, 22, 7, 90, 47, 73, 6].", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 501, + 507, + 513 + ], + "spans": [ + { + "bbox": [ + 106, + 501, + 507, + 513 + ], + "score": 1.0, + "content": "In particular, for video restoration, Cao et al. [4] propose the first transformer model for video SR,", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 480, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 480, + 525 + ], + "score": 1.0, + "content": "while Liang et al. [38] propose an unified framework for video SR, deblurring and denoising.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 506, + 627 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 506, + 540 + ], + "score": 1.0, + "content": "We note that some transformer-based works [101, 84] have tried to combine the concept of defor-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 506, + 552 + ], + "score": 1.0, + "content": "mation [15, 100] with the attention mechanism [77]. Zhu et al. 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Deformable convolution [15, 100] uses a learned weight", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "spans": [ + { + "bbox": [ + 106, + 487, + 505, + 499 + ], + "score": 1.0, + "content": "for feature aggregation, which can be seen as a special case of GDA, i.e., using different projection", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 496, + 506, + 510 + ], + "spans": [ + { + "bbox": [ + 105, + 496, + 135, + 510 + ], + "score": 1.0, + "content": "matrix", + "type": "text" + }, + { + "bbox": [ + 136, + 497, + 150, + 508 + ], + "score": 0.89, + "content": "P _ { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 150, + 496, + 506, + 510 + ], + "score": 1.0, + "content": "for different locations and then directly averaging the resulting features. Its parameter", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 507, + 505, + 521 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 273, + 521 + ], + "score": 1.0, + "content": "number and computation complexity are", + "type": "text" + }, + { + "bbox": [ + 273, + 507, + 297, + 518 + ], + "score": 0.88, + "content": "M C ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 507, + 316, + 521 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 316, + 507, + 355, + 520 + ], + "score": 0.92, + "content": "\\mathcal { O } ( M C ^ { 2 } )", + "type": "inline_equation" + }, + { + "bbox": [ + 356, + 507, + 505, + 521 + ], + "score": 1.0, + "content": ", respectively. In contrast, GDA uses", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 506, + 532 + ], + "score": 1.0, + "content": "the same projection matrix for all locations but generates dynamic weights to aggregate them. Its", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 529, + 506, + 543 + ], + "spans": [ + { + "bbox": [ + 105, + 529, + 308, + 543 + ], + "score": 1.0, + "content": "parameter number and computation complexity are", + "type": "text" + }, + { + "bbox": [ + 309, + 529, + 358, + 542 + ], + "score": 0.92, + "content": "( 3 + 2 R ) \\dot { C } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 358, + 529, + 376, + 543 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 376, + 530, + 474, + 542 + ], + "score": 0.9, + "content": "\\mathcal { O } ( ( 3 C + 2 R C + M ) C )", + "type": "inline_equation" + }, + { + "bbox": [ + 474, + 529, + 506, + 543 + ], + "score": 1.0, + "content": ", which", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 541, + 393, + 553 + ], + "spans": [ + { + "bbox": [ + 106, + 541, + 350, + 553 + ], + "score": 1.0, + "content": "are similar to deformable convolution when choosing proper", + "type": "text" + }, + { + "bbox": [ + 350, + 542, + 362, + 551 + ], + "score": 0.8, + "content": "M", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 541, + 380, + 553 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 380, + 542, + 389, + 551 + ], + "score": 0.82, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 389, + 541, + 393, + 553 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 474, + 506, + 553 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 567, + 191, + 581 + ], + "lines": [ + { + "bbox": [ + 104, + 565, + 193, + 584 + ], + "spans": [ + { + "bbox": [ + 104, + 565, + 193, + 584 + ], + "score": 1.0, + "content": "4 Experiments", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 35 + }, + { + "type": "title", + "bbox": [ + 107, + 592, + 215, + 604 + ], + "lines": [ + { + "bbox": [ + 104, + 590, + 217, + 608 + ], + "spans": [ + { + "bbox": [ + 104, + 590, + 217, + 608 + ], + "score": 1.0, + "content": "4.1 Experimental Setup", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 107, + 612, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "spans": [ + { + "bbox": [ + 105, + 613, + 505, + 624 + ], + "score": 1.0, + "content": "For shallow feature extraction and HQ frame reconstruction, we use 1 RSTB that has 2 swin", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 624, + 505, + 636 + ], + "score": 1.0, + "content": "transformer layers. For recurrent feature refinement, we use 4 refinement modules with a clip size", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 635, + 505, + 646 + ], + "score": 1.0, + "content": "of 2, each of which has 2 MRSTBs with 2 modified swin transformer layers. For both RSTB and", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 645, + 505, + 658 + ], + "spans": [ + { + "bbox": [ + 105, + 645, + 351, + 658 + ], + "score": 1.0, + "content": "MRSTB, spatial attention window size and head number are", + "type": "text" + }, + { + "bbox": [ + 351, + 645, + 375, + 656 + ], + "score": 0.9, + "content": "8 \\times 8", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 645, + 505, + 658 + ], + "score": 1.0, + "content": "and 6, respectively. We use 144", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 656, + 506, + 668 + ], + "score": 1.0, + "content": "channels for video SR and 192 channels for deblurring and denoising. In GDA, we use 12 deformable", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 506, + 680 + ], + "score": 1.0, + "content": "groups and 12 deformable heads with 9 candidate locations. We empirically project the query to", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 104, + 677, + 506, + 691 + ], + "spans": [ + { + "bbox": [ + 104, + 677, + 244, + 691 + ], + "score": 1.0, + "content": "a higher-dimensional space (e.g.,", + "type": "text" + }, + { + "bbox": [ + 245, + 678, + 259, + 688 + ], + "score": 0.53, + "content": "2 C", + "type": "inline_equation" + }, + { + "bbox": [ + 260, + 677, + 506, + 691 + ], + "score": 1.0, + "content": ") because we found it can improve the performance slightly", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 504, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 460, + 702 + ], + "score": 1.0, + "content": "and the parameter number of GDA is not a bottleneck. In training, we randomly crop", + "type": "text" + }, + { + "bbox": [ + 460, + 689, + 504, + 699 + ], + "score": 0.88, + "content": "2 5 6 \\times 2 5 6", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "HQ patches and use different video lengths for different datasets: 30 frames for REDS [53], 14", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "frames for Vimeo-90K [87], and 16 frames for DVD [63], GoPro [54] as well as DAVIS [31]. Adam", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "optimizer [33] with default setting is used to train the model for 600,000 iterations when the batch size", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 255, + 96 + ], + "score": 1.0, + "content": "is 8. The learning rate is initialized as", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 255, + 83, + 293, + 94 + ], + "score": 0.92, + "content": "4 \\times 1 0 ^ { - 4 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 293, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "and deceased with the Cosine Annealing scheme [52].", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "To stabilize training, we initialize SpyNet [58, 56] with pretrained weights, fix it for the first 30,000", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 293, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 268, + 118 + ], + "score": 1.0, + "content": "iterations and reduce its learning rate by", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 269, + 106, + 288, + 116 + ], + "score": 0.87, + "content": "7 5 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 289, + 105, + 293, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text", + "cross_page": true + } + ], + "index": 3 + } + ], + "index": 41.5, + "bbox_fs": [ + 104, + 613, + 506, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 72, + 505, + 117 + ], + "lines": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "spans": [ + { + "bbox": [ + 106, + 72, + 505, + 85 + ], + "score": 1.0, + "content": "optimizer [33] with default setting is used to train the model for 600,000 iterations when the batch size", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 83, + 506, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 255, + 96 + ], + "score": 1.0, + "content": "is 8. The learning rate is initialized as", + "type": "text" + }, + { + "bbox": [ + 255, + 83, + 293, + 94 + ], + "score": 0.92, + "content": "4 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 83, + 506, + 96 + ], + "score": 1.0, + "content": "and deceased with the Cosine Annealing scheme [52].", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 94, + 506, + 107 + ], + "score": 1.0, + "content": "To stabilize training, we initialize SpyNet [58, 56] with pretrained weights, fix it for the first 30,000", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 105, + 293, + 118 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 268, + 118 + ], + "score": 1.0, + "content": "iterations and reduce its learning rate by", + "type": "text" + }, + { + "bbox": [ + 269, + 106, + 288, + 116 + ], + "score": 0.87, + "content": "7 5 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 289, + 105, + 293, + 118 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 1.5 + }, + { + "type": "title", + "bbox": [ + 107, + 130, + 195, + 142 + ], + "lines": [ + { + "bbox": [ + 105, + 128, + 196, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 128, + 196, + 145 + ], + "score": 1.0, + "content": "4.2 Ablation Study", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 107, + 151, + 504, + 173 + ], + "lines": [ + { + "bbox": [ + 106, + 150, + 505, + 163 + ], + "spans": [ + { + "bbox": [ + 106, + 150, + 505, + 163 + ], + "score": 1.0, + "content": "To explore the effectiveness of different components, we conduct ablation studies on REDS [53] for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 161, + 482, + 175 + ], + "spans": [ + { + "bbox": [ + 105, + 161, + 482, + 175 + ], + "score": 1.0, + "content": "video SR. For efficiency, we reduce the MRSTB blocks by half and use 12 frames in training.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5 + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 360, + 189 + ], + "score": 1.0, + "content": "The impact of clip length. In RVRT, we divide the video into", + "type": "text" + }, + { + "bbox": [ + 360, + 177, + 370, + 187 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 177, + 506, + 189 + ], + "score": 1.0, + "content": "-frame clips. As shown in Table 1,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 188, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 506, + 200 + ], + "score": 1.0, + "content": "the performance rises when clip length is increased from 1 to 2. However, the performance saturates", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 129, + 212 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 199, + 158, + 209 + ], + "score": 0.88, + "content": "N = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 199, + 505, + 212 + ], + "score": 1.0, + "content": ", possibly due to large within-clip motions and inaccurate optical flow derivation. When", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "we directly estimate all optical flows (marked by ∗), the PSNR hits 32.21dB. Besides, to compare the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "temporal modelling ability, we hack the input LQ video (Clip 000 from REDS, 100 frames in total)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "by manually setting all pixels of the 50-th frame as zeros. As indicated in Fig. 4, on the one hand,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 135, + 253 + ], + "score": 0.89, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 243, + 444, + 255 + ], + "score": 1.0, + "content": "has a smaller performance drop and all its frames still have higher PSNR than", + "type": "text" + }, + { + "bbox": [ + 444, + 243, + 473, + 253 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "(equals", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 265 + ], + "score": 1.0, + "content": "to a recurrent model) after the attack, showing that RVRT can mitigate the noise amplification from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 417, + 277 + ], + "score": 1.0, + "content": "the hacked frame to the rest frames. One the other hand, the hacked frame of", + "type": "text" + }, + { + "bbox": [ + 417, + 265, + 446, + 275 + ], + "score": 0.9, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "has an impact", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 249, + 288 + ], + "score": 1.0, + "content": "on more neighbouring frames than", + "type": "text" + }, + { + "bbox": [ + 249, + 276, + 277, + 286 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 276, + 506, + 288 + ], + "score": 1.0, + "content": ", which means that RVRT can alleviate information loss", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 286, + 314, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 223, + 298 + ], + "score": 1.0, + "content": "and utilize more frames than", + "type": "text" + }, + { + "bbox": [ + 223, + 286, + 252, + 297 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 286, + 314, + 298 + ], + "score": 1.0, + "content": "for restoration.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 368 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "The impact of video alignment. The alignment of video clips plays a key role in our framework.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "We compare the proposed clip-to-clip guided deformable attention (GDA) with existing frame-to-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "frame alignment techniques by performing them frame by frame, followed by concatenation and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "channel reduction. As we can see from Table 2, GDA outperforms all existing methods when it is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 298, + 358 + ], + "score": 1.0, + "content": "used for frame-to-frame alignment (denoted as", + "type": "text" + }, + { + "bbox": [ + 298, + 345, + 326, + 356 + ], + "score": 0.43, + "content": "\\mathrm { G D A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "), and leads a further improvement when we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 356, + 298, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 298, + 369 + ], + "score": 1.0, + "content": "aggregate features directly from the whole clip.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "The impact of different components in GDA. We further conduct an ablation study on GDA in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "Table 3. As we can see, the optical flow guidance is critical for the model, leading to a PSNR gain of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "1.11dB. The update of optical flow in different layers can further improve the result. The channel", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "interaction in MLP also plays an important role, since the attention mechanism only aggregates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 415, + 194, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 194, + 428 + ], + "score": 1.0, + "content": "information spatially.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "The impact of deformable group and attention head. We also conduct experiments on different", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "group and head numbers in GDA. As shown in Table 4, when the deformable group rises, the PSNR", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "first rises and then keeps almost unchanged. 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Clip1233*
PSNR31.9832.1032.0732.21
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AlignmentWarping[87]TMSA [38]DCN [72]GDA*GDA
PSNR28.8830.4531.9332.0032.10
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Optical Flow GuidanceOptical Flow UpdateMLP厂√√√兴√
PSNR30.9932.0331.8332.10
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Deformable GroupAttention Head11661212122412362424
PSNR31.6332.0332.1032.1332.0332.11
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For efficiency, we reduce the MRSTB blocks by half and use 12 frames in training.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5.5, + "bbox_fs": [ + 105, + 150, + 505, + 175 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 177, + 505, + 298 + ], + "lines": [ + { + "bbox": [ + 106, + 177, + 506, + 189 + ], + "spans": [ + { + "bbox": [ + 106, + 177, + 360, + 189 + ], + "score": 1.0, + "content": "The impact of clip length. In RVRT, we divide the video into", + "type": "text" + }, + { + "bbox": [ + 360, + 177, + 370, + 187 + ], + "score": 0.82, + "content": "N", + "type": "inline_equation" + }, + { + "bbox": [ + 370, + 177, + 506, + 189 + ], + "score": 1.0, + "content": "-frame clips. As shown in Table 1,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 188, + 506, + 200 + ], + "spans": [ + { + "bbox": [ + 106, + 188, + 506, + 200 + ], + "score": 1.0, + "content": "the performance rises when clip length is increased from 1 to 2. However, the performance saturates", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 199, + 505, + 212 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 129, + 212 + ], + "score": 1.0, + "content": "when", + "type": "text" + }, + { + "bbox": [ + 130, + 199, + 158, + 209 + ], + "score": 0.88, + "content": "N = 3", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 199, + 505, + 212 + ], + "score": 1.0, + "content": ", possibly due to large within-clip motions and inaccurate optical flow derivation. When", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 210, + 505, + 222 + ], + "score": 1.0, + "content": "we directly estimate all optical flows (marked by ∗), the PSNR hits 32.21dB. Besides, to compare the", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "spans": [ + { + "bbox": [ + 105, + 220, + 506, + 233 + ], + "score": 1.0, + "content": "temporal modelling ability, we hack the input LQ video (Clip 000 from REDS, 100 frames in total)", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "spans": [ + { + "bbox": [ + 106, + 232, + 506, + 244 + ], + "score": 1.0, + "content": "by manually setting all pixels of the 50-th frame as zeros. As indicated in Fig. 4, on the one hand,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 107, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 107, + 243, + 135, + 253 + ], + "score": 0.89, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 136, + 243, + 444, + 255 + ], + "score": 1.0, + "content": "has a smaller performance drop and all its frames still have higher PSNR than", + "type": "text" + }, + { + "bbox": [ + 444, + 243, + 473, + 253 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "(equals", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 254, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 106, + 254, + 505, + 265 + ], + "score": 1.0, + "content": "to a recurrent model) after the attack, showing that RVRT can mitigate the noise amplification from", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 264, + 506, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 417, + 277 + ], + "score": 1.0, + "content": "the hacked frame to the rest frames. One the other hand, the hacked frame of", + "type": "text" + }, + { + "bbox": [ + 417, + 265, + 446, + 275 + ], + "score": 0.9, + "content": "N = 2", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 264, + 506, + 277 + ], + "score": 1.0, + "content": "has an impact", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 276, + 506, + 288 + ], + "spans": [ + { + "bbox": [ + 105, + 276, + 249, + 288 + ], + "score": 1.0, + "content": "on more neighbouring frames than", + "type": "text" + }, + { + "bbox": [ + 249, + 276, + 277, + 286 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 276, + 506, + 288 + ], + "score": 1.0, + "content": ", which means that RVRT can alleviate information loss", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 286, + 314, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 223, + 298 + ], + "score": 1.0, + "content": "and utilize more frames than", + "type": "text" + }, + { + "bbox": [ + 223, + 286, + 252, + 297 + ], + "score": 0.9, + "content": "N = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 252, + 286, + 314, + 298 + ], + "score": 1.0, + "content": "for restoration.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 177, + 506, + 298 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 302, + 505, + 368 + ], + "lines": [ + { + "bbox": [ + 106, + 301, + 506, + 314 + ], + "spans": [ + { + "bbox": [ + 106, + 301, + 506, + 314 + ], + "score": 1.0, + "content": "The impact of video alignment. The alignment of video clips plays a key role in our framework.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 312, + 506, + 326 + ], + "score": 1.0, + "content": "We compare the proposed clip-to-clip guided deformable attention (GDA) with existing frame-to-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "spans": [ + { + "bbox": [ + 105, + 323, + 506, + 336 + ], + "score": 1.0, + "content": "frame alignment techniques by performing them frame by frame, followed by concatenation and", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 335, + 505, + 347 + ], + "score": 1.0, + "content": "channel reduction. As we can see from Table 2, GDA outperforms all existing methods when it is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 345, + 506, + 358 + ], + "spans": [ + { + "bbox": [ + 105, + 345, + 298, + 358 + ], + "score": 1.0, + "content": "used for frame-to-frame alignment (denoted as", + "type": "text" + }, + { + "bbox": [ + 298, + 345, + 326, + 356 + ], + "score": 0.43, + "content": "\\mathrm { G D A } ^ { * }", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 345, + 506, + 358 + ], + "score": 1.0, + "content": "), and leads a further improvement when we", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 356, + 298, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 356, + 298, + 369 + ], + "score": 1.0, + "content": "aggregate features directly from the whole clip.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 301, + 506, + 369 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 371, + 505, + 426 + ], + "lines": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "spans": [ + { + "bbox": [ + 106, + 371, + 505, + 383 + ], + "score": 1.0, + "content": "The impact of different components in GDA. We further conduct an ablation study on GDA in", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "spans": [ + { + "bbox": [ + 106, + 382, + 505, + 394 + ], + "score": 1.0, + "content": "Table 3. As we can see, the optical flow guidance is critical for the model, leading to a PSNR gain of", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "spans": [ + { + "bbox": [ + 106, + 394, + 505, + 406 + ], + "score": 1.0, + "content": "1.11dB. The update of optical flow in different layers can further improve the result. The channel", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 105, + 403, + 505, + 417 + ], + "score": 1.0, + "content": "interaction in MLP also plays an important role, since the attention mechanism only aggregates", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 415, + 194, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 194, + 428 + ], + "score": 1.0, + "content": "information spatially.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 26, + "bbox_fs": [ + 105, + 371, + 505, + 428 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 430, + 505, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 429, + 506, + 443 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 506, + 443 + ], + "score": 1.0, + "content": "The impact of deformable group and attention head. We also conduct experiments on different", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 441, + 505, + 453 + ], + "score": 1.0, + "content": "group and head numbers in GDA. As shown in Table 4, when the deformable group rises, the PSNR", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "spans": [ + { + "bbox": [ + 106, + 453, + 505, + 464 + ], + "score": 1.0, + "content": "first rises and then keeps almost unchanged. Besides, double attention heads lead to slightly better", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "spans": [ + { + "bbox": [ + 106, + 464, + 505, + 476 + ], + "score": 1.0, + "content": "results at the expense of higher computation, but using too many heads has an adverse impact as the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 474, + 244, + 486 + ], + "spans": [ + { + "bbox": [ + 105, + 474, + 244, + 486 + ], + "score": 1.0, + "content": "head dimension may be too small.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31, + "bbox_fs": [ + 105, + 429, + 506, + 486 + ] + }, + { + "type": "table", + "bbox": [ + 109, + 519, + 250, + 538 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 504, + 243, + 514 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 502, + 243, + 516 + ], + "spans": [ + { + "bbox": [ + 105, + 502, + 243, + 516 + ], + "score": 1.0, + "content": "Table 1: Ablation study on clip length.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 34 + }, + { + "type": "table_body", + "bbox": [ + 109, + 519, + 250, + 538 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 109, + 519, + 250, + 538 + ], + "spans": [ + { + "bbox": [ + 109, + 519, + 250, + 538 + ], + "score": 0.929, + "html": "
Clip1233*
PSNR31.9832.1032.0732.21
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AlignmentWarping[87]TMSA [38]DCN [72]GDA*GDA
PSNR28.8830.4531.9332.0032.10
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Optical Flow GuidanceOptical Flow UpdateMLP厂√√√兴√
PSNR30.9932.0331.8332.10
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Deformable GroupAttention Head11661212122412362424
PSNR31.6332.0332.1032.1332.0332.11
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We additionally", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "test Vid4 [46] along with Vimeo-90K. For BD degradation, we train it on Vimeo-90K and test it on", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 432, + 507, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 507, + 445 + ], + "score": 1.0, + "content": "Vimeo-90K-T, Vid4, and UDM10 [89]. The comparisons with existing methods are shown in Table 5.", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "As we can see, RVRT achieves the best performance on REDS4 and Vid4 for both degradations.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 331, + 468 + ], + "score": 1.0, + "content": "Compared with the representative recurrent model Basic", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 331, + 455, + 363, + 466 + ], + "score": 0.8, + "content": "{ \\mathrm { V S R } } + +", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 363, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "[11], RVRT improves the PSNR by", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 198, + 478 + ], + "score": 1.0, + "content": "significant margins of", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 198, + 465, + 245, + 476 + ], + "score": 0.85, + "content": "\\mathbf { 0 . 2 { \\sim } 0 . 5 } \\mathbf { d B }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 245, + 465, + 506, + 478 + ], + "score": 1.0, + "content": ". Compared with the recent transformer-based model VRT [38],", + "type": "text", + "cross_page": true + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 477, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 506, + 488 + ], + "score": 1.0, + "content": "RVRT outperforms VRT on REDS4 and Vid4 by up to 0.36dB. The visual comparisons of different", + "type": "text", + "cross_page": true + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "methods are shown in Fig. 5. It is clear that RVRT generates sharp and clear HQ frames, while other", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 499, + 296, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 296, + 510 + ], + "score": 1.0, + "content": "methods fail to restore fine textures and details.", + "type": "text", + "cross_page": true + } + ], + "index": 17 + } + ], + "index": 50.5, + "bbox_fs": [ + 105, + 699, + 505, + 723 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 129, + 96, + 482, + 291 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 70, + 503, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 69, + 505, + 82 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 505, + 82 + ], + "score": 1.0, + "content": "Table 5: Quantitative comparison (average PSNR/SSIM) with state-of-the-art methods for video super-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 407, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 147, + 92 + ], + "score": 1.0, + "content": "resolution", + "type": "text" + }, + { + "bbox": [ + 147, + 81, + 166, + 91 + ], + "score": 0.84, + "content": "( \\times 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 81, + 407, + 92 + ], + "score": 1.0, + "content": "on REDS4 [53], Vimeo-90K-T [87], Vid4 [46] and UDM10 [89].", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 129, + 96, + 482, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 96, + 482, + 291 + ], + "spans": [ + { + "bbox": [ + 129, + 96, + 482, + 291 + ], + "score": 0.983, + "html": "
MethodBI degradationBD degradation
REDS4[53] (RGB channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)UDM10[89] (Y channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)
Bicubic26.14/0.729231.32/0.868423.78/0.634728.47/0.825331.30/0.868721.80/0.5246
SwinIR[39]29.05/0.826935.67/0.928725.68/0.749135.42/0.938034.12/0.916725.25/0.7262
SwinIR-ft [39]29.24/0.831935.89/0.930125.69/0.748836.76/0.946735.70/0.929325.62/0.7498
TOFlow [87]27.98/0.799033.08/0.905425.89/0.765136.26/0.943834.62/0.921225.85/0.7659
FRVSR[59]137.09/0.952235.64/0.931926.69/0.8103
DUF[29]28.63/0.825127.33/0.831938.48/0.960536.87/0.944727.38/0.8329
PFNL [89]29.63/0.850236.14/0.936326.73/0.802938.74/0.9627=27.16/0.8355
RBPN[23]30.09/0.859037.07/0.943527.12/0.818038.66/0.959637.20/0.945827.17/0.8205
MuCAN[36]30.88/0.875037.32/0.9465
RLSP[21]38.48/0.960636.49/0.940327.48/0.8388
TGA [27]38.74/0.962737.59/0.951627.63/0.8423
RSDN[26]39.35/0.965337.23/0.947127.92/0.8505
RRN[28]38.96/0.964427.69/0.8488
FDAN [45]39.91/0.968637.75/0.952227.88/0.8508
EDVR[80]31.09/0.880037.61/0.948927.35/0.826439.89/0.968637.81/0.952327.85/0.8503
GOVSR [88]40.14/0.971337.63/0.950328.41/0.8724
BasicVSR[9]31.42/0.890937.18/0.945027.24/0.825139.96/0.969437.53/0.949827.96/0.8553
IconVSR[9]31.67/0.894837.47/0.947627.39/0.827940.03/0.969437.84/0.952428.04/0.8570
VRT[38]32.19/0.900638.20/0.953027.93/0.842541.05/0.973738.72/0.958429.42/0.8795
BasicVSR++[11]32.39/0.906937.79/0.950027.79/0.840040.72/0.972238.21/0.955029.04/0.8753
RVRT (ours)32.75/0.911338.15/0.952727.99/0.846240.90/0.972938.59/0.957629.54/0.8810
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Method#Param (M)Memory (M)Runtime (ms)PSNR (dB)
BasicVSR++[11]7.32237732.39
BasicVSR++ [11]+RSTB[39]9.3102120132.61
EDVR[80]20.6353537831.09
VSRT[4]32.62748732831.19
VRT[38]35.6214924332.19
RVRT (ours)10.8105618332.75
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We additionally", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 422, + 506, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 506, + 434 + ], + "score": 1.0, + "content": "test Vid4 [46] along with Vimeo-90K. For BD degradation, we train it on Vimeo-90K and test it on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 432, + 507, + 445 + ], + "spans": [ + { + "bbox": [ + 105, + 432, + 507, + 445 + ], + "score": 1.0, + "content": "Vimeo-90K-T, Vid4, and UDM10 [89]. The comparisons with existing methods are shown in Table 5.", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "spans": [ + { + "bbox": [ + 106, + 444, + 506, + 456 + ], + "score": 1.0, + "content": "As we can see, RVRT achieves the best performance on REDS4 and Vid4 for both degradations.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 454, + 506, + 468 + ], + "spans": [ + { + "bbox": [ + 105, + 454, + 331, + 468 + ], + "score": 1.0, + "content": "Compared with the representative recurrent model Basic", + "type": "text" + }, + { + "bbox": [ + 331, + 455, + 363, + 466 + ], + "score": 0.8, + "content": "{ \\mathrm { V S R } } + +", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 454, + 506, + 468 + ], + "score": 1.0, + "content": "[11], RVRT improves the PSNR by", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 465, + 506, + 478 + ], + "spans": [ + { + "bbox": [ + 106, + 465, + 198, + 478 + ], + "score": 1.0, + "content": "significant margins of", + "type": "text" + }, + { + "bbox": [ + 198, + 465, + 245, + 476 + ], + "score": 0.85, + "content": "\\mathbf { 0 . 2 { \\sim } 0 . 5 } \\mathbf { d B }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 465, + 506, + 478 + ], + "score": 1.0, + "content": ". Compared with the recent transformer-based model VRT [38],", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 477, + 506, + 488 + ], + "spans": [ + { + "bbox": [ + 106, + 477, + 506, + 488 + ], + "score": 1.0, + "content": "RVRT outperforms VRT on REDS4 and Vid4 by up to 0.36dB. The visual comparisons of different", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "spans": [ + { + "bbox": [ + 105, + 487, + 506, + 500 + ], + "score": 1.0, + "content": "methods are shown in Fig. 5. It is clear that RVRT generates sharp and clear HQ frames, while other", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 106, + 499, + 296, + 510 + ], + "spans": [ + { + "bbox": [ + 106, + 499, + 296, + 510 + ], + "score": 1.0, + "content": "methods fail to restore fine textures and details.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 106, + 514, + 506, + 603 + ], + "lines": [ + { + "bbox": [ + 105, + 514, + 507, + 527 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 507, + 527 + ], + "score": 1.0, + "content": "We compare the model size, testing memory consumption and runtime of different models in Table 6.", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "spans": [ + { + "bbox": [ + 105, + 525, + 506, + 538 + ], + "score": 1.0, + "content": "Compared with representative parallel methods EDVR [80], VSRT [4] and VST [38], RVRT achieves", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 536, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 536, + 316, + 550 + ], + "score": 1.0, + "content": "significant performance gains with less than at least", + "type": "text" + }, + { + "bbox": [ + 317, + 537, + 338, + 547 + ], + "score": 0.86, + "content": "50 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 536, + 506, + 550 + ], + "score": 1.0, + "content": "of model parameters and testing memory", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 547, + 506, + 560 + ], + "spans": [ + { + "bbox": [ + 106, + 548, + 284, + 560 + ], + "score": 1.0, + "content": "usage. It also reduces the runtime by at least", + "type": "text" + }, + { + "bbox": [ + 284, + 547, + 304, + 558 + ], + "score": 0.87, + "content": "25 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 548, + 456, + 560 + ], + "score": 1.0, + "content": ". Compared the recurrent model Basic", + "type": "text" + }, + { + "bbox": [ + 456, + 548, + 484, + 558 + ], + "score": 0.4, + "content": "J \\mathrm { S R } + +", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "[11],", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "RVRT brings a PSNR improvement of 0.26dB. As for the inferiority of testing memory and runtime,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "score": 1.0, + "content": "we argue that it is mainly because the CNN layers are highly optimized on existing deep learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "frameworks. To prove it, we use the transformer-based RSTB blocks in RVRT to replace the CNN", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 171, + 603 + ], + "score": 1.0, + "content": "blocks in Basic", + "type": "text" + }, + { + "bbox": [ + 172, + 591, + 198, + 602 + ], + "score": 0.33, + "content": "/ \\mathrm { S R } { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 591, + 506, + 603 + ], + "score": 1.0, + "content": ", in which case it has similar memory usage and more runtime than our model.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "In addition, to better understand how guided deformable attention works, we visualize the predicted", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "offsets on the LQ frames and show the attention weight in Fig. 6. As we can see, multiple offsets", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "are predicted to select multiple sampled locations in the neighbourhood of the corresponding pixel.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "According to the feature similarity between the query feature and the sampled features, features of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 650, + 413, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 413, + 664 + ], + "score": 1.0, + "content": "different locations are aggregated by calculating a dynamic attention weight.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28 + }, + { + "type": "title", + "bbox": [ + 107, + 678, + 205, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 207, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 207, + 693 + ], + "score": 1.0, + "content": "4.4 Video Deblurring", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "For video deblurring, the model is trained and tested on two different datasets, DVD [63] and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "GoPro [54], with their official training/testing splits. As shown in Table 7 and 8, RVRT shows its", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32.5 + } + ], + "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": [ + 129, + 96, + 482, + 291 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 70, + 503, + 91 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 69, + 505, + 82 + ], + "spans": [ + { + "bbox": [ + 105, + 69, + 505, + 82 + ], + "score": 1.0, + "content": "Table 5: Quantitative comparison (average PSNR/SSIM) with state-of-the-art methods for video super-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 81, + 407, + 92 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 147, + 92 + ], + "score": 1.0, + "content": "resolution", + "type": "text" + }, + { + "bbox": [ + 147, + 81, + 166, + 91 + ], + "score": 0.84, + "content": "( \\times 4 )", + "type": "inline_equation" + }, + { + "bbox": [ + 166, + 81, + 407, + 92 + ], + "score": 1.0, + "content": "on REDS4 [53], Vimeo-90K-T [87], Vid4 [46] and UDM10 [89].", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 129, + 96, + 482, + 291 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 129, + 96, + 482, + 291 + ], + "spans": [ + { + "bbox": [ + 129, + 96, + 482, + 291 + ], + "score": 0.983, + "html": "
MethodBI degradationBD degradation
REDS4[53] (RGB channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)UDM10[89] (Y channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)
Bicubic26.14/0.729231.32/0.868423.78/0.634728.47/0.825331.30/0.868721.80/0.5246
SwinIR[39]29.05/0.826935.67/0.928725.68/0.749135.42/0.938034.12/0.916725.25/0.7262
SwinIR-ft [39]29.24/0.831935.89/0.930125.69/0.748836.76/0.946735.70/0.929325.62/0.7498
TOFlow [87]27.98/0.799033.08/0.905425.89/0.765136.26/0.943834.62/0.921225.85/0.7659
FRVSR[59]137.09/0.952235.64/0.931926.69/0.8103
DUF[29]28.63/0.825127.33/0.831938.48/0.960536.87/0.944727.38/0.8329
PFNL [89]29.63/0.850236.14/0.936326.73/0.802938.74/0.9627=27.16/0.8355
RBPN[23]30.09/0.859037.07/0.943527.12/0.818038.66/0.959637.20/0.945827.17/0.8205
MuCAN[36]30.88/0.875037.32/0.9465
RLSP[21]38.48/0.960636.49/0.940327.48/0.8388
TGA [27]38.74/0.962737.59/0.951627.63/0.8423
RSDN[26]39.35/0.965337.23/0.947127.92/0.8505
RRN[28]38.96/0.964427.69/0.8488
FDAN [45]39.91/0.968637.75/0.952227.88/0.8508
EDVR[80]31.09/0.880037.61/0.948927.35/0.826439.89/0.968637.81/0.952327.85/0.8503
GOVSR [88]40.14/0.971337.63/0.950328.41/0.8724
BasicVSR[9]31.42/0.890937.18/0.945027.24/0.825139.96/0.969437.53/0.949827.96/0.8553
IconVSR[9]31.67/0.894837.47/0.947627.39/0.827940.03/0.969437.84/0.952428.04/0.8570
VRT[38]32.19/0.900638.20/0.953027.93/0.842541.05/0.973738.72/0.958429.42/0.8795
BasicVSR++[11]32.39/0.906937.79/0.950027.79/0.840040.72/0.972238.21/0.955029.04/0.8753
RVRT (ours)32.75/0.911338.15/0.952727.99/0.846240.90/0.972938.59/0.957629.54/0.8810
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Method#Param (M)Memory (M)Runtime (ms)PSNR (dB)
BasicVSR++[11]7.32237732.39
BasicVSR++ [11]+RSTB[39]9.3102120132.61
EDVR[80]20.6353537831.09
VSRT[4]32.62748732831.19
VRT[38]35.6214924332.19
RVRT (ours)10.8105618332.75
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It also reduces the runtime by at least", + "type": "text" + }, + { + "bbox": [ + 284, + 547, + 304, + 558 + ], + "score": 0.87, + "content": "25 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 548, + 456, + 560 + ], + "score": 1.0, + "content": ". Compared the recurrent model Basic", + "type": "text" + }, + { + "bbox": [ + 456, + 548, + 484, + 558 + ], + "score": 0.4, + "content": "J \\mathrm { S R } + +", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 548, + 506, + 560 + ], + "score": 1.0, + "content": "[11],", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "spans": [ + { + "bbox": [ + 105, + 558, + 506, + 572 + ], + "score": 1.0, + "content": "RVRT brings a PSNR improvement of 0.26dB. As for the inferiority of testing memory and runtime,", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 105, + 568, + 506, + 583 + ], + "score": 1.0, + "content": "we argue that it is mainly because the CNN layers are highly optimized on existing deep learning", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 580, + 506, + 592 + ], + "spans": [ + { + "bbox": [ + 105, + 580, + 506, + 592 + ], + "score": 1.0, + "content": "frameworks. To prove it, we use the transformer-based RSTB blocks in RVRT to replace the CNN", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 591, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 106, + 591, + 171, + 603 + ], + "score": 1.0, + "content": "blocks in Basic", + "type": "text" + }, + { + "bbox": [ + 172, + 591, + 198, + 602 + ], + "score": 0.33, + "content": "/ \\mathrm { S R } { + + }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 591, + 506, + 603 + ], + "score": 1.0, + "content": ", in which case it has similar memory usage and more runtime than our model.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 21.5, + "bbox_fs": [ + 105, + 514, + 507, + 603 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 607, + 505, + 663 + ], + "lines": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "spans": [ + { + "bbox": [ + 105, + 606, + 505, + 620 + ], + "score": 1.0, + "content": "In addition, to better understand how guided deformable attention works, we visualize the predicted", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "spans": [ + { + "bbox": [ + 105, + 617, + 505, + 631 + ], + "score": 1.0, + "content": "offsets on the LQ frames and show the attention weight in Fig. 6. As we can see, multiple offsets", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "spans": [ + { + "bbox": [ + 105, + 628, + 506, + 642 + ], + "score": 1.0, + "content": "are predicted to select multiple sampled locations in the neighbourhood of the corresponding pixel.", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "spans": [ + { + "bbox": [ + 105, + 640, + 506, + 653 + ], + "score": 1.0, + "content": "According to the feature similarity between the query feature and the sampled features, features of", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 650, + 413, + 664 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 413, + 664 + ], + "score": 1.0, + "content": "different locations are aggregated by calculating a dynamic attention weight.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 28, + "bbox_fs": [ + 105, + 606, + 506, + 664 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 678, + 205, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 676, + 207, + 693 + ], + "spans": [ + { + "bbox": [ + 105, + 676, + 207, + 693 + ], + "score": 1.0, + "content": "4.4 Video Deblurring", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 700, + 503, + 722 + ], + "lines": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "For video deblurring, the model is trained and tested on two different datasets, DVD [63] and", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "GoPro [54], with their official training/testing splits. 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For fairness of comparison, following [70, 71], we train a non-blind", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 592, + 505, + 604 + ], + "spans": [ + { + "bbox": [ + 106, + 592, + 333, + 604 + ], + "score": 1.0, + "content": "additive white Gaussian denoising model for noise level", + "type": "text" + }, + { + "bbox": [ + 334, + 592, + 388, + 604 + ], + "score": 0.93, + "content": "\\sigma \\sim \\mathcal { U } ( 0 , 5 0 )", + "type": "inline_equation" + }, + { + "bbox": [ + 388, + 592, + 505, + 604 + ], + "score": 1.0, + "content": ". Similar to the case of video", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 602, + 506, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 602, + 233, + 616 + ], + "score": 1.0, + "content": "deblurring, there is a huge gap", + "type": "text" + }, + { + "bbox": [ + 234, + 603, + 293, + 614 + ], + "score": 0.81, + "content": "( \\mathbf { 0 . 6 0 } { \\sim } 2 . 3 7 \\mathbf { d B } ", + "type": "inline_equation" + }, + { + "bbox": [ + 293, + 602, + 506, + 616 + ], + "score": 1.0, + "content": ") between RVRT and most methods. Compared with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "spans": [ + { + "bbox": [ + 106, + 613, + 505, + 626 + ], + "score": 1.0, + "content": "VRT, RVRT has slightly better performance on large noise levels, with a smaller model size (12.8M", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 624, + 506, + 637 + ], + "spans": [ + { + "bbox": [ + 106, + 624, + 226, + 637 + ], + "score": 1.0, + "content": "v.s.18.4M) and less runtime", + "type": "text" + }, + { + "bbox": [ + 226, + 625, + 281, + 636 + ], + "score": 0.48, + "content": "( 0 . 2 \\mathrm { s } ~ \\nu . s . 1 . 5 \\mathrm { s } )", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 624, + 304, + 637 + ], + "score": 1.0, + "content": "on a", + "type": "text" + }, + { + "bbox": [ + 304, + 624, + 353, + 635 + ], + "score": 0.77, + "content": "1 2 8 0 \\times 7 2 0", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 624, + 506, + 637 + ], + "score": 1.0, + "content": "LQ input. The visual comparison is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 636, + 356, + 648 + ], + "spans": [ + { + "bbox": [ + 105, + 636, + 356, + 648 + ], + "score": 1.0, + "content": "provided in the supplementary material due to the space limit.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 19, + "bbox_fs": [ + 105, + 570, + 506, + 648 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 663, + 183, + 677 + ], + "lines": [ + { + "bbox": [ + 104, + 661, + 185, + 679 + ], + "spans": [ + { + "bbox": [ + 104, + 661, + 185, + 679 + ], + "score": 1.0, + "content": "5 Conclusion", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 107, + 689, + 505, + 722 + ], + "lines": [ + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 507, + 702 + ], + "score": 1.0, + "content": "In this paper, we proposed a recurrent video restoration transformer with guided deformable attention.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 700, + 505, + 712 + ], + "score": 1.0, + "content": "It is a globally recurrent model with locally parallel designs, which benefits from the advantages", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 106, + 711, + 505, + 723 + ], + "score": 1.0, + "content": "of both parallel methods and recurrent methods. 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MethodDBN[63]STFAN [99]STTN [32]SFE [86]EDVR[80]TSP [57]
PSNR30.0131.2431.6131.7131.8232.13
SSIM0.88770.93400.91600.91600.91600.9268
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PSNR SSIM32.31 0.926032.53 0.946832.80 0.935233.36 0.950034.24 0.965134.30 0.9655
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MethodSRN[69]MPRNet[91]MAXIM[73]IFI-RNN[55]ESTRNN[98]EDVR [80]
PSNRSSIM30.260.934232.660.959032.860.961031.050.911031.070.902331.540.9260
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Dataset0VLNB [1]DVDNet[70]FastDVDNet [71]PaCNet[75]VRT[38]RVRT (ours)
DAVIS102038.8535.6838.1338.7139.9740.8240.5738.05
35.7035.7736.8238.1538.05
3033.7334.0834.0434.7936.5236.57
405032.3231.1332.8631.8532.8233.3435.3235.47
31.8632.2034.3634.57
Set8102037.2633.7236.0836.4437.0637.8837.53
33.4933.4333.9435.0234.83
3031.7431.7931.6832.0533.3533.30
405030.3929.2430.5529.5630.4630.7032.1532.2131.33
29.5329.6631.22
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MethodBI degradationBD degradation
REDS4[53] (RGB channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)UDM10[89] (Y channel)Vimeo-90K-T[87] (Y channel)Vid4 [46] (Y channel)
Bicubic26.14/0.729231.32/0.868423.78/0.634728.47/0.825331.30/0.868721.80/0.5246
SwinIR[39]29.05/0.826935.67/0.928725.68/0.749135.42/0.938034.12/0.916725.25/0.7262
SwinIR-ft [39]29.24/0.831935.89/0.930125.69/0.748836.76/0.946735.70/0.929325.62/0.7498
TOFlow [87]27.98/0.799033.08/0.905425.89/0.765136.26/0.943834.62/0.921225.85/0.7659
FRVSR[59]137.09/0.952235.64/0.931926.69/0.8103
DUF[29]28.63/0.825127.33/0.831938.48/0.960536.87/0.944727.38/0.8329
PFNL [89]29.63/0.850236.14/0.936326.73/0.802938.74/0.9627=27.16/0.8355
RBPN[23]30.09/0.859037.07/0.943527.12/0.818038.66/0.959637.20/0.945827.17/0.8205
MuCAN[36]30.88/0.875037.32/0.9465
RLSP[21]38.48/0.960636.49/0.940327.48/0.8388
TGA [27]38.74/0.962737.59/0.951627.63/0.8423
RSDN[26]39.35/0.965337.23/0.947127.92/0.8505
RRN[28]38.96/0.964427.69/0.8488
FDAN [45]39.91/0.968637.75/0.952227.88/0.8508
EDVR[80]31.09/0.880037.61/0.948927.35/0.826439.89/0.968637.81/0.952327.85/0.8503
GOVSR [88]40.14/0.971337.63/0.950328.41/0.8724
BasicVSR[9]31.42/0.890937.18/0.945027.24/0.825139.96/0.969437.53/0.949827.96/0.8553
IconVSR[9]31.67/0.894837.47/0.947627.39/0.827940.03/0.969437.84/0.952428.04/0.8570
VRT[38]32.19/0.900638.20/0.953027.93/0.842541.05/0.973738.72/0.958429.42/0.8795
BasicVSR++[11]32.39/0.906937.79/0.950027.79/0.840040.72/0.972238.21/0.955029.04/0.8753
RVRT (ours)32.75/0.911338.15/0.952727.99/0.846240.90/0.972938.59/0.957629.54/0.8810
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Method#Param (M)Memory (M)Runtime (ms)PSNR (dB)
BasicVSR++[11]7.32237732.39
BasicVSR++ [11]+RSTB[39]9.3102120132.61
EDVR[80]20.6353537831.09
VSRT[4]32.62748732831.19
VRT[38]35.6214924332.19
RVRT (ours)10.8105618332.75
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Dataset0VLNB [1]DVDNet[70]FastDVDNet [71]PaCNet[75]VRT[38]RVRT (ours)
DAVIS102038.8535.6838.1338.7139.9740.8240.5738.05
35.7035.7736.8238.1538.05
3033.7334.0834.0434.7936.5236.57
405032.3231.1332.8631.8532.8233.3435.3235.47
31.8632.2034.3634.57
Set8102037.2633.7236.0836.4437.0637.8837.53
33.4933.4333.9435.0234.83
3031.7431.7931.6832.0533.3533.30
405030.3929.2430.5529.5630.4630.7032.1532.2131.33
29.5329.6631.22
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MethodDBN[63]STFAN [99]STTN [32]SFE [86]EDVR[80]TSP [57]
PSNR30.0131.2431.6131.7131.8232.13
SSIM0.88770.93400.91600.91600.91600.9268
MethodPVDNet [62]GSTA [64]ARVo[35]FGST [44]VRT[38]RVRT (ours)
PSNR SSIM32.31 0.926032.53 0.946832.80 0.935233.36 0.950034.24 0.965134.30 0.9655
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MethodSRN[69]MPRNet[91]MAXIM[73]IFI-RNN[55]ESTRNN[98]EDVR [80]
PSNRSSIM30.260.934232.660.959032.860.961031.050.911031.070.902331.540.9260
MethodTSP [57]PVDNet [62]GSTA [64]FGST[44]VRT[38]RVRT (ours)
PSNRSSIM31.670.927931.980.928032.100.960032.900.961034.810.972434.920.9738
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0000000000000000000000000000000000000000..8cb3c94ac6b2ea51fcedac4cb48967c5fbe9c2f2 --- /dev/null +++ b/parse/dev/Qx8lUU8CzQ/Qx8lUU8CzQ_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c716e8b4d6ce8f13f41bda87a1225f619a5e80219de9fd5cfa1887cdf8f5fef +size 12143532 diff --git a/parse/dev/RFGkzxMFqby/RFGkzxMFqby.md b/parse/dev/RFGkzxMFqby/RFGkzxMFqby.md new file mode 100644 index 0000000000000000000000000000000000000000..c6fe866c4a36eb43ba2b658aed0c25b44cf621c2 --- /dev/null +++ b/parse/dev/RFGkzxMFqby/RFGkzxMFqby.md @@ -0,0 +1,392 @@ +# ADVERSARIALLY TRAINED MODELS WITH TEST-TIME COVARIATE SHIFT ADAPTATION + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Existing defense models against adversarial examples typically provide either empirical or certified robustness. Adversarially trained models empirically demonstrate state-of-the-art defense while providing no robustness guarantees for large classifiers or higher-dimensional inputs. In contrast, a randomized smoothing framework provides state-of-the-art certification while significantly degrades the empirical performance against adversarial attacks. In this work, we propose a novel certification through adaptation technique that transforms an adversarially trained model into a randomized smoothing classifier during inference to provide certified robustness for $\ell _ { 2 }$ norm without affecting their empirical robustness against adversarial attacks. One advantage of our proposed technique is that it allows us to separately choose the appropriate noise level for certifying each test example during inference. It also leads to outperform the existing randomized smoothing models for $\ell _ { 2 }$ certification on CIFAR-10. Therefore, our work is a step towards bridging the gap between the empirical and certified robustness against adversarial examples by achieving both using the same classifier for the first time. + +# 1 INTRODUCTION + +Deep neural network (DNN) based models are found to be brittle to minor, adversarially-chosen perturbations for their inputs that remain undetectable to human eyes. A DNN classifier that correctly classifies an image $x$ , can be easily fooled by an adversarial attack to misclassify $x + \delta$ (Szegedy et al., 2014; Goodfellow et al., 2015; Madry et al., 2018). Here, $\delta$ is a minor adversarial perturbation such that the change between $x$ and $x + \delta$ remains imperceptible. + +Among the existing successful defense frameworks, adversarial training (AT) produces the best empirical robustness against the known adversarial attacks without providing any guarantee (Madry et al., 2018; Tramer & Boneh, 2019; Zhang et al., 2019; Rice et al., 2020; Gowal et al., 2020). It \` trains a DNN classifier using strong adversaries from a specific class of perturbation (e.g., a small $\ell _ { p }$ -norm) to provide robustness for the same perturbation types. Several certification techniques are proposed that can be applied to adversarially trained models to certifiably verify if the prediction of a test example, $x$ remains constant within its neighborhood (Wong & Kolter, 2018; Wang et al., 2018; Salman et al., $2 0 1 9 6$ ; Dvijotham et al., 2018; Gehr et al., 2018; Sheikholeslami et al., 2021). However, these certification techniques typically do not scale for larger networks (e.g., ResNet50) and datasets (e.g., IMAGENET). Hence, currently, we cannot guarantee that a more powerful, not yet known attack can not break these adversarially trained models. In fact, several recently proposed empirical defense models are later broken by stronger adaptive adversarial attacks, indicating the importance of investigating certified defenses with suitable robustness guarantees. + +In contrast to adversarial training, randomized smoothing provides a scalable $\ell _ { 2 }$ -certification framework for any classification model, which is robust against large isotropic Gaussian noise (Cohen et al., 2019; Salman et al., 2019a). However, the existing randomized smoothing-based certified models produce significantly lower empirical robustness compared to the AT models. On the other hand, this technique cannot be applied for AT models as they are not robust against such large random Gaussian noises in the standard settings. Towards this, we investigate to bridge the gap between the state-of-the-art empirical and certifiable robust models against adversarial examples. + +In this paper, we present a novel certification through adaptation framework to transform an AT model into a randomized smoothing framework during inference, providing $\ell _ { 2 }$ certification without any additional training or architectural modification. Our proposed certification technique consists of two steps: we first apply a covariate shift adaptation to a classifier against Gaussian noise during inference for each test example (Cariucci et al., 2017; Li et al., 2016). For our paper, we use the wellknown batch normalization adaptation. This process significantly boosts the performance of the AT models against the random isotropic Gaussian noises compared to the standard non-robust models. Hence, we can now directly apply the randomized smoothing based certification technique to provide $\ell _ { 2 }$ certification in the next step. Further, the existing randomized smoothing models require selecting the noise level at training time. In contrast, our proposed framework can separately choose the appropriate noise levels for different test examples during inference (Figure 4). Furthermore, we can also evaluate the input test examples without transforming the AT models to a randomized smoothing model, ensuring that their empirical performance remains unaffected. Therefore, we are the first to provide the test-time flexibility to obtain empirically robust predictions as well as certify their predictions using the same classifier for high-dimensional datasets to the best of our knowledge. Hence, we improve the reliability of AT models sensitive real-world applications. + +Table 1: CIFAR-10: Certified accuracy at various $\ell _ { 2 }$ radii and ACR scores. We train different models by varying the hyper-parameters for SmoothAdv, $\mathrm { \ A d v _ { 2 } }$ and $\mathbf { A d v } _ { \infty }$ (as in (Salman et al., 2019a)) and by choosing $\sigma = \{ 0 . 2 5 , 0 . 5 , 0 . 7 5 \}$ for test-time adaptation to obtain the maximum certified radii for each test example. See Table 5 and 6 (Appendix) for detailed results on both IMAGENET and CIFAR-10 respectively. We also present the best reported results for MARCER and Consistancy at $\sigma = 0 . 5$ , obtained from their respective papers. + +
l2Radius (CIFAR-10)0.250.50.751.01.251.51.752.0ACR
Baseline6.962.040.090.00.00.00.00.00.026
Randg =0.5 (Cohen et al.,2019)51.6840.3830.2520.8113.367.713.380.00.488
(Ours)Randg=0.5 +adaptation62.9152.2540.0625.5717.4310.675.461.920.657
SmoothAdvg=0.5 (Salman et al.,2019a)58.8249.6842.6837.5532.6427.5222.420.00.918
(Ours) SmoothAdvg=0.5+adaptation59.8950.441.7635.530.9226.120.2515.051.008
Advo (Rice et al.,2020)35.9529.4423.510.00.00.00.00.00.317
(Ours)Advo + adaptation67.9655.0643.2730.5524.6818.4912.118.450.903
Adv2 (Rice et al., 2020)41.8934.1526.70.00.00.00.00.00.359
(Ours) Adv2 +adaptation68.8458.7749.7137.7433.3728.8223.6518.231.198
MARCERg=0.5 (Zhai et al.,2020)60.053.046.038.029.019.012.00.00.726
Consistancyg=0.5 (Jeong& Shin,2020)48.945.141.337.833.929.925.20.00.726
+ +# Contributions:- + +1. We propose a novel certification through adaptation framework that can adapt an AT model during inference to provide certified robustness. Our experimental results on CIFAR-10 and IMAGENET demonstrate that the proposed certification framework can transform any AT model into a randomized smoothing classifier to provide certification for $\ell _ { 2 }$ norm, even when the model is learned using $\ell _ { \infty }$ -bounded adversaries (Table 1 & Figure 2). +2. One main advantage of our proposed framework is that it allows us to select appropriate noise levels for different test examples during inference. This leads to outperforming the existing state-of-the-art randomized smoothing models for $\ell _ { 2 }$ certification on CIFAR-10 using AT models (Table 1 & Figure 4). Further, we can provide certification at larger $\ell _ { 2 }$ radii for existing randomized smoothing models, improving their overall average certified radius (ACR). +3. Our results also indicate a strong correlation between empirical and certified robustness than previously believed (Cohen et al., 2019; Salman et al., $2 0 1 9 \mathrm { a }$ ; Tramer & Boneh, 2019). \` In particular, we observe that the empirically stronger AT models lead to better $\ell _ { 2 }$ certification performance (Figure 5). + +# 2 RELATED WORK + +Empirical Defenses and Adversarial Training. Existing defense models against adversarial attacks can be broadly classified into empirical and certified defenses. Empirical defenses demonstrate empirical robustness against adversarial attacks (Schott et al., 2019; Moosavi Dezfooli et al., 2019; Nandy et al., 2020; Mao et al., 2021). Adversarial training achieves the state-of-the-art empirical defense (Madry et al., 2018). It optimizes the following loss function for a DNN classifier, $f$ , to provide robustness within an $\epsilon$ -bounded threat model for an $\ell _ { p }$ norm, where the perturbations, $\delta \in \Delta$ are constrained as $\Delta = \{ \delta : | | \delta | | _ { p } \leq \epsilon \}$ : + +$$ +\operatorname* { m i n } _ { \theta } \mathbb { E } _ { ( x , y ) } [ \operatorname* { m a x } _ { \delta \in \Delta } \mathcal { L } ( f _ { \theta } ( x + \delta ) , y ) ] +$$ + +where, $\theta$ denotes the model parameters. $\mathcal { L }$ is the classification loss. + +The inner maximization in Eq. 1 is solved by producing adversarial examples using strong iterative adversaries, e.g., projected gradient descent $( P G D )$ attack (Kurakin et al., 2016; Madry et al., 2018). Wong et al. (2020) found that even a single-step fast gradient sign method (FGSM) attack-based AT models also achieves high empirical robustness (Goodfellow et al., 2015). Zhang et al. (2020) proposed to use the least adversaries for training. Recently Trades (Zhang et al., 2019), Adv-LLR (Qin et al., 2019) introduced additional regularizers to achieve higher empirical robustness by smoothing the loss surface. However, Rice et al. (2020) showed that the standard PGD based AT model with early-stopping criteria provides one of the best empirical defenses for a given perturbation type. Recent works also explored the importance of different hyper-parameters for adversarial training (Gowal et al., 2020; Pang et al., 2021) as well as incorporating additional data in a semi-supervised fashion (Carmon et al., 2019; Uesato et al., 2019) to further improve their empirical robustness. + +Certified Defenses. Empirical defenses demonstrate robustness only against the known adversaries without providing any guarantees. In fact, most empirical defenses proposed in the literature were later broken by stronger adversaries, highlighting the importance of certified defenses to provide robustness guarantees (Athalye et al., 2018; Uesato et al., 2018; Jalal et al., 2019). + +Several recent works proposed to train neural network models with provable robustness guarantees. These works include methods based on semi-definite relaxations (Raghunathan et al., 2018), linear relaxations and duality (Wong & Kolter, 2018; Wong et al., 2018), abstract interpretation (Mirman et al., 2018), and interval bound propagation (Gowal et al., 2018). Parallel to training a certified defense, several works also focus on certifying the already trained models (Tjeng et al., 2017; Gehr et al., 2018; Weng et al., 2018; Wang et al., 2018; Bunel et al., 2018). Recently Mueller et al. (2021) combined a small certification network with a large, empirically robust AT model using some selection criteria to boost overall benign accuracy along with empirical robustness for the certified framework. However, none of these techniques scale for large networks (e.g., ResNet50) or higher-dimensional datasets (e.g., IMAGENET). + +Randomized Smoothing for Certification. A randomized smoothing classifier is not a neural network. It uses a neural network as its base for classification. Randomized smoothing was initially proposed as a heuristic defense (Cao & Gong, 2017; Liu et al., 2018) and later shown to be certifiable (Lecuyer et al., 2019; Li et al., 2019). Recently, Cohen et al. (2019) and Salman et al. (2019a) separately provided a tight robustness guarantee for $\ell _ { 2 }$ -norm. Salman et al. (2019a) provides the current state-of-the-art $\ell _ { 2 }$ certification robustness by adversarially choosing the noise using an adaptive attack to train their base classifier. This framework is also analyzed for other $\ell _ { p }$ norms using different noise distributions as well (Li et al., 2019; Lee et al., 2019; Dvijotham et al., 2020; Yang et al., 2020). Salman et al. (2020) proposed to incorporate an additional denoising module as a preprocessing unit to convert a standard DNN classifier into a randomized smoothing model to provide non-trivial certified robustness. Notably, randomized smoothing is the only scalable certification framework. Further, it also achieves superior performance for different perturbation types. + +While achieving the state-of-the-art certification performance, randomized smoothing significantly degrades the empirical robustness against adversarial attacks (Lecuyer et al., 2019; Salman et al., 2019a; Cohen et al., 2019). Towards this, our proposed technique transforms an AT model into a randomized smoothing classifier without any additional training or architectural modification. Since AT models already provide the state-of-the-art empirical defense, we achieve both empirical and certified robustness against adversarial examples using the same classifier. + +# 3 PROPOSED METHODOLOGY + +In this section, we first present the background of the randomized smoothing technique and explain why it is not directly effective for AT models. Next, we present the existing test-time co-variate shift adaptation for domain adaptations and corruption robustness. Then, we present our proposed certification through adaptation framework that adapts a DNN model during inference to provide certified robustness without additional training or architectural modification. + +# 3.1 BACKGROUND ON RANDOMIZED SMOOTHING + +Consider a classifier $f$ that maps inputs in $\mathbb { R } ^ { d }$ to $\mathcal { V }$ classes. The randomized smoothing framework transforms the original base classifier $f$ into a new, smoothed classifier $g$ . In particular, for an + +input $x \in \mathbb { R } ^ { d }$ , the smoothed classifier $g$ returns the most probable class to be predicted by the base classifier $f$ under isotropic Gaussian noises of $x$ . That is, + +$$ +g ( x ) = a r g \operatorname* { m a x } _ { y \in \mathcal { V } } \mathbb { P } ( f ( x + \delta ) = = y ) \qquad { \mathrm { w h e r e } } , \delta \sim { \mathcal { N } } ( 0 , \sigma ^ { 2 } I ) . +$$ + +The noise level, $\sigma$ controls the trade-off between robustness and accuracy: Increasing $\sigma$ would improve the robustness of $g$ at higher $\ell _ { 2 }$ radii. However, it degrades the robustness at lower $\ell _ { 2 }$ radii as well as the benign accuracy. + +Cohen et al. (2019) presented a tight robustness guarantee based on the Neyman-Pearson lemma for the smoothed classifier $g$ and gave an efficient algorithm using Monte Carlo sampling for certifying of $g$ . We can also obtain this guarantee alternatively by explicitly computing the Lipschitz constant of the smoothed classifier as shown in (Salman et al., 2019a; Yang et al., 2020). The certification procedure is as follows: Suppose a base classifier $f$ classifies $\sqrt { ( x , \sigma ^ { 2 } I ) }$ to return the “most probable” class, $c _ { A }$ with probability $p _ { A } = \mathbb { P } ( f ( x + \delta ) = = c _ { A } )$ ) and the “runner-up” class $c _ { B }$ with probability $\begin{array} { r } { p _ { B } = \operatorname* { m a x } _ { y \neq c _ { A } } \mathbb { P } ( f ( x + \delta ) = = y } \end{array}$ ). Then, the smooth classifier, $g$ is certifiably robust around $x$ within an $\ell _ { 2 }$ radius of $R$ : + +$$ +R = { \frac { \sigma } { 2 } } \Bigl ( \Phi ^ { - 1 } ( p _ { A } ) - \Phi ^ { - 1 } ( p _ { B } ) \Bigr ) +$$ + +where, $\Phi ^ { - 1 }$ is the inverse of the standard Gaussian CDF. + +However, computing the exact values of $p _ { A }$ and $p _ { B }$ is not possible in practice when $f$ is a DNN. Cohen et al. (2019) addressed this problem using Monte Carlo sampling to estimate some $\underline { p _ { A } }$ and $\overline { { p _ { B } } }$ such that $p _ { A } \leq p _ { A }$ and ${ \overline { { p _ { B } } } } \geq p _ { B }$ with arbitrarily high probability. The certified radius for input $x$ is then computed by replacing $p _ { A }$ and $p _ { B }$ with $\underline { p _ { A } }$ and $\overline { { p _ { B } } }$ respectively in Eq. 3. + +As we can see in Equation 2 that the original base classifier, $f$ needs to be robust against large Gaussian noises to provide non-trivial robustness certification results. Otherwise, it leads to lower $p _ { A }$ and hence a lower certification of $R$ for the test examples. Existing randomized smoothing-based models applies custom-trained using explicit Gaussian noises to learn their original base classifier (Lecuyer et al., 2019; Cohen et al., 2019; Salman et al., 2019a; Zhai et al., 2020; Jeong & Shin, 2020). However, these models produce significantly lower empirical robustness compared to the AT models. Consequently, AT models are not robust against large Gaussian noises in the standard inference settings (see Table 2). Hence, we cannot directly use them as the base classifier for randomized smoothing. + +# 3.2 BACKGROUND ON COVARIATE SHIFT ADAPTATION + +Recent works on (Sun et al., 2017; Roy et al., 2019; Huang et al., 2018; Li et al., 2016) and corruption robustness (Schneider et al., 2020; Nado et al., 2020; Benz et al., 2021) demonstrate the importance of unsupervised covariate shift adaptation. We use adaptive batch-normalization (BN), one of the most popular and effective unsupervised covariate shift adaptation mechanisms. + +A BN layer computes the mean and variance of the hidden activation maps across the channels to normalize these activations to $\mathcal { N } ( 0 , 1 )$ before feeding into the next hidden layer (Ioffe & Szegedy, 2015). It reduces the dependencies among different hidden layers, improving the training efficiency for deep architectures. Hence, most of the recent DNN architectures frequently incorporate BN layers for complex machine learning tasks. However, the distributional shifts in the test examples lead to different activation statistics compared to the training examples. Hence, impacted by the covariate shift, the statistics estimated during training fail to normalize the activation tensors to $\mathcal { N } ( 0 , 1 )$ . As a result, it breaks the crucial assumption for the subsequent hidden layers to work. + +More formally, let $P _ { T } : \mathcal { X } \times \mathcal { Y } \mathbb { R } ^ { + }$ as the training distribution and $P _ { t } : \mathcal { X } \times \mathcal { Y } \mathbb { R } ^ { + }$ as the test distribution; where $x \in \mathcal { X }$ are inputs and $y \in \mathcal { V }$ are the corresponding class labels. There exists covariate shift between training and test distribution iff: $P _ { T } ( y | \bar { x } ) { = } P _ { t } ( \bar { y } | x )$ and $P _ { T } ( x ) \neq P _ { t } ( x )$ (Sugiyama & Kawanabe, 2012; Scholkopf et al., 2012). If the covariate shift only affects the first ¨ and second-order moments of the hidden layer feature activations, $f _ { h } ( x )$ , we can remove it using normalization (Schneider et al., 2020): + +$$ +P _ { T } \Big ( \frac { f _ { h } ( x ) - \mathbb { E } _ { T } [ f _ { h } ( x ) ] } { \sqrt { \mathbb { V } _ { T } [ f _ { h } ( x ) ] } } \Big ) P _ { T } ( x ) \approx P _ { t } \Big ( \frac { f _ { h } ( x ) - \mathbb { E } _ { t } [ f _ { h } ( x ) ] } { \sqrt { \mathbb { V } _ { t } [ f _ { h } ( x ) ] } } \Big ) P _ { t } ( x ) . +$$ + +Covariate shift adaptation using adaptive BN computes the BN statistics from the feature activations, $\mu _ { t } , s _ { t } ^ { 2 }$ , of the test batch. We can adapt them with the existing training statistics, $\mu _ { T } , s _ { T } ^ { 2 }$ , obtained using the training batches as (Cariucci et al., 2017; Li et al., 2016; Schneider et al., 2020): + +$$ +\overline { { \mu } } = \rho \cdot \mu _ { t } + ( 1 - \rho ) \cdot \mu _ { T } \quad \overline { { s } } = \rho \cdot s _ { t } + ( 1 - \rho ) \cdot s _ { T } +$$ + +where, $\rho \in [ 0 , 1 ]$ is the momentum. The choice of $\rho = 0$ is equivalent to the standard inference setup with a deterministic DNN classifier in the IID settings. We should choose $\rho = 1$ when receiving larger test batches as it can provide a better estimation of the test distributions. + +Assumptions for BN adaptation. It is noteworthy that these existing adaptive BN-based frameworks require a large set of test images from the same covariate shift to estimate the BN parameters. However, this assumption may not hold for several real-world applications, e.g., stateless web APIs. Also, these test images should be semantically diverse, preferably over multiple classes, to effectively estimate the test distributions. Hence, it further limits the practical usability of these frameworks for real-world applications, e.g., autonomous cars. + +In contrast to these models for domain adaptation and corruption robustness, our proposed certification framework against adversarial examples does not make any such assumptions. In this case, we already know the perturbation type on which we need to adapt the model to provide the certification. Hence, we can explicitly pre-select a diverse set of clean images, ${ \bf X } _ { b a t c h }$ and control the perturbations to adapt the models, addressing both of these limitations. + +# Algorithm 1: Steps for CERTIFICATION THROUGH ADAPTATION Algorithm + +Input: $f$ : classifier, $x _ { t e s t }$ : test example, $\sigma$ : desired noise-level, ${ \bf X } _ { b a t c h }$ : set of clean images (preselected from validation data or test stream). Output: Certifiably robust $\ell _ { 2 }$ radius of $R$ for $x _ { t e s t }$ . /\* Step 1: Adapt BN parameters using ${ \bf X } _ { b a t c h }$ with $\rho = 1$ (Eqn 5). \*/ 1 $\tilde { \mathbf { X } } _ { b a t c h } = [ x + \mathcal { N } ( 0 , \sigma I ) \ \forall \ x \in \ \mathbf { X } _ { b a t c h } ]$ // perturb ${ \bf X } _ { b a t c h }$ with desired noise. 2 $f _ { a d a p t } = \mathrm { C L O N E } ( f . t r a i n ( ) )$ // clone $f$ with train-mode. 3 $\underline { { \mathbf { \Pi } } } _ { - } = f _ { a d a p t } ( \tilde { \mathbf { X } } _ { b a t c h } )$ // forward pass for BN parameter adaptation. 4 fadapt.eval() // fix the parameters. /\* Step 2: Certify $x _ { t e s t }$ using Randomized Smoothing framework. \*/ 5 $g =$ GETRANDOMIZEDMODEL(fadapt) // Convert fadapt to randomized-smoothing classifier $g$ (Eqn 2). 6 $R = { \bf C E R T I F Y } ( g , x _ { t e s t } , \sigma )$ // Execute 3 for $\ell _ { 2 }$ certification. 7 return $R$ + +# 3.3 PROPOSED CERTIFICATION THROUGH ADAPTATION + +The robustness guarantee in Eq. 3 suggests that randomized smoothing gives a framework for certifying any classifier $f$ that is robust against large Gaussian noises. Previous works proposed customized training using explicit Gaussian noise augmentation for their training (Section 3.1). Subsequently, in Section 3.2 we note that robustness against random Gaussian noises of any classifier, $f$ can be improved by applying covariate shift adaptation using adaptive BN technique without any additional training. However, it modifies the original base classifier $f$ at each forward pass by recomputing the BN parameters. Since the certification guarantee in Eq. 3 is provided only for a fixed base classifier $f$ , we cannot directly apply adaptive BN to provide $\ell _ { 2 }$ certification using the randomized smoothing framework. This motivates us to propose a novel certification framework that applies the covariate shift adaptation using adaptive BN as an offline pre-processing step to improve the robustness against random Gaussian noises, addressing the above problem. + +Our proposed certification through adaptation framework consists of two steps: Given test image $x _ { t e s t }$ , we first apply the adaptive BN technique to achieve robustness against Gaussian perturbations. Recall that adaptive BN requires a large set of diverse test images to correctly re-estimate the batch-normalization statistics. However, to provide certification for $\ell _ { 2 }$ -norm, we only need to adapt our model against Gaussian perturbations. Hence, we can pre-select a sufficiently large set of diverse clean images, ${ \bf X } _ { b a t c h }$ and apply Gaussian perturbations to adapt our classifier, $f$ , as an offline pre-processing step to obtain $f _ { a d a p t }$ . Alternatively, when a large set of diverse test examples are available, we can also use them for our BN adaptation. The Gaussian noise samples should be drawn from the same isotropic Gaussian distribution ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } I )$ as we need to use for the certification process. Then, we freeze the model parameters and use the adapted model, $f _ { a d a p t }$ , as our base classifier to certify the test example, $x _ { t e s t }$ . Hence, the base classifier $f _ { a d a p t }$ remains fixed during calculating the certification radius $R$ (Equation 3). Our proposed certification through adaptation technique is presented in Algorithm 1. + +Advantages. The main advantage of our proposed framework is that we can adapt the classifier, $f$ at any noise level $\sigma$ as an offline pre-processing step, without any additional training (see Figure 4). As we can see in Equation 3, that we should select a large $\sigma$ to certify at a bigger $\ell _ { 2 }$ radius of $R$ . However, a test image that does not remain robust at higher $\sigma$ produces a lower value of $p _ { A }$ . It leads to reducing the overall certification radius, $R$ . Hence, providing the flexibility of choosing appropriate noise levels for different test examples allows us to improve the certification radius, $R$ . + +In contrast to our proposed framework, existing randomized smoothing frameworks cannot choose a different $\sigma$ at test-time since it typically degrades their overall certification performance. Hence, they need to fix $\sigma$ during training their base models or its components. + +Applicability. Our proposed certification through adaptation technique can be applied to any classification model, $f$ with batch-normalization layers. However, note that achieving high accuracy against large random Gaussian perturbations is only a necessary condition: a randomized smoothing classifier, $g$ requires to consistently predict the correct class to provide higher certification guarantees at larger radii. Hence, we achieve non-trivial $\ell _ { 2 }$ certification guarantees at very small $\ell _ { 2 }$ radii for standard non-robust DNN classifiers (see Appendix B.1). + +On the other hand, for existing randomized smoothing models, we achieve higher certification at larger $\ell _ { 2 }$ radii by adapting their base models with larger $\sigma$ , improving their overall average certified radius $( A C R )$ (Table 6 and 5 (Appendix)). However, we could not find any $\sigma$ to obtain a significant improvement at lower $\ell _ { 2 }$ radii. In contrast, AT models with our proposed offline adaptation technique significantly improve their performance against large Gaussian perturbations, providing non-trivial certification robustness. Experimentally we find that our proposed technique outperforms the state-of-the-art certification models for the $\ell _ { 2 }$ norm. + +Finally, while we focus on adaptive BN, there also exists other unsupervised covariate shift adaptation techniques such as self-supervised domain adaptation on single test examples (Sun et al., 2020), pseudo-labeling (French et al., 2017; Xie et al., 2020) etc. Wang et al. (2020) also proposed to update the normalization parameters by entropy minimization to improve the corruption robustness. Future studies may also explore these techniques for the offline pre-processing step. + +# 4 EXPERIMENTS + +Experimental setup. We use CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng et al., 2009) datasets for our experiments. For CIFAR-10, we use pre-activation ResNet18 and ResNet50 for IMAGENET (He et al., 2016a;b). Our AT models are trained using early stopping criteria (Rice et al., 2020) as follows: For IMAGENET, we use two AT models, $\bar { \mathrm { A d v } } _ { \infty } [ \ell _ { \infty } \overset { \_ } { \le } 4 / 2 5 5 ]$ and $\mathrm { A d v } _ { 2 } [ \ell _ { 2 } \leq 3 ]$ , learned at $\ell _ { \infty }$ and $\ell _ { 2 }$ threat models with threat boundaries of $4 / 2 5 5$ and 3 respectively. For CIFAR-10, we train multiple AT models with different threat boundaries. For example, we denote $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \leq 8 / 2 5 5 ]$ and $\mathrm { A d v _ { 2 } } [ \ell _ { 2 } \leq 1 ]$ as the AT models for $\ell _ { \infty }$ and $\ell _ { 2 }$ threat models, trained with threat boundaries of $8 / 2 5 5$ and 1, respectively. We compare with Baseline and $\mathrm { R a n d } _ { \sigma = 0 . 5 }$ models. Baseline models are trained using clean images. $\mathrm { R a n d } _ { \sigma = 0 . 5 }$ models are trained by augmenting random noise, sampled from isotropic Gaussian distribution, ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } I )$ with $\sigma = 0 . 5$ . We also compare with the current state-of-the-art certification models, SmoothAdv for CIFAR-10 (Salman et al., 2019a). Please refer to Appendix A for more details. + +# 4.1 PERFORMANCE UNDER GAUSSIAN NOISE. + +We first investigate the performance of different classification models under significantly larger Gaussian perturbations. It is a necessary condition to provide $\ell _ { 2 }$ robustness certification. In Table 2, we present the performance. We observe that when the test examples are sampled from IID settings as training distributions (i.e., $\sigma = 0$ for Baseline, $\mathbf { A d v } _ { \infty }$ , and $\mathrm { \ A d v _ { 2 } }$ and $\sigma = 0 . 5$ for $\mathrm { R a n d } _ { \sigma = 0 . 5 } )$ , these models produces the best results regardless of whether BN adaptation is applied. However, as we move away from the IID settings by increasing (or decreasing) $\sigma$ , the performance of all these models significantly degrades in the standard inference setup. In contrast, covariate shift adaptation using adaptive BN improves the performance for all models. In particular, AT models achieve significantly higher performance gain using adaptive BN than the non-robust baseline models at higher noise levels. For example, at $\sigma = 0 . 5$ , Baseline, $\mathrm { A d v _ { 2 } } [ \ell _ { 2 } \leq 3 ]$ and $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \leq 4 / 2 5 5 ]$ respectively achieve top-1 accuracy of $0 . 3 \%$ , $0 . 4 \%$ , and $0 . 9 \%$ for IMAGENET without using BN adaptation (Table 2 (a)). However, adaptive BN for $\mathrm { A d v } _ { 2 } [ \ell _ { 2 } \leq 3 ]$ and $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \leq 4 / 2 5 5 ]$ significantly improves the top-1 accuracy to $4 7 . 3 \%$ and $4 4 . 9 \%$ respectively. In contrast, the baseline model only achieves $7 . 7 \%$ accuracy. We observe similar results for CIFAR-10 in Table 2 (b). + +
(a) IMAGENET
Modelσ=0σ=0.25σ=0.5σ=0.75
Baseline75.2±0.011.8±0.220.3±0.010.1±0.0
+ adaptive BN74.4±0.0431.0±0.277.7±0.242.4±0.01
Advo∞≤4/255]62.8±0.03.9±0.030.4±0.010.2±0.01
+ adaptive BN60.8±0.1653.4±0.1544.9±0.0833.7±0.28
Adv2≤3]59.8±0.09.8±0.080.9±0.010.3±0.0
+adaptive BN58.3±0.0853.7±0.1447.3±0.1439.8±0.18
Rand g=0.522.0±0.032.8±0.1160.9±0.040.9±0.06
+ adaptive BN62.7±0.0362.3±0.1859.5±0.1151.4±0.27
+ +
(b) CIFAR-10
Modelσ=0g=0.25σ=0.5g=0.75
Baseline + adaptive BN95.2±0.010.9±0.8810.6±0.7610.5±1.19
95.0±0.5740.1±0.9722.0±0.8317.2±0.66
Advo≤8/255]82.1±0.040.2±4.5616.1±7.8512.2±5.23
+ adaptive BN81.6±0.9674.2±0.9562.4±0.6451.0±1.03
Adv2[≤1]81.6±0.047.5±5.121.5±7.7914.3±5.63
+ adaptive BN81.8±0.775.8±0.4364.9±0.7353.5±1.71
Rand g=0.566.7±0.069.1±1.0161.2±0.8425.9±1.41
+ adaptive BN74.0±2.173.0±2.0466.8±2.0156.7±0.94
+ +![](images/fd7d6a2b75a66244db051455b16e7df86066ba1cd535282d32891358d59c5cad.jpg) +Table 2: Top-1 accuracy of different classifiers under different levels of Gaussian noises augmented to the test images. We randomly shuffle test images and sample the noises and report $( m e a n \pm 2 \times s d )$ ) of five runs. +Figure 1: Visualizing loss-gradients produced by AT models as we apply different levels of Gaussian noises. + +Loss Gradients under Gaussian Noises. To further investigate the performance of AT models, we visualize the loss gradients for individual pixels of an image as we increase the Gaussian noise (i.e., $\sigma$ ) (Figure 1). Loss-gradients reflect the most relevant input pixels for classification predictions. Here, we scale, translate and clip the loss-gradient values without using any sophisticated techniques (as suggested in Tsipras et al. (2019)). At $\sigma = 0$ (i.e., for clean images), the loss-gradients from AT models align properly with perceptually relevant features (as observed previously (Tsipras et al., 2019; Etmann et al., 2019)). However, as we choose higher noise using $\sigma { = } 0 . 5$ and $\sigma { = } 0 . 7 5$ , the overall loss gradients become noisier. Specifically, AT models without adaptation produce sharper loss gradients (i.e., greater importance) even for background pixels. In contrast, test-time BN adaptation produces gradients for the pixels from the object of interest and suppress the gradients for background pixels (see Figure 1(c) and Figure 1(d)). Hence, they extract the required semantic information for correct classifications. It is interesting to note that $\mathbf { A d v } _ { 2 }$ produces significantly more human-aligned loss gradients compared to $\mathrm { \bf A d v _ { \infty } }$ . This behavior is also reflected in their classification (Table 2) and overall certification (Table 1) as we note that $\mathrm { \bf A d v } _ { 2 }$ overall produces much better performance compared to $\mathbf { A d v } _ { \infty }$ . + +# 4.2 CERTIFICATION USING RANDOMIZED SMOOTHING + +We now present the $\ell _ { 2 }$ certification results using the randomized smoothing framework as the backbone, as proposed in our Algorithm 1. We certify the test images with $9 9 . 9 \%$ probability. We estimate the class label probabilities of $g$ (in Equation 3) using Monte-Carlo sampling with 100, 000 noisy samples for each test image, as in Cohen et al. (2019); Salman et al. (2019a). We use the full test-set for CIFAR-10 and a sub-sample of 500 test images for IMAGENET (as in Cohen et al. (2019)). We provide the detailed results of certified accuracy along with average certified radius $( A C R )$ for several models, trained using different specifications and adapting with different $\sigma$ in Table 5 and 6 (Appendix). + +Certifying AT models. In Figure 2, we first demonstrate that AT models can provide non-trivial $\ell _ { 2 }$ certified robustness using our proposed framework for both IMAGENET and CIFAR-10 datasets. Here, we use $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \ \le \ 4 / 2 5 5 ]$ and $\mathrm { A d v _ { 2 } } [ \ell _ { 2 } ~ \le ~ 3 ]$ for ImageNet and $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \ \le \ 8 / 2 5 5 ]$ and $\mathrm { A d v _ { 2 } } [ \ell _ { 2 } ^ { \mathbf { \bar { \rho } } } \leq 1 ]$ for CIFAR-10 and use $\sigma =$ 0.5 for adaptation and certification using Algorithm 1. We compare with the certification results of Baseline, $\mathrm { \bf A d v _ { \infty } }$ and $\mathrm { \bf A d v } _ { 2 }$ models in the standard settings, without using any adaptation and certified at $\sigma = 0 . 2 5$ . We can see a significant boost of $\ell _ { 2 }$ certification results for + +![](images/c5a061ba1ff2a99c9a40493b61a6a37e31ba97df6aa5fd28e688702dff16fe89.jpg) +Figure 2: Certified top-1 accuracy at various $\ell _ { 2 }$ radii for (Left) IMAGENET using ResNet-50 and (Right) CIFAR-10 using preactivation ResNet-18. + +both $\mathbf { A d v } _ { \infty }$ and $\mathbf { A d v } _ { 2 }$ models using our proposed framework. Further, $\mathrm { \bf A d v _ { 2 } }$ models consistently achieve better performance compared to $\mathrm { \bf A d v _ { \infty } }$ in terms of certified accuracy. For CIFAR-10, both $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \le 8 / 2 5 5 ]$ and $\Delta \mathrm { d v } _ { 2 } [ \ell _ { 2 } \ \leq \ 1 ]$ outperform the standard randomized smoothing framework i.e., $\mathrm { R a n d } _ { \sigma = 0 . 5 }$ , certified using $\sigma = 0 . 5$ (Cohen et al., 2019). For IMAGENET, $\mathrm { A d v _ { 2 } } \mathrm { [ } \ell _ { 2 } \leq 3 \mathrm { ] }$ achieves better certified accuracy compared to Rand $\sigma { = } 0 . 5$ beyond $\ell _ { 2 }$ -radii of 1.5. + +![](images/db94ddc1389c47cb5064897a41747daceb067b81bb45cab9ea14b08efa67308e.jpg) +Figure 3: CIFAR-10: Certified top-1 accuracy achieved by (a) $\mathbf { A d v } _ { \infty }$ and (b) $\mathbf { A d v } _ { 2 }$ models (with test-time adaptive BN at $\sigma \ : = \ : 0 . 5$ ), learned at different threat boundaries. (c) Comparison with the state-of-the-art SmoothAdv models (Salman et al., 2019a), trained at $\sigma = 0 . 5$ using preactivation ResNet-18. + +Larger Threat Boundary for Better Certified Robustness. Learning AT models at a higher threat boundary improves the certification accuracy at higher $\ell _ { 2 }$ radii. We demonstrate this phenomena for both $\mathbf { A d v } _ { \infty }$ and $\mathrm { \bf A d v } _ { 2 }$ models in Figure 3(a) and 3(b) respectively for CIFAR-10. + +Figure 3(c) also compares the certified accuracy of $\mathrm { \bf A d v } _ { 2 }$ models with the existing state-of-the-art SmoothAdv models (Salman et al., 2019a). SmoothAdv utilizes adversarial training using an adaptive attack with $\ell _ { 2 }$ threat boundary of $\epsilon$ and Gaussian noises, ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } I )$ (See details in Appendix A). We set the noise to $\sigma = 0 . 5$ and vary $\epsilon$ for their training to compare with different SmoothAdv models in Figure 3(c). By adapting $\mathrm { \bf A d v } _ { 2 }$ models with $\sigma = 0 . 5$ at test-time using our proposed Algorithm 1, we already achieve similar performance as SmoothAdv. Moreover, unlike existing frameworks, we also provide test-time flexibility to adapt the same models using different $\sigma$ , without retraining, to improve their certification, as shown below. 2 + +![](images/ec244e7cfdafcee398f327c2168a3bc55d27ba680e0ba185b05e01c9378b0c33.jpg) +Figure 4: Certified accuracy at various $\ell _ { 2 }$ radii by varying $\sigma$ for test-time adaptation of the same models. Choosing large $\sigma$ degrades certification at lower $\ell _ { 2 }$ radii, while provides better performance at higher $\ell _ { 2 }$ radii. + +Flexibility of choosing different noise-level $\sigma$ for certification at test-time. Figure 4 presents the certification results as we vary $\sigma = \{ 0 . 2 5 , 0 . 5 , 0 . 7 5 \}$ for test-time adaptation of the same models using our Algorithm 1. We note that the choice of large $\sigma$ degrades certification at lower $\ell _ { 2 }$ radii while providing better performance for higher $\ell _ { 2 }$ radii. For each test example, we adapt the models with appropriate $\sigma$ that provides the maximum certified radius to obtain the upper envelope of the certification accuracy curves in Figure 4. This leads to the state-of-the-art certification performance for $\mathbf { A d v } _ { 2 }$ models, outperforming the existing SmoothAdv models for CIFAR-10 (Table 1). Further for randomized smoothing models (Figure 4(c)), we consistently provide certification at larger $\ell _ { 2 }$ radii by adapting using larger $\sigma$ values, improving their overall ACR scores (see Table 6 and 5 (Appendix)). Additional results using several models with different training setups are provided in Figure 7 and 8 (Appendix). + +Over-fitting reduces Certification. Rice et al. (2020) demonstrate that AT models overfit as we train without early stopping criteria. It degrades their empirical robustness against adversarial attacks. In Figure 5, we compare with the certification accuracy of such overfitted AT models, denoted as $\mathbf { A d v } ^ { o v e r f i t }$ . We observe that $\mathbf { A d v } ^ { o v e r f i t }$ models also degrade the certified robustness, in particular, at higher $\ell _ { 2 }$ radii, compared to their corresponding AT models with early + +![](images/bbc286f3639a94c0f5ab7b4d76878ed10e23a85fadbb5632f294e53274b0ccd8.jpg) +Figure 5: CIFAR-10: Comparing the certified accuracy of $\mathbf { A d v } _ { \infty }$ (Left) and $\mathrm { \bf A d v _ { 2 } }$ (Right) models with and without applying early-stopping criteria (denoted as Adv $_ { o v e r f i t }$ ). + +stopping criteria. These results also indicate that the empirical and certified robustness are closely related: improving empirical robustness also improves the certified robustness. + +# 5 CONCLUSION + +We propose a novel certification through adaptation algorithm that transforms adversarially trained models into a randomized smoothing classifier using test-time covariate shift adaptation to provide certified robustness for $\ell _ { 2 }$ norm. Unlike existing models using BN adaptation for different applications, our certification framework does not make any assumptions on the test examples. One main advantage of our proposed certification algorithm is to separately choose appropriate noise levels $\sigma$ during inference for each test example. We achieve the state-of-the-art $\ell _ { 2 }$ certification using $\mathrm { \ A d v _ { 2 } }$ models for CIFAR-10. Finally, while we mainly focus on $\ell _ { 2 }$ certification using Gaussian noise, we can also extend this framework for other types of perturbations as long as randomized smoothing works (e.g., uniform noise for $\ell _ { 1 }$ norm (Yang et al., 2020)) for different applications without any additional training. + +# 6 CODE OF ETHICS AND REPRODUCIBILITY + +Code of Ethics. Existing defense models can only provide either empirical or certified robustness against adversarial attacks for higher dimensional input domains. In this paper, we propose a solution to provide high performance to achieve both empirical or certified robustness. It allows us to improve the reliability and trustworthiness for large AI models for sensitive real-world applications. + +Reproducibility. The key results of our paper are presented using adversarially trained models. For CIFAR-10, we train the models using the codes provided in https://github.com/locuslab/robust overfitting (Rice et al., 2020). 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PMLR, 2020. + +# APPENDIX ORGANIZATION + +• Section A: Experimental setup. • Section B: Additional Results on Certification. • Section C:Performance against different corruptions + +A EXPERIMENTAL SETUP + +# A.1 IMPLEMENTATION DETAILS + +We present our experimental results on CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng et al., 2009) datasets. The descriptions of different models and training hyper-parameters are provided in the following: + +# A.1.1 CIFAR-10. + +We use pre-activation ResNet18 architecture (He et al., 2016b) for our experiments on CIFAR-10. We apply the SGD optimizer with a batch size of 128. We execute a total of 200 training epochs and apply a step-wise learning rate decay set initially at 0.1 and divided by 10 at 100 and 150 epochs, and weight decay $5 \times 1 0 ^ { - 4 }$ . + +AT models (Madry et al., 2018; Rice et al., 2020): Unless and otherwise specified, our AT models are learned using early stopping criteria as described in (Rice et al., 2020). We learn several AT models with different threat boundaries for our experiments. We denote them by specifying their corresponding threat model and threat boundaries. For example, $\mathrm { A d v } _ { 2 } [ \ell _ { 2 } \ \leq \ 1 . 5 ]$ denotes an AT model that is learned using PGD adversary with $\ell _ { 2 }$ threat model and a threat boundary of $\epsilon = 1 . 5$ , along with early-stopping criteria (Rice et al., 2020). We also learn AT models without using earlystopping criteria, as in (Madry et al., 2018) for our comparison in Figure 5. These models are denoted as Advoverf it. + +We use projected gradient descent $( P G D )$ adversarial attack (Madry et al., 2018) to train these AT models as follows: For $\mathrm { \bf A d v _ { \infty } }$ , we use 10 iterations and an $\ell _ { \infty }$ step size of $\epsilon / 4$ . For $\mathrm { \ A d v _ { 2 } }$ , we use 10 iterations and an $\ell _ { 2 }$ step size of $\epsilon / 8 . 5 $ . This is the same experimental setup as in (Rice et al., 2020)). We choose a small set of $1 , 0 0 0$ images from the CIFAR-10 test set for our validation. We apply the PGD attack with the same hyper-parameters for our validation during training. We save the best model using the early-stopping criteria (Rice et al., 2020). + +Randomized smoothing model by Cohen et al. (2019): We also train $\mathrm { R a n d } _ { \sigma = 0 . 5 }$ by training with augmented random noise, sampled from an isotropic Gaussian distribution ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } I )$ with $\sigma = 0 . 5$ . Here, we keep the same model architecture, learning rates, batch sizes, and other hyper-parameters as used to learn the AT models. + +Randomized smoothing model by Salman et al. (2019a): We also compare with the state-ofthe-art certification models, called ‘SmoothAdv’, by Salman et al. (2019a) for our experiments on $\ell _ { 2 }$ certification We train the SmoothAdv models by choosing random noise vectors followed by an adaptive adversarial attack with specified $\ell _ { 2 }$ threat boundary of $\epsilon$ at each iteration. The noise vectors are sampled from an isotropic Gaussian distribution $\mathcal { N } ( 0 , \bar { \sigma } ^ { 2 } I )$ . + +We note that the training hyper-parameter $\epsilon$ has the most significant impact on the certification curve for a SmoothAdv model (please refer to Table 7-15 of (Salman et al., 2019a) for more details). For our experiments, we train 4 different SmoothAdv models with $\epsilon =$ $\{ 0 . 2 5 , 0 . 5 , 1 , 2 \}$ and $\sigma ~ = ~ 0 . 5$ using adaptive PGD attack with 10 steps. We denote them as SmoothAd $\scriptstyle v _ { \sigma = 0 . 5 , \epsilon = 0 . 2 5 }$ , $\mathrm { S m o o t h A d v } _ { \sigma = 0 . 5 , \epsilon = 0 . 5 }$ , $\mathrm { S m o o t h A d v } _ { \sigma = 0 . 5 , \epsilon = 1 }$ and SmoothAd $\scriptstyle { \mathrm { \mathbf { U } } } _ { \sigma = 0 . 5 , \epsilon = 2 }$ respectively. We use the same training set-up and other hyper-parameters as specified in their Github: https://github.com/Hadisalman/smoothing-adversarial. + +# A.1.2 IMAGENET. + +We use ResNet50 architecture (He et al., 2016a) for IMAGENET. We obtain the Baseline and $\mathrm { R a n d } _ { \sigma = 0 . 5 }$ models from (Cohen et al., $2 0 1 9 ) ^ { 3 }$ . These models are trained using Gaussian augmented noises, sampled from isotropic Gaussian distribution ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } I )$ with $\sigma = 0 . 0$ (i.e., no noise) and $\sigma = 0 . 5$ respectively. + +The AT models i.e., $\mathrm { A d v } _ { \infty } [ \ell _ { \infty } \leq 4 / 2 5 5 ]$ and $\mathrm { A d v } _ { 2 } [ \ell _ { 2 } \leq 3 ]$ are learned for $\ell _ { \infty }$ and $\ell _ { 2 }$ threat models with threat boundary of $4 / 2 5 5$ and 3, respectively. We use the publicly available models provided by Rice et al. (2020) 4. These models are fine-tuned using PGD-based adversarial training with early stopping criteria, originally provided by Engstrom et al. (2019) 5. + +We resize the input images to $2 5 6 \times 2 6 5$ pixels and crop $2 2 4 \times 2 2 4$ pixels from the center. For our experiments on certification, we use a set of 500 test images by choosing at most 1 sample for each class. + +# A.2 CHOICE OF TEST-TIME ADAPTIVE BN HYPER-PARAMETERS + +BN adaptation technique is controlled by two hyper-parameters, i.e., the test batch-size and momentum $( \rho )$ (see Equation 5) to update the statistics of the batch-normalization layers. Assuming that the test images are obtained independently from the same test distribution, we can efficiently compute the BN statistics from these images. The hyper-parameter $\rho \in [ 0 , 1 ]$ controls the tread-off between pre-computed training statistics and test statistics. We can obtain a better estimation of the test distribution from a large test batch. Hence, we can choose a higher value of $\rho$ . + +Here, we compare the top-1 test accuracy of AT models under Gaussian augmented noise with $\sigma = 0 . 5$ for different choices of $\rho$ and the batch size. We skip the standard baseline models from our analysis and refer to the previous works (Schneider et al., 2020; Nado et al., 2020) that analyzed the effects of these hyper-parameters for the standard baseline DNN classifiers. + +
(b) CIFAR-10
pAdvoAdv2
0.0 (No adaptation)16.1±7.8521.5±7.79
0.145.1±0.4946.9±0.48
0.359.2±0.4260.8±0.33
0.562.4±0.2764.4±0.6
0.762.8±0.5264.9±0.31
0.962.8±0.7164.9±0.31
1.0 (Full adaptation)62.4±0.6464.9±0.73
+ +Table 3: Top-1 accuracy using fixed test batch-size $= 5 1 2$ for AT models under Gaussian augmented noise with $\sigma = 0 . 5$ for different choices of momentum, $\rho$ during inference. We randomly shuffle the test images to report $( m e a n + 2 \times s d )$ of 5 different runs. + +
(a)IMAGENET
pAdvoAdv2
0.0 (No adaptation)0.4±0.010.9±0.01
0.12.1±0.047.7±0.09
0.320.6±0.1636.6±0.09
0.541.1±0.0945.5±0.13
0.743.5±0.1446.7±0.13
0.944.2±0.1246.8±0.13
1.0 (Full adaptation)44.8±0.1347.2±0.14
+ +Momentum $( \rho )$ . We first investigate the effect of momentum $( \rho )$ as we choose a large batch size of 512. In Table 3, we present the performance of AT models for different values of $\rho$ . Recall that, $\rho = 1$ denotes full adaptation (Equation 5). Here, we completely ignore the training statistics and recompute the BN statistics using the test batches. In contrast, $\rho = 0$ represents no adaptation, i.e., the same as the standard ‘deterministic’ inference setup. In this case, we use the previously computed BN statistics obtained during training. + +We observe that for IMAGENET (Table 3 [Left]) the performance started converging at $\rho = 0 . 7$ . For CIFAR-10 (Table 3 [Right]), the convergence started even earlier at $\rho = 0 . 5$ . + +Batch Size. Next, we investigate the minimum size of the test batches to choose $\rho = 1$ (i.e., fulladaptation). In Table 4, we fix $\rho = 1$ and vary the test batch sizes as we evaluate these AT models. We observe that the performance of these models started improving even when we are using the test batches of size 8. The performance further improves as we choose larger sizes of test batches. We can see that their performance started converging as we choose the test batches of size 64 for IMAGENET. On the other hand, the convergence started much earlier for CIFAR-10. + +
(b) CIFAR-10
Batch SizeAdvoAdv2
w/o BNadapt16.1±7.8521.5±7.79
857.2±1.2359.5±0.38
1660.2±0.7962.3±0.87
3261.5±0.4663.6±0.55
6462.3±0.564.0±0.38
12862.7±0.6864.4±0.53
25662.7±0.6864.9±0.48
51262.4±0.6464.9±0.73
+ +Table 4: Top-1 accuracy using fixed $\rho = 1$ for AT models under Gaussian augmented noise with $\sigma = 0 . 5$ for different size of test batches during inference. We randomly shuffle the test images to report $( m e a n + 2 \times s . d . )$ of 5 different runs. + +
(a)IMAGENET
Batch SizeAdvoAdv2
w/oBNadapt0.4±0.010.9±0.01
811.5±0.229.1±0.15
1628.1±0.2226.7±0.14
3237.1±0.2437.6±0.2
6441.4±0.2642.9±0.12
12843.3±0.1545.4±0.13
25644.4±0.2146.7±0.07
51244.8±0.1347.2±0.14
+ +# B ADDITIONAL RESULTS ON CERTIFICATION + +![](images/a72edb939dd233d7ef46f6ab157fc07a509c87a2ab1938860a9bf0bb78e26b1d.jpg) +Figure 6: $\ell _ { 2 }$ Certification for standard non-robust classifiers. For CIFAR-10, we observe that, even after adaptation, the baseline produces lower certification compared to $\mathrm { A d v _ { 2 } } [ \ell _ { 2 } \leq 1 ]$ model without any adaptation. + +# B.1 $\ell _ { 2 }$ CERTIFICATION FOR STANDARD NON-ROBUST CLASSIFIERS + +In Figure 6, we present the $\ell _ { 2 }$ certification results for standard non-robust classification models using our proposed Algorithm 1. In Table 2, we note that the adaptive BN technique can also significantly improve the performance of a non-robust model at lower noise levels, $\sigma$ . In particular, for CIFAR-10 dataset, Baseline models using adaptation achieve similar performance as $\mathrm { A d v _ { 2 } } [ \ell _ { 2 } \leq$ 1] without BN adaptation, while produces significantly lower $\ell _ { 2 }$ certification robustness. This is because, Baseline models, even after adaptation cannot consistently predict the same class to provide higher certified robustness at larger $\ell _ { 2 }$ radii. As a result, we can only improve the certified robustness at smaller $\ell _ { 2 }$ radii. + +![](images/2c20dffba6826dfc30c2b7b6b9c8e8057f5321d3caea73d0268f2a64ebc189bb.jpg) +Figure 7: IMAGENET: Certified top-1 accuracy at various $\ell _ { 2 }$ radii as we vary the noise-level, $\sigma$ at test-time using proposed Algorithm 1. $\mathbf { A d v } _ { \infty }$ and $\mathbf { A d v } _ { 2 }$ models are as defined in experimental set-up (section 4). Refer to Table 5 for complete results of all models and different settings. + +![](images/0e4a0c6dfa810b8c74282bb378bde050de58369c2b0ff45a101ae3c2c7280b30.jpg) +Figure 8: CIFAR-10: Certified top-1 accuracy at various $\ell _ { 2 }$ radii as we vary the noise-level, $\sigma$ at test-time using proposed Algorithm 1. Refer to Table 6 for complete results of all models and different settings. + +
IMAGENET
ModelBN adaptionCertification0.5l2 Radius 1.251.75
0.250.751.01.52.02.252.52.75ACR
Baseline=atσ=0.257.84.83.00.00.00.00.0 0.00.00.00.01 0.054
Advo[l∞o ≤4/255]at σ = 0.25 at σ = 0.50at g = 0.2550.046.441.60.0 0.00.00.00.00.00.00.00.445 0.607
at σ=0.75at σ = 0.50 at σ=0.7543.6 31.639.435.831.4 27.623.418.20.00.00.00.00.443
26.422.418.6 16.814.411.89.47.65.63.6
Advo[loo≤4/255]+adapt[BestRadij(Ours)50.046.441.631.4 27.623.418.29.47.65.63.60.759
at σ = 0.250.480
Adv2[l2 ≤3.00]at σ= 0.50at σ = 0.2553.2 47.050.2 43.046.8 39.00.0 0.00.00.00.00.00.00.00.711
at σ=0.75at g = 0.50 at g=0.7537.832.236.4 32.830.8 20.227.00.00.0 14.20.0 12.00.0 9.60.639
Adv2l2≤3.00] +adapt[Best Radii](Ours)53.228.426.0 22.4 32.830.819.0 27.017.4 17.414.212.09.60.930
50.246.836.4
Randg=0.5 Cohen et al. (2019)=at σ=0.5060.854.447.839.034.2 29.023.80.00.00.00.00.809
at σ = 0.25at g = 0.2559.853.646.60.0 0.00.00.00.00.00.00.00.507
+ adaptationat σ= 0.50at σ= 0.5058.651.043.837.432.2 27.422.40.00.00.00.00.768
at σ=0.75at σ=0.7548.641.636.631.226.2 22.418.616.812.88.65.40.720
Randg=0.5+adapt[BestRadii](Ours)22.416.8
59.853.646.637.432.227.412.88.65.40.973
+ +Table 5: IMAGENET: Certified top-1 accuracy at various $\ell _ { 2 }$ radii as we vary $\sigma$ for BN adaptation and certification along with average certified radii (ACR). We use ResNet50 for IMAGENET. Each gray block is corresponding to one classification model while the rows are corresponding to its certification performances as we choose different noise levels for adaptations and certifications. The Best Radii are obtained by selecting the highest radius for each test example as we adapt the models with different noise levels, $\sigma$ . + +
CIFAR-10
ModelBN adaptionCertification0.250.50.75l2Radius 1.0 1.251.51.752.0ACR
Baseline0.0 0.00.00.0 0.00.026
Advo[∞ ≤ 4/255]at σ = 0.25 at σ = 0.50 at σ = 0.75at σ = 0.25 at σ = 0.50 at σ = 0.7567.96 47.34 26.8950.46 31.83 15.9231.96 18.78 8.440.0 9.98 4.310.0 4.44 2.030.0 1.62 0.28 0.79 0.230.0 0.0 0.00.080.485 0.350 0.146
Advo[lo ≤8/255]at σ =0.25 at σ = 0.50 at σ =0.75at g =0.25 at σ = 0.50 at σ =0.7566.43 53.65 39.9655.06 42.91 30.7642.86 32.58 22.010.0 22.68 14.640.0 14.24 8.840.0 7.88 4.810.0 2.940.0 0.0 1.150.527 0.515 0.352
Advoo[∞ ≤12/255]atσ =0.25 at σ = 0.50 at σ = 0.75at σ= 0.25 at σ = 0.50 atσ=0.7560.52 51.53 42.6152.42 43.94 35.5643.27 36.41 28.470.0 28.69 22.390.0 21.25 16.690.0 14.532.29 0.0 8.030.0 0.0 4.430.499 0.581 0.482
Adv[l∞o ≤ 16/255]at σ =0.25 at σ = 0.50 at σ =0.75atg=0.25 at σ = 0.50 atσ =0.7553.75 48.07 42.05 67.9647.57 42.51 36.42 55.0641.18 36.54 31.24 43.270.0 30.55 26.05 30.550.0 24.68 20.7411.58 0.0 18.49 16.15 12.017.42 0.0 12.110.0 0.0 8.450.454 0.598 0.557 0.903
Advo + adapt [Best Radii] (Ours) 24.6818.49 12.11 8.45 0.0
Adv2[l2 ≤0.50]at σ = 0.25 at σ = 0.50 atg =0.75at σ = 0.25 at σ = 0.50 atσ =0.7568.84 48.81 27.3854.04 33.82 16.1537.13 20.95 9.230.0 11.5 4.560.0 5.64 2.060.0 2.29 0.62 0.91 0.330.0 0.0 0.080.518 0.382 0.153
Adv2[l2 ≤ 1.00]at σ = 0.25 atσ =0.50 atσ =0.75at σ = 0.25 at σ = 0.50 atg =0.7568.02 56.45 43.0458.54 46.24 33.0846.98 35.6 24.810.0 26.89 17.680.0 18.73 11.390.0 11.37 6.60.0 5.41 3.570.0 0.0 1.950.551 0.580 0.405
Adv2[l2 ≤1.25]at g =0.25 at σ =0.50 atσ =0.75at σ= 0.25 at σ = 0.50 at σ = 0.7567.13 57.73 46.5458.77 48.8 37.5349.43 39.64 29.350.0 31.07 22.00.0 22.61 15.620.0 15.82 10.510.0 8.96 6.550.0 0.0 3.680.557 0.647 0.496
Adv2[l2 ≤ 1.50]at σ = 0.25 at σ =0.50 at σ=0.75at g = 0.25 at σ = 0.50 atg =0.7564.21 56.55 47.7357.13 49.19 40.8949.71 41.72 33.780.0 34.47 27.220.0 27.36 20.780.0 20.23 15.190.0 12.98 10.510.0 0.0 6.770.543 0.689 0.585
Adv2[l2 ≤ 2.00]at σ= 0.25 at σ =0.50 at σ =0.75at σ = 0.25 at σ = 0.50 at σ = 0.7560.4 54.27 47.9654.71 48.89 42.5448.35 43.1 37.040.0 37.34 31.650.0 31.52 26.170.0 25.74 21.120.0 19.14 16.590.0 0.0 12.440.523 0.731 0.698
Adv2[l2 ≤ 2.25]at σ = 0.25 at σ = 0.50 at σ = 0.75atg=0.25 at σ = 0.50 at σ = 0.7557.08 52.1 46.4552.5 46.99 41.7147.11 42.26 36.750.0 36.9 31.880.0 31.58 26.950.0 26.08 22.330.0 20.03 17.820.0 0.0 13.550.504 0.724 0.713
Adv2[l2 ≤ 2.50]at σ =0.25 at σ = 0.50 atσ=0.75at σ = 0.25 at σ = 0.50 at σ =0.7554.88 50.53 45.9550.79 46.26 41.8946.29 41.84 37.530.0 37.74 33.550.0 33.2 29.310.0 28.69 25.270.0 23.34 20.980.0 0.0 17.280.487 0.734 0.765
Adv2[2 ≤ 3.00]at σ =0.25 atσ =0.50 at σ = 0.75at σ =0.25 at σ = 0.50 atσ =0.7553.82 49.41 45.3749.69 45.57 41.5445.04 41.52 37.750.0 37.43 33.490.0 33.37 29.350.0 28.82 25.620.0 23.65 21.830.0 0.0 18.230.475 0.720 0.771 1.198
Adv2+adapt[Best Radii](Ours) 68.84 58.77
Randg=0.5at g= 0.50 at σ = 0.2551.68 62.9140.38 52.2549.71 30.25 40.0637.74 20.81 0.033.37 13.36 0.028.82 7.71 0.023.65 3.38 0.018.23 0.0 0.00.488 0.497 0.575
at σ = 0.25 + adaptation at σ =0.50 at σ = 0.50
Randg=0.5 +adapt[Best Radi] (Ours)atσ =0.75at σ =0.7557.58 46.4 62.9146.46 35.63 52.2535.5 26.06 40.0625.57 18.17 25.5717.43 11.61 17.4310.67 5.46 6.86 3.64 10.67 5.460.0 1.92 1.920.427 0.657 0.609
SmoothAdvg=0.5,=0.25 at g= 0.50
+ adaptationat σ = 0.25 at σ = 0.50at σ = 0.25 at g = 0.5057.8 58.74 54.047.63 48.29 42.9137.41 36.7 32.6327.88 0.0 23.620.33 13.53 0.0 16.088.03 0.0 0.0 9.93 5.50.0 0.0 0.00.464 0.535 0.390
SmoothAdvg=0.5,∈=0.50at σ = 0.75 at σ = 0.25at σ = 0.75 atσ =0.50 at σ = 0.2543.15 58.82 59.8932.29 49.68 50.423.61 40.35 39.9916.41 31.93 0.010.94 24.18 0.06.65 17.05 0.03.88 10.57 0.02.16 0.0 0.00.661 0.483 0.592
+ adaptation
SmoothAdvg=0.5,∈=1.0at σ = 0.50 at σ = 0.75at σ = 0.50 at σ =0.75 atg=0.5055.73 46.25 56.5345.79 36.72 49.5336.6 28.2 41.3827.4 20.9 34.6320.03 14.73 27.8113.48 9.46 21.227.85 5.78 14.410.0 3.32 0.00.470 0.691 0.467
+ adaptation- at σ = 0.25 at σ = 0.50at σ = 0.25 at σ = 0.50 at σ = 0.7557.13 53.56 47.1748.38 45.79 39.439.7 37.640.0 30.060.0 23.12 18.930.0 17.27 13.70.0 11.14 9.260.0 0.0 5.80.620 0.545
at σ = 0.75
SmoothAdvg=0.5,∈=2.0at σ = 0.25 at σ =0.50atg =0.50 at σ = 0.2552.82 52.2347.67 47.2431.8 42.68 41.7624.74 37.55 0.032.64 0.027.52 0.022.42 0.00.0 0.00.732 0.451 0.692
+ adaptation at σ =0.75at σ = 0.50 atσ =0.7550.28 46.745.24 41.7940.39 37.1435.5 33.0530.92 28.2826.1 23.8820.25 19.340.0 15.050.727
SmoothAdvg=0.5[BestRadii]58.82 59.8949.68 50.442.68 41.7637.55 35.532.64 30.9227.52 26.122.42 20.250.0 15.050.918 1.008
SmoothAdvg=0.5 + adapt [Best Radii] (Ours) MARCERg=0.5 (Zhai et al.,2020) 60.0 53.0
+ +Table 6: CIFAR-10: Certified top-1 accuracy at various $\ell _ { 2 }$ radii as we vary $\sigma$ for test-time BN adaptation along with average certified radii (ACR) for individual settings. Each gray block is corresponding to one classification model while the rows are corresponding to its certification performances as we choose different noise levels for adaptations and certifications. The Best Radii are obtained by training different models with varying hyper-parameters and adapting them with different noise levels during inference. We also present the best reported results for MARCER (Zhai et al., 2020) and Consistancy Jeong & Shin (2020) at $\sigma = 0 . 5$ , obtained from the respective papers. + +# C PERFORMANCE AGAINST DIFFERENT CORRUPTIONS + +We mainly focus on $\ell _ { 2 }$ certification using Gaussian noise in this paper. However, we note that randomized smoothing techniques have been also applied to provide certifications for other perturbation 19 + +types as well (e.g., random uniform noise for $\ell _ { 1 }$ norm (Yang et al., 2020)). Consequently, we can apply our proposed Algorithm 1 to adapt an AT model for any given perturbation types without any additional training for different applications. + +Further, Hendrycks & Dietterich (2019) recently introduced ImageNet-C and CIFAR10-C datasets by algorithmically generated random corruptions from noise, blur, weather, and digital categories with 5 different severity levels for each corruption. Several recent works demonstrated that adaptive BN techniques can significantly improve the performance of any classifier (including AT models) against different random corruptions. Further, – also demonstrated the effectiveness of AT models even without applying any adaptation. Hence, our proposed certification framework for AT models is a step forward towards further improving the reliability of sensitive real-world applications. \ No newline at end of file diff --git a/parse/dev/RFGkzxMFqby/RFGkzxMFqby_content_list.json b/parse/dev/RFGkzxMFqby/RFGkzxMFqby_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..b06bdc118ffef91b90f6d944ea9a27ab22def2a5 --- /dev/null +++ b/parse/dev/RFGkzxMFqby/RFGkzxMFqby_content_list.json @@ -0,0 +1,2104 @@ +[ + { + "type": "text", + "text": "ADVERSARIALLY TRAINED MODELS WITH TEST-TIME COVARIATE SHIFT ADAPTATION ", + "text_level": 1, + "bbox": [ + 176, + 98, + 823, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 171, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Existing defense models against adversarial examples typically provide either empirical or certified robustness. Adversarially trained models empirically demonstrate state-of-the-art defense while providing no robustness guarantees for large classifiers or higher-dimensional inputs. In contrast, a randomized smoothing framework provides state-of-the-art certification while significantly degrades the empirical performance against adversarial attacks. In this work, we propose a novel certification through adaptation technique that transforms an adversarially trained model into a randomized smoothing classifier during inference to provide certified robustness for $\\ell _ { 2 }$ norm without affecting their empirical robustness against adversarial attacks. One advantage of our proposed technique is that it allows us to separately choose the appropriate noise level for certifying each test example during inference. It also leads to outperform the existing randomized smoothing models for $\\ell _ { 2 }$ certification on CIFAR-10. Therefore, our work is a step towards bridging the gap between the empirical and certified robustness against adversarial examples by achieving both using the same classifier for the first time. ", + "bbox": [ + 233, + 265, + 764, + 472 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 496, + 336, + 512 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Deep neural network (DNN) based models are found to be brittle to minor, adversarially-chosen perturbations for their inputs that remain undetectable to human eyes. A DNN classifier that correctly classifies an image $x$ , can be easily fooled by an adversarial attack to misclassify $x + \\delta$ (Szegedy et al., 2014; Goodfellow et al., 2015; Madry et al., 2018). Here, $\\delta$ is a minor adversarial perturbation such that the change between $x$ and $x + \\delta$ remains imperceptible. ", + "bbox": [ + 174, + 526, + 823, + 597 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Among the existing successful defense frameworks, adversarial training (AT) produces the best empirical robustness against the known adversarial attacks without providing any guarantee (Madry et al., 2018; Tramer & Boneh, 2019; Zhang et al., 2019; Rice et al., 2020; Gowal et al., 2020). It \\` trains a DNN classifier using strong adversaries from a specific class of perturbation (e.g., a small $\\ell _ { p }$ -norm) to provide robustness for the same perturbation types. Several certification techniques are proposed that can be applied to adversarially trained models to certifiably verify if the prediction of a test example, $x$ remains constant within its neighborhood (Wong & Kolter, 2018; Wang et al., 2018; Salman et al., $2 0 1 9 6$ ; Dvijotham et al., 2018; Gehr et al., 2018; Sheikholeslami et al., 2021). However, these certification techniques typically do not scale for larger networks (e.g., ResNet50) and datasets (e.g., IMAGENET). Hence, currently, we cannot guarantee that a more powerful, not yet known attack can not break these adversarially trained models. In fact, several recently proposed empirical defense models are later broken by stronger adaptive adversarial attacks, indicating the importance of investigating certified defenses with suitable robustness guarantees. ", + "bbox": [ + 174, + 603, + 825, + 784 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In contrast to adversarial training, randomized smoothing provides a scalable $\\ell _ { 2 }$ -certification framework for any classification model, which is robust against large isotropic Gaussian noise (Cohen et al., 2019; Salman et al., 2019a). However, the existing randomized smoothing-based certified models produce significantly lower empirical robustness compared to the AT models. On the other hand, this technique cannot be applied for AT models as they are not robust against such large random Gaussian noises in the standard settings. Towards this, we investigate to bridge the gap between the state-of-the-art empirical and certifiable robust models against adversarial examples. ", + "bbox": [ + 174, + 791, + 823, + 888 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we present a novel certification through adaptation framework to transform an AT model into a randomized smoothing framework during inference, providing $\\ell _ { 2 }$ certification without any additional training or architectural modification. Our proposed certification technique consists of two steps: we first apply a covariate shift adaptation to a classifier against Gaussian noise during inference for each test example (Cariucci et al., 2017; Li et al., 2016). For our paper, we use the wellknown batch normalization adaptation. This process significantly boosts the performance of the AT models against the random isotropic Gaussian noises compared to the standard non-robust models. Hence, we can now directly apply the randomized smoothing based certification technique to provide $\\ell _ { 2 }$ certification in the next step. Further, the existing randomized smoothing models require selecting the noise level at training time. In contrast, our proposed framework can separately choose the appropriate noise levels for different test examples during inference (Figure 4). Furthermore, we can also evaluate the input test examples without transforming the AT models to a randomized smoothing model, ensuring that their empirical performance remains unaffected. Therefore, we are the first to provide the test-time flexibility to obtain empirically robust predictions as well as certify their predictions using the same classifier for high-dimensional datasets to the best of our knowledge. Hence, we improve the reliability of AT models sensitive real-world applications. ", + "bbox": [ + 174, + 895, + 823, + 924 + ], + "page_idx": 0 + }, + { + "type": "table", + "img_path": "images/f6ba45d1041dd41f0661d176679810dee2d1d275dc417c8fd3df06f5568cfecb.jpg", + "table_caption": [ + "Table 1: CIFAR-10: Certified accuracy at various $\\ell _ { 2 }$ radii and ACR scores. We train different models by varying the hyper-parameters for SmoothAdv, $\\mathrm { \\ A d v _ { 2 } }$ and $\\mathbf { A d v } _ { \\infty }$ (as in (Salman et al., 2019a)) and by choosing $\\sigma = \\{ 0 . 2 5 , 0 . 5 , 0 . 7 5 \\}$ for test-time adaptation to obtain the maximum certified radii for each test example. See Table 5 and 6 (Appendix) for detailed results on both IMAGENET and CIFAR-10 respectively. We also present the best reported results for MARCER and Consistancy at $\\sigma = 0 . 5$ , obtained from their respective papers. " + ], + "table_footnote": [], + "table_body": "
l2Radius (CIFAR-10)0.250.50.751.01.251.51.752.0ACR
Baseline6.962.040.090.00.00.00.00.00.026
Randg =0.5 (Cohen et al.,2019)51.6840.3830.2520.8113.367.713.380.00.488
(Ours)Randg=0.5 +adaptation62.9152.2540.0625.5717.4310.675.461.920.657
SmoothAdvg=0.5 (Salman et al.,2019a)58.8249.6842.6837.5532.6427.5222.420.00.918
(Ours) SmoothAdvg=0.5+adaptation59.8950.441.7635.530.9226.120.2515.051.008
Advo (Rice et al.,2020)35.9529.4423.510.00.00.00.00.00.317
(Ours)Advo + adaptation67.9655.0643.2730.5524.6818.4912.118.450.903
Adv2 (Rice et al., 2020)41.8934.1526.70.00.00.00.00.00.359
(Ours) Adv2 +adaptation68.8458.7749.7137.7433.3728.8223.6518.231.198
MARCERg=0.5 (Zhai et al.,2020)60.053.046.038.029.019.012.00.00.726
Consistancyg=0.5 (Jeong& Shin,2020)48.945.141.337.833.929.925.20.00.726
", + "bbox": [ + 238, + 101, + 753, + 213 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 309, + 825, + 502 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Contributions:- ", + "text_level": 1, + "bbox": [ + 174, + 511, + 272, + 523 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We propose a novel certification through adaptation framework that can adapt an AT model during inference to provide certified robustness. Our experimental results on CIFAR-10 and IMAGENET demonstrate that the proposed certification framework can transform any AT model into a randomized smoothing classifier to provide certification for $\\ell _ { 2 }$ norm, even when the model is learned using $\\ell _ { \\infty }$ -bounded adversaries (Table 1 & Figure 2). \n2. One main advantage of our proposed framework is that it allows us to select appropriate noise levels for different test examples during inference. This leads to outperforming the existing state-of-the-art randomized smoothing models for $\\ell _ { 2 }$ certification on CIFAR-10 using AT models (Table 1 & Figure 4). Further, we can provide certification at larger $\\ell _ { 2 }$ radii for existing randomized smoothing models, improving their overall average certified radius (ACR). \n3. Our results also indicate a strong correlation between empirical and certified robustness than previously believed (Cohen et al., 2019; Salman et al., $2 0 1 9 \\mathrm { a }$ ; Tramer & Boneh, 2019). \\` In particular, we observe that the empirically stronger AT models lead to better $\\ell _ { 2 }$ certification performance (Figure 5). ", + "bbox": [ + 212, + 536, + 825, + 755 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORK ", + "text_level": 1, + "bbox": [ + 174, + 773, + 344, + 791 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Empirical Defenses and Adversarial Training. Existing defense models against adversarial attacks can be broadly classified into empirical and certified defenses. Empirical defenses demonstrate empirical robustness against adversarial attacks (Schott et al., 2019; Moosavi Dezfooli et al., 2019; Nandy et al., 2020; Mao et al., 2021). Adversarial training achieves the state-of-the-art empirical defense (Madry et al., 2018). It optimizes the following loss function for a DNN classifier, $f$ , to provide robustness within an $\\epsilon$ -bounded threat model for an $\\ell _ { p }$ norm, where the perturbations, $\\delta \\in \\Delta$ are constrained as $\\Delta = \\{ \\delta : | | \\delta | | _ { p } \\leq \\epsilon \\}$ : ", + "bbox": [ + 173, + 799, + 825, + 898 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/bb136b79e7d02c31aca49a57107971424a025ea42cb59d00f8bede63f3248bf4.jpg", + "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { ( x , y ) } [ \\operatorname* { m a x } _ { \\delta \\in \\Delta } \\mathcal { L } ( f _ { \\theta } ( x + \\delta ) , y ) ]\n$$", + "text_format": "latex", + "bbox": [ + 388, + 904, + 611, + 928 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "where, $\\theta$ denotes the model parameters. $\\mathcal { L }$ is the classification loss. ", + "bbox": [ + 174, + 103, + 611, + 117 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The inner maximization in Eq. 1 is solved by producing adversarial examples using strong iterative adversaries, e.g., projected gradient descent $( P G D )$ attack (Kurakin et al., 2016; Madry et al., 2018). Wong et al. (2020) found that even a single-step fast gradient sign method (FGSM) attack-based AT models also achieves high empirical robustness (Goodfellow et al., 2015). Zhang et al. (2020) proposed to use the least adversaries for training. Recently Trades (Zhang et al., 2019), Adv-LLR (Qin et al., 2019) introduced additional regularizers to achieve higher empirical robustness by smoothing the loss surface. However, Rice et al. (2020) showed that the standard PGD based AT model with early-stopping criteria provides one of the best empirical defenses for a given perturbation type. Recent works also explored the importance of different hyper-parameters for adversarial training (Gowal et al., 2020; Pang et al., 2021) as well as incorporating additional data in a semi-supervised fashion (Carmon et al., 2019; Uesato et al., 2019) to further improve their empirical robustness. ", + "bbox": [ + 174, + 125, + 825, + 279 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Certified Defenses. Empirical defenses demonstrate robustness only against the known adversaries without providing any guarantees. In fact, most empirical defenses proposed in the literature were later broken by stronger adversaries, highlighting the importance of certified defenses to provide robustness guarantees (Athalye et al., 2018; Uesato et al., 2018; Jalal et al., 2019). ", + "bbox": [ + 174, + 285, + 825, + 340 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Several recent works proposed to train neural network models with provable robustness guarantees. These works include methods based on semi-definite relaxations (Raghunathan et al., 2018), linear relaxations and duality (Wong & Kolter, 2018; Wong et al., 2018), abstract interpretation (Mirman et al., 2018), and interval bound propagation (Gowal et al., 2018). Parallel to training a certified defense, several works also focus on certifying the already trained models (Tjeng et al., 2017; Gehr et al., 2018; Weng et al., 2018; Wang et al., 2018; Bunel et al., 2018). Recently Mueller et al. (2021) combined a small certification network with a large, empirically robust AT model using some selection criteria to boost overall benign accuracy along with empirical robustness for the certified framework. However, none of these techniques scale for large networks (e.g., ResNet50) or higher-dimensional datasets (e.g., IMAGENET). ", + "bbox": [ + 174, + 347, + 825, + 487 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Randomized Smoothing for Certification. A randomized smoothing classifier is not a neural network. It uses a neural network as its base for classification. Randomized smoothing was initially proposed as a heuristic defense (Cao & Gong, 2017; Liu et al., 2018) and later shown to be certifiable (Lecuyer et al., 2019; Li et al., 2019). Recently, Cohen et al. (2019) and Salman et al. (2019a) separately provided a tight robustness guarantee for $\\ell _ { 2 }$ -norm. Salman et al. (2019a) provides the current state-of-the-art $\\ell _ { 2 }$ certification robustness by adversarially choosing the noise using an adaptive attack to train their base classifier. This framework is also analyzed for other $\\ell _ { p }$ norms using different noise distributions as well (Li et al., 2019; Lee et al., 2019; Dvijotham et al., 2020; Yang et al., 2020). Salman et al. (2020) proposed to incorporate an additional denoising module as a preprocessing unit to convert a standard DNN classifier into a randomized smoothing model to provide non-trivial certified robustness. Notably, randomized smoothing is the only scalable certification framework. Further, it also achieves superior performance for different perturbation types. ", + "bbox": [ + 174, + 493, + 825, + 660 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "While achieving the state-of-the-art certification performance, randomized smoothing significantly degrades the empirical robustness against adversarial attacks (Lecuyer et al., 2019; Salman et al., 2019a; Cohen et al., 2019). Towards this, our proposed technique transforms an AT model into a randomized smoothing classifier without any additional training or architectural modification. Since AT models already provide the state-of-the-art empirical defense, we achieve both empirical and certified robustness against adversarial examples using the same classifier. ", + "bbox": [ + 174, + 667, + 825, + 751 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 PROPOSED METHODOLOGY ", + "text_level": 1, + "bbox": [ + 176, + 758, + 434, + 775 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In this section, we first present the background of the randomized smoothing technique and explain why it is not directly effective for AT models. Next, we present the existing test-time co-variate shift adaptation for domain adaptations and corruption robustness. Then, we present our proposed certification through adaptation framework that adapts a DNN model during inference to provide certified robustness without additional training or architectural modification. ", + "bbox": [ + 174, + 784, + 825, + 853 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3.1 BACKGROUND ON RANDOMIZED SMOOTHING ", + "text_level": 1, + "bbox": [ + 174, + 869, + 532, + 883 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Consider a classifier $f$ that maps inputs in $\\mathbb { R } ^ { d }$ to $\\mathcal { V }$ classes. The randomized smoothing framework transforms the original base classifier $f$ into a new, smoothed classifier $g$ . In particular, for an ", + "bbox": [ + 176, + 895, + 823, + 924 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "input $x \\in \\mathbb { R } ^ { d }$ , the smoothed classifier $g$ returns the most probable class to be predicted by the base classifier $f$ under isotropic Gaussian noises of $x$ . That is, ", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/d642bd88882c6a455c63a1b0d67fe8bf16b6e506ca496fa45998b8629dbb1b5d.jpg", + "text": "$$\ng ( x ) = a r g \\operatorname* { m a x } _ { y \\in \\mathcal { V } } \\mathbb { P } ( f ( x + \\delta ) = = y ) \\qquad { \\mathrm { w h e r e } } , \\delta \\sim { \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I ) .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 138, + 712, + 162 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "The noise level, $\\sigma$ controls the trade-off between robustness and accuracy: Increasing $\\sigma$ would improve the robustness of $g$ at higher $\\ell _ { 2 }$ radii. However, it degrades the robustness at lower $\\ell _ { 2 }$ radii as well as the benign accuracy. ", + "bbox": [ + 174, + 170, + 825, + 212 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Cohen et al. (2019) presented a tight robustness guarantee based on the Neyman-Pearson lemma for the smoothed classifier $g$ and gave an efficient algorithm using Monte Carlo sampling for certifying of $g$ . We can also obtain this guarantee alternatively by explicitly computing the Lipschitz constant of the smoothed classifier as shown in (Salman et al., 2019a; Yang et al., 2020). The certification procedure is as follows: Suppose a base classifier $f$ classifies $\\sqrt { ( x , \\sigma ^ { 2 } I ) }$ to return the “most probable” class, $c _ { A }$ with probability $p _ { A } = \\mathbb { P } ( f ( x + \\delta ) = = c _ { A } )$ ) and the “runner-up” class $c _ { B }$ with probability $\\begin{array} { r } { p _ { B } = \\operatorname* { m a x } _ { y \\neq c _ { A } } \\mathbb { P } ( f ( x + \\delta ) = = y } \\end{array}$ ). Then, the smooth classifier, $g$ is certifiably robust around $x$ within an $\\ell _ { 2 }$ radius of $R$ : ", + "bbox": [ + 173, + 218, + 825, + 330 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/66490bfbc623a772adee01f3b171007c4f09ba689095ba461f0ca6a229eb61ff.jpg", + "text": "$$\nR = { \\frac { \\sigma } { 2 } } \\Bigl ( \\Phi ^ { - 1 } ( p _ { A } ) - \\Phi ^ { - 1 } ( p _ { B } ) \\Bigr )\n$$", + "text_format": "latex", + "bbox": [ + 392, + 335, + 606, + 364 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where, $\\Phi ^ { - 1 }$ is the inverse of the standard Gaussian CDF. ", + "bbox": [ + 173, + 371, + 545, + 386 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "However, computing the exact values of $p _ { A }$ and $p _ { B }$ is not possible in practice when $f$ is a DNN. Cohen et al. (2019) addressed this problem using Monte Carlo sampling to estimate some $\\underline { p _ { A } }$ and $\\overline { { p _ { B } } }$ such that $p _ { A } \\leq p _ { A }$ and ${ \\overline { { p _ { B } } } } \\geq p _ { B }$ with arbitrarily high probability. The certified radius for input $x$ is then computed by replacing $p _ { A }$ and $p _ { B }$ with $\\underline { p _ { A } }$ and $\\overline { { p _ { B } } }$ respectively in Eq. 3. ", + "bbox": [ + 173, + 392, + 825, + 450 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "As we can see in Equation 2 that the original base classifier, $f$ needs to be robust against large Gaussian noises to provide non-trivial robustness certification results. Otherwise, it leads to lower $p _ { A }$ and hence a lower certification of $R$ for the test examples. Existing randomized smoothing-based models applies custom-trained using explicit Gaussian noises to learn their original base classifier (Lecuyer et al., 2019; Cohen et al., 2019; Salman et al., 2019a; Zhai et al., 2020; Jeong & Shin, 2020). However, these models produce significantly lower empirical robustness compared to the AT models. Consequently, AT models are not robust against large Gaussian noises in the standard inference settings (see Table 2). Hence, we cannot directly use them as the base classifier for randomized smoothing. ", + "bbox": [ + 173, + 455, + 825, + 580 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "3.2 BACKGROUND ON COVARIATE SHIFT ADAPTATION", + "text_level": 1, + "bbox": [ + 174, + 598, + 566, + 612 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Recent works on (Sun et al., 2017; Roy et al., 2019; Huang et al., 2018; Li et al., 2016) and corruption robustness (Schneider et al., 2020; Nado et al., 2020; Benz et al., 2021) demonstrate the importance of unsupervised covariate shift adaptation. We use adaptive batch-normalization (BN), one of the most popular and effective unsupervised covariate shift adaptation mechanisms. ", + "bbox": [ + 174, + 623, + 825, + 680 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "A BN layer computes the mean and variance of the hidden activation maps across the channels to normalize these activations to $\\mathcal { N } ( 0 , 1 )$ before feeding into the next hidden layer (Ioffe & Szegedy, 2015). It reduces the dependencies among different hidden layers, improving the training efficiency for deep architectures. Hence, most of the recent DNN architectures frequently incorporate BN layers for complex machine learning tasks. However, the distributional shifts in the test examples lead to different activation statistics compared to the training examples. Hence, impacted by the covariate shift, the statistics estimated during training fail to normalize the activation tensors to $\\mathcal { N } ( 0 , 1 )$ . As a result, it breaks the crucial assumption for the subsequent hidden layers to work. ", + "bbox": [ + 173, + 686, + 825, + 799 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "More formally, let $P _ { T } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R } ^ { + }$ as the training distribution and $P _ { t } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R } ^ { + }$ as the test distribution; where $x \\in \\mathcal { X }$ are inputs and $y \\in \\mathcal { V }$ are the corresponding class labels. There exists covariate shift between training and test distribution iff: $P _ { T } ( y | \\bar { x } ) { = } P _ { t } ( \\bar { y } | x )$ and $P _ { T } ( x ) \\neq P _ { t } ( x )$ (Sugiyama & Kawanabe, 2012; Scholkopf et al., 2012). If the covariate shift only affects the first ¨ and second-order moments of the hidden layer feature activations, $f _ { h } ( x )$ , we can remove it using normalization (Schneider et al., 2020): ", + "bbox": [ + 174, + 804, + 825, + 888 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/5ae0dfd2aa4d58ccac8c953a76996e3b1a2a9e9cf8bcbe00d8ebdd6cba21304e.jpg", + "text": "$$\nP _ { T } \\Big ( \\frac { f _ { h } ( x ) - \\mathbb { E } _ { T } [ f _ { h } ( x ) ] } { \\sqrt { \\mathbb { V } _ { T } [ f _ { h } ( x ) ] } } \\Big ) P _ { T } ( x ) \\approx P _ { t } \\Big ( \\frac { f _ { h } ( x ) - \\mathbb { E } _ { t } [ f _ { h } ( x ) ] } { \\sqrt { \\mathbb { V } _ { t } [ f _ { h } ( x ) ] } } \\Big ) P _ { t } ( x ) .\n$$", + "text_format": "latex", + "bbox": [ + 294, + 895, + 704, + 928 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Covariate shift adaptation using adaptive BN computes the BN statistics from the feature activations, $\\mu _ { t } , s _ { t } ^ { 2 }$ , of the test batch. We can adapt them with the existing training statistics, $\\mu _ { T } , s _ { T } ^ { 2 }$ , obtained using the training batches as (Cariucci et al., 2017; Li et al., 2016; Schneider et al., 2020): ", + "bbox": [ + 174, + 103, + 825, + 146 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/e6745a30c421bc620e09cb1c762342005c18b45aefa56c281e52cc85eeaec2dc.jpg", + "text": "$$\n\\overline { { \\mu } } = \\rho \\cdot \\mu _ { t } + ( 1 - \\rho ) \\cdot \\mu _ { T } \\quad \\overline { { s } } = \\rho \\cdot s _ { t } + ( 1 - \\rho ) \\cdot s _ { T }\n$$", + "text_format": "latex", + "bbox": [ + 323, + 152, + 673, + 170 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where, $\\rho \\in [ 0 , 1 ]$ is the momentum. The choice of $\\rho = 0$ is equivalent to the standard inference setup with a deterministic DNN classifier in the IID settings. We should choose $\\rho = 1$ when receiving larger test batches as it can provide a better estimation of the test distributions. ", + "bbox": [ + 174, + 176, + 823, + 219 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Assumptions for BN adaptation. It is noteworthy that these existing adaptive BN-based frameworks require a large set of test images from the same covariate shift to estimate the BN parameters. However, this assumption may not hold for several real-world applications, e.g., stateless web APIs. Also, these test images should be semantically diverse, preferably over multiple classes, to effectively estimate the test distributions. Hence, it further limits the practical usability of these frameworks for real-world applications, e.g., autonomous cars. ", + "bbox": [ + 173, + 226, + 825, + 310 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In contrast to these models for domain adaptation and corruption robustness, our proposed certification framework against adversarial examples does not make any such assumptions. In this case, we already know the perturbation type on which we need to adapt the model to provide the certification. Hence, we can explicitly pre-select a diverse set of clean images, ${ \\bf X } _ { b a t c h }$ and control the perturbations to adapt the models, addressing both of these limitations. ", + "bbox": [ + 173, + 316, + 825, + 386 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Algorithm 1: Steps for CERTIFICATION THROUGH ADAPTATION Algorithm ", + "text_level": 1, + "bbox": [ + 174, + 400, + 683, + 414 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Input: $f$ : classifier, $x _ { t e s t }$ : test example, $\\sigma$ : desired noise-level, ${ \\bf X } _ { b a t c h }$ : set of clean images (preselected from validation data or test stream). Output: Certifiably robust $\\ell _ { 2 }$ radius of $R$ for $x _ { t e s t }$ . /\\* Step 1: Adapt BN parameters using ${ \\bf X } _ { b a t c h }$ with $\\rho = 1$ (Eqn 5). \\*/ 1 $\\tilde { \\mathbf { X } } _ { b a t c h } = [ x + \\mathcal { N } ( 0 , \\sigma I ) \\ \\forall \\ x \\in \\ \\mathbf { X } _ { b a t c h } ]$ // perturb ${ \\bf X } _ { b a t c h }$ with desired noise. 2 $f _ { a d a p t } = \\mathrm { C L O N E } ( f . t r a i n ( ) )$ // clone $f$ with train-mode. 3 $\\underline { { \\mathbf { \\Pi } } } _ { - } = f _ { a d a p t } ( \\tilde { \\mathbf { X } } _ { b a t c h } )$ // forward pass for BN parameter adaptation. 4 fadapt.eval() // fix the parameters. /\\* Step 2: Certify $x _ { t e s t }$ using Randomized Smoothing framework. \\*/ 5 $g =$ GETRANDOMIZEDMODEL(fadapt) // Convert fadapt to randomized-smoothing classifier $g$ (Eqn 2). 6 $R = { \\bf C E R T I F Y } ( g , x _ { t e s t } , \\sigma )$ // Execute 3 for $\\ell _ { 2 }$ certification. 7 return $R$ ", + "bbox": [ + 160, + 417, + 802, + 619 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.3 PROPOSED CERTIFICATION THROUGH ADAPTATION", + "text_level": 1, + "bbox": [ + 173, + 640, + 570, + 655 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The robustness guarantee in Eq. 3 suggests that randomized smoothing gives a framework for certifying any classifier $f$ that is robust against large Gaussian noises. Previous works proposed customized training using explicit Gaussian noise augmentation for their training (Section 3.1). Subsequently, in Section 3.2 we note that robustness against random Gaussian noises of any classifier, $f$ can be improved by applying covariate shift adaptation using adaptive BN technique without any additional training. However, it modifies the original base classifier $f$ at each forward pass by recomputing the BN parameters. Since the certification guarantee in Eq. 3 is provided only for a fixed base classifier $f$ , we cannot directly apply adaptive BN to provide $\\ell _ { 2 }$ certification using the randomized smoothing framework. This motivates us to propose a novel certification framework that applies the covariate shift adaptation using adaptive BN as an offline pre-processing step to improve the robustness against random Gaussian noises, addressing the above problem. ", + "bbox": [ + 173, + 666, + 825, + 819 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our proposed certification through adaptation framework consists of two steps: Given test image $x _ { t e s t }$ , we first apply the adaptive BN technique to achieve robustness against Gaussian perturbations. Recall that adaptive BN requires a large set of diverse test images to correctly re-estimate the batch-normalization statistics. However, to provide certification for $\\ell _ { 2 }$ -norm, we only need to adapt our model against Gaussian perturbations. Hence, we can pre-select a sufficiently large set of diverse clean images, ${ \\bf X } _ { b a t c h }$ and apply Gaussian perturbations to adapt our classifier, $f$ , as an offline pre-processing step to obtain $f _ { a d a p t }$ . Alternatively, when a large set of diverse test examples are available, we can also use them for our BN adaptation. The Gaussian noise samples should be drawn from the same isotropic Gaussian distribution ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )$ as we need to use for the certification process. Then, we freeze the model parameters and use the adapted model, $f _ { a d a p t }$ , as our base classifier to certify the test example, $x _ { t e s t }$ . Hence, the base classifier $f _ { a d a p t }$ remains fixed during calculating the certification radius $R$ (Equation 3). Our proposed certification through adaptation technique is presented in Algorithm 1. ", + "bbox": [ + 174, + 827, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 825, + 188 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Advantages. The main advantage of our proposed framework is that we can adapt the classifier, $f$ at any noise level $\\sigma$ as an offline pre-processing step, without any additional training (see Figure 4). As we can see in Equation 3, that we should select a large $\\sigma$ to certify at a bigger $\\ell _ { 2 }$ radius of $R$ . However, a test image that does not remain robust at higher $\\sigma$ produces a lower value of $p _ { A }$ . It leads to reducing the overall certification radius, $R$ . Hence, providing the flexibility of choosing appropriate noise levels for different test examples allows us to improve the certification radius, $R$ . ", + "bbox": [ + 174, + 198, + 825, + 281 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In contrast to our proposed framework, existing randomized smoothing frameworks cannot choose a different $\\sigma$ at test-time since it typically degrades their overall certification performance. Hence, they need to fix $\\sigma$ during training their base models or its components. ", + "bbox": [ + 176, + 289, + 821, + 330 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Applicability. Our proposed certification through adaptation technique can be applied to any classification model, $f$ with batch-normalization layers. However, note that achieving high accuracy against large random Gaussian perturbations is only a necessary condition: a randomized smoothing classifier, $g$ requires to consistently predict the correct class to provide higher certification guarantees at larger radii. Hence, we achieve non-trivial $\\ell _ { 2 }$ certification guarantees at very small $\\ell _ { 2 }$ radii for standard non-robust DNN classifiers (see Appendix B.1). ", + "bbox": [ + 174, + 342, + 825, + 425 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "On the other hand, for existing randomized smoothing models, we achieve higher certification at larger $\\ell _ { 2 }$ radii by adapting their base models with larger $\\sigma$ , improving their overall average certified radius $( A C R )$ (Table 6 and 5 (Appendix)). However, we could not find any $\\sigma$ to obtain a significant improvement at lower $\\ell _ { 2 }$ radii. In contrast, AT models with our proposed offline adaptation technique significantly improve their performance against large Gaussian perturbations, providing non-trivial certification robustness. Experimentally we find that our proposed technique outperforms the state-of-the-art certification models for the $\\ell _ { 2 }$ norm. ", + "bbox": [ + 174, + 433, + 825, + 530 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Finally, while we focus on adaptive BN, there also exists other unsupervised covariate shift adaptation techniques such as self-supervised domain adaptation on single test examples (Sun et al., 2020), pseudo-labeling (French et al., 2017; Xie et al., 2020) etc. Wang et al. (2020) also proposed to update the normalization parameters by entropy minimization to improve the corruption robustness. Future studies may also explore these techniques for the offline pre-processing step. ", + "bbox": [ + 174, + 536, + 825, + 607 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 622, + 326, + 637 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Experimental setup. We use CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng et al., 2009) datasets for our experiments. For CIFAR-10, we use pre-activation ResNet18 and ResNet50 for IMAGENET (He et al., 2016a;b). Our AT models are trained using early stopping criteria (Rice et al., 2020) as follows: For IMAGENET, we use two AT models, $\\bar { \\mathrm { A d v } } _ { \\infty } [ \\ell _ { \\infty } \\overset { \\_ } { \\le } 4 / 2 5 5 ]$ and $\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\leq 3 ]$ , learned at $\\ell _ { \\infty }$ and $\\ell _ { 2 }$ threat models with threat boundaries of $4 / 2 5 5$ and 3 respectively. For CIFAR-10, we train multiple AT models with different threat boundaries. For example, we denote $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 8 / 2 5 5 ]$ and $\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq 1 ]$ as the AT models for $\\ell _ { \\infty }$ and $\\ell _ { 2 }$ threat models, trained with threat boundaries of $8 / 2 5 5$ and 1, respectively. We compare with Baseline and $\\mathrm { R a n d } _ { \\sigma = 0 . 5 }$ models. Baseline models are trained using clean images. $\\mathrm { R a n d } _ { \\sigma = 0 . 5 }$ models are trained by augmenting random noise, sampled from isotropic Gaussian distribution, ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )$ with $\\sigma = 0 . 5$ . We also compare with the current state-of-the-art certification models, SmoothAdv for CIFAR-10 (Salman et al., 2019a). Please refer to Appendix A for more details. ", + "bbox": [ + 173, + 647, + 825, + 814 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4.1 PERFORMANCE UNDER GAUSSIAN NOISE. ", + "text_level": 1, + "bbox": [ + 173, + 832, + 506, + 845 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We first investigate the performance of different classification models under significantly larger Gaussian perturbations. It is a necessary condition to provide $\\ell _ { 2 }$ robustness certification. In Table 2, we present the performance. We observe that when the test examples are sampled from IID settings as training distributions (i.e., $\\sigma = 0$ for Baseline, $\\mathbf { A d v } _ { \\infty }$ , and $\\mathrm { \\ A d v _ { 2 } }$ and $\\sigma = 0 . 5$ for $\\mathrm { R a n d } _ { \\sigma = 0 . 5 } )$ , these models produces the best results regardless of whether BN adaptation is applied. However, as we move away from the IID settings by increasing (or decreasing) $\\sigma$ , the performance of all these models significantly degrades in the standard inference setup. In contrast, covariate shift adaptation using adaptive BN improves the performance for all models. In particular, AT models achieve significantly higher performance gain using adaptive BN than the non-robust baseline models at higher noise levels. For example, at $\\sigma = 0 . 5$ , Baseline, $\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq 3 ]$ and $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 4 / 2 5 5 ]$ respectively achieve top-1 accuracy of $0 . 3 \\%$ , $0 . 4 \\%$ , and $0 . 9 \\%$ for IMAGENET without using BN adaptation (Table 2 (a)). However, adaptive BN for $\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\leq 3 ]$ and $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 4 / 2 5 5 ]$ significantly improves the top-1 accuracy to $4 7 . 3 \\%$ and $4 4 . 9 \\%$ respectively. In contrast, the baseline model only achieves $7 . 7 \\%$ accuracy. We observe similar results for CIFAR-10 in Table 2 (b). ", + "bbox": [ + 178, + 857, + 820, + 886 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/5a0c9ca305744cc6bc22d418723d3100f6a3a50be977e1ff8fafd2f69f54d3a1.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
(a) IMAGENET
Modelσ=0σ=0.25σ=0.5σ=0.75
Baseline75.2±0.011.8±0.220.3±0.010.1±0.0
+ adaptive BN74.4±0.0431.0±0.277.7±0.242.4±0.01
Advo∞≤4/255]62.8±0.03.9±0.030.4±0.010.2±0.01
+ adaptive BN60.8±0.1653.4±0.1544.9±0.0833.7±0.28
Adv2≤3]59.8±0.09.8±0.080.9±0.010.3±0.0
+adaptive BN58.3±0.0853.7±0.1447.3±0.1439.8±0.18
Rand g=0.522.0±0.032.8±0.1160.9±0.040.9±0.06
+ adaptive BN62.7±0.0362.3±0.1859.5±0.1151.4±0.27
", + "bbox": [ + 196, + 102, + 483, + 193 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/01dec90f7cd3b7ae557f624774c0fef8cebeceef0f55f1679ddf368c84e10fc9.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
(b) CIFAR-10
Modelσ=0g=0.25σ=0.5g=0.75
Baseline + adaptive BN95.2±0.010.9±0.8810.6±0.7610.5±1.19
95.0±0.5740.1±0.9722.0±0.8317.2±0.66
Advo≤8/255]82.1±0.040.2±4.5616.1±7.8512.2±5.23
+ adaptive BN81.6±0.9674.2±0.9562.4±0.6451.0±1.03
Adv2[≤1]81.6±0.047.5±5.121.5±7.7914.3±5.63
+ adaptive BN81.8±0.775.8±0.4364.9±0.7353.5±1.71
Rand g=0.566.7±0.069.1±1.0161.2±0.8425.9±1.41
+ adaptive BN74.0±2.173.0±2.0466.8±2.0156.7±0.94
", + "bbox": [ + 509, + 103, + 797, + 194 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 251, + 825, + 416 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/fd7d6a2b75a66244db051455b16e7df86066ba1cd535282d32891358d59c5cad.jpg", + "image_caption": [ + "Table 2: Top-1 accuracy of different classifiers under different levels of Gaussian noises augmented to the test images. We randomly shuffle test images and sample the noises and report $( m e a n \\pm 2 \\times s d )$ ) of five runs. ", + "Figure 1: Visualizing loss-gradients produced by AT models as we apply different levels of Gaussian noises. " + ], + "image_footnote": [], + "bbox": [ + 199, + 438, + 799, + 705 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Loss Gradients under Gaussian Noises. To further investigate the performance of AT models, we visualize the loss gradients for individual pixels of an image as we increase the Gaussian noise (i.e., $\\sigma$ ) (Figure 1). Loss-gradients reflect the most relevant input pixels for classification predictions. Here, we scale, translate and clip the loss-gradient values without using any sophisticated techniques (as suggested in Tsipras et al. (2019)). At $\\sigma = 0$ (i.e., for clean images), the loss-gradients from AT models align properly with perceptually relevant features (as observed previously (Tsipras et al., 2019; Etmann et al., 2019)). However, as we choose higher noise using $\\sigma { = } 0 . 5$ and $\\sigma { = } 0 . 7 5$ , the overall loss gradients become noisier. Specifically, AT models without adaptation produce sharper loss gradients (i.e., greater importance) even for background pixels. In contrast, test-time BN adaptation produces gradients for the pixels from the object of interest and suppress the gradients for background pixels (see Figure 1(c) and Figure 1(d)). Hence, they extract the required semantic information for correct classifications. It is interesting to note that $\\mathbf { A d v } _ { 2 }$ produces significantly more human-aligned loss gradients compared to $\\mathrm { \\bf A d v _ { \\infty } }$ . This behavior is also reflected in their classification (Table 2) and overall certification (Table 1) as we note that $\\mathrm { \\bf A d v } _ { 2 }$ overall produces much better performance compared to $\\mathbf { A d v } _ { \\infty }$ . ", + "bbox": [ + 173, + 743, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "4.2 CERTIFICATION USING RANDOMIZED SMOOTHING ", + "text_level": 1, + "bbox": [ + 174, + 159, + 563, + 174 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We now present the $\\ell _ { 2 }$ certification results using the randomized smoothing framework as the backbone, as proposed in our Algorithm 1. We certify the test images with $9 9 . 9 \\%$ probability. We estimate the class label probabilities of $g$ (in Equation 3) using Monte-Carlo sampling with 100, 000 noisy samples for each test image, as in Cohen et al. (2019); Salman et al. (2019a). We use the full test-set for CIFAR-10 and a sub-sample of 500 test images for IMAGENET (as in Cohen et al. (2019)). We provide the detailed results of certified accuracy along with average certified radius $( A C R )$ for several models, trained using different specifications and adapting with different $\\sigma$ in Table 5 and 6 (Appendix). ", + "bbox": [ + 173, + 189, + 825, + 301 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Certifying AT models. In Figure 2, we first demonstrate that AT models can provide non-trivial $\\ell _ { 2 }$ certified robustness using our proposed framework for both IMAGENET and CIFAR-10 datasets. Here, we use $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\ \\le \\ 4 / 2 5 5 ]$ and $\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } ~ \\le ~ 3 ]$ for ImageNet and $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\ \\le \\ 8 / 2 5 5 ]$ and $\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } ^ { \\mathbf { \\bar { \\rho } } } \\leq 1 ]$ for CIFAR-10 and use $\\sigma =$ 0.5 for adaptation and certification using Algorithm 1. We compare with the certification results of Baseline, $\\mathrm { \\bf A d v _ { \\infty } }$ and $\\mathrm { \\bf A d v } _ { 2 }$ models in the standard settings, without using any adaptation and certified at $\\sigma = 0 . 2 5$ . We can see a significant boost of $\\ell _ { 2 }$ certification results for ", + "bbox": [ + 174, + 308, + 482, + 501 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/c5a061ba1ff2a99c9a40493b61a6a37e31ba97df6aa5fd28e688702dff16fe89.jpg", + "image_caption": [ + "Figure 2: Certified top-1 accuracy at various $\\ell _ { 2 }$ radii for (Left) IMAGENET using ResNet-50 and (Right) CIFAR-10 using preactivation ResNet-18. " + ], + "image_footnote": [], + "bbox": [ + 501, + 320, + 816, + 433 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "both $\\mathbf { A d v } _ { \\infty }$ and $\\mathbf { A d v } _ { 2 }$ models using our proposed framework. Further, $\\mathrm { \\bf A d v _ { 2 } }$ models consistently achieve better performance compared to $\\mathrm { \\bf A d v _ { \\infty } }$ in terms of certified accuracy. For CIFAR-10, both $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\le 8 / 2 5 5 ]$ and $\\Delta \\mathrm { d v } _ { 2 } [ \\ell _ { 2 } \\ \\leq \\ 1 ]$ outperform the standard randomized smoothing framework i.e., $\\mathrm { R a n d } _ { \\sigma = 0 . 5 }$ , certified using $\\sigma = 0 . 5$ (Cohen et al., 2019). For IMAGENET, $\\mathrm { A d v _ { 2 } } \\mathrm { [ } \\ell _ { 2 } \\leq 3 \\mathrm { ] }$ achieves better certified accuracy compared to Rand $\\sigma { = } 0 . 5$ beyond $\\ell _ { 2 }$ -radii of 1.5. ", + "bbox": [ + 174, + 502, + 825, + 571 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/db94ddc1389c47cb5064897a41747daceb067b81bb45cab9ea14b08efa67308e.jpg", + "image_caption": [ + "Figure 3: CIFAR-10: Certified top-1 accuracy achieved by (a) $\\mathbf { A d v } _ { \\infty }$ and (b) $\\mathbf { A d v } _ { 2 }$ models (with test-time adaptive BN at $\\sigma \\ : = \\ : 0 . 5$ ), learned at different threat boundaries. (c) Comparison with the state-of-the-art SmoothAdv models (Salman et al., 2019a), trained at $\\sigma = 0 . 5$ using preactivation ResNet-18. " + ], + "image_footnote": [], + "bbox": [ + 200, + 594, + 790, + 727 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Larger Threat Boundary for Better Certified Robustness. Learning AT models at a higher threat boundary improves the certification accuracy at higher $\\ell _ { 2 }$ radii. We demonstrate this phenomena for both $\\mathbf { A d v } _ { \\infty }$ and $\\mathrm { \\bf A d v } _ { 2 }$ models in Figure 3(a) and 3(b) respectively for CIFAR-10. ", + "bbox": [ + 174, + 790, + 821, + 833 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Figure 3(c) also compares the certified accuracy of $\\mathrm { \\bf A d v } _ { 2 }$ models with the existing state-of-the-art SmoothAdv models (Salman et al., 2019a). SmoothAdv utilizes adversarial training using an adaptive attack with $\\ell _ { 2 }$ threat boundary of $\\epsilon$ and Gaussian noises, ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )$ (See details in Appendix A). We set the noise to $\\sigma = 0 . 5$ and vary $\\epsilon$ for their training to compare with different SmoothAdv models in Figure 3(c). By adapting $\\mathrm { \\bf A d v } _ { 2 }$ models with $\\sigma = 0 . 5$ at test-time using our proposed Algorithm 1, we already achieve similar performance as SmoothAdv. Moreover, unlike existing frameworks, we also provide test-time flexibility to adapt the same models using different $\\sigma$ , without retraining, to improve their certification, as shown below. 2 ", + "bbox": [ + 174, + 840, + 825, + 924 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/ec244e7cfdafcee398f327c2168a3bc55d27ba680e0ba185b05e01c9378b0c33.jpg", + "image_caption": [ + "Figure 4: Certified accuracy at various $\\ell _ { 2 }$ radii by varying $\\sigma$ for test-time adaptation of the same models. Choosing large $\\sigma$ degrades certification at lower $\\ell _ { 2 }$ radii, while provides better performance at higher $\\ell _ { 2 }$ radii. " + ], + "image_footnote": [], + "bbox": [ + 181, + 148, + 810, + 281 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Flexibility of choosing different noise-level $\\sigma$ for certification at test-time. Figure 4 presents the certification results as we vary $\\sigma = \\{ 0 . 2 5 , 0 . 5 , 0 . 7 5 \\}$ for test-time adaptation of the same models using our Algorithm 1. We note that the choice of large $\\sigma$ degrades certification at lower $\\ell _ { 2 }$ radii while providing better performance for higher $\\ell _ { 2 }$ radii. For each test example, we adapt the models with appropriate $\\sigma$ that provides the maximum certified radius to obtain the upper envelope of the certification accuracy curves in Figure 4. This leads to the state-of-the-art certification performance for $\\mathbf { A d v } _ { 2 }$ models, outperforming the existing SmoothAdv models for CIFAR-10 (Table 1). Further for randomized smoothing models (Figure 4(c)), we consistently provide certification at larger $\\ell _ { 2 }$ radii by adapting using larger $\\sigma$ values, improving their overall ACR scores (see Table 6 and 5 (Appendix)). Additional results using several models with different training setups are provided in Figure 7 and 8 (Appendix). ", + "bbox": [ + 174, + 329, + 825, + 481 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Over-fitting reduces Certification. Rice et al. (2020) demonstrate that AT models overfit as we train without early stopping criteria. It degrades their empirical robustness against adversarial attacks. In Figure 5, we compare with the certification accuracy of such overfitted AT models, denoted as $\\mathbf { A d v } ^ { o v e r f i t }$ . We observe that $\\mathbf { A d v } ^ { o v e r f i t }$ models also degrade the certified robustness, in particular, at higher $\\ell _ { 2 }$ radii, compared to their corresponding AT models with early ", + "bbox": [ + 174, + 489, + 436, + 669 + ], + "page_idx": 8 + }, + { + "type": "image", + "img_path": "images/bbc286f3639a94c0f5ab7b4d76878ed10e23a85fadbb5632f294e53274b0ccd8.jpg", + "image_caption": [ + "Figure 5: CIFAR-10: Comparing the certified accuracy of $\\mathbf { A d v } _ { \\infty }$ (Left) and $\\mathrm { \\bf A d v _ { 2 } }$ (Right) models with and without applying early-stopping criteria (denoted as Adv $_ { o v e r f i t }$ ). " + ], + "image_footnote": [], + "bbox": [ + 454, + 492, + 816, + 606 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "stopping criteria. These results also indicate that the empirical and certified robustness are closely related: improving empirical robustness also improves the certified robustness. ", + "bbox": [ + 171, + 670, + 821, + 696 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 714, + 318, + 729 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We propose a novel certification through adaptation algorithm that transforms adversarially trained models into a randomized smoothing classifier using test-time covariate shift adaptation to provide certified robustness for $\\ell _ { 2 }$ norm. Unlike existing models using BN adaptation for different applications, our certification framework does not make any assumptions on the test examples. One main advantage of our proposed certification algorithm is to separately choose appropriate noise levels $\\sigma$ during inference for each test example. We achieve the state-of-the-art $\\ell _ { 2 }$ certification using $\\mathrm { \\ A d v _ { 2 } }$ models for CIFAR-10. Finally, while we mainly focus on $\\ell _ { 2 }$ certification using Gaussian noise, we can also extend this framework for other types of perturbations as long as randomized smoothing works (e.g., uniform noise for $\\ell _ { 1 }$ norm (Yang et al., 2020)) for different applications without any additional training. ", + "bbox": [ + 174, + 742, + 825, + 881 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 CODE OF ETHICS AND REPRODUCIBILITY ", + "text_level": 1, + "bbox": [ + 174, + 102, + 552, + 118 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Code of Ethics. Existing defense models can only provide either empirical or certified robustness against adversarial attacks for higher dimensional input domains. In this paper, we propose a solution to provide high performance to achieve both empirical or certified robustness. It allows us to improve the reliability and trustworthiness for large AI models for sensitive real-world applications. ", + "bbox": [ + 174, + 133, + 825, + 189 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Reproducibility. The key results of our paper are presented using adversarially trained models. For CIFAR-10, we train the models using the codes provided in https://github.com/locuslab/robust overfitting (Rice et al., 2020). For IMAGENET, we obtained the already trained AT models from https://github.com/locuslab/robust overfitting (Rice et al., 2020). Please refer to Appendix A for more details. We have provided the codes for our certification algorithms in the supplementary materials for reproducing the results of our paper. ", + "bbox": [ + 174, + 205, + 825, + 289 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 176, + 310, + 285, + 325 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Anish Athalye, Nicholas Carlini, and David Wagner. 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Towards fast computation of certified robustness for relu networks. In ICML, 2018. ", + "bbox": [ + 176, + 215, + 823, + 258 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Eric Wong and Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. In ICML, 2018. ", + "bbox": [ + 171, + 268, + 825, + 297 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Eric Wong, Frank R Schmidt, Jan Hendrik Metzen, and J Zico Kolter. Scaling provable adversarial defenses. NeurIPS, 2018. ", + "bbox": [ + 173, + 305, + 823, + 334 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Eric Wong, Leslie Rice, and J Zico Kolter. Fast is better than free: Revisiting adversarial training. ICLR, 2020. ", + "bbox": [ + 171, + 343, + 821, + 372 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Qizhe Xie, Minh-Thang Luong, Eduard Hovy, and Quoc V Le. Self-training with noisy student improves imagenet classification. In CVPR, 2020. ", + "bbox": [ + 174, + 381, + 821, + 410 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Greg Yang, Tony Duan, Edward Hu, Hadi Salman, Ilya Razenshteyn, and Jerry Li. Randomized smoothing of all shapes and sizes. arXiv, 2020. ", + "bbox": [ + 174, + 419, + 821, + 448 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Runtian Zhai, Chen Dan, Di He, Huan Zhang, Boqing Gong, Pradeep Ravikumar, Cho-Jui Hsieh, and Liwei Wang. Macer: Attack-free and scalable robust training via maximizing certified radius. In ICLR, 2020. ", + "bbox": [ + 174, + 457, + 821, + 500 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Hongyang Zhang et al. Theoretically principled trade-off between robustness and accuracy. In ICML, 2019. ", + "bbox": [ + 173, + 508, + 821, + 537 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "Jingfeng Zhang, Xilie Xu, Bo Han, Gang Niu, Lizhen Cui, Masashi Sugiyama, and Mohan Kankanhalli. Attacks which do not kill training make adversarial learning stronger. In ICML. PMLR, 2020. ", + "bbox": [ + 173, + 546, + 825, + 588 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "APPENDIX ORGANIZATION ", + "text_level": 1, + "bbox": [ + 176, + 102, + 400, + 118 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "• Section A: Experimental setup. • Section B: Additional Results on Certification. • Section C:Performance against different corruptions ", + "bbox": [ + 217, + 140, + 601, + 215 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A EXPERIMENTAL SETUP ", + "bbox": [ + 176, + 246, + 403, + 263 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1 IMPLEMENTATION DETAILS ", + "text_level": 1, + "bbox": [ + 176, + 285, + 410, + 299 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We present our experimental results on CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng et al., 2009) datasets. The descriptions of different models and training hyper-parameters are provided in the following: ", + "bbox": [ + 174, + 315, + 825, + 358 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1.1 CIFAR-10. ", + "text_level": 1, + "bbox": [ + 176, + 385, + 312, + 400 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We use pre-activation ResNet18 architecture (He et al., 2016b) for our experiments on CIFAR-10. We apply the SGD optimizer with a batch size of 128. We execute a total of 200 training epochs and apply a step-wise learning rate decay set initially at 0.1 and divided by 10 at 100 and 150 epochs, and weight decay $5 \\times 1 0 ^ { - 4 }$ . ", + "bbox": [ + 174, + 415, + 825, + 470 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "AT models (Madry et al., 2018; Rice et al., 2020): Unless and otherwise specified, our AT models are learned using early stopping criteria as described in (Rice et al., 2020). We learn several AT models with different threat boundaries for our experiments. We denote them by specifying their corresponding threat model and threat boundaries. For example, $\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\ \\leq \\ 1 . 5 ]$ denotes an AT model that is learned using PGD adversary with $\\ell _ { 2 }$ threat model and a threat boundary of $\\epsilon = 1 . 5$ , along with early-stopping criteria (Rice et al., 2020). We also learn AT models without using earlystopping criteria, as in (Madry et al., 2018) for our comparison in Figure 5. These models are denoted as Advoverf it. ", + "bbox": [ + 174, + 477, + 825, + 588 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We use projected gradient descent $( P G D )$ adversarial attack (Madry et al., 2018) to train these AT models as follows: For $\\mathrm { \\bf A d v _ { \\infty } }$ , we use 10 iterations and an $\\ell _ { \\infty }$ step size of $\\epsilon / 4$ . For $\\mathrm { \\ A d v _ { 2 } }$ , we use 10 iterations and an $\\ell _ { 2 }$ step size of $\\epsilon / 8 . 5 $ . This is the same experimental setup as in (Rice et al., 2020)). We choose a small set of $1 , 0 0 0$ images from the CIFAR-10 test set for our validation. We apply the PGD attack with the same hyper-parameters for our validation during training. We save the best model using the early-stopping criteria (Rice et al., 2020). ", + "bbox": [ + 174, + 595, + 825, + 680 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Randomized smoothing model by Cohen et al. (2019): We also train $\\mathrm { R a n d } _ { \\sigma = 0 . 5 }$ by training with augmented random noise, sampled from an isotropic Gaussian distribution ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )$ with $\\sigma = 0 . 5$ . Here, we keep the same model architecture, learning rates, batch sizes, and other hyper-parameters as used to learn the AT models. ", + "bbox": [ + 174, + 686, + 823, + 742 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "Randomized smoothing model by Salman et al. (2019a): We also compare with the state-ofthe-art certification models, called ‘SmoothAdv’, by Salman et al. (2019a) for our experiments on $\\ell _ { 2 }$ certification We train the SmoothAdv models by choosing random noise vectors followed by an adaptive adversarial attack with specified $\\ell _ { 2 }$ threat boundary of $\\epsilon$ at each iteration. The noise vectors are sampled from an isotropic Gaussian distribution $\\mathcal { N } ( 0 , \\bar { \\sigma } ^ { 2 } I )$ . ", + "bbox": [ + 174, + 750, + 825, + 819 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "We note that the training hyper-parameter $\\epsilon$ has the most significant impact on the certification curve for a SmoothAdv model (please refer to Table 7-15 of (Salman et al., 2019a) for more details). For our experiments, we train 4 different SmoothAdv models with $\\epsilon =$ $\\{ 0 . 2 5 , 0 . 5 , 1 , 2 \\}$ and $\\sigma ~ = ~ 0 . 5$ using adaptive PGD attack with 10 steps. We denote them as SmoothAd $\\scriptstyle v _ { \\sigma = 0 . 5 , \\epsilon = 0 . 2 5 }$ , $\\mathrm { S m o o t h A d v } _ { \\sigma = 0 . 5 , \\epsilon = 0 . 5 }$ , $\\mathrm { S m o o t h A d v } _ { \\sigma = 0 . 5 , \\epsilon = 1 }$ and SmoothAd $\\scriptstyle { \\mathrm { \\mathbf { U } } } _ { \\sigma = 0 . 5 , \\epsilon = 2 }$ respectively. We use the same training set-up and other hyper-parameters as specified in their Github: https://github.com/Hadisalman/smoothing-adversarial. ", + "bbox": [ + 174, + 827, + 825, + 924 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "A.1.2 IMAGENET. ", + "text_level": 1, + "bbox": [ + 176, + 103, + 313, + 117 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We use ResNet50 architecture (He et al., 2016a) for IMAGENET. We obtain the Baseline and $\\mathrm { R a n d } _ { \\sigma = 0 . 5 }$ models from (Cohen et al., $2 0 1 9 ) ^ { 3 }$ . These models are trained using Gaussian augmented noises, sampled from isotropic Gaussian distribution ${ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )$ with $\\sigma = 0 . 0$ (i.e., no noise) and $\\sigma = 0 . 5$ respectively. ", + "bbox": [ + 174, + 127, + 825, + 184 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "The AT models i.e., $\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 4 / 2 5 5 ]$ and $\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\leq 3 ]$ are learned for $\\ell _ { \\infty }$ and $\\ell _ { 2 }$ threat models with threat boundary of $4 / 2 5 5$ and 3, respectively. We use the publicly available models provided by Rice et al. (2020) 4. These models are fine-tuned using PGD-based adversarial training with early stopping criteria, originally provided by Engstrom et al. (2019) 5. ", + "bbox": [ + 173, + 190, + 825, + 247 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We resize the input images to $2 5 6 \\times 2 6 5$ pixels and crop $2 2 4 \\times 2 2 4$ pixels from the center. For our experiments on certification, we use a set of 500 test images by choosing at most 1 sample for each class. ", + "bbox": [ + 176, + 253, + 823, + 295 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "A.2 CHOICE OF TEST-TIME ADAPTIVE BN HYPER-PARAMETERS ", + "text_level": 1, + "bbox": [ + 174, + 314, + 635, + 329 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "BN adaptation technique is controlled by two hyper-parameters, i.e., the test batch-size and momentum $( \\rho )$ (see Equation 5) to update the statistics of the batch-normalization layers. Assuming that the test images are obtained independently from the same test distribution, we can efficiently compute the BN statistics from these images. The hyper-parameter $\\rho \\in [ 0 , 1 ]$ controls the tread-off between pre-computed training statistics and test statistics. We can obtain a better estimation of the test distribution from a large test batch. Hence, we can choose a higher value of $\\rho$ . ", + "bbox": [ + 174, + 340, + 825, + 424 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Here, we compare the top-1 test accuracy of AT models under Gaussian augmented noise with $\\sigma = 0 . 5$ for different choices of $\\rho$ and the batch size. We skip the standard baseline models from our analysis and refer to the previous works (Schneider et al., 2020; Nado et al., 2020) that analyzed the effects of these hyper-parameters for the standard baseline DNN classifiers. ", + "bbox": [ + 174, + 431, + 825, + 487 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/eb1c173c93a9cd50739a743f2106d0a827da3af3fe3e97e30c1789b3f5bf699b.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
(b) CIFAR-10
pAdvoAdv2
0.0 (No adaptation)16.1±7.8521.5±7.79
0.145.1±0.4946.9±0.48
0.359.2±0.4260.8±0.33
0.562.4±0.2764.4±0.6
0.762.8±0.5264.9±0.31
0.962.8±0.7164.9±0.31
1.0 (Full adaptation)62.4±0.6464.9±0.73
", + "bbox": [ + 514, + 502, + 777, + 621 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/3990c4577cef1e94e6ab701f89e247cf0bdecd375d59e9675eb9d0982fc90a01.jpg", + "table_caption": [ + "Table 3: Top-1 accuracy using fixed test batch-size $= 5 1 2$ for AT models under Gaussian augmented noise with $\\sigma = 0 . 5$ for different choices of momentum, $\\rho$ during inference. We randomly shuffle the test images to report $( m e a n + 2 \\times s d )$ of 5 different runs. " + ], + "table_footnote": [], + "table_body": "
(a)IMAGENET
pAdvoAdv2
0.0 (No adaptation)0.4±0.010.9±0.01
0.12.1±0.047.7±0.09
0.320.6±0.1636.6±0.09
0.541.1±0.0945.5±0.13
0.743.5±0.1446.7±0.13
0.944.2±0.1246.8±0.13
1.0 (Full adaptation)44.8±0.1347.2±0.14
", + "bbox": [ + 215, + 502, + 478, + 621 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Momentum $( \\rho )$ . We first investigate the effect of momentum $( \\rho )$ as we choose a large batch size of 512. In Table 3, we present the performance of AT models for different values of $\\rho$ . Recall that, $\\rho = 1$ denotes full adaptation (Equation 5). Here, we completely ignore the training statistics and recompute the BN statistics using the test batches. In contrast, $\\rho = 0$ represents no adaptation, i.e., the same as the standard ‘deterministic’ inference setup. In this case, we use the previously computed BN statistics obtained during training. ", + "bbox": [ + 173, + 693, + 825, + 776 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "We observe that for IMAGENET (Table 3 [Left]) the performance started converging at $\\rho = 0 . 7$ . For CIFAR-10 (Table 3 [Right]), the convergence started even earlier at $\\rho = 0 . 5$ . ", + "bbox": [ + 174, + 784, + 821, + 813 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "Batch Size. Next, we investigate the minimum size of the test batches to choose $\\rho = 1$ (i.e., fulladaptation). In Table 4, we fix $\\rho = 1$ and vary the test batch sizes as we evaluate these AT models. We observe that the performance of these models started improving even when we are using the test batches of size 8. The performance further improves as we choose larger sizes of test batches. We can see that their performance started converging as we choose the test batches of size 64 for IMAGENET. On the other hand, the convergence started much earlier for CIFAR-10. ", + "bbox": [ + 176, + 828, + 823, + 871 + ], + "page_idx": 14 + }, + { + "type": "table", + "img_path": "images/cf29ed71fe841d19110c881cc30ea430c925467685fa421c9c6b82784f383449.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
(b) CIFAR-10
Batch SizeAdvoAdv2
w/o BNadapt16.1±7.8521.5±7.79
857.2±1.2359.5±0.38
1660.2±0.7962.3±0.87
3261.5±0.4663.6±0.55
6462.3±0.564.0±0.38
12862.7±0.6864.4±0.53
25662.7±0.6864.9±0.48
51262.4±0.6464.9±0.73
", + "bbox": [ + 517, + 102, + 732, + 223 + ], + "page_idx": 15 + }, + { + "type": "table", + "img_path": "images/58cefb42f3d0e1c954553e2f2b83a1ffc801b1c2bce7f68a0adb49ab22d226e8.jpg", + "table_caption": [ + "Table 4: Top-1 accuracy using fixed $\\rho = 1$ for AT models under Gaussian augmented noise with $\\sigma = 0 . 5$ for different size of test batches during inference. We randomly shuffle the test images to report $( m e a n + 2 \\times s . d . )$ of 5 different runs. " + ], + "table_footnote": [], + "table_body": "
(a)IMAGENET
Batch SizeAdvoAdv2
w/oBNadapt0.4±0.010.9±0.01
811.5±0.229.1±0.15
1628.1±0.2226.7±0.14
3237.1±0.2437.6±0.2
6441.4±0.2642.9±0.12
12843.3±0.1545.4±0.13
25644.4±0.2146.7±0.07
51244.8±0.1347.2±0.14
", + "bbox": [ + 264, + 102, + 482, + 223 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 300, + 826, + 342 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B ADDITIONAL RESULTS ON CERTIFICATION ", + "text_level": 1, + "bbox": [ + 173, + 363, + 563, + 378 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/a72edb939dd233d7ef46f6ab157fc07a509c87a2ab1938860a9bf0bb78e26b1d.jpg", + "image_caption": [ + "Figure 6: $\\ell _ { 2 }$ Certification for standard non-robust classifiers. For CIFAR-10, we observe that, even after adaptation, the baseline produces lower certification compared to $\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq 1 ]$ model without any adaptation. " + ], + "image_footnote": [], + "bbox": [ + 204, + 400, + 789, + 608 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "B.1 $\\ell _ { 2 }$ CERTIFICATION FOR STANDARD NON-ROBUST CLASSIFIERS ", + "text_level": 1, + "bbox": [ + 174, + 675, + 643, + 689 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "In Figure 6, we present the $\\ell _ { 2 }$ certification results for standard non-robust classification models using our proposed Algorithm 1. In Table 2, we note that the adaptive BN technique can also significantly improve the performance of a non-robust model at lower noise levels, $\\sigma$ . In particular, for CIFAR-10 dataset, Baseline models using adaptation achieve similar performance as $\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq$ 1] without BN adaptation, while produces significantly lower $\\ell _ { 2 }$ certification robustness. This is because, Baseline models, even after adaptation cannot consistently predict the same class to provide higher certified robustness at larger $\\ell _ { 2 }$ radii. As a result, we can only improve the certified robustness at smaller $\\ell _ { 2 }$ radii. ", + "bbox": [ + 173, + 700, + 825, + 813 + ], + "page_idx": 15 + }, + { + "type": "image", + "img_path": "images/2c20dffba6826dfc30c2b7b6b9c8e8057f5321d3caea73d0268f2a64ebc189bb.jpg", + "image_caption": [ + "Figure 7: IMAGENET: Certified top-1 accuracy at various $\\ell _ { 2 }$ radii as we vary the noise-level, $\\sigma$ at test-time using proposed Algorithm 1. $\\mathbf { A d v } _ { \\infty }$ and $\\mathbf { A d v } _ { 2 }$ models are as defined in experimental set-up (section 4). Refer to Table 5 for complete results of all models and different settings. " + ], + "image_footnote": [], + "bbox": [ + 202, + 103, + 789, + 520 + ], + "page_idx": 16 + }, + { + "type": "image", + "img_path": "images/0e4a0c6dfa810b8c74282bb378bde050de58369c2b0ff45a101ae3c2c7280b30.jpg", + "image_caption": [ + "Figure 8: CIFAR-10: Certified top-1 accuracy at various $\\ell _ { 2 }$ radii as we vary the noise-level, $\\sigma$ at test-time using proposed Algorithm 1. Refer to Table 6 for complete results of all models and different settings. " + ], + "image_footnote": [], + "bbox": [ + 173, + 103, + 816, + 520 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/2c6f7250896cc5ee2fae5970dd445ed7d67ae44e26e8b3984ce4ab9e26792847.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
IMAGENET
ModelBN adaptionCertification0.5l2 Radius 1.251.75
0.250.751.01.52.02.252.52.75ACR
Baseline=atσ=0.257.84.83.00.00.00.00.0 0.00.00.00.01 0.054
Advo[l∞o ≤4/255]at σ = 0.25 at σ = 0.50at g = 0.2550.046.441.60.0 0.00.00.00.00.00.00.00.445 0.607
at σ=0.75at σ = 0.50 at σ=0.7543.6 31.639.435.831.4 27.623.418.20.00.00.00.00.443
26.422.418.6 16.814.411.89.47.65.63.6
Advo[loo≤4/255]+adapt[BestRadij(Ours)50.046.441.631.4 27.623.418.29.47.65.63.60.759
at σ = 0.250.480
Adv2[l2 ≤3.00]at σ= 0.50at σ = 0.2553.2 47.050.2 43.046.8 39.00.0 0.00.00.00.00.00.00.00.711
at σ=0.75at g = 0.50 at g=0.7537.832.236.4 32.830.8 20.227.00.00.0 14.20.0 12.00.0 9.60.639
Adv2l2≤3.00] +adapt[Best Radii](Ours)53.228.426.0 22.4 32.830.819.0 27.017.4 17.414.212.09.60.930
50.246.836.4
Randg=0.5 Cohen et al. (2019)=at σ=0.5060.854.447.839.034.2 29.023.80.00.00.00.00.809
at σ = 0.25at g = 0.2559.853.646.60.0 0.00.00.00.00.00.00.00.507
+ adaptationat σ= 0.50at σ= 0.5058.651.043.837.432.2 27.422.40.00.00.00.00.768
at σ=0.75at σ=0.7548.641.636.631.226.2 22.418.616.812.88.65.40.720
Randg=0.5+adapt[BestRadii](Ours)22.416.8
59.853.646.637.432.227.412.88.65.40.973
", + "bbox": [ + 197, + 597, + 800, + 776 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Table 5: IMAGENET: Certified top-1 accuracy at various $\\ell _ { 2 }$ radii as we vary $\\sigma$ for BN adaptation and certification along with average certified radii (ACR). We use ResNet50 for IMAGENET. Each gray block is corresponding to one classification model while the rows are corresponding to its certification performances as we choose different noise levels for adaptations and certifications. The Best Radii are obtained by selecting the highest radius for each test example as we adapt the models with different noise levels, $\\sigma$ . ", + "bbox": [ + 173, + 787, + 825, + 871 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/0dfcd3a18b937a2fff5cae90819ce8c40205dbe69414517034a264d5dfb4ae01.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
CIFAR-10
ModelBN adaptionCertification0.250.50.75l2Radius 1.0 1.251.51.752.0ACR
Baseline0.0 0.00.00.0 0.00.026
Advo[∞ ≤ 4/255]at σ = 0.25 at σ = 0.50 at σ = 0.75at σ = 0.25 at σ = 0.50 at σ = 0.7567.96 47.34 26.8950.46 31.83 15.9231.96 18.78 8.440.0 9.98 4.310.0 4.44 2.030.0 1.62 0.28 0.79 0.230.0 0.0 0.00.080.485 0.350 0.146
Advo[lo ≤8/255]at σ =0.25 at σ = 0.50 at σ =0.75at g =0.25 at σ = 0.50 at σ =0.7566.43 53.65 39.9655.06 42.91 30.7642.86 32.58 22.010.0 22.68 14.640.0 14.24 8.840.0 7.88 4.810.0 2.940.0 0.0 1.150.527 0.515 0.352
Advoo[∞ ≤12/255]atσ =0.25 at σ = 0.50 at σ = 0.75at σ= 0.25 at σ = 0.50 atσ=0.7560.52 51.53 42.6152.42 43.94 35.5643.27 36.41 28.470.0 28.69 22.390.0 21.25 16.690.0 14.532.29 0.0 8.030.0 0.0 4.430.499 0.581 0.482
Adv[l∞o ≤ 16/255]at σ =0.25 at σ = 0.50 at σ =0.75atg=0.25 at σ = 0.50 atσ =0.7553.75 48.07 42.05 67.9647.57 42.51 36.42 55.0641.18 36.54 31.24 43.270.0 30.55 26.05 30.550.0 24.68 20.7411.58 0.0 18.49 16.15 12.017.42 0.0 12.110.0 0.0 8.450.454 0.598 0.557 0.903
Advo + adapt [Best Radii] (Ours) 24.6818.49 12.11 8.45 0.0
Adv2[l2 ≤0.50]at σ = 0.25 at σ = 0.50 atg =0.75at σ = 0.25 at σ = 0.50 atσ =0.7568.84 48.81 27.3854.04 33.82 16.1537.13 20.95 9.230.0 11.5 4.560.0 5.64 2.060.0 2.29 0.62 0.91 0.330.0 0.0 0.080.518 0.382 0.153
Adv2[l2 ≤ 1.00]at σ = 0.25 atσ =0.50 atσ =0.75at σ = 0.25 at σ = 0.50 atg =0.7568.02 56.45 43.0458.54 46.24 33.0846.98 35.6 24.810.0 26.89 17.680.0 18.73 11.390.0 11.37 6.60.0 5.41 3.570.0 0.0 1.950.551 0.580 0.405
Adv2[l2 ≤1.25]at g =0.25 at σ =0.50 atσ =0.75at σ= 0.25 at σ = 0.50 at σ = 0.7567.13 57.73 46.5458.77 48.8 37.5349.43 39.64 29.350.0 31.07 22.00.0 22.61 15.620.0 15.82 10.510.0 8.96 6.550.0 0.0 3.680.557 0.647 0.496
Adv2[l2 ≤ 1.50]at σ = 0.25 at σ =0.50 at σ=0.75at g = 0.25 at σ = 0.50 atg =0.7564.21 56.55 47.7357.13 49.19 40.8949.71 41.72 33.780.0 34.47 27.220.0 27.36 20.780.0 20.23 15.190.0 12.98 10.510.0 0.0 6.770.543 0.689 0.585
Adv2[l2 ≤ 2.00]at σ= 0.25 at σ =0.50 at σ =0.75at σ = 0.25 at σ = 0.50 at σ = 0.7560.4 54.27 47.9654.71 48.89 42.5448.35 43.1 37.040.0 37.34 31.650.0 31.52 26.170.0 25.74 21.120.0 19.14 16.590.0 0.0 12.440.523 0.731 0.698
Adv2[l2 ≤ 2.25]at σ = 0.25 at σ = 0.50 at σ = 0.75atg=0.25 at σ = 0.50 at σ = 0.7557.08 52.1 46.4552.5 46.99 41.7147.11 42.26 36.750.0 36.9 31.880.0 31.58 26.950.0 26.08 22.330.0 20.03 17.820.0 0.0 13.550.504 0.724 0.713
Adv2[l2 ≤ 2.50]at σ =0.25 at σ = 0.50 atσ=0.75at σ = 0.25 at σ = 0.50 at σ =0.7554.88 50.53 45.9550.79 46.26 41.8946.29 41.84 37.530.0 37.74 33.550.0 33.2 29.310.0 28.69 25.270.0 23.34 20.980.0 0.0 17.280.487 0.734 0.765
Adv2[2 ≤ 3.00]at σ =0.25 atσ =0.50 at σ = 0.75at σ =0.25 at σ = 0.50 atσ =0.7553.82 49.41 45.3749.69 45.57 41.5445.04 41.52 37.750.0 37.43 33.490.0 33.37 29.350.0 28.82 25.620.0 23.65 21.830.0 0.0 18.230.475 0.720 0.771 1.198
Adv2+adapt[Best Radii](Ours) 68.84 58.77
Randg=0.5at g= 0.50 at σ = 0.2551.68 62.9140.38 52.2549.71 30.25 40.0637.74 20.81 0.033.37 13.36 0.028.82 7.71 0.023.65 3.38 0.018.23 0.0 0.00.488 0.497 0.575
at σ = 0.25 + adaptation at σ =0.50 at σ = 0.50
Randg=0.5 +adapt[Best Radi] (Ours)atσ =0.75at σ =0.7557.58 46.4 62.9146.46 35.63 52.2535.5 26.06 40.0625.57 18.17 25.5717.43 11.61 17.4310.67 5.46 6.86 3.64 10.67 5.460.0 1.92 1.920.427 0.657 0.609
SmoothAdvg=0.5,=0.25 at g= 0.50
+ adaptationat σ = 0.25 at σ = 0.50at σ = 0.25 at g = 0.5057.8 58.74 54.047.63 48.29 42.9137.41 36.7 32.6327.88 0.0 23.620.33 13.53 0.0 16.088.03 0.0 0.0 9.93 5.50.0 0.0 0.00.464 0.535 0.390
SmoothAdvg=0.5,∈=0.50at σ = 0.75 at σ = 0.25at σ = 0.75 atσ =0.50 at σ = 0.2543.15 58.82 59.8932.29 49.68 50.423.61 40.35 39.9916.41 31.93 0.010.94 24.18 0.06.65 17.05 0.03.88 10.57 0.02.16 0.0 0.00.661 0.483 0.592
+ adaptation
SmoothAdvg=0.5,∈=1.0at σ = 0.50 at σ = 0.75at σ = 0.50 at σ =0.75 atg=0.5055.73 46.25 56.5345.79 36.72 49.5336.6 28.2 41.3827.4 20.9 34.6320.03 14.73 27.8113.48 9.46 21.227.85 5.78 14.410.0 3.32 0.00.470 0.691 0.467
+ adaptation- at σ = 0.25 at σ = 0.50at σ = 0.25 at σ = 0.50 at σ = 0.7557.13 53.56 47.1748.38 45.79 39.439.7 37.640.0 30.060.0 23.12 18.930.0 17.27 13.70.0 11.14 9.260.0 0.0 5.80.620 0.545
at σ = 0.75
SmoothAdvg=0.5,∈=2.0at σ = 0.25 at σ =0.50atg =0.50 at σ = 0.2552.82 52.2347.67 47.2431.8 42.68 41.7624.74 37.55 0.032.64 0.027.52 0.022.42 0.00.0 0.00.732 0.451 0.692
+ adaptation at σ =0.75at σ = 0.50 atσ =0.7550.28 46.745.24 41.7940.39 37.1435.5 33.0530.92 28.2826.1 23.8820.25 19.340.0 15.050.727
SmoothAdvg=0.5[BestRadii]58.82 59.8949.68 50.442.68 41.7637.55 35.532.64 30.9227.52 26.122.42 20.250.0 15.050.918 1.008
SmoothAdvg=0.5 + adapt [Best Radii] (Ours) MARCERg=0.5 (Zhai et al.,2020) 60.0 53.0
", + "bbox": [ + 205, + 98, + 789, + 773 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "Table 6: CIFAR-10: Certified top-1 accuracy at various $\\ell _ { 2 }$ radii as we vary $\\sigma$ for test-time BN adaptation along with average certified radii (ACR) for individual settings. Each gray block is corresponding to one classification model while the rows are corresponding to its certification performances as we choose different noise levels for adaptations and certifications. The Best Radii are obtained by training different models with varying hyper-parameters and adapting them with different noise levels during inference. We also present the best reported results for MARCER (Zhai et al., 2020) and Consistancy Jeong & Shin (2020) at $\\sigma = 0 . 5$ , obtained from the respective papers. ", + "bbox": [ + 173, + 782, + 825, + 872 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "C PERFORMANCE AGAINST DIFFERENT CORRUPTIONS ", + "text_level": 1, + "bbox": [ + 174, + 893, + 637, + 909 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "We mainly focus on $\\ell _ { 2 }$ certification using Gaussian noise in this paper. However, we note that randomized smoothing techniques have been also applied to provide certifications for other perturbation 19 ", + "bbox": [ + 174, + 921, + 823, + 959 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "types as well (e.g., random uniform noise for $\\ell _ { 1 }$ norm (Yang et al., 2020)). Consequently, we can apply our proposed Algorithm 1 to adapt an AT model for any given perturbation types without any additional training for different applications. ", + "bbox": [ + 176, + 103, + 823, + 146 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "Further, Hendrycks & Dietterich (2019) recently introduced ImageNet-C and CIFAR10-C datasets by algorithmically generated random corruptions from noise, blur, weather, and digital categories with 5 different severity levels for each corruption. Several recent works demonstrated that adaptive BN techniques can significantly improve the performance of any classifier (including AT models) against different random corruptions. Further, – also demonstrated the effectiveness of AT models even without applying any adaptation. Hence, our proposed certification framework for AT models is a step forward towards further improving the reliability of sensitive real-world applications. ", + "bbox": [ + 174, + 152, + 825, + 251 + ], + "page_idx": 19 + } +] \ No newline at end of file diff --git a/parse/dev/RFGkzxMFqby/RFGkzxMFqby_middle.json b/parse/dev/RFGkzxMFqby/RFGkzxMFqby_middle.json new file mode 100644 index 0000000000000000000000000000000000000000..f3a167a841bdb4e38b6ba7ece7fdaa7e5817c411 --- /dev/null +++ b/parse/dev/RFGkzxMFqby/RFGkzxMFqby_middle.json @@ -0,0 +1,55259 @@ +{ + "pdf_info": [ + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 78, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 78, + 505, + 97 + ], + "spans": [ + { + "bbox": [ + 106, + 78, + 505, + 97 + ], + "score": 1.0, + "content": "ADVERSARIALLY TRAINED MODELS WITH TEST-TIME", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 97, + 345, + 119 + ], + "spans": [ + { + "bbox": [ + 105, + 97, + 345, + 119 + ], + "score": 1.0, + "content": "COVARIATE SHIFT ADAPTATION", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 112, + 136, + 244, + 157 + ], + "lines": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "spans": [ + { + "bbox": [ + 113, + 136, + 201, + 147 + ], + "score": 1.0, + "content": "Anonymous authors", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "spans": [ + { + "bbox": [ + 112, + 146, + 245, + 159 + ], + "score": 1.0, + "content": "Paper under double-blind review", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2.5 + }, + { + "type": "title", + "bbox": [ + 278, + 186, + 333, + 199 + ], + "lines": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "spans": [ + { + "bbox": [ + 276, + 186, + 335, + 200 + ], + "score": 1.0, + "content": "ABSTRACT", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 143, + 210, + 468, + 374 + ], + "lines": [ + { + "bbox": [ + 141, + 210, + 468, + 222 + ], + "spans": [ + { + "bbox": [ + 141, + 210, + 468, + 222 + ], + "score": 1.0, + "content": "Existing defense models against adversarial examples typically provide either em-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "spans": [ + { + "bbox": [ + 141, + 221, + 469, + 234 + ], + "score": 1.0, + "content": "pirical or certified robustness. 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It `", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "trains a DNN classifier using strong adversaries from a specific class of perturbation (e.g., a small", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 116, + 535 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "-norm) to provide robustness for the same perturbation types. Several certification techniques are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "proposed that can be applied to adversarially trained models to certifiably verify if the prediction", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 180, + 558 + ], + "score": 1.0, + "content": "of a test example,", + "type": "text" + }, + { + "bbox": [ + 181, + 546, + 188, + 555 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "remains constant within its neighborhood (Wong & Kolter, 2018; Wang et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 189, + 568 + ], + "score": 1.0, + "content": "2018; Salman et al.,", + "type": "text" + }, + { + "bbox": [ + 189, + 555, + 216, + 566 + ], + "score": 0.38, + "content": "2 0 1 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "; Dvijotham et al., 2018; Gehr et al., 2018; Sheikholeslami et al., 2021).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "However, these certification techniques typically do not scale for larger networks (e.g., ResNet50)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "and datasets (e.g., IMAGENET). Hence, currently, we cannot guarantee that a more powerful, not", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "yet known attack can not break these adversarially trained models. In fact, several recently proposed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "empirical defense models are later broken by stronger adaptive adversarial attacks, indicating the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 610, + 436, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 436, + 623 + ], + "score": 1.0, + "content": "importance of investigating certified defenses with suitable robustness guarantees.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 414, + 639 + ], + "score": 1.0, + "content": "In contrast to adversarial training, randomized smoothing provides a scalable", + "type": "text" + }, + { + "bbox": [ + 414, + 627, + 424, + 638 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "-certification frame-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "work for any classification model, which is robust against large isotropic Gaussian noise (Cohen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "et al., 2019; Salman et al., 2019a). However, the existing randomized smoothing-based certified", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "models produce significantly lower empirical robustness compared to the AT models. On the other", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "hand, this technique cannot be applied for AT models as they are not robust against such large ran-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 680, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 695 + ], + "score": 1.0, + "content": "dom Gaussian noises in the standard settings. 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A DNN classifier that correctly", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 439, + 505, + 453 + ], + "spans": [ + { + "bbox": [ + 105, + 439, + 185, + 453 + ], + "score": 1.0, + "content": "classifies an image", + "type": "text" + }, + { + "bbox": [ + 185, + 442, + 192, + 450 + ], + "score": 0.67, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 192, + 439, + 439, + 453 + ], + "score": 1.0, + "content": ", can be easily fooled by an adversarial attack to misclassify", + "type": "text" + }, + { + "bbox": [ + 439, + 440, + 464, + 450 + ], + "score": 0.9, + "content": "x + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 439, + 505, + 453 + ], + "score": 1.0, + "content": "(Szegedy", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 104, + 449, + 506, + 464 + ], + "spans": [ + { + "bbox": [ + 104, + 449, + 356, + 464 + ], + "score": 1.0, + "content": "et al., 2014; Goodfellow et al., 2015; Madry et al., 2018). Here,", + "type": "text" + }, + { + "bbox": [ + 357, + 451, + 363, + 460 + ], + "score": 0.78, + "content": "\\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 449, + 506, + 464 + ], + "score": 1.0, + "content": "is a minor adversarial perturbation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 461, + 369, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 461, + 225, + 475 + ], + "score": 1.0, + "content": "such that the change between", + "type": "text" + }, + { + "bbox": [ + 226, + 464, + 233, + 471 + ], + "score": 0.73, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 233, + 461, + 250, + 475 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 251, + 462, + 275, + 472 + ], + "score": 0.91, + "content": "x + \\delta", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 461, + 369, + 475 + ], + "score": 1.0, + "content": "remains imperceptible.", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 23, + "bbox_fs": [ + 104, + 417, + 506, + 475 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 478, + 505, + 621 + ], + "lines": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "spans": [ + { + "bbox": [ + 106, + 479, + 505, + 491 + ], + "score": 1.0, + "content": "Among the existing successful defense frameworks, adversarial training (AT) produces the best", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "spans": [ + { + "bbox": [ + 105, + 489, + 505, + 502 + ], + "score": 1.0, + "content": "empirical robustness against the known adversarial attacks without providing any guarantee (Madry", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 505, + 513 + ], + "score": 1.0, + "content": "et al., 2018; Tramer & Boneh, 2019; Zhang et al., 2019; Rice et al., 2020; Gowal et al., 2020). It `", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "spans": [ + { + "bbox": [ + 105, + 510, + 505, + 524 + ], + "score": 1.0, + "content": "trains a DNN classifier using strong adversaries from a specific class of perturbation (e.g., a small", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 106, + 523, + 116, + 535 + ], + "score": 0.87, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "-norm) to provide robustness for the same perturbation types. Several certification techniques are", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "spans": [ + { + "bbox": [ + 105, + 533, + 505, + 546 + ], + "score": 1.0, + "content": "proposed that can be applied to adversarially trained models to certifiably verify if the prediction", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 543, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 180, + 558 + ], + "score": 1.0, + "content": "of a test example,", + "type": "text" + }, + { + "bbox": [ + 181, + 546, + 188, + 555 + ], + "score": 0.71, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 188, + 543, + 505, + 558 + ], + "score": 1.0, + "content": "remains constant within its neighborhood (Wong & Kolter, 2018; Wang et al.,", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 554, + 505, + 568 + ], + "spans": [ + { + "bbox": [ + 105, + 554, + 189, + 568 + ], + "score": 1.0, + "content": "2018; Salman et al.,", + "type": "text" + }, + { + "bbox": [ + 189, + 555, + 216, + 566 + ], + "score": 0.38, + "content": "2 0 1 9 6", + "type": "inline_equation" + }, + { + "bbox": [ + 216, + 554, + 505, + 568 + ], + "score": 1.0, + "content": "; Dvijotham et al., 2018; Gehr et al., 2018; Sheikholeslami et al., 2021).", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "However, these certification techniques typically do not scale for larger networks (e.g., ResNet50)", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 506, + 590 + ], + "score": 1.0, + "content": "and datasets (e.g., IMAGENET). Hence, currently, we cannot guarantee that a more powerful, not", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "yet known attack can not break these adversarially trained models. In fact, several recently proposed", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 506, + 613 + ], + "score": 1.0, + "content": "empirical defense models are later broken by stronger adaptive adversarial attacks, indicating the", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 610, + 436, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 436, + 623 + ], + "score": 1.0, + "content": "importance of investigating certified defenses with suitable robustness guarantees.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 32, + "bbox_fs": [ + 105, + 479, + 506, + 623 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 627, + 504, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 627, + 505, + 639 + ], + "spans": [ + { + "bbox": [ + 106, + 627, + 414, + 639 + ], + "score": 1.0, + "content": "In contrast to adversarial training, randomized smoothing provides a scalable", + "type": "text" + }, + { + "bbox": [ + 414, + 627, + 424, + 638 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 424, + 627, + 505, + 639 + ], + "score": 1.0, + "content": "-certification frame-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 505, + 650 + ], + "score": 1.0, + "content": "work for any classification model, which is robust against large isotropic Gaussian noise (Cohen", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 648, + 506, + 662 + ], + "score": 1.0, + "content": "et al., 2019; Salman et al., 2019a). However, the existing randomized smoothing-based certified", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 505, + 672 + ], + "score": 1.0, + "content": "models produce significantly lower empirical robustness compared to the AT models. On the other", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "spans": [ + { + "bbox": [ + 105, + 670, + 505, + 684 + ], + "score": 1.0, + "content": "hand, this technique cannot be applied for AT models as they are not robust against such large ran-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 680, + 506, + 695 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 506, + 695 + ], + "score": 1.0, + "content": "dom Gaussian noises in the standard settings. Towards this, we investigate to bridge the gap between", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 106, + 693, + 460, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 460, + 705 + ], + "score": 1.0, + "content": "the state-of-the-art empirical and certifiable robust models against adversarial examples.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 627, + 506, + 705 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 709, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 505, + 722 + ], + "score": 1.0, + "content": "In this paper, we present a novel certification through adaptation framework to transform an AT", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 410, + 733 + ], + "score": 1.0, + "content": "model into a randomized smoothing framework during inference, providing", + "type": "text" + }, + { + "bbox": [ + 411, + 721, + 421, + 732 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 421, + 720, + 505, + 733 + ], + "score": 1.0, + "content": "certification without", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "any additional training or architectural modification. Our proposed certification technique consists", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "spans": [ + { + "bbox": [ + 105, + 255, + 505, + 270 + ], + "score": 1.0, + "content": "of two steps: we first apply a covariate shift adaptation to a classifier against Gaussian noise during", + "type": "text", + "cross_page": true + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "spans": [ + { + "bbox": [ + 105, + 267, + 505, + 280 + ], + "score": 1.0, + "content": "inference for each test example (Cariucci et al., 2017; Li et al., 2016). For our paper, we use the well-", + "type": "text", + "cross_page": true + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "spans": [ + { + "bbox": [ + 105, + 277, + 505, + 290 + ], + "score": 1.0, + "content": "known batch normalization adaptation. This process significantly boosts the performance of the AT", + "type": "text", + "cross_page": true + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "models against the random isotropic Gaussian noises compared to the standard non-robust models.", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "Hence, we can now directly apply the randomized smoothing based certification technique to provide", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 309, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 117, + 322 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 117, + 309, + 506, + 325 + ], + "score": 1.0, + "content": "certification in the next step. 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Furthermore, we", + "type": "text", + "cross_page": true + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 344, + 506, + 356 + ], + "score": 1.0, + "content": "can also evaluate the input test examples without transforming the AT models to a randomized", + "type": "text", + "cross_page": true + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 354, + 506, + 367 + ], + "score": 1.0, + "content": "smoothing model, ensuring that their empirical performance remains unaffected. 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l2Radius (CIFAR-10)0.250.50.751.01.251.51.752.0ACR
Baseline6.962.040.090.00.00.00.00.00.026
Randg =0.5 (Cohen et al.,2019)51.6840.3830.2520.8113.367.713.380.00.488
(Ours)Randg=0.5 +adaptation62.9152.2540.0625.5717.4310.675.461.920.657
SmoothAdvg=0.5 (Salman et al.,2019a)58.8249.6842.6837.5532.6427.5222.420.00.918
(Ours) SmoothAdvg=0.5+adaptation59.8950.441.7635.530.9226.120.2515.051.008
Advo (Rice et al.,2020)35.9529.4423.510.00.00.00.00.00.317
(Ours)Advo + adaptation67.9655.0643.2730.5524.6818.4912.118.450.903
Adv2 (Rice et al., 2020)41.8934.1526.70.00.00.00.00.00.359
(Ours) Adv2 +adaptation68.8458.7749.7137.7433.3728.8223.6518.231.198
MARCERg=0.5 (Zhai et al.,2020)60.053.046.038.029.019.012.00.00.726
Consistancyg=0.5 (Jeong& Shin,2020)48.945.141.337.833.929.925.20.00.726
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See", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 506, + 215 + ], + "score": 1.0, + "content": "Table 5 and 6 (Appendix) for detailed results on both IMAGENET and CIFAR-10 respectively. We also present", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 212, + 489, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 317, + 225 + ], + "score": 1.0, + "content": "the best reported results for MARCER and Consistancy at", + "type": "text" + }, + { + "bbox": [ + 317, + 214, + 348, + 223 + ], + "score": 0.88, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 348, + 212, + 489, + 225 + ], + "score": 1.0, + "content": ", obtained from their respective papers.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 245, + 505, + 398 + ], + "lines": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "spans": [ + { + "bbox": [ + 105, + 245, + 505, + 258 + ], + "score": 1.0, + "content": "any additional training or architectural modification. 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This process significantly boosts the performance of the AT", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "spans": [ + { + "bbox": [ + 105, + 289, + 505, + 301 + ], + "score": 1.0, + "content": "models against the random isotropic Gaussian noises compared to the standard non-robust models.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "spans": [ + { + "bbox": [ + 105, + 299, + 505, + 313 + ], + "score": 1.0, + "content": "Hence, we can now directly apply the randomized smoothing based certification technique to provide", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 309, + 506, + 325 + ], + "spans": [ + { + "bbox": [ + 106, + 311, + 117, + 322 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 309, + 506, + 325 + ], + "score": 1.0, + "content": "certification in the next step. 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Our experimental results on CIFAR-10", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 447, + 505, + 460 + ], + "spans": [ + { + "bbox": [ + 141, + 447, + 505, + 460 + ], + "score": 1.0, + "content": "and IMAGENET demonstrate that the proposed certification framework can transform any", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 141, + 458, + 505, + 470 + ], + "spans": [ + { + "bbox": [ + 141, + 458, + 447, + 470 + ], + "score": 1.0, + "content": "AT model into a randomized smoothing classifier to provide certification for", + "type": "text" + }, + { + "bbox": [ + 447, + 459, + 457, + 469 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 457, + 458, + 505, + 470 + ], + "score": 1.0, + "content": "norm, even", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 469, + 460, + 482 + ], + "spans": [ + { + "bbox": [ + 142, + 469, + 273, + 482 + ], + "score": 1.0, + "content": "when the model is learned using", + "type": "text" + }, + { + "bbox": [ + 273, + 469, + 287, + 480 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 287, + 469, + 460, + 482 + ], + "score": 1.0, + "content": "-bounded adversaries (Table 1 & Figure 2).", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 130, + 483, + 505, + 496 + ], + "spans": [ + { + "bbox": [ + 130, + 483, + 505, + 496 + ], + "score": 1.0, + "content": "2. One main advantage of our proposed framework is that it allows us to select appropriate", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 141, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "noise levels for different test examples during inference. 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l2Radius (CIFAR-10)0.250.50.751.01.251.51.752.0ACR
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It optimizes the following loss function for a DNN classifier,", + "type": "text" + }, + { + "bbox": [ + 465, + 678, + 472, + 690 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 473, + 677, + 505, + 691 + ], + "score": 1.0, + "content": ", to pro-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 689, + 504, + 702 + ], + "spans": [ + { + "bbox": [ + 105, + 689, + 212, + 701 + ], + "score": 1.0, + "content": "vide robustness within an", + "type": "text" + }, + { + "bbox": [ + 212, + 691, + 217, + 699 + ], + "score": 0.67, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 689, + 336, + 701 + ], + "score": 1.0, + "content": "-bounded threat model for an", + "type": "text" + }, + { + "bbox": [ + 337, + 689, + 347, + 702 + ], + "score": 0.89, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 689, + 475, + 701 + ], + "score": 1.0, + "content": "norm, where the perturbations,", + "type": "text" + }, + { + "bbox": [ + 476, + 689, + 504, + 699 + ], + "score": 0.89, + "content": "\\delta \\in \\Delta", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 699, + 269, + 713 + ], + "spans": [ + { + "bbox": [ + 105, + 699, + 180, + 713 + ], + "score": 1.0, + "content": "are constrained as", + "type": "text" + }, + { + "bbox": [ + 181, + 700, + 264, + 712 + ], + "score": 0.92, + "content": "\\Delta = \\{ \\delta : | | \\delta | | _ { p } \\leq \\epsilon \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 264, + 699, + 269, + 713 + ], + "score": 1.0, + "content": ":", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42, + "bbox_fs": [ + 105, + 633, + 505, + 713 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 238, + 716, + 374, + 735 + ], + "lines": [ + { + "bbox": [ + 238, + 716, + 374, + 735 + ], + "spans": [ + { + "bbox": [ + 238, + 716, + 374, + 735 + ], + "score": 0.93, + "content": "\\operatorname* { m i n } _ { \\theta } \\mathbb { E } _ { ( x , y ) } [ \\operatorname* { m a x } _ { \\delta \\in \\Delta } \\mathcal { L } ( f _ { \\theta } ( x + \\delta ) , y ) ]", + "type": "interline_equation", + "image_path": "bb136b79e7d02c31aca49a57107971424a025ea42cb59d00f8bede63f3248bf4.jpg" + } + ] + } + ], + "index": 46, + "virtual_lines": [ + { + "bbox": [ + 238, + 716, + 374, + 735 + ], + "spans": [], + "index": 46 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 374, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 376, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 136, + 96 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 136, + 83, + 142, + 92 + ], + "score": 0.67, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 81, + 267, + 96 + ], + "score": 1.0, + "content": "denotes the model parameters.", + "type": "text" + }, + { + "bbox": [ + 267, + 83, + 276, + 92 + ], + "score": 0.73, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 81, + 376, + 96 + ], + "score": 1.0, + "content": "is the classification loss.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 98, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 506, + 113 + ], + "score": 1.0, + "content": "The inner maximization in Eq. 1 is solved by producing adversarial examples using strong iterative", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 282, + 123 + ], + "score": 1.0, + "content": "adversaries, e.g., projected gradient descent", + "type": "text" + }, + { + "bbox": [ + 283, + 111, + 311, + 122 + ], + "score": 0.48, + "content": "( P G D )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "attack (Kurakin et al., 2016; Madry et al., 2018).", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "Wong et al. (2020) found that even a single-step fast gradient sign method (FGSM) attack-based AT", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "models also achieves high empirical robustness (Goodfellow et al., 2015). Zhang et al. (2020) pro-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "posed to use the least adversaries for training. Recently Trades (Zhang et al., 2019), Adv-LLR (Qin", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "et al., 2019) introduced additional regularizers to achieve higher empirical robustness by smoothing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "the loss surface. However, Rice et al. (2020) showed that the standard PGD based AT model with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "score": 1.0, + "content": "early-stopping criteria provides one of the best empirical defenses for a given perturbation type.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 104, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "Recent works also explored the importance of different hyper-parameters for adversarial training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "(Gowal et al., 2020; Pang et al., 2021) as well as incorporating additional data in a semi-supervised", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 489, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 489, + 222 + ], + "score": 1.0, + "content": "fashion (Carmon et al., 2019; Uesato et al., 2019) to further improve their empirical robustness.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Certified Defenses. Empirical defenses demonstrate robustness only against the known adversaries", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "without providing any guarantees. In fact, most empirical defenses proposed in the literature were", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "later broken by stronger adversaries, highlighting the importance of certified defenses to provide", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 435, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 435, + 271 + ], + "score": 1.0, + "content": "robustness guarantees (Athalye et al., 2018; Uesato et al., 2018; Jalal et al., 2019).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5 + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "Several recent works proposed to train neural network models with provable robustness guarantees.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "These works include methods based on semi-definite relaxations (Raghunathan et al., 2018), linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "relaxations and duality (Wong & Kolter, 2018; Wong et al., 2018), abstract interpretation (Mirman", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "et al., 2018), and interval bound propagation (Gowal et al., 2018). Parallel to training a certified", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "defense, several works also focus on certifying the already trained models (Tjeng et al., 2017; Gehr", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "et al., 2018; Weng et al., 2018; Wang et al., 2018; Bunel et al., 2018). Recently Mueller et al.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 356 + ], + "score": 1.0, + "content": "(2021) combined a small certification network with a large, empirically robust AT model using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "some selection criteria to boost overall benign accuracy along with empirical robustness for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "certified framework. However, none of these techniques scale for large networks (e.g., ResNet50)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 308, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 308, + 386 + ], + "score": 1.0, + "content": "or higher-dimensional datasets (e.g., IMAGENET).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5 + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "Randomized Smoothing for Certification. A randomized smoothing classifier is not a neural", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "score": 1.0, + "content": "network. It uses a neural network as its base for classification. Randomized smoothing was initially", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "proposed as a heuristic defense (Cao & Gong, 2017; Liu et al., 2018) and later shown to be certifi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "able (Lecuyer et al., 2019; Li et al., 2019). Recently, Cohen et al. (2019) and Salman et al. (2019a)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 320, + 448 + ], + "score": 1.0, + "content": "separately provided a tight robustness guarantee for", + "type": "text" + }, + { + "bbox": [ + 320, + 435, + 330, + 446 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "-norm. Salman et al. (2019a) provides the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 198, + 459 + ], + "score": 1.0, + "content": "current state-of-the-art", + "type": "text" + }, + { + "bbox": [ + 198, + 447, + 208, + 457 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "certification robustness by adversarially choosing the noise using an adap-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 441, + 471 + ], + "score": 1.0, + "content": "tive attack to train their base classifier. This framework is also analyzed for other", + "type": "text" + }, + { + "bbox": [ + 441, + 457, + 451, + 469 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "norms using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "different noise distributions as well (Li et al., 2019; Lee et al., 2019; Dvijotham et al., 2020; Yang", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "et al., 2020). Salman et al. (2020) proposed to incorporate an additional denoising module as a pre-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "processing unit to convert a standard DNN classifier into a randomized smoothing model to provide", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "non-trivial certified robustness. Notably, randomized smoothing is the only scalable certification", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 511, + 468, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 468, + 525 + ], + "score": 1.0, + "content": "framework. Further, it also achieves superior performance for different perturbation types.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 504, + 541 + ], + "score": 1.0, + "content": "While achieving the state-of-the-art certification performance, randomized smoothing significantly", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "degrades the empirical robustness against adversarial attacks (Lecuyer et al., 2019; Salman et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "2019a; Cohen et al., 2019). Towards this, our proposed technique transforms an AT model into a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "randomized smoothing classifier without any additional training or architectural modification. Since", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "AT models already provide the state-of-the-art empirical defense, we achieve both empirical and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 585, + 404, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 404, + 596 + ], + "score": 1.0, + "content": "certified robustness against adversarial examples using the same classifier.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5 + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 266, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 268, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 268, + 616 + ], + "score": 1.0, + "content": "3 PROPOSED METHODOLOGY", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 504, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 504, + 632 + ], + "score": 1.0, + "content": "In this section, we first present the background of the randomized smoothing technique and explain", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "why it is not directly effective for AT models. Next, we present the existing test-time co-variate", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 642, + 505, + 656 + ], + "score": 1.0, + "content": "shift adaptation for domain adaptations and corruption robustness. 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The randomized smoothing framework", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 720, + 505, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 720, + 264, + 733 + ], + "score": 1.0, + "content": "transforms the original base classifier", + "type": "text" + }, + { + "bbox": [ + 265, + 721, + 272, + 732 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 272, + 720, + 406, + 733 + ], + "score": 1.0, + "content": "into a new, smoothed classifier", + "type": "text" + }, + { + "bbox": [ + 406, + 723, + 412, + 732 + ], + "score": 0.75, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 413, + 720, + 505, + 733 + ], + "score": 1.0, + "content": ". In particular, for an", + "type": "text" + } + ], + "index": 52 + } + ], + "index": 51.5 + } + ], + "page_idx": 2, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 303, + 751, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 301, + 750, + 310, + 762 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 12, + "width": 9 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 374, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 81, + 376, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 81, + 136, + 96 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 136, + 83, + 142, + 92 + ], + "score": 0.67, + "content": "\\theta", + "type": "inline_equation" + }, + { + "bbox": [ + 143, + 81, + 267, + 96 + ], + "score": 1.0, + "content": "denotes the model parameters.", + "type": "text" + }, + { + "bbox": [ + 267, + 83, + 276, + 92 + ], + "score": 0.73, + "content": "\\mathcal { L }", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 81, + 376, + 96 + ], + "score": 1.0, + "content": "is the classification loss.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0, + "bbox_fs": [ + 106, + 81, + 376, + 96 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 99, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 98, + 506, + 113 + ], + "spans": [ + { + "bbox": [ + 105, + 98, + 506, + 113 + ], + "score": 1.0, + "content": "The inner maximization in Eq. 1 is solved by producing adversarial examples using strong iterative", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 110, + 505, + 123 + ], + "spans": [ + { + "bbox": [ + 105, + 110, + 282, + 123 + ], + "score": 1.0, + "content": "adversaries, e.g., projected gradient descent", + "type": "text" + }, + { + "bbox": [ + 283, + 111, + 311, + 122 + ], + "score": 0.48, + "content": "( P G D )", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 110, + 505, + 123 + ], + "score": 1.0, + "content": "attack (Kurakin et al., 2016; Madry et al., 2018).", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "spans": [ + { + "bbox": [ + 105, + 121, + 506, + 135 + ], + "score": 1.0, + "content": "Wong et al. (2020) found that even a single-step fast gradient sign method (FGSM) attack-based AT", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "spans": [ + { + "bbox": [ + 105, + 132, + 505, + 145 + ], + "score": 1.0, + "content": "models also achieves high empirical robustness (Goodfellow et al., 2015). Zhang et al. (2020) pro-", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "spans": [ + { + "bbox": [ + 105, + 142, + 505, + 156 + ], + "score": 1.0, + "content": "posed to use the least adversaries for training. Recently Trades (Zhang et al., 2019), Adv-LLR (Qin", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "et al., 2019) introduced additional regularizers to achieve higher empirical robustness by smoothing", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 165, + 505, + 178 + ], + "score": 1.0, + "content": "the loss surface. However, Rice et al. (2020) showed that the standard PGD based AT model with", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "spans": [ + { + "bbox": [ + 105, + 175, + 505, + 190 + ], + "score": 1.0, + "content": "early-stopping criteria provides one of the best empirical defenses for a given perturbation type.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 186, + 506, + 201 + ], + "spans": [ + { + "bbox": [ + 104, + 186, + 506, + 201 + ], + "score": 1.0, + "content": "Recent works also explored the importance of different hyper-parameters for adversarial training", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "(Gowal et al., 2020; Pang et al., 2021) as well as incorporating additional data in a semi-supervised", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 489, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 489, + 222 + ], + "score": 1.0, + "content": "fashion (Carmon et al., 2019; Uesato et al., 2019) to further improve their empirical robustness.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 6, + "bbox_fs": [ + 104, + 98, + 506, + 222 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 505, + 270 + ], + "lines": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "spans": [ + { + "bbox": [ + 105, + 225, + 505, + 238 + ], + "score": 1.0, + "content": "Certified Defenses. Empirical defenses demonstrate robustness only against the known adversaries", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 250 + ], + "score": 1.0, + "content": "without providing any guarantees. In fact, most empirical defenses proposed in the literature were", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 248, + 505, + 261 + ], + "score": 1.0, + "content": "later broken by stronger adversaries, highlighting the importance of certified defenses to provide", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 259, + 435, + 271 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 435, + 271 + ], + "score": 1.0, + "content": "robustness guarantees (Athalye et al., 2018; Uesato et al., 2018; Jalal et al., 2019).", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 13.5, + "bbox_fs": [ + 105, + 225, + 506, + 271 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 275, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "score": 1.0, + "content": "Several recent works proposed to train neural network models with provable robustness guarantees.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "spans": [ + { + "bbox": [ + 105, + 286, + 505, + 299 + ], + "score": 1.0, + "content": "These works include methods based on semi-definite relaxations (Raghunathan et al., 2018), linear", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "spans": [ + { + "bbox": [ + 105, + 298, + 505, + 311 + ], + "score": 1.0, + "content": "relaxations and duality (Wong & Kolter, 2018; Wong et al., 2018), abstract interpretation (Mirman", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 505, + 321 + ], + "score": 1.0, + "content": "et al., 2018), and interval bound propagation (Gowal et al., 2018). Parallel to training a certified", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "spans": [ + { + "bbox": [ + 105, + 320, + 505, + 333 + ], + "score": 1.0, + "content": "defense, several works also focus on certifying the already trained models (Tjeng et al., 2017; Gehr", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "spans": [ + { + "bbox": [ + 105, + 331, + 505, + 343 + ], + "score": 1.0, + "content": "et al., 2018; Weng et al., 2018; Wang et al., 2018; Bunel et al., 2018). Recently Mueller et al.", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 340, + 506, + 356 + ], + "spans": [ + { + "bbox": [ + 105, + 340, + 506, + 356 + ], + "score": 1.0, + "content": "(2021) combined a small certification network with a large, empirically robust AT model using", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 505, + 365 + ], + "score": 1.0, + "content": "some selection criteria to boost overall benign accuracy along with empirical robustness for the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 105, + 363, + 505, + 376 + ], + "score": 1.0, + "content": "certified framework. However, none of these techniques scale for large networks (e.g., ResNet50)", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 106, + 375, + 308, + 386 + ], + "spans": [ + { + "bbox": [ + 106, + 375, + 308, + 386 + ], + "score": 1.0, + "content": "or higher-dimensional datasets (e.g., IMAGENET).", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 275, + 506, + 386 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 391, + 505, + 523 + ], + "lines": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 403 + ], + "score": 1.0, + "content": "Randomized Smoothing for Certification. A randomized smoothing classifier is not a neural", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 504, + 414 + ], + "score": 1.0, + "content": "network. It uses a neural network as its base for classification. Randomized smoothing was initially", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "spans": [ + { + "bbox": [ + 105, + 414, + 505, + 425 + ], + "score": 1.0, + "content": "proposed as a heuristic defense (Cao & Gong, 2017; Liu et al., 2018) and later shown to be certifi-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 105, + 424, + 505, + 437 + ], + "score": 1.0, + "content": "able (Lecuyer et al., 2019; Li et al., 2019). Recently, Cohen et al. (2019) and Salman et al. (2019a)", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 435, + 505, + 448 + ], + "spans": [ + { + "bbox": [ + 105, + 435, + 320, + 448 + ], + "score": 1.0, + "content": "separately provided a tight robustness guarantee for", + "type": "text" + }, + { + "bbox": [ + 320, + 435, + 330, + 446 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 435, + 505, + 448 + ], + "score": 1.0, + "content": "-norm. Salman et al. (2019a) provides the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 446, + 505, + 459 + ], + "spans": [ + { + "bbox": [ + 105, + 446, + 198, + 459 + ], + "score": 1.0, + "content": "current state-of-the-art", + "type": "text" + }, + { + "bbox": [ + 198, + 447, + 208, + 457 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 209, + 446, + 505, + 459 + ], + "score": 1.0, + "content": "certification robustness by adversarially choosing the noise using an adap-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 456, + 506, + 471 + ], + "spans": [ + { + "bbox": [ + 105, + 456, + 441, + 471 + ], + "score": 1.0, + "content": "tive attack to train their base classifier. This framework is also analyzed for other", + "type": "text" + }, + { + "bbox": [ + 441, + 457, + 451, + 469 + ], + "score": 0.88, + "content": "\\ell _ { p }", + "type": "inline_equation" + }, + { + "bbox": [ + 451, + 456, + 506, + 471 + ], + "score": 1.0, + "content": "norms using", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "spans": [ + { + "bbox": [ + 105, + 466, + 506, + 482 + ], + "score": 1.0, + "content": "different noise distributions as well (Li et al., 2019; Lee et al., 2019; Dvijotham et al., 2020; Yang", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "spans": [ + { + "bbox": [ + 105, + 478, + 505, + 492 + ], + "score": 1.0, + "content": "et al., 2020). Salman et al. (2020) proposed to incorporate an additional denoising module as a pre-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "spans": [ + { + "bbox": [ + 104, + 489, + 506, + 503 + ], + "score": 1.0, + "content": "processing unit to convert a standard DNN classifier into a randomized smoothing model to provide", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "spans": [ + { + "bbox": [ + 105, + 501, + 506, + 514 + ], + "score": 1.0, + "content": "non-trivial certified robustness. Notably, randomized smoothing is the only scalable certification", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 511, + 468, + 525 + ], + "spans": [ + { + "bbox": [ + 105, + 511, + 468, + 525 + ], + "score": 1.0, + "content": "framework. Further, it also achieves superior performance for different perturbation types.", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 31.5, + "bbox_fs": [ + 104, + 391, + 506, + 525 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 529, + 505, + 595 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 504, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 504, + 541 + ], + "score": 1.0, + "content": "While achieving the state-of-the-art certification performance, randomized smoothing significantly", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "spans": [ + { + "bbox": [ + 105, + 540, + 505, + 552 + ], + "score": 1.0, + "content": "degrades the empirical robustness against adversarial attacks (Lecuyer et al., 2019; Salman et al.,", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "spans": [ + { + "bbox": [ + 105, + 551, + 506, + 564 + ], + "score": 1.0, + "content": "2019a; Cohen et al., 2019). Towards this, our proposed technique transforms an AT model into a", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "spans": [ + { + "bbox": [ + 105, + 562, + 505, + 573 + ], + "score": 1.0, + "content": "randomized smoothing classifier without any additional training or architectural modification. Since", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "spans": [ + { + "bbox": [ + 105, + 573, + 505, + 585 + ], + "score": 1.0, + "content": "AT models already provide the state-of-the-art empirical defense, we achieve both empirical and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 585, + 404, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 585, + 404, + 596 + ], + "score": 1.0, + "content": "certified robustness against adversarial examples using the same classifier.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40.5, + "bbox_fs": [ + 105, + 528, + 506, + 596 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 601, + 266, + 614 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 268, + 616 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 268, + 616 + ], + "score": 1.0, + "content": "3 PROPOSED METHODOLOGY", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 505, + 676 + ], + "lines": [ + { + "bbox": [ + 105, + 620, + 504, + 632 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 504, + 632 + ], + "score": 1.0, + "content": "In this section, we first present the background of the randomized smoothing technique and explain", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 505, + 644 + ], + "score": 1.0, + "content": "why it is not directly effective for AT models. 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Otherwise, it leads to lower", + "type": "text" + }, + { + "bbox": [ + 491, + 374, + 504, + 384 + ], + "score": 0.76, + "content": "p _ { A }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 245, + 396 + ], + "score": 1.0, + "content": "and hence a lower certification of", + "type": "text" + }, + { + "bbox": [ + 245, + 384, + 255, + 393 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "for the test examples. Existing randomized smoothing-based", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "models applies custom-trained using explicit Gaussian noises to learn their original base classifier", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "(Lecuyer et al., 2019; Cohen et al., 2019; Salman et al., 2019a; Zhai et al., 2020; Jeong & Shin,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "2020). However, these models produce significantly lower empirical robustness compared to the AT", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "models. Consequently, AT models are not robust against large Gaussian noises in the standard infer-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "ence settings (see Table 2). Hence, we cannot directly use them as the base classifier for randomized", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 154, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 154, + 463 + ], + "score": 1.0, + "content": "smoothing.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24 + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 347, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 348, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 348, + 487 + ], + "score": 1.0, + "content": "3.2 BACKGROUND ON COVARIATE SHIFT ADAPTATION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Recent works on (Sun et al., 2017; Roy et al., 2019; Huang et al., 2018; Li et al., 2016) and corruption", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "robustness (Schneider et al., 2020; Nado et al., 2020; Benz et al., 2021) demonstrate the importance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "of unsupervised covariate shift adaptation. We use adaptive batch-normalization (BN), one of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 427, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 427, + 540 + ], + "score": 1.0, + "content": "most popular and effective unsupervised covariate shift adaptation mechanisms.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5 + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "score": 1.0, + "content": "A BN layer computes the mean and variance of the hidden activation maps across the channels to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 228, + 569 + ], + "score": 1.0, + "content": "normalize these activations to", + "type": "text" + }, + { + "bbox": [ + 229, + 555, + 262, + 568 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "before feeding into the next hidden layer (Ioffe & Szegedy,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "2015). It reduces the dependencies among different hidden layers, improving the training efficiency", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "for deep architectures. Hence, most of the recent DNN architectures frequently incorporate BN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "layers for complex machine learning tasks. However, the distributional shifts in the test examples", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "lead to different activation statistics compared to the training examples. 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As a result, it breaks the crucial assumption for the subsequent hidden layers to work.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5 + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 183, + 650 + ], + "score": 1.0, + "content": "More formally, let", + "type": "text" + }, + { + "bbox": [ + 183, + 638, + 267, + 649 + ], + "score": 0.9, + "content": "P _ { T } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 638, + 396, + 650 + ], + "score": 1.0, + "content": "as the training distribution and", + "type": "text" + }, + { + "bbox": [ + 396, + 638, + 477, + 649 + ], + "score": 0.91, + "content": "P _ { t } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 200, + 662 + ], + "score": 1.0, + "content": "test distribution; where", + "type": "text" + }, + { + "bbox": [ + 200, + 650, + 228, + 659 + ], + "score": 0.89, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 649, + 288, + 662 + ], + "score": 1.0, + "content": "are inputs and", + "type": "text" + }, + { + "bbox": [ + 288, + 650, + 314, + 660 + ], + "score": 0.92, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "are the corresponding class labels. 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The certification", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 217, + 505, + 230 + ], + "spans": [ + { + "bbox": [ + 104, + 218, + 308, + 230 + ], + "score": 1.0, + "content": "procedure is as follows: Suppose a base classifier", + "type": "text" + }, + { + "bbox": [ + 308, + 218, + 316, + 229 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 218, + 356, + 230 + ], + "score": 1.0, + "content": "classifies", + "type": "text" + }, + { + "bbox": [ + 356, + 217, + 400, + 230 + ], + "score": 0.93, + "content": "\\sqrt { ( x , \\sigma ^ { 2 } I ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 400, + 218, + 505, + 230 + ], + "score": 1.0, + "content": "to return the “most prob-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 228, + 505, + 241 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 158, + 241 + ], + "score": 1.0, + "content": "able” class,", + "type": "text" + }, + { + "bbox": [ + 158, + 230, + 170, + 240 + ], + "score": 0.84, + "content": "c _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 171, + 228, + 240, + 241 + ], + "score": 1.0, + "content": "with probability", + "type": "text" + }, + { + "bbox": [ + 241, + 229, + 355, + 240 + ], + "score": 0.88, + "content": "p _ { A } = \\mathbb { P } ( f ( x + \\delta ) = = c _ { A } )", + "type": "inline_equation" + }, + { + "bbox": [ + 355, + 228, + 470, + 241 + ], + "score": 1.0, + "content": ") and the “runner-up” class", + "type": "text" + }, + { + "bbox": [ + 470, + 230, + 483, + 240 + ], + "score": 0.84, + "content": "c _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 483, + 228, + 505, + 241 + ], + "score": 1.0, + "content": "with", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 240, + 506, + 253 + ], + "spans": [ + { + "bbox": [ + 105, + 240, + 153, + 253 + ], + "score": 1.0, + "content": "probability", + "type": "text" + }, + { + "bbox": [ + 153, + 240, + 296, + 252 + ], + "score": 0.91, + "content": "\\begin{array} { r } { p _ { B } = \\operatorname* { m a x } _ { y \\neq c _ { A } } \\mathbb { P } ( f ( x + \\delta ) = = y } \\end{array}", + "type": "inline_equation" + }, + { + "bbox": [ + 297, + 240, + 417, + 253 + ], + "score": 1.0, + "content": "). 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(2019) addressed this problem using Monte Carlo sampling to estimate some", + "type": "text" + }, + { + "bbox": [ + 473, + 325, + 487, + 335 + ], + "score": 0.84, + "content": "\\underline { p _ { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 487, + 322, + 505, + 335 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 106, + 334, + 506, + 346 + ], + "spans": [ + { + "bbox": [ + 106, + 334, + 120, + 345 + ], + "score": 0.86, + "content": "\\overline { { p _ { B } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 120, + 334, + 159, + 346 + ], + "score": 1.0, + "content": "such that", + "type": "text" + }, + { + "bbox": [ + 159, + 334, + 197, + 345 + ], + "score": 0.91, + "content": "p _ { A } \\leq p _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 334, + 215, + 346 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 216, + 334, + 254, + 345 + ], + "score": 0.91, + "content": "{ \\overline { { p _ { B } } } } \\geq p _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 254, + 334, + 506, + 346 + ], + "score": 1.0, + "content": "with arbitrarily high probability. The certified radius for input", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 106, + 344, + 435, + 358 + ], + "spans": [ + { + "bbox": [ + 106, + 347, + 113, + 354 + ], + "score": 0.77, + "content": "x", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 344, + 237, + 358 + ], + "score": 1.0, + "content": "is then computed by replacing", + "type": "text" + }, + { + "bbox": [ + 237, + 347, + 250, + 356 + ], + "score": 0.86, + "content": "p _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 250, + 344, + 268, + 358 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 268, + 346, + 281, + 356 + ], + "score": 0.86, + "content": "p _ { B }", + "type": "inline_equation" + }, + { + "bbox": [ + 281, + 344, + 302, + 358 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 302, + 346, + 316, + 357 + ], + "score": 0.87, + "content": "\\underline { p _ { A } }", + "type": "inline_equation" + }, + { + "bbox": [ + 316, + 344, + 333, + 358 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 334, + 345, + 347, + 356 + ], + "score": 0.89, + "content": "\\overline { { p _ { B } } }", + "type": "inline_equation" + }, + { + "bbox": [ + 347, + 344, + 435, + 358 + ], + "score": 1.0, + "content": "respectively in Eq. 3.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 311, + 506, + 358 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 361, + 505, + 460 + ], + "lines": [ + { + "bbox": [ + 106, + 361, + 505, + 374 + ], + "spans": [ + { + "bbox": [ + 106, + 361, + 344, + 374 + ], + "score": 1.0, + "content": "As we can see in Equation 2 that the original base classifier,", + "type": "text" + }, + { + "bbox": [ + 345, + 362, + 352, + 373 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 352, + 361, + 505, + 374 + ], + "score": 1.0, + "content": "needs to be robust against large Gaus-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 104, + 371, + 504, + 387 + ], + "spans": [ + { + "bbox": [ + 104, + 371, + 491, + 387 + ], + "score": 1.0, + "content": "sian noises to provide non-trivial robustness certification results. Otherwise, it leads to lower", + "type": "text" + }, + { + "bbox": [ + 491, + 374, + 504, + 384 + ], + "score": 0.76, + "content": "p _ { A }", + "type": "inline_equation" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 383, + 505, + 396 + ], + "spans": [ + { + "bbox": [ + 105, + 383, + 245, + 396 + ], + "score": 1.0, + "content": "and hence a lower certification of", + "type": "text" + }, + { + "bbox": [ + 245, + 384, + 255, + 393 + ], + "score": 0.81, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 255, + 383, + 505, + 396 + ], + "score": 1.0, + "content": "for the test examples. Existing randomized smoothing-based", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "spans": [ + { + "bbox": [ + 105, + 394, + 506, + 407 + ], + "score": 1.0, + "content": "models applies custom-trained using explicit Gaussian noises to learn their original base classifier", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 417 + ], + "score": 1.0, + "content": "(Lecuyer et al., 2019; Cohen et al., 2019; Salman et al., 2019a; Zhai et al., 2020; Jeong & Shin,", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "spans": [ + { + "bbox": [ + 105, + 415, + 505, + 428 + ], + "score": 1.0, + "content": "2020). However, these models produce significantly lower empirical robustness compared to the AT", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "spans": [ + { + "bbox": [ + 106, + 428, + 505, + 439 + ], + "score": 1.0, + "content": "models. Consequently, AT models are not robust against large Gaussian noises in the standard infer-", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "spans": [ + { + "bbox": [ + 105, + 438, + 506, + 451 + ], + "score": 1.0, + "content": "ence settings (see Table 2). Hence, we cannot directly use them as the base classifier for randomized", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 448, + 154, + 463 + ], + "spans": [ + { + "bbox": [ + 105, + 448, + 154, + 463 + ], + "score": 1.0, + "content": "smoothing.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 24, + "bbox_fs": [ + 104, + 361, + 506, + 463 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 474, + 347, + 485 + ], + "lines": [ + { + "bbox": [ + 106, + 474, + 348, + 487 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 348, + 487 + ], + "score": 1.0, + "content": "3.2 BACKGROUND ON COVARIATE SHIFT ADAPTATION", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 29 + }, + { + "type": "text", + "bbox": [ + 107, + 494, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Recent works on (Sun et al., 2017; Roy et al., 2019; Huang et al., 2018; Li et al., 2016) and corruption", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "spans": [ + { + "bbox": [ + 105, + 506, + 505, + 518 + ], + "score": 1.0, + "content": "robustness (Schneider et al., 2020; Nado et al., 2020; Benz et al., 2021) demonstrate the importance", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 529 + ], + "score": 1.0, + "content": "of unsupervised covariate shift adaptation. We use adaptive batch-normalization (BN), one of the", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 528, + 427, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 528, + 427, + 540 + ], + "score": 1.0, + "content": "most popular and effective unsupervised covariate shift adaptation mechanisms.", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 31.5, + "bbox_fs": [ + 105, + 495, + 505, + 540 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 544, + 505, + 633 + ], + "lines": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "spans": [ + { + "bbox": [ + 106, + 545, + 504, + 556 + ], + "score": 1.0, + "content": "A BN layer computes the mean and variance of the hidden activation maps across the channels to", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 104, + 555, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 104, + 555, + 228, + 569 + ], + "score": 1.0, + "content": "normalize these activations to", + "type": "text" + }, + { + "bbox": [ + 229, + 555, + 262, + 568 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 262, + 555, + 505, + 569 + ], + "score": 1.0, + "content": "before feeding into the next hidden layer (Ioffe & Szegedy,", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 579 + ], + "score": 1.0, + "content": "2015). It reduces the dependencies among different hidden layers, improving the training efficiency", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 505, + 590 + ], + "score": 1.0, + "content": "for deep architectures. Hence, most of the recent DNN architectures frequently incorporate BN", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "layers for complex machine learning tasks. However, the distributional shifts in the test examples", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 104, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "lead to different activation statistics compared to the training examples. Hence, impacted by the", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 611, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 106, + 611, + 505, + 622 + ], + "score": 1.0, + "content": "covariate shift, the statistics estimated during training fail to normalize the activation tensors to", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 621, + 487, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 621, + 139, + 633 + ], + "score": 0.92, + "content": "\\mathcal { N } ( 0 , 1 )", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 622, + 487, + 634 + ], + "score": 1.0, + "content": ". As a result, it breaks the crucial assumption for the subsequent hidden layers to work.", + "type": "text" + } + ], + "index": 41 + } + ], + "index": 37.5, + "bbox_fs": [ + 104, + 545, + 505, + 634 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 637, + 505, + 704 + ], + "lines": [ + { + "bbox": [ + 106, + 638, + 505, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 638, + 183, + 650 + ], + "score": 1.0, + "content": "More formally, let", + "type": "text" + }, + { + "bbox": [ + 183, + 638, + 267, + 649 + ], + "score": 0.9, + "content": "P _ { T } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 638, + 396, + 650 + ], + "score": 1.0, + "content": "as the training distribution and", + "type": "text" + }, + { + "bbox": [ + 396, + 638, + 477, + 649 + ], + "score": 0.91, + "content": "P _ { t } : \\mathcal { X } \\times \\mathcal { Y } \\mathbb { R } ^ { + }", + "type": "inline_equation" + }, + { + "bbox": [ + 478, + 638, + 505, + 650 + ], + "score": 1.0, + "content": "as the", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 649, + 505, + 662 + ], + "spans": [ + { + "bbox": [ + 105, + 649, + 200, + 662 + ], + "score": 1.0, + "content": "test distribution; where", + "type": "text" + }, + { + "bbox": [ + 200, + 650, + 228, + 659 + ], + "score": 0.89, + "content": "x \\in \\mathcal { X }", + "type": "inline_equation" + }, + { + "bbox": [ + 228, + 649, + 288, + 662 + ], + "score": 1.0, + "content": "are inputs and", + "type": "text" + }, + { + "bbox": [ + 288, + 650, + 314, + 660 + ], + "score": 0.92, + "content": "y \\in \\mathcal { V }", + "type": "inline_equation" + }, + { + "bbox": [ + 315, + 649, + 505, + 662 + ], + "score": 1.0, + "content": "are the corresponding class labels. There exists", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 660, + 504, + 673 + ], + "spans": [ + { + "bbox": [ + 105, + 660, + 342, + 673 + ], + "score": 1.0, + "content": "covariate shift between training and test distribution iff:", + "type": "text" + }, + { + "bbox": [ + 343, + 660, + 414, + 672 + ], + "score": 0.93, + "content": "P _ { T } ( y | \\bar { x } ) { = } P _ { t } ( \\bar { y } | x )", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 660, + 435, + 673 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 435, + 660, + 504, + 672 + ], + "score": 0.92, + "content": "P _ { T } ( x ) \\neq P _ { t } ( x )", + "type": "inline_equation" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "spans": [ + { + "bbox": [ + 105, + 671, + 506, + 683 + ], + "score": 1.0, + "content": "(Sugiyama & Kawanabe, 2012; Scholkopf et al., 2012). If the covariate shift only affects the first ¨", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 680, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 378, + 696 + ], + "score": 1.0, + "content": "and second-order moments of the hidden layer feature activations,", + "type": "text" + }, + { + "bbox": [ + 379, + 682, + 403, + 694 + ], + "score": 0.92, + "content": "f _ { h } ( x )", + "type": "inline_equation" + }, + { + "bbox": [ + 403, + 680, + 506, + 696 + ], + "score": 1.0, + "content": ", we can remove it using", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 693, + 263, + 705 + ], + "spans": [ + { + "bbox": [ + 105, + 693, + 263, + 705 + ], + "score": 1.0, + "content": "normalization (Schneider et al., 2020):", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 44.5, + "bbox_fs": [ + 105, + 638, + 506, + 705 + ] + }, + { + "type": "interline_equation", + "bbox": [ + 180, + 709, + 431, + 735 + ], + "lines": [ + { + "bbox": [ + 180, + 709, + 431, + 735 + ], + "spans": [ + { + "bbox": [ + 180, + 709, + 431, + 735 + ], + "score": 0.92, + "content": "P _ { T } \\Big ( \\frac { f _ { h } ( x ) - \\mathbb { E } _ { T } [ f _ { h } ( x ) ] } { \\sqrt { \\mathbb { V } _ { T } [ f _ { h } ( x ) ] } } \\Big ) P _ { T } ( x ) \\approx P _ { t } \\Big ( \\frac { f _ { h } ( x ) - \\mathbb { E } _ { t } [ f _ { h } ( x ) ] } { \\sqrt { \\mathbb { V } _ { t } [ f _ { h } ( x ) ] } } \\Big ) P _ { t } ( x ) .", + "type": "interline_equation", + "image_path": "5ae0dfd2aa4d58ccac8c953a76996e3b1a2a9e9cf8bcbe00d8ebdd6cba21304e.jpg" + } + ] + } + ], + "index": 48, + "virtual_lines": [ + { + "bbox": [ + 180, + 709, + 431, + 735 + ], + "spans": [], + "index": 48 + } + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 116 + ], + "lines": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 105, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "Covariate shift adaptation using adaptive BN computes the BN statistics from the feature activations,", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 93, + 506, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 93, + 132, + 105 + ], + "score": 0.62, + "content": "\\mu _ { t } , s _ { t } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 132, + 93, + 433, + 106 + ], + "score": 1.0, + "content": ", of the test batch. We can adapt them with the existing training statistics,", + "type": "text" + }, + { + "bbox": [ + 434, + 93, + 464, + 105 + ], + "score": 0.34, + "content": "\\mu _ { T } , s _ { T } ^ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 93, + 506, + 106 + ], + "score": 1.0, + "content": ", obtained", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 105, + 466, + 116 + ], + "spans": [ + { + "bbox": [ + 105, + 105, + 466, + 116 + ], + "score": 1.0, + "content": "using the training batches as (Cariucci et al., 2017; Li et al., 2016; Schneider et al., 2020):", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "interline_equation", + "bbox": [ + 198, + 121, + 412, + 135 + ], + "lines": [ + { + "bbox": [ + 198, + 121, + 412, + 135 + ], + "spans": [ + { + "bbox": [ + 198, + 121, + 412, + 135 + ], + "score": 0.89, + "content": "\\overline { { \\mu } } = \\rho \\cdot \\mu _ { t } + ( 1 - \\rho ) \\cdot \\mu _ { T } \\quad \\overline { { s } } = \\rho \\cdot s _ { t } + ( 1 - \\rho ) \\cdot s _ { T }", + "type": "interline_equation", + "image_path": "e6745a30c421bc620e09cb1c762342005c18b45aefa56c281e52cc85eeaec2dc.jpg" + } + ] + } + ], + "index": 3, + "virtual_lines": [ + { + "bbox": [ + 198, + 121, + 412, + 135 + ], + "spans": [], + "index": 3 + } + ] + }, + { + "type": "text", + "bbox": [ + 107, + 140, + 504, + 174 + ], + "lines": [ + { + "bbox": [ + 105, + 139, + 506, + 154 + ], + "spans": [ + { + "bbox": [ + 105, + 139, + 135, + 154 + ], + "score": 1.0, + "content": "where,", + "type": "text" + }, + { + "bbox": [ + 136, + 140, + 174, + 152 + ], + "score": 0.92, + "content": "\\rho \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 139, + 305, + 154 + ], + "score": 1.0, + "content": "is the momentum. The choice of", + "type": "text" + }, + { + "bbox": [ + 305, + 141, + 330, + 152 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 330, + 139, + 506, + 154 + ], + "score": 1.0, + "content": "is equivalent to the standard inference setup", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 150, + 506, + 165 + ], + "spans": [ + { + "bbox": [ + 105, + 150, + 411, + 165 + ], + "score": 1.0, + "content": "with a deterministic DNN classifier in the IID settings. We should choose", + "type": "text" + }, + { + "bbox": [ + 412, + 152, + 439, + 163 + ], + "score": 0.92, + "content": "\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 439, + 150, + 506, + 165 + ], + "score": 1.0, + "content": "when receiving", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 162, + 422, + 174 + ], + "spans": [ + { + "bbox": [ + 106, + 162, + 422, + 174 + ], + "score": 1.0, + "content": "larger test batches as it can provide a better estimation of the test distributions.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 106, + 179, + 505, + 246 + ], + "lines": [ + { + "bbox": [ + 106, + 180, + 505, + 191 + ], + "spans": [ + { + "bbox": [ + 106, + 180, + 505, + 191 + ], + "score": 1.0, + "content": "Assumptions for BN adaptation. It is noteworthy that these existing adaptive BN-based frame-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "spans": [ + { + "bbox": [ + 106, + 191, + 505, + 202 + ], + "score": 1.0, + "content": "works require a large set of test images from the same covariate shift to estimate the BN parame-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 105, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "ters. However, this assumption may not hold for several real-world applications, e.g., stateless web", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 211, + 506, + 225 + ], + "score": 1.0, + "content": "APIs. Also, these test images should be semantically diverse, preferably over multiple classes, to", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "spans": [ + { + "bbox": [ + 106, + 223, + 505, + 235 + ], + "score": 1.0, + "content": "effectively estimate the test distributions. Hence, it further limits the practical usability of these", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 234, + 359, + 246 + ], + "spans": [ + { + "bbox": [ + 106, + 234, + 359, + 246 + ], + "score": 1.0, + "content": "frameworks for real-world applications, e.g., autonomous cars.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 106, + 251, + 505, + 306 + ], + "lines": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "spans": [ + { + "bbox": [ + 105, + 250, + 505, + 263 + ], + "score": 1.0, + "content": "In contrast to these models for domain adaptation and corruption robustness, our proposed certifi-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 261, + 505, + 275 + ], + "spans": [ + { + "bbox": [ + 106, + 261, + 505, + 275 + ], + "score": 1.0, + "content": "cation framework against adversarial examples does not make any such assumptions. In this case,", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 273, + 505, + 286 + ], + "score": 1.0, + "content": "we already know the perturbation type on which we need to adapt the model to provide the certi-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 284, + 505, + 297 + ], + "spans": [ + { + "bbox": [ + 106, + 284, + 410, + 297 + ], + "score": 1.0, + "content": "fication. Hence, we can explicitly pre-select a diverse set of clean images,", + "type": "text" + }, + { + "bbox": [ + 410, + 284, + 440, + 295 + ], + "score": 0.91, + "content": "{ \\bf X } _ { b a t c h }", + "type": "inline_equation" + }, + { + "bbox": [ + 440, + 284, + 505, + 297 + ], + "score": 1.0, + "content": "and control the", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 295, + 390, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 295, + 390, + 308 + ], + "score": 1.0, + "content": "perturbations to adapt the models, addressing both of these limitations.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 15 + }, + { + "type": "title", + "bbox": [ + 107, + 317, + 418, + 328 + ], + "lines": [ + { + "bbox": [ + 106, + 315, + 419, + 330 + ], + "spans": [ + { + "bbox": [ + 106, + 315, + 419, + 330 + ], + "score": 1.0, + "content": "Algorithm 1: Steps for CERTIFICATION THROUGH ADAPTATION Algorithm", + "type": "text" + } + ], + "index": 18 + } + ], + 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desired noise.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 97, + 392, + 488, + 406 + ], + "spans": [ + { + "bbox": [ + 97, + 392, + 107, + 406 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 108, + 393, + 219, + 405 + ], + "score": 0.8, + "content": "f _ { a d a p t } = \\mathrm { C L O N E } ( f . t r a i n ( ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 392, + 390, + 405 + ], + "score": 1.0, + "content": "// clone", + "type": "text" + }, + { + "bbox": [ + 390, + 394, + 398, + 404 + ], + "score": 0.37, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 392, + 488, + 405 + ], + "score": 1.0, + "content": "with train-mode.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 96, + 405, + 489, + 420 + ], + "spans": [ + { + "bbox": [ + 96, + 405, + 107, + 420 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 107, + 405, + 186, + 418 + ], + "score": 0.62, + "content": "\\underline { { \\mathbf { \\Pi } } } _ { - } = f _ { a d a p t } ( \\tilde { \\mathbf { X } } _ { b a t c h } )", + "type": "inline_equation" + }, + { + "bbox": [ + 251, + 406, + 489, + 419 + ], + "score": 1.0, + "content": "// forward pass for BN parameter adaptation.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 96, + 415, + 489, + 431 + ], + "spans": [ + { + "bbox": [ + 96, + 415, + 164, + 431 + ], + "score": 1.0, + "content": "4 fadapt.eval()", + "type": "text" + }, + { + "bbox": [ + 367, + 417, + 489, + 430 + ], + "score": 1.0, + "content": "// fix the parameters.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 432, + 488, + 445 + ], + "spans": [ + { + "bbox": [ + 106, + 432, + 164, + 445 + ], + "score": 1.0, + "content": "/* Step 2:", + "type": "text" + }, + { + "bbox": [ + 169, + 433, + 213, + 445 + ], + "score": 1.0, + "content": "Certify", + "type": "text" + }, + { + "bbox": [ + 214, + 435, + 234, + 444 + ], + "score": 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Previous works proposed cus-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "tomized training using explicit Gaussian noise augmentation for their training (Section 3.1). Sub-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 505, + 575 + ], + "score": 1.0, + "content": "sequently, in Section 3.2 we note that robustness against random Gaussian noises of any classifier,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 107, + 573, + 114, + 584 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "can be improved by applying covariate shift adaptation using adaptive BN technique without any", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 383, + 596 + ], + "score": 1.0, + "content": "additional training. 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This motivates us to propose a novel certification framework that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "applies the covariate shift adaptation using adaptive BN as an offline pre-processing step to improve", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 639, + 421, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 421, + 650 + ], + "score": 1.0, + "content": "the robustness against random Gaussian noises, addressing the above problem.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Our proposed certification through adaptation framework consists of two steps: Given test image", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 127, + 677 + ], + "score": 0.86, + "content": "x _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 666, + 506, + 678 + ], + "score": 1.0, + "content": ", we first apply the adaptive BN technique to achieve robustness against Gaussian perturba-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "tions. 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"\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 369, + 442, + 381 + ], + "score": 1.0, + "content": "(Eqn 5).", + "type": "text" + }, + { + "bbox": [ + 478, + 371, + 487, + 378 + ], + "score": 1.0, + "content": "*/", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 97, + 379, + 489, + 395 + ], + "spans": [ + { + "bbox": [ + 97, + 379, + 105, + 395 + ], + "score": 1.0, + "content": "1", + "type": "text" + }, + { + "bbox": [ + 105, + 380, + 276, + 393 + ], + "score": 0.75, + "content": "\\tilde { \\mathbf { X } } _ { b a t c h } = [ x + \\mathcal { N } ( 0 , \\sigma I ) \\ \\forall \\ x \\in \\ \\mathbf { X } _ { b a t c h } ]", + "type": "inline_equation" + }, + { + "bbox": [ + 292, + 381, + 353, + 394 + ], + "score": 1.0, + "content": "// perturb", + "type": "text" + }, + { + "bbox": [ + 353, + 382, + 381, + 393 + ], + "score": 0.82, + "content": "{ \\bf X } _ { b a t c h }", + "type": "inline_equation" + }, + { + "bbox": [ + 381, + 381, + 489, + 394 + ], + "score": 1.0, + "content": "with desired noise.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 97, + 392, + 488, + 406 + ], + "spans": [ + { + "bbox": [ + 97, + 392, + 107, + 406 + ], + "score": 1.0, + "content": "2", + "type": "text" + }, + { + "bbox": [ + 108, + 393, + 219, + 405 + ], + "score": 0.8, + "content": "f _ { a d a p t } = \\mathrm { C L O N E } ( f . t r a i n ( ) )", + "type": "inline_equation" + }, + { + "bbox": [ + 340, + 392, + 390, + 405 + ], + "score": 1.0, + "content": "// clone", + "type": "text" + }, + { + "bbox": [ + 390, + 394, + 398, + 404 + ], + "score": 0.37, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 392, + 488, + 405 + ], + "score": 1.0, + "content": "with train-mode.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 96, + 405, + 489, + 420 + ], + "spans": [ + { + "bbox": [ + 96, + 405, + 107, + 420 + ], + "score": 1.0, + "content": "3", + "type": "text" + }, + { + "bbox": [ + 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], + "index": 30 + }, + { + "bbox": [ + 96, + 475, + 142, + 488 + ], + "spans": [ + { + "bbox": [ + 96, + 475, + 133, + 488 + ], + "score": 1.0, + "content": "7 return", + "type": "text" + }, + { + "bbox": [ + 133, + 477, + 142, + 487 + ], + "score": 0.4, + "content": "R", + "type": "inline_equation" + } + ], + "index": 31 + } + ], + "index": 25, + "bbox_fs": [ + 95, + 329, + 490, + 488 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 507, + 349, + 519 + ], + "lines": [ + { + "bbox": [ + 105, + 507, + 350, + 520 + ], + "spans": [ + { + "bbox": [ + 105, + 507, + 350, + 520 + ], + "score": 1.0, + "content": "3.3 PROPOSED CERTIFICATION THROUGH ADAPTATION", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 106, + 528, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 106, + 528, + 505, + 541 + ], + "score": 1.0, + "content": "The robustness guarantee in Eq. 3 suggests that randomized smoothing gives a framework for cer-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 539, + 506, + 552 + ], + "spans": [ + { + "bbox": [ + 106, + 539, + 191, + 552 + ], + "score": 1.0, + "content": "tifying any classifier", + "type": "text" + }, + { + "bbox": [ + 192, + 540, + 199, + 551 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 539, + 506, + 552 + ], + "score": 1.0, + "content": "that is robust against large Gaussian noises. Previous works proposed cus-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 105, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "tomized training using explicit Gaussian noise augmentation for their training (Section 3.1). Sub-", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 104, + 560, + 505, + 575 + ], + "spans": [ + { + "bbox": [ + 104, + 560, + 505, + 575 + ], + "score": 1.0, + "content": "sequently, in Section 3.2 we note that robustness against random Gaussian noises of any classifier,", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 107, + 572, + 505, + 586 + ], + "spans": [ + { + "bbox": [ + 107, + 573, + 114, + 584 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 572, + 505, + 586 + ], + "score": 1.0, + "content": "can be improved by applying covariate shift adaptation using adaptive BN technique without any", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 583, + 505, + 596 + ], + "spans": [ + { + "bbox": [ + 106, + 583, + 383, + 596 + ], + "score": 1.0, + "content": "additional training. However, it modifies the original base classifier", + "type": "text" + }, + { + "bbox": [ + 383, + 583, + 391, + 594 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 391, + 583, + 505, + 596 + ], + "score": 1.0, + "content": "at each forward pass by re-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 594, + 505, + 606 + ], + "score": 1.0, + "content": "computing the BN parameters. Since the certification guarantee in Eq. 3 is provided only for a fixed", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 605, + 506, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 605, + 166, + 618 + ], + "score": 1.0, + "content": "base classifier", + "type": "text" + }, + { + "bbox": [ + 167, + 605, + 174, + 617 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 174, + 605, + 382, + 618 + ], + "score": 1.0, + "content": ", we cannot directly apply adaptive BN to provide", + "type": "text" + }, + { + "bbox": [ + 382, + 605, + 393, + 616 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 393, + 605, + 506, + 618 + ], + "score": 1.0, + "content": "certification using the ran-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "spans": [ + { + "bbox": [ + 105, + 615, + 506, + 630 + ], + "score": 1.0, + "content": "domized smoothing framework. This motivates us to propose a novel certification framework that", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 506, + 641 + ], + "score": 1.0, + "content": "applies the covariate shift adaptation using adaptive BN as an offline pre-processing step to improve", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 639, + 421, + 650 + ], + "spans": [ + { + "bbox": [ + 106, + 639, + 421, + 650 + ], + "score": 1.0, + "content": "the robustness against random Gaussian noises, addressing the above problem.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 38, + "bbox_fs": [ + 104, + 528, + 506, + 650 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 504, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 669 + ], + "score": 1.0, + "content": "Our proposed certification through adaptation framework consists of two steps: Given test image", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 107, + 666, + 506, + 678 + ], + "spans": [ + { + "bbox": [ + 107, + 667, + 127, + 677 + ], + "score": 0.86, + "content": "x _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 128, + 666, + 506, + 678 + ], + "score": 1.0, + "content": ", we first apply the adaptive BN technique to achieve robustness against Gaussian perturba-", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "tions. Recall that adaptive BN requires a large set of diverse test images to correctly re-estimate", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 398, + 700 + ], + "score": 1.0, + "content": "the batch-normalization statistics. However, to provide certification for", + "type": "text" + }, + { + "bbox": [ + 398, + 688, + 408, + 699 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 408, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "-norm, we only need to", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 506, + 712 + ], + "score": 1.0, + "content": "adapt our model against Gaussian perturbations. Hence, we can pre-select a sufficiently large set", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 708, + 506, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 708, + 207, + 723 + ], + "score": 1.0, + "content": "of diverse clean images,", + "type": "text" + }, + { + "bbox": [ + 208, + 710, + 237, + 721 + ], + "score": 0.9, + "content": "{ \\bf X } _ { b a t c h }", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 708, + 470, + 723 + ], + "score": 1.0, + "content": "and apply Gaussian perturbations to adapt our classifier,", + "type": "text" + }, + { + "bbox": [ + 470, + 710, + 477, + 721 + ], + "score": 0.83, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 477, + 708, + 506, + 723 + ], + "score": 1.0, + "content": ", as an", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 720, + 506, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 252, + 734 + ], + "score": 1.0, + "content": "offline pre-processing step to obtain", + "type": "text" + }, + { + "bbox": [ + 253, + 721, + 279, + 733 + ], + "score": 0.89, + "content": "f _ { a d a p t }", + "type": "inline_equation" + }, + { + "bbox": [ + 279, + 720, + 506, + 734 + ], + "score": 1.0, + "content": ". Alternatively, when a large set of diverse test examples", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "are available, we can also use them for our BN adaptation. The Gaussian noise samples should be", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 319, + 106 + ], + "score": 1.0, + "content": "drawn from the same isotropic Gaussian distribution", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 319, + 93, + 363, + 105 + ], + "score": 0.94, + "content": "{ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 363, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "as we need to use for the certifica-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 428, + 117 + ], + "score": 1.0, + "content": "tion process. Then, we freeze the model parameters and use the adapted model,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 428, + 105, + 454, + 117 + ], + "score": 0.9, + "content": "f _ { a d a p t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 454, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ", as our base", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 257, + 129 + ], + "score": 1.0, + "content": "classifier to certify the test example,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 257, + 117, + 278, + 127 + ], + "score": 0.89, + "content": "x _ { t e s t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 278, + 114, + 390, + 129 + ], + "score": 1.0, + "content": ". Hence, the base classifier", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 390, + 115, + 417, + 128 + ], + "score": 0.91, + "content": "f _ { a d a p t }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 417, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "remains fixed during", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 246, + 139 + ], + "score": 1.0, + "content": "calculating the certification radius", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 247, + 127, + 256, + 137 + ], + "score": 0.79, + "content": "R", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 256, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "(Equation 3). Our proposed certification through adaptation", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 262, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 262, + 149 + ], + "score": 1.0, + "content": "technique is presented in Algorithm 1.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 47, + "bbox_fs": [ + 105, + 654, + 506, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [ + { + "bbox": [ + 105, + 83, + 505, + 94 + ], + "spans": [ + { + "bbox": [ + 105, + 83, + 505, + 94 + ], + "score": 1.0, + "content": "are available, we can also use them for our BN adaptation. The Gaussian noise samples should be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 92, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 92, + 319, + 106 + ], + "score": 1.0, + "content": "drawn from the same isotropic Gaussian distribution", + "type": "text" + }, + { + "bbox": [ + 319, + 93, + 363, + 105 + ], + "score": 0.94, + "content": "{ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 363, + 92, + 505, + 106 + ], + "score": 1.0, + "content": "as we need to use for the certifica-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 428, + 117 + ], + "score": 1.0, + "content": "tion process. Then, we freeze the model parameters and use the adapted model,", + "type": "text" + }, + { + "bbox": [ + 428, + 105, + 454, + 117 + ], + "score": 0.9, + "content": "f _ { a d a p t }", + "type": "inline_equation" + }, + { + "bbox": [ + 454, + 104, + 505, + 117 + ], + "score": 1.0, + "content": ", as our base", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 114, + 506, + 129 + ], + "spans": [ + { + "bbox": [ + 105, + 114, + 257, + 129 + ], + "score": 1.0, + "content": "classifier to certify the test example,", + "type": "text" + }, + { + "bbox": [ + 257, + 117, + 278, + 127 + ], + "score": 0.89, + "content": "x _ { t e s t }", + "type": "inline_equation" + }, + { + "bbox": [ + 278, + 114, + 390, + 129 + ], + "score": 1.0, + "content": ". Hence, the base classifier", + "type": "text" + }, + { + "bbox": [ + 390, + 115, + 417, + 128 + ], + "score": 0.91, + "content": "f _ { a d a p t }", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 114, + 506, + 129 + ], + "score": 1.0, + "content": "remains fixed during", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 505, + 139 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 246, + 139 + ], + "score": 1.0, + "content": "calculating the certification radius", + "type": "text" + }, + { + "bbox": [ + 247, + 127, + 256, + 137 + ], + "score": 0.79, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 126, + 505, + 139 + ], + "score": 1.0, + "content": "(Equation 3). Our proposed certification through adaptation", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 138, + 262, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 138, + 262, + 149 + ], + "score": 1.0, + "content": "technique is presented in Algorithm 1.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 107, + 157, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "score": 1.0, + "content": "Advantages. The main advantage of our proposed framework is that we can adapt the classifier,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 114, + 180 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 168, + 187, + 181 + ], + "score": 1.0, + "content": "at any noise level", + "type": "text" + }, + { + "bbox": [ + 187, + 170, + 195, + 178 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "as an offline pre-processing step, without any additional training (see Figure", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 360, + 192 + ], + "score": 1.0, + "content": "4). As we can see in Equation 3, that we should select a large", + "type": "text" + }, + { + "bbox": [ + 361, + 182, + 368, + 189 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 179, + 455, + 192 + ], + "score": 1.0, + "content": "to certify at a bigger", + "type": "text" + }, + { + "bbox": [ + 455, + 180, + 465, + 190 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "radius of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 189, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 107, + 191, + 115, + 200 + ], + "score": 0.69, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 189, + 370, + 204 + ], + "score": 1.0, + "content": ". However, a test image that does not remain robust at higher", + "type": "text" + }, + { + "bbox": [ + 370, + 192, + 378, + 200 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 189, + 488, + 204 + ], + "score": 1.0, + "content": "produces a lower value of", + "type": "text" + }, + { + "bbox": [ + 488, + 192, + 501, + 202 + ], + "score": 0.86, + "content": "p _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 189, + 506, + 204 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 199, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 312, + 215 + ], + "score": 1.0, + "content": "It leads to reducing the overall certification radius,", + "type": "text" + }, + { + "bbox": [ + 312, + 202, + 321, + 211 + ], + "score": 0.66, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 199, + 506, + 215 + ], + "score": 1.0, + "content": ". Hence, providing the flexibility of choosing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 503, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 489, + 225 + ], + "score": 1.0, + "content": "appropriate noise levels for different test examples allows us to improve the certification radius,", + "type": "text" + }, + { + "bbox": [ + 489, + 213, + 498, + 222 + ], + "score": 0.77, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 212, + 503, + 225 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 108, + 229, + 503, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "In contrast to our proposed framework, existing randomized smoothing frameworks cannot choose", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 150, + 254 + ], + "score": 1.0, + "content": "a different", + "type": "text" + }, + { + "bbox": [ + 151, + 242, + 158, + 250 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 239, + 506, + 254 + ], + "score": 1.0, + "content": "at test-time since it typically degrades their overall certification performance. Hence,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 250, + 389, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 170, + 264 + ], + "score": 1.0, + "content": "they need to fix", + "type": "text" + }, + { + "bbox": [ + 171, + 253, + 178, + 261 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 250, + 389, + 264 + ], + "score": 1.0, + "content": "during training their base models or its components.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13 + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "score": 1.0, + "content": "Applicability. Our proposed certification through adaptation technique can be applied to any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 190, + 296 + ], + "score": 1.0, + "content": "classification model,", + "type": "text" + }, + { + "bbox": [ + 190, + 283, + 198, + 294 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "with batch-normalization layers. However, note that achieving high accuracy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "score": 1.0, + "content": "against large random Gaussian perturbations is only a necessary condition: a randomized smoothing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 146, + 318 + ], + "score": 1.0, + "content": "classifier,", + "type": "text" + }, + { + "bbox": [ + 146, + 306, + 153, + 316 + ], + "score": 0.73, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "requires to consistently predict the correct class to provide higher certification guarantees", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 293, + 327 + ], + "score": 1.0, + "content": "at larger radii. Hence, we achieve non-trivial", + "type": "text" + }, + { + "bbox": [ + 293, + 315, + 303, + 326 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 315, + 457, + 327 + ], + "score": 1.0, + "content": "certification guarantees at very small", + "type": "text" + }, + { + "bbox": [ + 457, + 316, + 468, + 326 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "radii for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 336, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 336, + 338 + ], + "score": 1.0, + "content": "standard non-robust DNN classifiers (see Appendix B.1).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "On the other hand, for existing randomized smoothing models, we achieve higher certification at", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 132, + 367 + ], + "score": 1.0, + "content": "larger", + "type": "text" + }, + { + "bbox": [ + 132, + 354, + 142, + 365 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 353, + 330, + 367 + ], + "score": 1.0, + "content": "radii by adapting their base models with larger", + "type": "text" + }, + { + "bbox": [ + 330, + 356, + 338, + 364 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 353, + 505, + 367 + ], + "score": 1.0, + "content": ", improving their overall average certified", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 135, + 377 + ], + "score": 1.0, + "content": "radius", + "type": "text" + }, + { + "bbox": [ + 135, + 365, + 162, + 376 + ], + "score": 0.39, + "content": "( A C R )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 364, + 417, + 377 + ], + "score": 1.0, + "content": "(Table 6 and 5 (Appendix)). However, we could not find any", + "type": "text" + }, + { + "bbox": [ + 418, + 367, + 425, + 375 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "to obtain a signifi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 218, + 388 + ], + "score": 1.0, + "content": "cant improvement at lower", + "type": "text" + }, + { + "bbox": [ + 219, + 376, + 229, + 387 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "radii. In contrast, AT models with our proposed offline adaptation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "technique significantly improve their performance against large Gaussian perturbations, providing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "non-trivial certification robustness. Experimentally we find that our proposed technique outperforms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 408, + 330, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 292, + 420 + ], + "score": 1.0, + "content": "the state-of-the-art certification models for the", + "type": "text" + }, + { + "bbox": [ + 293, + 409, + 303, + 420 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 408, + 330, + 420 + ], + "score": 1.0, + "content": "norm.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Finally, while we focus on adaptive BN, there also exists other unsupervised covariate shift adapta-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "tion techniques such as self-supervised domain adaptation on single test examples (Sun et al., 2020),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "pseudo-labeling (French et al., 2017; Xie et al., 2020) etc. Wang et al. (2020) also proposed to up-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "date the normalization parameters by entropy minimization to improve the corruption robustness.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 442, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 442, + 484 + ], + "score": 1.0, + "content": "Future studies may also explore these techniques for the offline pre-processing step.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 200, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 201, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 201, + 507 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Experimental setup. We use CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "2009) datasets for our experiments. For CIFAR-10, we use pre-activation ResNet18 and ResNet50", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "for IMAGENET (He et al., 2016a;b). Our AT models are trained using early stopping criteria", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 400, + 559 + ], + "score": 1.0, + "content": "(Rice et al., 2020) as follows: For IMAGENET, we use two AT models,", + "type": "text" + }, + { + "bbox": [ + 401, + 546, + 486, + 558 + ], + "score": 0.91, + "content": "\\bar { \\mathrm { A d v } } _ { \\infty } [ \\ell _ { \\infty } \\overset { \\_ } { \\le } 4 / 2 5 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 162, + 568 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\leq 3 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 557, + 205, + 569 + ], + "score": 1.0, + "content": ", learned at", + "type": "text" + }, + { + "bbox": [ + 206, + 557, + 219, + 568 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 557, + 236, + 569 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 557, + 246, + 568 + ], + "score": 0.89, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 557, + 403, + 569 + ], + "score": 1.0, + "content": "threat models with threat boundaries of", + "type": "text" + }, + { + "bbox": [ + 403, + 557, + 429, + 568 + ], + "score": 0.4, + "content": "4 / 2 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "and 3 respectively.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "For CIFAR-10, we train multiple AT models with different threat boundaries. For example, we de-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 126, + 591 + ], + "score": 1.0, + "content": "note", + "type": "text" + }, + { + "bbox": [ + 127, + 579, + 209, + 591 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 8 / 2 5 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 578, + 228, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 579, + 284, + 591 + ], + "score": 0.92, + "content": "\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 578, + 372, + 591 + ], + "score": 1.0, + "content": "as the AT models for", + "type": "text" + }, + { + "bbox": [ + 372, + 579, + 386, + 590 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 578, + 403, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 404, + 579, + 414, + 590 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "threat models, trained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 207, + 603 + ], + "score": 1.0, + "content": "with threat boundaries of", + "type": "text" + }, + { + "bbox": [ + 207, + 590, + 233, + 601 + ], + "score": 0.35, + "content": "8 / 2 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 588, + 437, + 603 + ], + "score": 1.0, + "content": "and 1, respectively. We compare with Baseline and", + "type": "text" + }, + { + "bbox": [ + 437, + 590, + 480, + 601 + ], + "score": 0.56, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "mod-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 320, + 614 + ], + "score": 1.0, + "content": "els. Baseline models are trained using clean images.", + "type": "text" + }, + { + "bbox": [ + 320, + 601, + 364, + 612 + ], + "score": 0.84, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "models are trained by augmenting", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 358, + 623 + ], + "score": 1.0, + "content": "random noise, sampled from isotropic Gaussian distribution,", + "type": "text" + }, + { + "bbox": [ + 358, + 611, + 401, + 623 + ], + "score": 0.92, + "content": "{ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 611, + 425, + 623 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 425, + 612, + 462, + 622 + ], + "score": 0.89, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 611, + 505, + 623 + ], + "score": 1.0, + "content": ". We also", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "compare with the current state-of-the-art certification models, SmoothAdv for CIFAR-10 (Salman", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 346, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 346, + 646 + ], + "score": 1.0, + "content": "et al., 2019a). Please refer to Appendix A for more details.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5 + }, + { + "type": "title", + "bbox": [ + 106, + 659, + 310, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 311, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 311, + 671 + ], + "score": 1.0, + "content": "4.1 PERFORMANCE UNDER GAUSSIAN NOISE.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 109, + 679, + 502, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "We first investigate the performance of different classification models under significantly larger", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 351, + 703 + ], + "score": 1.0, + "content": "Gaussian perturbations. It is a necessary condition to provide", + "type": "text" + }, + { + "bbox": [ + 352, + 691, + 361, + 702 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "robustness certification. In Table 2,", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 47.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 712, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 119, + 710, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 119, + 710, + 221, + 723 + ], + "score": 1.0, + "content": "1For IMAGENET, we obtain", + "type": "text" + }, + { + "bbox": [ + 221, + 712, + 245, + 721 + ], + "score": 0.76, + "content": "\\mathbf { A d v } _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 245, + 710, + 261, + 723 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 261, + 712, + 282, + 721 + ], + "score": 0.4, + "content": "\\mathbf { A d v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 710, + 505, + 723 + ], + "score": 1.0, + "content": "from https://github.com/locuslab/robust overfitting and Base-", + "type": "text" + } + ] + }, + { + "bbox": [ + 105, + 720, + 369, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 157, + 734 + ], + "score": 1.0, + "content": "line and Rand", + "type": "text" + }, + { + "bbox": [ + 157, + 722, + 178, + 732 + ], + "score": 0.8, + "content": "\\sigma { = } 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 720, + 369, + 734 + ], + "score": 1.0, + "content": "models from https://github.com/locuslab/smoothing.", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 302, + 752, + 309, + 760 + ], + "lines": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "spans": [ + { + "bbox": [ + 302, + 751, + 310, + 762 + ], + "score": 1.0, + "content": "6", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "spans": [ + { + "bbox": [ + 106, + 25, + 309, + 39 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 505, + 149 + ], + "lines": [], + "index": 2.5, + "bbox_fs": [ + 105, + 83, + 506, + 149 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 107, + 157, + 505, + 223 + ], + "lines": [ + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "spans": [ + { + "bbox": [ + 105, + 157, + 506, + 170 + ], + "score": 1.0, + "content": "Advantages. The main advantage of our proposed framework is that we can adapt the classifier,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 168, + 506, + 181 + ], + "spans": [ + { + "bbox": [ + 106, + 169, + 114, + 180 + ], + "score": 0.84, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 114, + 168, + 187, + 181 + ], + "score": 1.0, + "content": "at any noise level", + "type": "text" + }, + { + "bbox": [ + 187, + 170, + 195, + 178 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 195, + 168, + 506, + 181 + ], + "score": 1.0, + "content": "as an offline pre-processing step, without any additional training (see Figure", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 179, + 506, + 192 + ], + "spans": [ + { + "bbox": [ + 105, + 179, + 360, + 192 + ], + "score": 1.0, + "content": "4). As we can see in Equation 3, that we should select a large", + "type": "text" + }, + { + "bbox": [ + 361, + 182, + 368, + 189 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 369, + 179, + 455, + 192 + ], + "score": 1.0, + "content": "to certify at a bigger", + "type": "text" + }, + { + "bbox": [ + 455, + 180, + 465, + 190 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 465, + 179, + 506, + 192 + ], + "score": 1.0, + "content": "radius of", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 107, + 189, + 506, + 204 + ], + "spans": [ + { + "bbox": [ + 107, + 191, + 115, + 200 + ], + "score": 0.69, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 116, + 189, + 370, + 204 + ], + "score": 1.0, + "content": ". However, a test image that does not remain robust at higher", + "type": "text" + }, + { + "bbox": [ + 370, + 192, + 378, + 200 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 378, + 189, + 488, + 204 + ], + "score": 1.0, + "content": "produces a lower value of", + "type": "text" + }, + { + "bbox": [ + 488, + 192, + 501, + 202 + ], + "score": 0.86, + "content": "p _ { A }", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 189, + 506, + 204 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 199, + 506, + 215 + ], + "spans": [ + { + "bbox": [ + 104, + 199, + 312, + 215 + ], + "score": 1.0, + "content": "It leads to reducing the overall certification radius,", + "type": "text" + }, + { + "bbox": [ + 312, + 202, + 321, + 211 + ], + "score": 0.66, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 322, + 199, + 506, + 215 + ], + "score": 1.0, + "content": ". Hence, providing the flexibility of choosing", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 212, + 503, + 225 + ], + "spans": [ + { + "bbox": [ + 105, + 212, + 489, + 225 + ], + "score": 1.0, + "content": "appropriate noise levels for different test examples allows us to improve the certification radius,", + "type": "text" + }, + { + "bbox": [ + 489, + 213, + 498, + 222 + ], + "score": 0.77, + "content": "R", + "type": "inline_equation" + }, + { + "bbox": [ + 499, + 212, + 503, + 225 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 104, + 157, + 506, + 225 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 229, + 503, + 262 + ], + "lines": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 105, + 228, + 505, + 243 + ], + "score": 1.0, + "content": "In contrast to our proposed framework, existing randomized smoothing frameworks cannot choose", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 239, + 506, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 239, + 150, + 254 + ], + "score": 1.0, + "content": "a different", + "type": "text" + }, + { + "bbox": [ + 151, + 242, + 158, + 250 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 159, + 239, + 506, + 254 + ], + "score": 1.0, + "content": "at test-time since it typically degrades their overall certification performance. Hence,", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 250, + 389, + 264 + ], + "spans": [ + { + "bbox": [ + 106, + 250, + 170, + 264 + ], + "score": 1.0, + "content": "they need to fix", + "type": "text" + }, + { + "bbox": [ + 171, + 253, + 178, + 261 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 178, + 250, + 389, + 264 + ], + "score": 1.0, + "content": "during training their base models or its components.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13, + "bbox_fs": [ + 105, + 228, + 506, + 264 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 271, + 505, + 337 + ], + "lines": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 270, + 505, + 285 + ], + "score": 1.0, + "content": "Applicability. Our proposed certification through adaptation technique can be applied to any", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 281, + 505, + 296 + ], + "spans": [ + { + "bbox": [ + 105, + 281, + 190, + 296 + ], + "score": 1.0, + "content": "classification model,", + "type": "text" + }, + { + "bbox": [ + 190, + 283, + 198, + 294 + ], + "score": 0.85, + "content": "f", + "type": "inline_equation" + }, + { + "bbox": [ + 198, + 281, + 505, + 296 + ], + "score": 1.0, + "content": "with batch-normalization layers. However, note that achieving high accuracy", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "spans": [ + { + "bbox": [ + 105, + 293, + 506, + 307 + ], + "score": 1.0, + "content": "against large random Gaussian perturbations is only a necessary condition: a randomized smoothing", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 303, + 506, + 318 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 146, + 318 + ], + "score": 1.0, + "content": "classifier,", + "type": "text" + }, + { + "bbox": [ + 146, + 306, + 153, + 316 + ], + "score": 0.73, + "content": "g", + "type": "inline_equation" + }, + { + "bbox": [ + 153, + 303, + 506, + 318 + ], + "score": 1.0, + "content": "requires to consistently predict the correct class to provide higher certification guarantees", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 315, + 505, + 327 + ], + "spans": [ + { + "bbox": [ + 105, + 315, + 293, + 327 + ], + "score": 1.0, + "content": "at larger radii. Hence, we achieve non-trivial", + "type": "text" + }, + { + "bbox": [ + 293, + 315, + 303, + 326 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 304, + 315, + 457, + 327 + ], + "score": 1.0, + "content": "certification guarantees at very small", + "type": "text" + }, + { + "bbox": [ + 457, + 316, + 468, + 326 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 468, + 315, + 505, + 327 + ], + "score": 1.0, + "content": "radii for", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 326, + 336, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 326, + 336, + 338 + ], + "score": 1.0, + "content": "standard non-robust DNN classifiers (see Appendix B.1).", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 270, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 343, + 505, + 420 + ], + "lines": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "spans": [ + { + "bbox": [ + 106, + 343, + 505, + 355 + ], + "score": 1.0, + "content": "On the other hand, for existing randomized smoothing models, we achieve higher certification at", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 353, + 505, + 367 + ], + "spans": [ + { + "bbox": [ + 105, + 353, + 132, + 367 + ], + "score": 1.0, + "content": "larger", + "type": "text" + }, + { + "bbox": [ + 132, + 354, + 142, + 365 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 353, + 330, + 367 + ], + "score": 1.0, + "content": "radii by adapting their base models with larger", + "type": "text" + }, + { + "bbox": [ + 330, + 356, + 338, + 364 + ], + "score": 0.74, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 338, + 353, + 505, + 367 + ], + "score": 1.0, + "content": ", improving their overall average certified", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 364, + 505, + 377 + ], + "spans": [ + { + "bbox": [ + 105, + 364, + 135, + 377 + ], + "score": 1.0, + "content": "radius", + "type": "text" + }, + { + "bbox": [ + 135, + 365, + 162, + 376 + ], + "score": 0.39, + "content": "( A C R )", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 364, + 417, + 377 + ], + "score": 1.0, + "content": "(Table 6 and 5 (Appendix)). However, we could not find any", + "type": "text" + }, + { + "bbox": [ + 418, + 367, + 425, + 375 + ], + "score": 0.75, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 426, + 364, + 505, + 377 + ], + "score": 1.0, + "content": "to obtain a signifi-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 376, + 505, + 388 + ], + "spans": [ + { + "bbox": [ + 105, + 376, + 218, + 388 + ], + "score": 1.0, + "content": "cant improvement at lower", + "type": "text" + }, + { + "bbox": [ + 219, + 376, + 229, + 387 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 229, + 376, + 505, + 388 + ], + "score": 1.0, + "content": "radii. In contrast, AT models with our proposed offline adaptation", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "spans": [ + { + "bbox": [ + 105, + 387, + 505, + 399 + ], + "score": 1.0, + "content": "technique significantly improve their performance against large Gaussian perturbations, providing", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "spans": [ + { + "bbox": [ + 105, + 398, + 505, + 411 + ], + "score": 1.0, + "content": "non-trivial certification robustness. Experimentally we find that our proposed technique outperforms", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 408, + 330, + 420 + ], + "spans": [ + { + "bbox": [ + 106, + 408, + 292, + 420 + ], + "score": 1.0, + "content": "the state-of-the-art certification models for the", + "type": "text" + }, + { + "bbox": [ + 293, + 409, + 303, + 420 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 303, + 408, + 330, + 420 + ], + "score": 1.0, + "content": "norm.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24, + "bbox_fs": [ + 105, + 343, + 505, + 420 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 425, + 505, + 481 + ], + "lines": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "spans": [ + { + "bbox": [ + 105, + 425, + 505, + 438 + ], + "score": 1.0, + "content": "Finally, while we focus on adaptive BN, there also exists other unsupervised covariate shift adapta-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "spans": [ + { + "bbox": [ + 106, + 437, + 505, + 449 + ], + "score": 1.0, + "content": "tion techniques such as self-supervised domain adaptation on single test examples (Sun et al., 2020),", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "spans": [ + { + "bbox": [ + 105, + 447, + 505, + 462 + ], + "score": 1.0, + "content": "pseudo-labeling (French et al., 2017; Xie et al., 2020) etc. Wang et al. (2020) also proposed to up-", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "spans": [ + { + "bbox": [ + 105, + 458, + 505, + 472 + ], + "score": 1.0, + "content": "date the normalization parameters by entropy minimization to improve the corruption robustness.", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 105, + 469, + 442, + 484 + ], + "spans": [ + { + "bbox": [ + 105, + 469, + 442, + 484 + ], + "score": 1.0, + "content": "Future studies may also explore these techniques for the offline pre-processing step.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 105, + 425, + 505, + 484 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 493, + 200, + 505 + ], + "lines": [ + { + "bbox": [ + 105, + 492, + 201, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 492, + 201, + 507 + ], + "score": 1.0, + "content": "4 EXPERIMENTS", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 33 + }, + { + "type": "text", + "bbox": [ + 106, + 513, + 505, + 645 + ], + "lines": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "spans": [ + { + "bbox": [ + 106, + 513, + 505, + 525 + ], + "score": 1.0, + "content": "Experimental setup. We use CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng et al.,", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "spans": [ + { + "bbox": [ + 105, + 523, + 505, + 536 + ], + "score": 1.0, + "content": "2009) datasets for our experiments. For CIFAR-10, we use pre-activation ResNet18 and ResNet50", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "spans": [ + { + "bbox": [ + 105, + 534, + 505, + 547 + ], + "score": 1.0, + "content": "for IMAGENET (He et al., 2016a;b). Our AT models are trained using early stopping criteria", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 545, + 506, + 559 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 400, + 559 + ], + "score": 1.0, + "content": "(Rice et al., 2020) as follows: For IMAGENET, we use two AT models,", + "type": "text" + }, + { + "bbox": [ + 401, + 546, + 486, + 558 + ], + "score": 0.91, + "content": "\\bar { \\mathrm { A d v } } _ { \\infty } [ \\ell _ { \\infty } \\overset { \\_ } { \\le } 4 / 2 5 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 486, + 545, + 506, + 559 + ], + "score": 1.0, + "content": "and", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 106, + 556, + 162, + 568 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\leq 3 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 162, + 557, + 205, + 569 + ], + "score": 1.0, + "content": ", learned at", + "type": "text" + }, + { + "bbox": [ + 206, + 557, + 219, + 568 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 220, + 557, + 236, + 569 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 236, + 557, + 246, + 568 + ], + "score": 0.89, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 246, + 557, + 403, + 569 + ], + "score": 1.0, + "content": "threat models with threat boundaries of", + "type": "text" + }, + { + "bbox": [ + 403, + 557, + 429, + 568 + ], + "score": 0.4, + "content": "4 / 2 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 429, + 557, + 505, + 569 + ], + "score": 1.0, + "content": "and 3 respectively.", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "For CIFAR-10, we train multiple AT models with different threat boundaries. For example, we de-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 104, + 578, + 506, + 591 + ], + "spans": [ + { + "bbox": [ + 104, + 578, + 126, + 591 + ], + "score": 1.0, + "content": "note", + "type": "text" + }, + { + "bbox": [ + 127, + 579, + 209, + 591 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 8 / 2 5 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 210, + 578, + 228, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 228, + 579, + 284, + 591 + ], + "score": 0.92, + "content": "\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 285, + 578, + 372, + 591 + ], + "score": 1.0, + "content": "as the AT models for", + "type": "text" + }, + { + "bbox": [ + 372, + 579, + 386, + 590 + ], + "score": 0.9, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 386, + 578, + 403, + 591 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 404, + 579, + 414, + 590 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 415, + 578, + 506, + 591 + ], + "score": 1.0, + "content": "threat models, trained", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 588, + 506, + 603 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 207, + 603 + ], + "score": 1.0, + "content": "with threat boundaries of", + "type": "text" + }, + { + "bbox": [ + 207, + 590, + 233, + 601 + ], + "score": 0.35, + "content": "8 / 2 5 5", + "type": "inline_equation" + }, + { + "bbox": [ + 234, + 588, + 437, + 603 + ], + "score": 1.0, + "content": "and 1, respectively. We compare with Baseline and", + "type": "text" + }, + { + "bbox": [ + 437, + 590, + 480, + 601 + ], + "score": 0.56, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 481, + 588, + 506, + 603 + ], + "score": 1.0, + "content": "mod-", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 600, + 505, + 614 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 320, + 614 + ], + "score": 1.0, + "content": "els. Baseline models are trained using clean images.", + "type": "text" + }, + { + "bbox": [ + 320, + 601, + 364, + 612 + ], + "score": 0.84, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 365, + 600, + 505, + 614 + ], + "score": 1.0, + "content": "models are trained by augmenting", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 358, + 623 + ], + "score": 1.0, + "content": "random noise, sampled from isotropic Gaussian distribution,", + "type": "text" + }, + { + "bbox": [ + 358, + 611, + 401, + 623 + ], + "score": 0.92, + "content": "{ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 402, + 611, + 425, + 623 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 425, + 612, + 462, + 622 + ], + "score": 0.89, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 462, + 611, + 505, + 623 + ], + "score": 1.0, + "content": ". We also", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 622, + 505, + 634 + ], + "score": 1.0, + "content": "compare with the current state-of-the-art certification models, SmoothAdv for CIFAR-10 (Salman", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 633, + 346, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 346, + 646 + ], + "score": 1.0, + "content": "et al., 2019a). Please refer to Appendix A for more details.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 39.5, + "bbox_fs": [ + 104, + 513, + 506, + 646 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 659, + 310, + 670 + ], + "lines": [ + { + "bbox": [ + 105, + 658, + 311, + 671 + ], + "spans": [ + { + "bbox": [ + 105, + 658, + 311, + 671 + ], + "score": 1.0, + "content": "4.1 PERFORMANCE UNDER GAUSSIAN NOISE.", + "type": "text" + } + ], + "index": 46 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 109, + 679, + 502, + 702 + ], + "lines": [ + { + "bbox": [ + 106, + 678, + 505, + 693 + ], + "spans": [ + { + "bbox": [ + 106, + 678, + 505, + 693 + ], + "score": 1.0, + "content": "We first investigate the performance of different classification models under significantly larger", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 690, + 505, + 703 + ], + "spans": [ + { + "bbox": [ + 106, + 690, + 351, + 703 + ], + "score": 1.0, + "content": "Gaussian perturbations. It is a necessary condition to provide", + "type": "text" + }, + { + "bbox": [ + 352, + 691, + 361, + 702 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 362, + 690, + 505, + 703 + ], + "score": 1.0, + "content": "robustness certification. In Table 2,", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 199, + 505, + 213 + ], + "spans": [ + { + "bbox": [ + 106, + 199, + 505, + 213 + ], + "score": 1.0, + "content": "we present the performance. We observe that when the test examples are sampled from IID settings", + "type": "text", + "cross_page": true + } + ], + "index": 12 + }, + { + "bbox": [ + 104, + 209, + 505, + 223 + ], + "spans": [ + { + "bbox": [ + 104, + 209, + 226, + 223 + ], + "score": 1.0, + "content": "as training distributions (i.e.,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 226, + 210, + 254, + 220 + ], + "score": 0.91, + "content": "\\sigma = 0", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 254, + 209, + 310, + 223 + ], + "score": 1.0, + "content": "for Baseline,", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 311, + 210, + 337, + 221 + ], + "score": 0.9, + "content": "\\mathbf { A d v } _ { \\infty }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 337, + 209, + 359, + 223 + ], + "score": 1.0, + "content": ", and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 360, + 210, + 382, + 221 + ], + "score": 0.88, + "content": "\\mathrm { \\ A d v _ { 2 } }", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 383, + 209, + 401, + 223 + ], + "score": 1.0, + "content": "and", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 402, + 210, + 437, + 221 + ], + "score": 0.9, + "content": "\\sigma = 0 . 5", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 438, + 209, + 453, + 223 + ], + "score": 1.0, + "content": "for", + "type": "text", + "cross_page": true + }, + { + "bbox": [ + 454, + 210, + 500, + 222 + ], + "score": 0.8, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 } )", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 500, + 209, + 505, + 223 + ], + "score": 1.0, + "content": ",", + "type": "text", + "cross_page": true + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "spans": [ + { + "bbox": [ + 105, + 219, + 506, + 234 + ], + "score": 1.0, + "content": "these models produces the best results regardless of whether BN adaptation is applied. 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In contrast, the baseline model only achieves", + "type": "text", + "cross_page": true + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 318, + 399, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 129, + 330 + ], + "score": 0.86, + "content": "7 . 7 \\%", + "type": "inline_equation", + "cross_page": true + }, + { + "bbox": [ + 129, + 318, + 399, + 332 + ], + "score": 1.0, + "content": "accuracy. We observe similar results for CIFAR-10 in Table 2 (b).", + "type": "text", + "cross_page": true + } + ], + "index": 23 + } + ], + "index": 47.5, + "bbox_fs": [ + 106, + 678, + 505, + 703 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 120, + 81, + 296, + 153 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 120, + 81, + 296, + 153 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 120, + 81, + 296, + 153 + ], + "spans": [ + { + "bbox": [ + 120, + 81, + 296, + 153 + ], + "score": 0.97, + "html": "
(a) IMAGENET
Modelσ=0σ=0.25σ=0.5σ=0.75
Baseline75.2±0.011.8±0.220.3±0.010.1±0.0
+ adaptive BN74.4±0.0431.0±0.277.7±0.242.4±0.01
Advo∞≤4/255]62.8±0.03.9±0.030.4±0.010.2±0.01
+ adaptive BN60.8±0.1653.4±0.1544.9±0.0833.7±0.28
Adv2≤3]59.8±0.09.8±0.080.9±0.010.3±0.0
+adaptive BN58.3±0.0853.7±0.1447.3±0.1439.8±0.18
Rand g=0.522.0±0.032.8±0.1160.9±0.040.9±0.06
+ adaptive BN62.7±0.0362.3±0.1859.5±0.1151.4±0.27
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(b) CIFAR-10
Modelσ=0g=0.25σ=0.5g=0.75
Baseline + adaptive BN95.2±0.010.9±0.8810.6±0.7610.5±1.19
95.0±0.5740.1±0.9722.0±0.8317.2±0.66
Advo≤8/255]82.1±0.040.2±4.5616.1±7.8512.2±5.23
+ adaptive BN81.6±0.9674.2±0.9562.4±0.6451.0±1.03
Adv2[≤1]81.6±0.047.5±5.121.5±7.7914.3±5.63
+ adaptive BN81.8±0.775.8±0.4364.9±0.7353.5±1.71
Rand g=0.566.7±0.069.1±1.0161.2±0.8425.9±1.41
+ adaptive BN74.0±2.173.0±2.0466.8±2.0156.7±0.94
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However, as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 232, + 505, + 245 + ], + "spans": [ + { + "bbox": [ + 105, + 232, + 378, + 245 + ], + "score": 1.0, + "content": "we move away from the IID settings by increasing (or decreasing)", + "type": "text" + }, + { + "bbox": [ + 378, + 234, + 385, + 242 + ], + "score": 0.77, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 385, + 232, + 505, + 245 + ], + "score": 1.0, + "content": ", the performance of all these", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "spans": [ + { + "bbox": [ + 106, + 243, + 505, + 255 + ], + "score": 1.0, + "content": "models significantly degrades in the standard inference setup. In contrast, covariate shift adaptation", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 268 + ], + "score": 1.0, + "content": "using adaptive BN improves the performance for all models. In particular, AT models achieve sig-", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "spans": [ + { + "bbox": [ + 105, + 265, + 505, + 277 + ], + "score": 1.0, + "content": "nificantly higher performance gain using adaptive BN than the non-robust baseline models at higher", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 275, + 505, + 289 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 222, + 289 + ], + "score": 1.0, + "content": "noise levels. 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However, adaptive BN for", + "type": "text" + }, + { + "bbox": [ + 256, + 298, + 311, + 309 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\leq 3 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 311, + 297, + 329, + 312 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 330, + 298, + 411, + 309 + ], + "score": 0.92, + "content": "\\mathrm { A d v } _ { \\infty } [ \\ell _ { \\infty } \\leq 4 / 2 5 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 412, + 297, + 506, + 312 + ], + "score": 1.0, + "content": "significantly improves", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 308, + 505, + 322 + ], + "spans": [ + { + "bbox": [ + 105, + 308, + 194, + 322 + ], + "score": 1.0, + "content": "the top-1 accuracy to", + "type": "text" + }, + { + "bbox": [ + 194, + 308, + 221, + 319 + ], + "score": 0.89, + "content": "4 7 . 3 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 308, + 240, + 322 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 240, + 309, + 267, + 319 + ], + "score": 0.88, + "content": "4 4 . 9 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 268, + 308, + 505, + 322 + ], + "score": 1.0, + "content": "respectively. In contrast, the baseline model only achieves", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 106, + 318, + 399, + 332 + ], + "spans": [ + { + "bbox": [ + 106, + 319, + 129, + 330 + ], + "score": 0.86, + "content": "7 . 7 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 129, + 318, + 399, + 332 + ], + "score": 1.0, + "content": "accuracy. We observe similar results for CIFAR-10 in Table 2 (b).", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 17.5 + }, + { + "type": "image", + "bbox": [ + 122, + 347, + 489, + 559 + ], + "blocks": [ + { + "type": "image_caption", + "bbox": [ + 105, + 155, + 505, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 153, + 505, + 166 + ], + "spans": [ + { + "bbox": [ + 106, + 153, + 505, + 166 + ], + "score": 1.0, + "content": "Table 2: Top-1 accuracy of different classifiers under different levels of Gaussian noises augmented to the test", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 164, + 494, + 176 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 377, + 176 + ], + "score": 1.0, + "content": "images. We randomly shuffle test images and sample the noises and report", + "type": "text" + }, + { + "bbox": [ + 377, + 165, + 446, + 175 + ], + "score": 0.84, + "content": "( m e a n \\pm 2 \\times s d )", + "type": "inline_equation" + }, + { + "bbox": [ + 446, + 164, + 494, + 176 + ], + "score": 1.0, + "content": ") of five runs.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "image_body", + "bbox": [ + 122, + 347, + 489, + 559 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 122, + 347, + 489, + 559 + ], + "spans": [ + { + "bbox": [ + 122, + 347, + 489, + 559 + ], + "score": 0.978, + "type": "image", + "image_path": "fd7d6a2b75a66244db051455b16e7df86066ba1cd535282d32891358d59c5cad.jpg" + } + ] + } + ], + "index": 25, + "virtual_lines": [ + { + "bbox": [ + 122, + 347, + 489, + 417.6666666666667 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 122, + 417.6666666666667, + 489, + 488.33333333333337 + ], + "spans": [], + "index": 25 + }, + { + "bbox": [ + 122, + 488.33333333333337, + 489, + 559.0 + ], + "spans": [], + "index": 26 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 564, + 501, + 575 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 562, + 503, + 576 + ], + "spans": [ + { + "bbox": [ + 108, + 562, + 503, + 576 + ], + "score": 1.0, + "content": "Figure 1: Visualizing loss-gradients produced by AT models as we apply different levels of Gaussian noises.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 27 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 106, + 589, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 588, + 505, + 601 + ], + "score": 1.0, + "content": "Loss Gradients under Gaussian Noises. 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Loss-gradients reflect the most relevant input pixels for classification predictions.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "Here, we scale, translate and clip the loss-gradient values without using any sophisticated techniques", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 281, + 646 + ], + "score": 1.0, + "content": "(as suggested in Tsipras et al. (2019)). At", + "type": "text" + }, + { + "bbox": [ + 281, + 633, + 310, + 643 + ], + "score": 0.89, + "content": "\\sigma = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 310, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "(i.e., for clean images), the loss-gradients from", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "AT models align properly with perceptually relevant features (as observed previously (Tsipras et al.,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 655, + 506, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 655, + 408, + 668 + ], + "score": 1.0, + "content": "2019; Etmann et al., 2019)). 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Hence, they extract the required semantic in-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 709, + 505, + 723 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 366, + 723 + ], + "score": 1.0, + "content": "formation for correct classifications. 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(a) IMAGENET
Modelσ=0σ=0.25σ=0.5σ=0.75
Baseline75.2±0.011.8±0.220.3±0.010.1±0.0
+ adaptive BN74.4±0.0431.0±0.277.7±0.242.4±0.01
Advo∞≤4/255]62.8±0.03.9±0.030.4±0.010.2±0.01
+ adaptive BN60.8±0.1653.4±0.1544.9±0.0833.7±0.28
Adv2≤3]59.8±0.09.8±0.080.9±0.010.3±0.0
+adaptive BN58.3±0.0853.7±0.1447.3±0.1439.8±0.18
Rand g=0.522.0±0.032.8±0.1160.9±0.040.9±0.06
+ adaptive BN62.7±0.0362.3±0.1859.5±0.1151.4±0.27
", + "type": "table", + "image_path": "5a0c9ca305744cc6bc22d418723d3100f6a3a50be977e1ff8fafd2f69f54d3a1.jpg" + } + ] + } + ], + "index": 4, + "virtual_lines": [ + { + "bbox": [ + 120, + 81, + 296, + 95.4 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 120, + 95.4, + 296, + 109.80000000000001 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 120, + 109.80000000000001, + 296, + 124.20000000000002 + ], + "spans": [], + "index": 4 + }, + { + "bbox": [ + 120, + 124.20000000000002, + 296, + 138.60000000000002 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 120, + 138.60000000000002, + 296, + 153.00000000000003 + ], + "spans": [], + "index": 8 + } + ] + } + ], + "index": 4 + }, + { + "type": "table", + "bbox": [ + 312, + 82, + 488, + 154 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 312, + 82, + 488, + 154 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 312, + 82, + 488, + 154 + ], + "spans": [ + { + "bbox": [ + 312, + 82, + 488, + 154 + ], + "score": 0.967, + "html": "
(b) CIFAR-10
Modelσ=0g=0.25σ=0.5g=0.75
Baseline + adaptive BN95.2±0.010.9±0.8810.6±0.7610.5±1.19
95.0±0.5740.1±0.9722.0±0.8317.2±0.66
Advo≤8/255]82.1±0.040.2±4.5616.1±7.8512.2±5.23
+ adaptive BN81.6±0.9674.2±0.9562.4±0.6451.0±1.03
Adv2[≤1]81.6±0.047.5±5.121.5±7.7914.3±5.63
+ adaptive BN81.8±0.775.8±0.4364.9±0.7353.5±1.71
Rand g=0.566.7±0.069.1±1.0161.2±0.8425.9±1.41
+ adaptive BN74.0±2.173.0±2.0466.8±2.0156.7±0.94
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Loss-gradients reflect the most relevant input pixels for classification predictions.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 104, + 620, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 620, + 506, + 636 + ], + "score": 1.0, + "content": "Here, we scale, translate and clip the loss-gradient values without using any sophisticated techniques", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 106, + 633, + 281, + 646 + ], + "score": 1.0, + "content": "(as suggested in Tsipras et al. (2019)). 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We have provided the codes for our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 217, + 488, + 231 + ], + "spans": [ + { + "bbox": [ + 105, + 217, + 488, + 231 + ], + "score": 1.0, + "content": "certification algorithms in the supplementary materials for reproducing the results of our paper.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 7.5 + }, + { + "type": "title", + "bbox": [ + 108, + 246, + 175, + 258 + ], + "lines": [ + { + "bbox": [ + 106, + 246, + 176, + 259 + ], + "spans": [ + { + "bbox": [ + 106, + 246, + 176, + 259 + ], + "score": 1.0, + "content": "REFERENCES", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 105, + 261, + 506, + 733 + ], + "lines": [ + { + "bbox": [ + 106, + 265, + 506, + 279 + ], + "spans": [ + { + "bbox": [ + 106, + 265, + 506, + 279 + ], + "score": 1.0, + "content": "Anish Athalye, Nicholas Carlini, and David Wagner. 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Macer: Attack-free and scalable robust training via maximizing certified radius.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 116, + 385, + 178, + 396 + ], + "spans": [ + { + "bbox": [ + 116, + 385, + 178, + 396 + ], + "score": 1.0, + "content": "In ICLR, 2020.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20, + "bbox_fs": [ + 105, + 362, + 505, + 396 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 403, + 503, + 426 + ], + "lines": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 506, + 416 + ], + "score": 1.0, + "content": "Hongyang Zhang et al. Theoretically principled trade-off between robustness and accuracy. In", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 115, + 412, + 171, + 427 + ], + "spans": [ + { + "bbox": [ + 115, + 412, + 171, + 427 + ], + "score": 1.0, + "content": "ICML, 2019.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 22.5, + "bbox_fs": [ + 105, + 402, + 506, + 427 + ] + }, + { + "type": "text", + "bbox": [ + 106, + 433, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 446 + ], + "score": 1.0, + "content": "Jingfeng Zhang, Xilie Xu, Bo Han, Gang Niu, Lizhen Cui, Masashi Sugiyama, and Mohan Kankan-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 115, + 443, + 505, + 457 + ], + "spans": [ + { + "bbox": [ + 115, + 443, + 505, + 457 + ], + "score": 1.0, + "content": "halli. Attacks which do not kill training make adversarial learning stronger. In ICML. PMLR,", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 115, + 454, + 142, + 466 + ], + "spans": [ + { + "bbox": [ + 115, + 454, + 142, + 466 + ], + "score": 1.0, + "content": "2020.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25, + "bbox_fs": [ + 105, + 433, + 505, + 466 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 245, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 246, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 246, + 96 + ], + "score": 1.0, + "content": "APPENDIX ORGANIZATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 133, + 111, + 368, + 171 + ], + "lines": [ + { + "bbox": [ + 133, + 111, + 277, + 127 + ], + "spans": [ + { + "bbox": [ + 133, + 111, + 277, + 127 + ], + "score": 1.0, + "content": "• Section A: Experimental setup.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 132, + 133, + 342, + 150 + ], + "spans": [ + { + "bbox": [ + 132, + 133, + 342, + 150 + ], + "score": 1.0, + "content": "• Section B: Additional Results on Certification.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 132, + 157, + 369, + 174 + ], + "spans": [ + { + "bbox": [ + 132, + 157, + 369, + 174 + ], + "score": 1.0, + "content": "• Section C:Performance against different corruptions", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 108, + 195, + 247, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 248, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 248, + 210 + ], + "score": 1.0, + "content": "A EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "title", + "bbox": [ + 108, + 226, + 251, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 252, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 252, + 239 + ], + "score": 1.0, + "content": "A.1 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 264 + ], + "score": 1.0, + "content": "We present our experimental results on CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "et al., 2009) datasets. The descriptions of different models and training hyper-parameters are pro-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 271, + 200, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 200, + 286 + ], + "score": 1.0, + "content": "vided in the following:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 108, + 305, + 191, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 193, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 193, + 320 + ], + "score": 1.0, + "content": "A.1.1 CIFAR-10.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "score": 1.0, + "content": "We use pre-activation ResNet18 architecture (He et al., 2016b) for our experiments on CIFAR-10.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "We apply the SGD optimizer with a batch size of 128. We execute a total of 200 training epochs and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "apply a step-wise learning rate decay set initially at 0.1 and divided by 10 at 100 and 150 epochs,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 361, + 221, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 178, + 374 + ], + "score": 1.0, + "content": "and weight decay", + "type": "text" + }, + { + "bbox": [ + 179, + 361, + 217, + 372 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 362, + 221, + 374 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5 + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "AT models (Madry et al., 2018; Rice et al., 2020): Unless and otherwise specified, our AT", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "models are learned using early stopping criteria as described in (Rice et al., 2020). We learn several", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "AT models with different threat boundaries for our experiments. We denote them by specifying their", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 374, + 424 + ], + "score": 1.0, + "content": "corresponding threat model and threat boundaries. For example,", + "type": "text" + }, + { + "bbox": [ + 375, + 412, + 441, + 424 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\ \\leq \\ 1 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "denotes an AT", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 302, + 435 + ], + "score": 1.0, + "content": "model that is learned using PGD adversary with", + "type": "text" + }, + { + "bbox": [ + 303, + 423, + 312, + 434 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 423, + 469, + 435 + ], + "score": 1.0, + "content": "threat model and a threat boundary of", + "type": "text" + }, + { + "bbox": [ + 469, + 423, + 501, + 433 + ], + "score": 0.89, + "content": "\\epsilon = 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 423, + 505, + 435 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "along with early-stopping criteria (Rice et al., 2020). We also learn AT models without using early-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "stopping criteria, as in (Madry et al., 2018) for our comparison in Figure 5. These models are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 455, + 200, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 200, + 467 + ], + "score": 1.0, + "content": "denoted as Advoverf it.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5 + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 249, + 484 + ], + "score": 1.0, + "content": "We use projected gradient descent", + "type": "text" + }, + { + "bbox": [ + 249, + 473, + 276, + 483 + ], + "score": 0.35, + "content": "( P G D )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "adversarial attack (Madry et al., 2018) to train these AT", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 199, + 496 + ], + "score": 1.0, + "content": "models as follows: For", + "type": "text" + }, + { + "bbox": [ + 200, + 484, + 226, + 495 + ], + "score": 0.88, + "content": "\\mathrm { \\bf A d v _ { \\infty } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 483, + 339, + 496 + ], + "score": 1.0, + "content": ", we use 10 iterations and an", + "type": "text" + }, + { + "bbox": [ + 339, + 484, + 353, + 495 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 483, + 401, + 496 + ], + "score": 1.0, + "content": "step size of", + "type": "text" + }, + { + "bbox": [ + 401, + 484, + 416, + 495 + ], + "score": 0.85, + "content": "\\epsilon / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 483, + 436, + 496 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 437, + 484, + 459, + 495 + ], + "score": 0.87, + "content": "\\mathrm { \\ A d v _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 483, + 506, + 496 + ], + "score": 1.0, + "content": ", we use 10", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 174, + 507 + ], + "score": 1.0, + "content": "iterations and an", + "type": "text" + }, + { + "bbox": [ + 174, + 495, + 185, + 505 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 493, + 232, + 507 + ], + "score": 1.0, + "content": "step size of", + "type": "text" + }, + { + "bbox": [ + 233, + 495, + 256, + 506 + ], + "score": 0.88, + "content": "\\epsilon / 8 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 493, + 505, + 507 + ], + "score": 1.0, + "content": ". This is the same experimental setup as in (Rice et al., 2020)).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 504, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 210, + 519 + ], + "score": 1.0, + "content": "We choose a small set of", + "type": "text" + }, + { + "bbox": [ + 210, + 506, + 236, + 517 + ], + "score": 0.3, + "content": "1 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 504, + 505, + 519 + ], + "score": 1.0, + "content": "images from the CIFAR-10 test set for our validation. We apply", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "the PGD attack with the same hyper-parameters for our validation during training. We save the best", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 527, + 339, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 339, + 539 + ], + "score": 1.0, + "content": "model using the early-stopping criteria (Rice et al., 2020).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5 + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 393, + 557 + ], + "score": 1.0, + "content": "Randomized smoothing model by Cohen et al. (2019): We also train", + "type": "text" + }, + { + "bbox": [ + 394, + 545, + 437, + 555 + ], + "score": 0.73, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "by training with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 403, + 567 + ], + "score": 1.0, + "content": "augmented random noise, sampled from an isotropic Gaussian distribution", + "type": "text" + }, + { + "bbox": [ + 403, + 555, + 447, + 567 + ], + "score": 0.92, + "content": "{ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 555, + 468, + 567 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 468, + 556, + 501, + 566 + ], + "score": 0.88, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 555, + 505, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Here, we keep the same model architecture, learning rates, batch sizes, and other hyper-parameters", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 234, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 234, + 588 + ], + "score": 1.0, + "content": "as used to learn the AT models.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5 + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "Randomized smoothing model by Salman et al. (2019a): We also compare with the state-of-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 505, + 618 + ], + "score": 1.0, + "content": "the-art certification models, called ‘SmoothAdv’, by Salman et al. (2019a) for our experiments on", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 106, + 616, + 505, + 628 + ], + "spans": [ + { + "bbox": [ + 106, + 616, + 117, + 627 + ], + "score": 0.85, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 117, + 616, + 505, + 628 + ], + "score": 1.0, + "content": "certification We train the SmoothAdv models by choosing random noise vectors followed by an", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 626, + 505, + 640 + ], + "spans": [ + { + "bbox": [ + 105, + 626, + 271, + 640 + ], + "score": 1.0, + "content": "adaptive adversarial attack with specified", + "type": "text" + }, + { + "bbox": [ + 271, + 627, + 281, + 638 + ], + "score": 0.86, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 626, + 358, + 640 + ], + "score": 1.0, + "content": "threat boundary of", + "type": "text" + }, + { + "bbox": [ + 358, + 629, + 363, + 637 + ], + "score": 0.74, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 364, + 626, + 505, + 640 + ], + "score": 1.0, + "content": "at each iteration. The noise vectors", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 637, + 362, + 650 + ], + "spans": [ + { + "bbox": [ + 105, + 637, + 315, + 650 + ], + "score": 1.0, + "content": "are sampled from an isotropic Gaussian distribution", + "type": "text" + }, + { + "bbox": [ + 315, + 637, + 358, + 650 + ], + "score": 0.93, + "content": "\\mathcal { N } ( 0 , \\bar { \\sigma } ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 359, + 637, + 362, + 650 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 34 + }, + { + "type": "text", + "bbox": [ + 107, + 655, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 654, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 292, + 668 + ], + "score": 1.0, + "content": "We note that the training hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 293, + 657, + 299, + 665 + ], + "score": 0.62, + "content": "\\epsilon", + "type": "inline_equation" + }, + { + "bbox": [ + 299, + 654, + 505, + 668 + ], + "score": 1.0, + "content": "has the most significant impact on the certifi-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "cation curve for a SmoothAdv model (please refer to Table 7-15 of (Salman et al., 2019a)", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 104, + 676, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 104, + 676, + 483, + 690 + ], + "score": 1.0, + "content": "for more details). For our experiments, we train 4 different SmoothAdv models with", + "type": "text" + }, + { + "bbox": [ + 483, + 678, + 505, + 688 + ], + "score": 0.85, + "content": "\\epsilon =", + "type": "inline_equation" + } + ], + "index": 39 + }, + { + "bbox": [ + 107, + 687, + 506, + 701 + ], + "spans": [ + { + "bbox": [ + 107, + 687, + 172, + 700 + ], + "score": 0.9, + "content": "\\{ 0 . 2 5 , 0 . 5 , 1 , 2 \\}", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 687, + 193, + 701 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 193, + 688, + 234, + 698 + ], + "score": 0.89, + "content": "\\sigma ~ = ~ 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 235, + 687, + 506, + 701 + ], + "score": 1.0, + "content": "using adaptive PGD attack with 10 steps. 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We use the same training set-up and other hyper-parameters as specified in their Github:", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 326, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 326, + 733 + ], + "score": 1.0, + "content": "https://github.com/Hadisalman/smoothing-adversarial.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40 + } + ], + "page_idx": 13, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 301, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 764 + ], + "score": 1.0, + "content": "", + "type": "text", + "height": 14, + "width": 14 + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 81, + 245, + 94 + ], + "lines": [ + { + "bbox": [ + 106, + 80, + 246, + 96 + ], + "spans": [ + { + "bbox": [ + 106, + 80, + 246, + 96 + ], + "score": 1.0, + "content": "APPENDIX ORGANIZATION", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 133, + 111, + 368, + 171 + ], + "lines": [ + { + "bbox": [ + 133, + 111, + 277, + 127 + ], + "spans": [ + { + "bbox": [ + 133, + 111, + 277, + 127 + ], + "score": 1.0, + "content": "• Section A: Experimental setup.", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 132, + 133, + 342, + 150 + ], + "spans": [ + { + "bbox": [ + 132, + 133, + 342, + 150 + ], + "score": 1.0, + "content": "• Section B: Additional Results on Certification.", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 132, + 157, + 369, + 174 + ], + "spans": [ + { + "bbox": [ + 132, + 157, + 369, + 174 + ], + "score": 1.0, + "content": "• Section C:Performance against different corruptions", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 2, + "bbox_fs": [ + 132, + 111, + 369, + 174 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 195, + 247, + 209 + ], + "lines": [ + { + "bbox": [ + 105, + 194, + 248, + 210 + ], + "spans": [ + { + "bbox": [ + 105, + 194, + 248, + 210 + ], + "score": 1.0, + "content": "A EXPERIMENTAL SETUP", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 194, + 248, + 210 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 226, + 251, + 237 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 252, + 239 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 252, + 239 + ], + "score": 1.0, + "content": "A.1 IMPLEMENTATION DETAILS", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 250, + 505, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 249, + 505, + 264 + ], + "spans": [ + { + "bbox": [ + 105, + 249, + 505, + 264 + ], + "score": 1.0, + "content": "We present our experimental results on CIFAR-10 (Krizhevsky et al., 2009) and IMAGENET (Deng", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "spans": [ + { + "bbox": [ + 105, + 261, + 505, + 274 + ], + "score": 1.0, + "content": "et al., 2009) datasets. The descriptions of different models and training hyper-parameters are pro-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 271, + 200, + 286 + ], + "spans": [ + { + "bbox": [ + 106, + 271, + 200, + 286 + ], + "score": 1.0, + "content": "vided in the following:", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 7, + "bbox_fs": [ + 105, + 249, + 505, + 286 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 305, + 191, + 317 + ], + "lines": [ + { + "bbox": [ + 105, + 304, + 193, + 320 + ], + "spans": [ + { + "bbox": [ + 105, + 304, + 193, + 320 + ], + "score": 1.0, + "content": "A.1.1 CIFAR-10.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 107, + 329, + 505, + 373 + ], + "lines": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "spans": [ + { + "bbox": [ + 106, + 329, + 504, + 341 + ], + "score": 1.0, + "content": "We use pre-activation ResNet18 architecture (He et al., 2016b) for our experiments on CIFAR-10.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "spans": [ + { + "bbox": [ + 106, + 340, + 505, + 352 + ], + "score": 1.0, + "content": "We apply the SGD optimizer with a batch size of 128. We execute a total of 200 training epochs and", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "spans": [ + { + "bbox": [ + 105, + 351, + 506, + 363 + ], + "score": 1.0, + "content": "apply a step-wise learning rate decay set initially at 0.1 and divided by 10 at 100 and 150 epochs,", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 361, + 221, + 374 + ], + "spans": [ + { + "bbox": [ + 105, + 362, + 178, + 374 + ], + "score": 1.0, + "content": "and weight decay", + "type": "text" + }, + { + "bbox": [ + 179, + 361, + 217, + 372 + ], + "score": 0.92, + "content": "5 \\times 1 0 ^ { - 4 }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 362, + 221, + 374 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 11.5, + "bbox_fs": [ + 105, + 329, + 506, + 374 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 378, + 505, + 466 + ], + "lines": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "spans": [ + { + "bbox": [ + 106, + 379, + 505, + 391 + ], + "score": 1.0, + "content": "AT models (Madry et al., 2018; Rice et al., 2020): Unless and otherwise specified, our AT", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 390, + 505, + 402 + ], + "score": 1.0, + "content": "models are learned using early stopping criteria as described in (Rice et al., 2020). We learn several", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 401, + 505, + 413 + ], + "score": 1.0, + "content": "AT models with different threat boundaries for our experiments. We denote them by specifying their", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 412, + 505, + 424 + ], + "spans": [ + { + "bbox": [ + 105, + 412, + 374, + 424 + ], + "score": 1.0, + "content": "corresponding threat model and threat boundaries. For example,", + "type": "text" + }, + { + "bbox": [ + 375, + 412, + 441, + 424 + ], + "score": 0.91, + "content": "\\mathrm { A d v } _ { 2 } [ \\ell _ { 2 } \\ \\leq \\ 1 . 5 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 442, + 412, + 505, + 424 + ], + "score": 1.0, + "content": "denotes an AT", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 423, + 505, + 435 + ], + "spans": [ + { + "bbox": [ + 105, + 423, + 302, + 435 + ], + "score": 1.0, + "content": "model that is learned using PGD adversary with", + "type": "text" + }, + { + "bbox": [ + 303, + 423, + 312, + 434 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 313, + 423, + 469, + 435 + ], + "score": 1.0, + "content": "threat model and a threat boundary of", + "type": "text" + }, + { + "bbox": [ + 469, + 423, + 501, + 433 + ], + "score": 0.89, + "content": "\\epsilon = 1 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 423, + 505, + 435 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 505, + 447 + ], + "score": 1.0, + "content": "along with early-stopping criteria (Rice et al., 2020). We also learn AT models without using early-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "spans": [ + { + "bbox": [ + 105, + 444, + 506, + 457 + ], + "score": 1.0, + "content": "stopping criteria, as in (Madry et al., 2018) for our comparison in Figure 5. These models are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 455, + 200, + 467 + ], + "spans": [ + { + "bbox": [ + 105, + 455, + 200, + 467 + ], + "score": 1.0, + "content": "denoted as Advoverf it.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 17.5, + "bbox_fs": [ + 105, + 379, + 506, + 467 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 472, + 505, + 539 + ], + "lines": [ + { + "bbox": [ + 106, + 472, + 505, + 484 + ], + "spans": [ + { + "bbox": [ + 106, + 472, + 249, + 484 + ], + "score": 1.0, + "content": "We use projected gradient descent", + "type": "text" + }, + { + "bbox": [ + 249, + 473, + 276, + 483 + ], + "score": 0.35, + "content": "( P G D )", + "type": "inline_equation" + }, + { + "bbox": [ + 276, + 472, + 505, + 484 + ], + "score": 1.0, + "content": "adversarial attack (Madry et al., 2018) to train these AT", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 199, + 496 + ], + "score": 1.0, + "content": "models as follows: For", + "type": "text" + }, + { + "bbox": [ + 200, + 484, + 226, + 495 + ], + "score": 0.88, + "content": "\\mathrm { \\bf A d v _ { \\infty } }", + "type": "inline_equation" + }, + { + "bbox": [ + 226, + 483, + 339, + 496 + ], + "score": 1.0, + "content": ", we use 10 iterations and an", + "type": "text" + }, + { + "bbox": [ + 339, + 484, + 353, + 495 + ], + "score": 0.89, + "content": "\\ell _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 354, + 483, + 401, + 496 + ], + "score": 1.0, + "content": "step size of", + "type": "text" + }, + { + "bbox": [ + 401, + 484, + 416, + 495 + ], + "score": 0.85, + "content": "\\epsilon / 4", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 483, + 436, + 496 + ], + "score": 1.0, + "content": ". For", + "type": "text" + }, + { + "bbox": [ + 437, + 484, + 459, + 495 + ], + "score": 0.87, + "content": "\\mathrm { \\ A d v _ { 2 } }", + "type": "inline_equation" + }, + { + "bbox": [ + 460, + 483, + 506, + 496 + ], + "score": 1.0, + "content": ", we use 10", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 493, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 493, + 174, + 507 + ], + "score": 1.0, + "content": "iterations and an", + "type": "text" + }, + { + "bbox": [ + 174, + 495, + 185, + 505 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 185, + 493, + 232, + 507 + ], + "score": 1.0, + "content": "step size of", + "type": "text" + }, + { + "bbox": [ + 233, + 495, + 256, + 506 + ], + "score": 0.88, + "content": "\\epsilon / 8 . 5 ", + "type": "inline_equation" + }, + { + "bbox": [ + 256, + 493, + 505, + 507 + ], + "score": 1.0, + "content": ". This is the same experimental setup as in (Rice et al., 2020)).", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 504, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 504, + 210, + 519 + ], + "score": 1.0, + "content": "We choose a small set of", + "type": "text" + }, + { + "bbox": [ + 210, + 506, + 236, + 517 + ], + "score": 0.3, + "content": "1 , 0 0 0", + "type": "inline_equation" + }, + { + "bbox": [ + 237, + 504, + 505, + 519 + ], + "score": 1.0, + "content": "images from the CIFAR-10 test set for our validation. We apply", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 515, + 506, + 529 + ], + "score": 1.0, + "content": "the PGD attack with the same hyper-parameters for our validation during training. We save the best", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 106, + 527, + 339, + 539 + ], + "spans": [ + { + "bbox": [ + 106, + 527, + 339, + 539 + ], + "score": 1.0, + "content": "model using the early-stopping criteria (Rice et al., 2020).", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 24.5, + "bbox_fs": [ + 105, + 472, + 506, + 539 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 544, + 504, + 588 + ], + "lines": [ + { + "bbox": [ + 105, + 543, + 506, + 557 + ], + "spans": [ + { + "bbox": [ + 105, + 543, + 393, + 557 + ], + "score": 1.0, + "content": "Randomized smoothing model by Cohen et al. (2019): We also train", + "type": "text" + }, + { + "bbox": [ + 394, + 545, + 437, + 555 + ], + "score": 0.73, + "content": "\\mathrm { R a n d } _ { \\sigma = 0 . 5 }", + "type": "inline_equation" + }, + { + "bbox": [ + 437, + 543, + 506, + 557 + ], + "score": 1.0, + "content": "by training with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 555, + 505, + 567 + ], + "spans": [ + { + "bbox": [ + 105, + 555, + 403, + 567 + ], + "score": 1.0, + "content": "augmented random noise, sampled from an isotropic Gaussian distribution", + "type": "text" + }, + { + "bbox": [ + 403, + 555, + 447, + 567 + ], + "score": 0.92, + "content": "{ \\mathcal { N } } ( 0 , \\sigma ^ { 2 } I )", + "type": "inline_equation" + }, + { + "bbox": [ + 447, + 555, + 468, + 567 + ], + "score": 1.0, + "content": "with", + "type": "text" + }, + { + "bbox": [ + 468, + 556, + 501, + 566 + ], + "score": 0.88, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 501, + 555, + 505, + 567 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "spans": [ + { + "bbox": [ + 105, + 566, + 505, + 578 + ], + "score": 1.0, + "content": "Here, we keep the same model architecture, learning rates, batch sizes, and other hyper-parameters", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 105, + 577, + 234, + 588 + ], + "spans": [ + { + "bbox": [ + 105, + 577, + 234, + 588 + ], + "score": 1.0, + "content": "as used to learn the AT models.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 29.5, + "bbox_fs": [ + 105, + 543, + 506, + 588 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 594, + 505, + 649 + ], + "lines": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 504, + 606 + ], + "score": 1.0, + "content": "Randomized smoothing model by Salman et al. 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We use the same training set-up and other hyper-parameters as specified in their Github:", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 106, + 721, + 326, + 733 + ], + "spans": [ + { + "bbox": [ + 106, + 721, + 326, + 733 + ], + "score": 1.0, + "content": "https://github.com/Hadisalman/smoothing-adversarial.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 40, + "bbox_fs": [ + 104, + 654, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 108, + 82, + 192, + 93 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 194, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 194, + 95 + ], + "score": 1.0, + "content": "A.1.2 IMAGENET.", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 107, + 101, + 505, + 146 + ], + "lines": [ + { + "bbox": [ + 106, + 101, + 505, + 113 + ], + "spans": [ + { + "bbox": [ + 106, + 101, + 505, + 113 + ], + "score": 1.0, + "content": "We use ResNet50 architecture (He et al., 2016a) for IMAGENET. 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We use the publicly available models provided", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 187 + ], + "score": 1.0, + "content": "by Rice et al. (2020) 4. These models are fine-tuned using PGD-based adversarial training with early", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 370, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 370, + 197 + ], + "score": 1.0, + "content": "stopping criteria, originally provided by Engstrom et al. (2019) 5.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 108, + 201, + 504, + 234 + ], + "lines": [ + { + "bbox": [ + 106, + 201, + 505, + 214 + ], + "spans": [ + { + "bbox": [ + 106, + 201, + 227, + 214 + ], + "score": 1.0, + "content": "We resize the input images to", + "type": "text" + }, + { + "bbox": [ + 227, + 202, + 271, + 212 + ], + "score": 0.88, + "content": "2 5 6 \\times 2 6 5", + "type": "inline_equation" + }, + { + "bbox": [ + 271, + 201, + 335, + 214 + ], + "score": 1.0, + "content": "pixels and crop", + "type": "text" + }, + { + "bbox": [ + 335, + 202, + 379, + 212 + ], + "score": 0.89, + "content": "2 2 4 \\times 2 2 4", + "type": "inline_equation" + }, + { + "bbox": [ + 380, + 201, + 505, + 214 + ], + "score": 1.0, + "content": "pixels from the center. For our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "experiments on certification, we use a set of 500 test images by choosing at most 1 sample for each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 131, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 131, + 235 + ], + "score": 1.0, + "content": "class.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10 + }, + { + "type": "title", + "bbox": [ + 107, + 249, + 389, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 390, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 390, + 261 + ], + "score": 1.0, + "content": "A.2 CHOICE OF TEST-TIME ADAPTIVE BN HYPER-PARAMETERS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "BN adaptation technique is controlled by two hyper-parameters, i.e., the test batch-size and mo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 141, + 295 + ], + "score": 1.0, + "content": "mentum", + "type": "text" + }, + { + "bbox": [ + 142, + 282, + 154, + 293 + ], + "score": 0.65, + "content": "( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 280, + 506, + 295 + ], + "score": 1.0, + "content": "(see Equation 5) to update the statistics of the batch-normalization layers. Assuming", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "that the test images are obtained independently from the same test distribution, we can efficiently", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 378, + 316 + ], + "score": 1.0, + "content": "compute the BN statistics from these images. The hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 378, + 303, + 417, + 315 + ], + "score": 0.92, + "content": "\\rho \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "controls the tread-off", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "between pre-computed training statistics and test statistics. We can obtain a better estimation of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 436, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 426, + 338 + ], + "score": 1.0, + "content": "test distribution from a large test batch. Hence, we can choose a higher value of", + "type": "text" + }, + { + "bbox": [ + 426, + 327, + 432, + 337 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 325, + 436, + 338 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "Here, we compare the top-1 test accuracy of AT models under Gaussian augmented noise with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 142, + 363 + ], + "score": 0.88, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 353, + 239, + 365 + ], + "score": 1.0, + "content": "for different choices of", + "type": "text" + }, + { + "bbox": [ + 240, + 355, + 246, + 365 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "and the batch size. We skip the standard baseline models from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "our analysis and refer to the previous works (Schneider et al., 2020; Nado et al., 2020) that analyzed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 424, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 424, + 387 + ], + "score": 1.0, + "content": "the effects of these hyper-parameters for the standard baseline DNN classifiers.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5 + }, + { + "type": "table", + "bbox": [ + 315, + 398, + 476, + 492 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 315, + 398, + 476, + 492 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 315, + 398, + 476, + 492 + ], + "spans": [ + { + "bbox": [ + 315, + 398, + 476, + 492 + ], + "score": 0.705, + "html": "
(b) CIFAR-10
pAdvoAdv2
0.0 (No adaptation)16.1±7.8521.5±7.79
0.145.1±0.4946.9±0.48
0.359.2±0.4260.8±0.33
0.562.4±0.2764.4±0.6
0.762.8±0.5264.9±0.31
0.962.8±0.7164.9±0.31
1.0 (Full adaptation)62.4±0.6464.9±0.73
", + "type": "table", + "image_path": "eb1c173c93a9cd50739a743f2106d0a827da3af3fe3e97e30c1789b3f5bf699b.jpg" + } + ] + } + ], + "index": 30, + "virtual_lines": [ + { + "bbox": [ + 315, + 398, + 476, + 411.42857142857144 + ], + "spans": [], + "index": 24 + }, + { + "bbox": [ + 315, + 411.42857142857144, + 476, + 424.8571428571429 + ], + "spans": [], + "index": 26 + }, + { + "bbox": [ + 315, + 424.8571428571429, + 476, + 438.28571428571433 + ], + "spans": [], + "index": 28 + }, + { + "bbox": [ + 315, + 438.28571428571433, + 476, + 451.7142857142858 + ], + "spans": [], + "index": 30 + }, + { + "bbox": [ + 315, + 451.7142857142858, + 476, + 465.1428571428572 + ], + "spans": [], + "index": 32 + }, + { + "bbox": [ + 315, + 465.1428571428572, + 476, + 478.57142857142867 + ], + "spans": [], + "index": 34 + }, + { + "bbox": [ + 315, + 478.57142857142867, + 476, + 492.0000000000001 + ], + "spans": [], + "index": 36 + } + ] + } + ], + "index": 30 + }, + { + "type": "table", + "bbox": [ + 132, + 398, + 293, + 492 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 132, + 398, + 293, + 492 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 132, + 398, + 293, + 492 + ], + "spans": [ + { + "bbox": [ + 132, + 398, + 293, + 492 + ], + "score": 0.803, + "html": "
(a)IMAGENET
pAdvoAdv2
0.0 (No adaptation)0.4±0.010.9±0.01
0.12.1±0.047.7±0.09
0.320.6±0.1636.6±0.09
0.541.1±0.0945.5±0.13
0.743.5±0.1446.7±0.13
0.944.2±0.1246.8±0.13
1.0 (Full adaptation)44.8±0.1347.2±0.14
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We randomly shuffle the test images to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 518, + 267, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 132, + 529 + ], + "score": 1.0, + "content": "report", + "type": "text" + }, + { + "bbox": [ + 132, + 518, + 196, + 528 + ], + "score": 0.88, + "content": "( m e a n + 2 \\times s d )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 518, + 267, + 529 + ], + "score": 1.0, + "content": "of 5 different runs.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 159, + 561 + ], + "score": 1.0, + "content": "Momentum", + "type": "text" + }, + { + "bbox": [ + 159, + 550, + 172, + 561 + ], + "score": 0.66, + "content": "( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 550, + 364, + 561 + ], + "score": 1.0, + "content": ". 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Recall that,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 572, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 133, + 583 + ], + "score": 0.9, + "content": "\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 572, + 506, + 583 + ], + "score": 1.0, + "content": "denotes full adaptation (Equation 5). Here, we completely ignore the training statistics and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 369, + 595 + ], + "score": 1.0, + "content": "recompute the BN statistics using the test batches. 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We use the publicly available models provided", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 172, + 505, + 187 + ], + "spans": [ + { + "bbox": [ + 105, + 172, + 505, + 187 + ], + "score": 1.0, + "content": "by Rice et al. (2020) 4. These models are fine-tuned using PGD-based adversarial training with early", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 185, + 370, + 197 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 370, + 197 + ], + "score": 1.0, + "content": "stopping criteria, originally provided by Engstrom et al. 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For our", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "spans": [ + { + "bbox": [ + 106, + 212, + 505, + 224 + ], + "score": 1.0, + "content": "experiments on certification, we use a set of 500 test images by choosing at most 1 sample for each", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 223, + 131, + 235 + ], + "spans": [ + { + "bbox": [ + 105, + 223, + 131, + 235 + ], + "score": 1.0, + "content": "class.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10, + "bbox_fs": [ + 105, + 201, + 505, + 235 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 249, + 389, + 261 + ], + "lines": [ + { + "bbox": [ + 106, + 249, + 390, + 261 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 390, + 261 + ], + "score": 1.0, + "content": "A.2 CHOICE OF TEST-TIME ADAPTIVE BN HYPER-PARAMETERS", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 107, + 270, + 505, + 336 + ], + "lines": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "spans": [ + { + "bbox": [ + 106, + 270, + 505, + 282 + ], + "score": 1.0, + "content": "BN adaptation technique is controlled by two hyper-parameters, i.e., the test batch-size and mo-", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 280, + 506, + 295 + ], + "spans": [ + { + "bbox": [ + 105, + 280, + 141, + 295 + ], + "score": 1.0, + "content": "mentum", + "type": "text" + }, + { + "bbox": [ + 142, + 282, + 154, + 293 + ], + "score": 0.65, + "content": "( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 155, + 280, + 506, + 295 + ], + "score": 1.0, + "content": "(see Equation 5) to update the statistics of the batch-normalization layers. Assuming", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "spans": [ + { + "bbox": [ + 106, + 293, + 505, + 304 + ], + "score": 1.0, + "content": "that the test images are obtained independently from the same test distribution, we can efficiently", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 303, + 506, + 316 + ], + "spans": [ + { + "bbox": [ + 105, + 303, + 378, + 316 + ], + "score": 1.0, + "content": "compute the BN statistics from these images. The hyper-parameter", + "type": "text" + }, + { + "bbox": [ + 378, + 303, + 417, + 315 + ], + "score": 0.92, + "content": "\\rho \\in [ 0 , 1 ]", + "type": "inline_equation" + }, + { + "bbox": [ + 417, + 303, + 506, + 316 + ], + "score": 1.0, + "content": "controls the tread-off", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "spans": [ + { + "bbox": [ + 105, + 314, + 505, + 326 + ], + "score": 1.0, + "content": "between pre-computed training statistics and test statistics. We can obtain a better estimation of the", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 436, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 426, + 338 + ], + "score": 1.0, + "content": "test distribution from a large test batch. Hence, we can choose a higher value of", + "type": "text" + }, + { + "bbox": [ + 426, + 327, + 432, + 337 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 433, + 325, + 436, + 338 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 15.5, + "bbox_fs": [ + 105, + 270, + 506, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 342, + 505, + 386 + ], + "lines": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "spans": [ + { + "bbox": [ + 105, + 342, + 505, + 354 + ], + "score": 1.0, + "content": "Here, we compare the top-1 test accuracy of AT models under Gaussian augmented noise with", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 106, + 353, + 506, + 365 + ], + "spans": [ + { + "bbox": [ + 106, + 353, + 142, + 363 + ], + "score": 0.88, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 142, + 353, + 239, + 365 + ], + "score": 1.0, + "content": "for different choices of", + "type": "text" + }, + { + "bbox": [ + 240, + 355, + 246, + 365 + ], + "score": 0.8, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 247, + 353, + 506, + 365 + ], + "score": 1.0, + "content": "and the batch size. We skip the standard baseline models from", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "spans": [ + { + "bbox": [ + 106, + 364, + 505, + 376 + ], + "score": 1.0, + "content": "our analysis and refer to the previous works (Schneider et al., 2020; Nado et al., 2020) that analyzed", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 375, + 424, + 387 + ], + "spans": [ + { + "bbox": [ + 105, + 375, + 424, + 387 + ], + "score": 1.0, + "content": "the effects of these hyper-parameters for the standard baseline DNN classifiers.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 342, + 506, + 387 + ] + }, + { + "type": "table", + "bbox": [ + 315, + 398, + 476, + 492 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 315, + 398, + 476, + 492 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 315, + 398, + 476, + 492 + ], + "spans": [ + { + "bbox": [ + 315, + 398, + 476, + 492 + ], + "score": 0.705, + "html": "
(b) CIFAR-10
pAdvoAdv2
0.0 (No adaptation)16.1±7.8521.5±7.79
0.145.1±0.4946.9±0.48
0.359.2±0.4260.8±0.33
0.562.4±0.2764.4±0.6
0.762.8±0.5264.9±0.31
0.962.8±0.7164.9±0.31
1.0 (Full adaptation)62.4±0.6464.9±0.73
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(a)IMAGENET
pAdvoAdv2
0.0 (No adaptation)0.4±0.010.9±0.01
0.12.1±0.047.7±0.09
0.320.6±0.1636.6±0.09
0.541.1±0.0945.5±0.13
0.743.5±0.1446.7±0.13
0.944.2±0.1246.8±0.13
1.0 (Full adaptation)44.8±0.1347.2±0.14
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We randomly shuffle the test images to", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 518, + 267, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 132, + 529 + ], + "score": 1.0, + "content": "report", + "type": "text" + }, + { + "bbox": [ + 132, + 518, + 196, + 528 + ], + "score": 0.88, + "content": "( m e a n + 2 \\times s d )", + "type": "inline_equation" + }, + { + "bbox": [ + 196, + 518, + 267, + 529 + ], + "score": 1.0, + "content": "of 5 different runs.", + "type": "text" + } + ], + "index": 39 + } + ], + "index": 38 + } + ], + "index": 33.5 + }, + { + "type": "text", + "bbox": [ + 106, + 549, + 505, + 615 + ], + "lines": [ + { + "bbox": [ + 106, + 550, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 159, + 561 + ], + "score": 1.0, + "content": "Momentum", + "type": "text" + }, + { + "bbox": [ + 159, + 550, + 172, + 561 + ], + "score": 0.66, + "content": "( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 172, + 550, + 364, + 561 + ], + "score": 1.0, + "content": ". We first investigate the effect of momentum", + "type": "text" + }, + { + "bbox": [ + 364, + 550, + 377, + 561 + ], + "score": 0.7, + "content": "( \\rho )", + "type": "inline_equation" + }, + { + "bbox": [ + 377, + 550, + 505, + 561 + ], + "score": 1.0, + "content": "as we choose a large batch size", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 560, + 504, + 572 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 445, + 572 + ], + "score": 1.0, + "content": "of 512. In Table 3, we present the performance of AT models for different values of", + "type": "text" + }, + { + "bbox": [ + 446, + 562, + 452, + 572 + ], + "score": 0.79, + "content": "\\rho", + "type": "inline_equation" + }, + { + "bbox": [ + 452, + 560, + 504, + 572 + ], + "score": 1.0, + "content": ". Recall that,", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 572, + 506, + 583 + ], + "spans": [ + { + "bbox": [ + 106, + 572, + 133, + 583 + ], + "score": 0.9, + "content": "\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 133, + 572, + 506, + 583 + ], + "score": 1.0, + "content": "denotes full adaptation (Equation 5). Here, we completely ignore the training statistics and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 582, + 505, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 582, + 369, + 595 + ], + "score": 1.0, + "content": "recompute the BN statistics using the test batches. In contrast,", + "type": "text" + }, + { + "bbox": [ + 369, + 583, + 398, + 594 + ], + "score": 0.91, + "content": "\\rho = 0", + "type": "inline_equation" + }, + { + "bbox": [ + 398, + 582, + 505, + 595 + ], + "score": 1.0, + "content": "represents no adaptation,", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "spans": [ + { + "bbox": [ + 105, + 593, + 505, + 606 + ], + "score": 1.0, + "content": "i.e., the same as the standard ‘deterministic’ inference setup. In this case, we use the previously", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 604, + 303, + 617 + ], + "spans": [ + { + "bbox": [ + 105, + 604, + 303, + 617 + ], + "score": 1.0, + "content": "computed BN statistics obtained during training.", + "type": "text" + } + ], + "index": 45 + } + ], + "index": 42.5, + "bbox_fs": [ + 105, + 550, + 506, + 617 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 621, + 503, + 644 + ], + "lines": [ + { + "bbox": [ + 106, + 620, + 505, + 634 + ], + "spans": [ + { + "bbox": [ + 106, + 620, + 452, + 634 + ], + "score": 1.0, + "content": "We observe that for IMAGENET (Table 3 [Left]) the performance started converging at", + "type": "text" + }, + { + "bbox": [ + 452, + 621, + 485, + 633 + ], + "score": 0.89, + "content": "\\rho = 0 . 7", + "type": "inline_equation" + }, + { + "bbox": [ + 485, + 620, + 505, + 634 + ], + "score": 1.0, + "content": ". For", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 632, + 418, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 381, + 645 + ], + "score": 1.0, + "content": "CIFAR-10 (Table 3 [Right]), the convergence started even earlier at", + "type": "text" + }, + { + "bbox": [ + 381, + 633, + 414, + 644 + ], + "score": 0.9, + "content": "\\rho = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 414, + 632, + 418, + 645 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 47 + } + ], + "index": 46.5, + "bbox_fs": [ + 106, + 620, + 505, + 645 + ] + }, + { + "type": "text", + "bbox": [ + 108, + 656, + 504, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 438, + 668 + ], + "score": 1.0, + "content": "Batch Size. Next, we investigate the minimum size of the test batches to choose", + "type": "text" + }, + { + "bbox": [ + 439, + 657, + 464, + 668 + ], + "score": 0.91, + "content": "\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 464, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "(i.e., full-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 106, + 668, + 504, + 679 + ], + "spans": [ + { + "bbox": [ + 106, + 668, + 230, + 678 + ], + "score": 1.0, + "content": "adaptation). In Table 4, we fix", + "type": "text" + }, + { + "bbox": [ + 231, + 668, + 256, + 679 + ], + "score": 0.91, + "content": "\\rho = 1", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 668, + 504, + 678 + ], + "score": 1.0, + "content": "and vary the test batch sizes as we evaluate these AT models.", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 691 + ], + "score": 1.0, + "content": "We observe that the performance of these models started improving even when we are using the", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "test batches of size 8. The performance further improves as we choose larger sizes of test batches.", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "We can see that their performance started converging as we choose the test batches of size 64 for", + "type": "text", + "cross_page": true + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 259, + 449, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 449, + 272 + ], + "score": 1.0, + "content": "IMAGENET. On the other hand, the convergence started much earlier for CIFAR-10.", + "type": "text", + "cross_page": true + } + ], + "index": 9 + } + ], + "index": 49, + "bbox_fs": [ + 105, + 655, + 505, + 691 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 317, + 81, + 448, + 177 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 317, + 81, + 448, + 177 + ], + "group_id": 1, + "lines": [ + { + "bbox": [ + 317, + 81, + 448, + 177 + ], + "spans": [ + { + "bbox": [ + 317, + 81, + 448, + 177 + ], + "score": 0.786, + "html": "
(b) CIFAR-10
Batch SizeAdvoAdv2
w/o BNadapt16.1±7.8521.5±7.79
857.2±1.2359.5±0.38
1660.2±0.7962.3±0.87
3261.5±0.4663.6±0.55
6462.3±0.564.0±0.38
12862.7±0.6864.4±0.53
25662.7±0.6864.9±0.48
51262.4±0.6464.9±0.73
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(a)IMAGENET
Batch SizeAdvoAdv2
w/oBNadapt0.4±0.010.9±0.01
811.5±0.229.1±0.15
1628.1±0.2226.7±0.14
3237.1±0.2437.6±0.2
6441.4±0.2642.9±0.12
12843.3±0.1545.4±0.13
25644.4±0.2146.7±0.07
51244.8±0.1347.2±0.14
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We randomly shuffle the test images to report", + "type": "text" + }, + { + "bbox": [ + 437, + 196, + 503, + 206 + ], + "score": 0.85, + "content": "( m e a n + 2 \\times s . d . )", + "type": "inline_equation" + }, + { + "bbox": [ + 503, + 195, + 505, + 208 + ], + "score": 0.0, + "content": "", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 205, + 176, + 217 + ], + "spans": [ + { + "bbox": [ + 105, + 205, + 176, + 217 + ], + "score": 1.0, + "content": "of 5 different runs.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 5 + } + ], + "index": 3.0 + }, + { + "type": "text", + "bbox": [ + 106, + 238, + 506, + 271 + ], + "lines": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "spans": [ + { + "bbox": [ + 105, + 237, + 505, + 251 + ], + "score": 1.0, + "content": "test batches of size 8. The performance further improves as we choose larger sizes of test batches.", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "spans": [ + { + "bbox": [ + 106, + 249, + 506, + 262 + ], + "score": 1.0, + "content": "We can see that their performance started converging as we choose the test batches of size 64 for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 259, + 449, + 272 + ], + "spans": [ + { + "bbox": [ + 105, + 259, + 449, + 272 + ], + "score": 1.0, + "content": "IMAGENET. 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In particular,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 459, + 601 + ], + "score": 1.0, + "content": "for CIFAR-10 dataset, Baseline models using adaptation achieve similar performance as", + "type": "text" + }, + { + "bbox": [ + 459, + 588, + 505, + 600 + ], + "score": 0.92, + "content": "\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 360, + 611 + ], + "score": 1.0, + "content": "1] without BN adaptation, while produces significantly lower", + "type": "text" + }, + { + "bbox": [ + 360, + 599, + 371, + 610 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "certification robustness. This is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "because, Baseline models, even after adaptation cannot consistently predict the same class to provide", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 246, + 634 + ], + "score": 1.0, + "content": "higher certified robustness at larger", + "type": "text" + }, + { + "bbox": [ + 247, + 621, + 257, + 632 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "radii. 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(b) CIFAR-10
Batch SizeAdvoAdv2
w/o BNadapt16.1±7.8521.5±7.79
857.2±1.2359.5±0.38
1660.2±0.7962.3±0.87
3261.5±0.4663.6±0.55
6462.3±0.564.0±0.38
12862.7±0.6864.4±0.53
25662.7±0.6864.9±0.48
51262.4±0.6464.9±0.73
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(a)IMAGENET
Batch SizeAdvoAdv2
w/oBNadapt0.4±0.010.9±0.01
811.5±0.229.1±0.15
1628.1±0.2226.7±0.14
3237.1±0.2437.6±0.2
6441.4±0.2642.9±0.12
12843.3±0.1545.4±0.13
25644.4±0.2146.7±0.07
51244.8±0.1347.2±0.14
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In particular,", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 587, + 505, + 601 + ], + "spans": [ + { + "bbox": [ + 105, + 587, + 459, + 601 + ], + "score": 1.0, + "content": "for CIFAR-10 dataset, Baseline models using adaptation achieve similar performance as", + "type": "text" + }, + { + "bbox": [ + 459, + 588, + 505, + 600 + ], + "score": 0.92, + "content": "\\mathrm { A d v _ { 2 } } [ \\ell _ { 2 } \\leq", + "type": "inline_equation" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 598, + 506, + 611 + ], + "spans": [ + { + "bbox": [ + 105, + 598, + 360, + 611 + ], + "score": 1.0, + "content": "1] without BN adaptation, while produces significantly lower", + "type": "text" + }, + { + "bbox": [ + 360, + 599, + 371, + 610 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 371, + 598, + 506, + 611 + ], + "score": 1.0, + "content": "certification robustness. This is", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "because, Baseline models, even after adaptation cannot consistently predict the same class to provide", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 246, + 634 + ], + "score": 1.0, + "content": "higher certified robustness at larger", + "type": "text" + }, + { + "bbox": [ + 247, + 621, + 257, + 632 + ], + "score": 0.88, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 257, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "radii. As a result, we can only improve the certified robustness", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 632, + 182, + 643 + ], + "spans": [ + { + "bbox": [ + 105, + 632, + 147, + 643 + ], + "score": 1.0, + "content": "at smaller", + "type": "text" + }, + { + "bbox": [ + 148, + 632, + 158, + 643 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 158, + 632, + 182, + 643 + ], + "score": 1.0, + "content": "radii.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 20.5, + "bbox_fs": [ + 105, + 555, + 506, + 643 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "image", + "bbox": [ + 124, + 82, + 483, + 412 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 82, + 483, + 412 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 82, + 483, + 412 + ], + "spans": [ + { + "bbox": [ + 124, + 82, + 483, + 412 + ], + "score": 0.975, + "type": "image", + "image_path": "2c20dffba6826dfc30c2b7b6b9c8e8057f5321d3caea73d0268f2a64ebc189bb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 82, + 483, + 192.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 192.0, + 483, + 302.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 302.0, + 483, + 412.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 422, + 504, + 453 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 324, + 434 + ], + "score": 1.0, + "content": "Figure 7: IMAGENET: Certified top-1 accuracy at various", + "type": "text" + }, + { + "bbox": [ + 324, + 423, + 334, + 433 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 422, + 453, + 434 + ], + "score": 1.0, + "content": "radii as we vary the noise-level,", + "type": "text" + }, + { + "bbox": [ + 454, + 425, + 461, + 432 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "at test-time", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 433, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 213, + 444 + ], + "score": 1.0, + "content": "using proposed Algorithm 1.", + "type": "text" + }, + { + "bbox": [ + 213, + 433, + 237, + 443 + ], + "score": 0.77, + "content": "\\mathbf { A d v } _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 433, + 254, + 444 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 433, + 275, + 443 + ], + "score": 0.78, + "content": "\\mathbf { A d v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 433, + 505, + 444 + ], + "score": 1.0, + "content": "models are as defined in experimental set-up (section 4). Refer", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 442, + 347, + 455 + ], + "spans": [ + { + "bbox": [ + 105, + 442, + 347, + 455 + ], + "score": 1.0, + "content": "to Table 5 for complete results of all models and different settings.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + } + ], + "index": 2.5 + } + ], + "page_idx": 16, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 312, + 764 + ], + "score": 1.0, + "content": "17", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 124, + 82, + 483, + 412 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 124, + 82, + 483, + 412 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 124, + 82, + 483, + 412 + ], + "spans": [ + { + "bbox": [ + 124, + 82, + 483, + 412 + ], + "score": 0.975, + "type": "image", + "image_path": "2c20dffba6826dfc30c2b7b6b9c8e8057f5321d3caea73d0268f2a64ebc189bb.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 124, + 82, + 483, + 192.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 124, + 192.0, + 483, + 302.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 124, + 302.0, + 483, + 412.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 422, + 504, + 453 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 422, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 422, + 324, + 434 + ], + "score": 1.0, + "content": "Figure 7: IMAGENET: Certified top-1 accuracy at various", + "type": "text" + }, + { + "bbox": [ + 324, + 423, + 334, + 433 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 334, + 422, + 453, + 434 + ], + "score": 1.0, + "content": "radii as we vary the noise-level,", + "type": "text" + }, + { + "bbox": [ + 454, + 425, + 461, + 432 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 422, + 505, + 434 + ], + "score": 1.0, + "content": "at test-time", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 433, + 505, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 213, + 444 + ], + "score": 1.0, + "content": "using proposed Algorithm 1.", + "type": "text" + }, + { + "bbox": [ + 213, + 433, + 237, + 443 + ], + "score": 0.77, + "content": "\\mathbf { A d v } _ { \\infty }", + "type": "inline_equation" + }, + { + "bbox": [ + 238, + 433, + 254, + 444 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 254, + 433, + 275, + 443 + ], + "score": 0.78, + "content": "\\mathbf { A d v } _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 275, + 433, + 505, + 444 + ], + "score": 1.0, + "content": "models are as defined in experimental set-up (section 4). 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IMAGENET
ModelBN adaptionCertification0.5l2 Radius 1.251.75
0.250.751.01.52.02.252.52.75ACR
Baseline=atσ=0.257.84.83.00.00.00.00.0 0.00.00.00.01 0.054
Advo[l∞o ≤4/255]at σ = 0.25 at σ = 0.50at g = 0.2550.046.441.60.0 0.00.00.00.00.00.00.00.445 0.607
at σ=0.75at σ = 0.50 at σ=0.7543.6 31.639.435.831.4 27.623.418.20.00.00.00.00.443
26.422.418.6 16.814.411.89.47.65.63.6
Advo[loo≤4/255]+adapt[BestRadij(Ours)50.046.441.631.4 27.623.418.29.47.65.63.60.759
at σ = 0.250.480
Adv2[l2 ≤3.00]at σ= 0.50at σ = 0.2553.2 47.050.2 43.046.8 39.00.0 0.00.00.00.00.00.00.00.711
at σ=0.75at g = 0.50 at g=0.7537.832.236.4 32.830.8 20.227.00.00.0 14.20.0 12.00.0 9.60.639
Adv2l2≤3.00] +adapt[Best Radii](Ours)53.228.426.0 22.4 32.830.819.0 27.017.4 17.414.212.09.60.930
50.246.836.4
Randg=0.5 Cohen et al. (2019)=at σ=0.5060.854.447.839.034.2 29.023.80.00.00.00.00.809
at σ = 0.25at g = 0.2559.853.646.60.0 0.00.00.00.00.00.00.00.507
+ adaptationat σ= 0.50at σ= 0.5058.651.043.837.432.2 27.422.40.00.00.00.00.768
at σ=0.75at σ=0.7548.641.636.631.226.2 22.418.616.812.88.65.40.720
Randg=0.5+adapt[BestRadii](Ours)22.416.8
59.853.646.637.432.227.412.88.65.40.973
", + "type": "table", + "image_path": "2c6f7250896cc5ee2fae5970dd445ed7d67ae44e26e8b3984ce4ab9e26792847.jpg" + } + ] + } + ], + "index": 6, + "virtual_lines": [ + { + "bbox": [ + 121, + 473, + 490, + 520.3333333333334 + ], + "spans": [], + "index": 5 + }, + { + "bbox": [ + 121, + 520.3333333333334, + 490, + 567.6666666666667 + ], + "spans": [], + "index": 6 + }, + { + "bbox": [ + 121, + 567.6666666666667, + 490, + 615.0000000000001 + ], + "spans": [], + "index": 7 + } + ] + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 106, + 624, + 505, + 690 + ], + "lines": [ + { + "bbox": [ + 105, + 623, + 505, + 636 + ], + "spans": [ + { + "bbox": [ + 105, + 623, + 340, + 636 + ], + "score": 1.0, + "content": "Table 5: IMAGENET: Certified top-1 accuracy at various", + "type": "text" + }, + { + "bbox": [ + 340, + 624, + 351, + 635 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 351, + 623, + 420, + 636 + ], + "score": 1.0, + "content": "radii as we vary", + "type": "text" + }, + { + "bbox": [ + 420, + 626, + 428, + 634 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 428, + 623, + 505, + 636 + ], + "score": 1.0, + "content": "for BN adaptation", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 634, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 634, + 505, + 646 + ], + "score": 1.0, + "content": "and certification along with average certified radii (ACR). We use ResNet50 for IMAGENET. Each", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 645, + 506, + 658 + ], + "spans": [ + { + "bbox": [ + 104, + 645, + 506, + 658 + ], + "score": 1.0, + "content": "gray block is corresponding to one classification model while the rows are corresponding to its", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 106, + 657, + 504, + 668 + ], + "spans": [ + { + "bbox": [ + 106, + 657, + 504, + 668 + ], + "score": 1.0, + "content": "certification performances as we choose different noise levels for adaptations and certifications. The", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "spans": [ + { + "bbox": [ + 105, + 667, + 505, + 680 + ], + "score": 1.0, + "content": "Best Radii are obtained by selecting the highest radius for each test example as we adapt the models", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 677, + 226, + 691 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 214, + 691 + ], + "score": 1.0, + "content": "with different noise levels,", + "type": "text" + }, + { + "bbox": [ + 215, + 681, + 222, + 689 + ], + "score": 0.68, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 222, + 677, + 226, + 691 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 10.5 + } + ], + "page_idx": 17, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 106, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "spans": [ + { + "bbox": [ + 106, + 26, + 309, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 300, + 751, + 311, + 760 + ], + "lines": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "spans": [ + { + "bbox": [ + 299, + 750, + 313, + 763 + ], + "score": 1.0, + "content": "18", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "image", + "bbox": [ + 106, + 82, + 500, + 412 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 106, + 82, + 500, + 412 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 82, + 500, + 412 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 500, + 412 + ], + "score": 0.975, + "type": "image", + "image_path": "0e4a0c6dfa810b8c74282bb378bde050de58369c2b0ff45a101ae3c2c7280b30.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 106, + 82, + 500, + 192.0 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 106, + 192.0, + 500, + 302.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 106, + 302.0, + 500, + 412.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "image_caption", + "bbox": [ + 106, + 422, + 504, + 444 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 106, + 423, + 505, + 434 + ], + "spans": [ + { + "bbox": [ + 106, + 423, + 323, + 434 + ], + "score": 1.0, + "content": "Figure 8: CIFAR-10: Certified top-1 accuracy at various", + "type": "text" + }, + { + "bbox": [ + 324, + 423, + 333, + 433 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 333, + 423, + 453, + 434 + ], + "score": 1.0, + "content": "radii as we vary the noise-level,", + "type": "text" + }, + { + "bbox": [ + 453, + 425, + 461, + 432 + ], + "score": 0.76, + "content": "\\sigma", + "type": "inline_equation" + }, + { + "bbox": [ + 461, + 423, + 505, + 434 + ], + "score": 1.0, + "content": "at test-time", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 433, + 475, + 444 + ], + "spans": [ + { + "bbox": [ + 105, + 433, + 475, + 444 + ], + "score": 1.0, + "content": "using proposed Algorithm 1. Refer to Table 6 for complete results of all models and different settings.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "table", + "bbox": [ + 121, + 473, + 490, + 615 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 121, + 473, + 490, + 615 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 121, + 473, + 490, + 615 + ], + "spans": [ + { + "bbox": [ + 121, + 473, + 490, + 615 + ], + "score": 0.974, + "html": "
IMAGENET
ModelBN adaptionCertification0.5l2 Radius 1.251.75
0.250.751.01.52.02.252.52.75ACR
Baseline=atσ=0.257.84.83.00.00.00.00.0 0.00.00.00.01 0.054
Advo[l∞o ≤4/255]at σ = 0.25 at σ = 0.50at g = 0.2550.046.441.60.0 0.00.00.00.00.00.00.00.445 0.607
at σ=0.75at σ = 0.50 at σ=0.7543.6 31.639.435.831.4 27.623.418.20.00.00.00.00.443
26.422.418.6 16.814.411.89.47.65.63.6
Advo[loo≤4/255]+adapt[BestRadij(Ours)50.046.441.631.4 27.623.418.29.47.65.63.60.759
at σ = 0.250.480
Adv2[l2 ≤3.00]at σ= 0.50at σ = 0.2553.2 47.050.2 43.046.8 39.00.0 0.00.00.00.00.00.00.00.711
at σ=0.75at g = 0.50 at g=0.7537.832.236.4 32.830.8 20.227.00.00.0 14.20.0 12.00.0 9.60.639
Adv2l2≤3.00] +adapt[Best Radii](Ours)53.228.426.0 22.4 32.830.819.0 27.017.4 17.414.212.09.60.930
50.246.836.4
Randg=0.5 Cohen et al. (2019)=at σ=0.5060.854.447.839.034.2 29.023.80.00.00.00.00.809
at σ = 0.25at g = 0.2559.853.646.60.0 0.00.00.00.00.00.00.00.507
+ adaptationat σ= 0.50at σ= 0.5058.651.043.837.432.2 27.422.40.00.00.00.00.768
at σ=0.75at σ=0.7548.641.636.631.226.2 22.418.616.812.88.65.40.720
Randg=0.5+adapt[BestRadii](Ours)22.416.8
59.853.646.637.432.227.412.88.65.40.973
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CIFAR-10
ModelBN adaptionCertification0.250.50.75l2Radius 1.0 1.251.51.752.0ACR
Baseline0.0 0.00.00.0 0.00.026
Advo[∞ ≤ 4/255]at σ = 0.25 at σ = 0.50 at σ = 0.75at σ = 0.25 at σ = 0.50 at σ = 0.7567.96 47.34 26.8950.46 31.83 15.9231.96 18.78 8.440.0 9.98 4.310.0 4.44 2.030.0 1.62 0.28 0.79 0.230.0 0.0 0.00.080.485 0.350 0.146
Advo[lo ≤8/255]at σ =0.25 at σ = 0.50 at σ =0.75at g =0.25 at σ = 0.50 at σ =0.7566.43 53.65 39.9655.06 42.91 30.7642.86 32.58 22.010.0 22.68 14.640.0 14.24 8.840.0 7.88 4.810.0 2.940.0 0.0 1.150.527 0.515 0.352
Advoo[∞ ≤12/255]atσ =0.25 at σ = 0.50 at σ = 0.75at σ= 0.25 at σ = 0.50 atσ=0.7560.52 51.53 42.6152.42 43.94 35.5643.27 36.41 28.470.0 28.69 22.390.0 21.25 16.690.0 14.532.29 0.0 8.030.0 0.0 4.430.499 0.581 0.482
Adv[l∞o ≤ 16/255]at σ =0.25 at σ = 0.50 at σ =0.75atg=0.25 at σ = 0.50 atσ =0.7553.75 48.07 42.05 67.9647.57 42.51 36.42 55.0641.18 36.54 31.24 43.270.0 30.55 26.05 30.550.0 24.68 20.7411.58 0.0 18.49 16.15 12.017.42 0.0 12.110.0 0.0 8.450.454 0.598 0.557 0.903
Advo + adapt [Best Radii] (Ours) 24.6818.49 12.11 8.45 0.0
Adv2[l2 ≤0.50]at σ = 0.25 at σ = 0.50 atg =0.75at σ = 0.25 at σ = 0.50 atσ =0.7568.84 48.81 27.3854.04 33.82 16.1537.13 20.95 9.230.0 11.5 4.560.0 5.64 2.060.0 2.29 0.62 0.91 0.330.0 0.0 0.080.518 0.382 0.153
Adv2[l2 ≤ 1.00]at σ = 0.25 atσ =0.50 atσ =0.75at σ = 0.25 at σ = 0.50 atg =0.7568.02 56.45 43.0458.54 46.24 33.0846.98 35.6 24.810.0 26.89 17.680.0 18.73 11.390.0 11.37 6.60.0 5.41 3.570.0 0.0 1.950.551 0.580 0.405
Adv2[l2 ≤1.25]at g =0.25 at σ =0.50 atσ =0.75at σ= 0.25 at σ = 0.50 at σ = 0.7567.13 57.73 46.5458.77 48.8 37.5349.43 39.64 29.350.0 31.07 22.00.0 22.61 15.620.0 15.82 10.510.0 8.96 6.550.0 0.0 3.680.557 0.647 0.496
Adv2[l2 ≤ 1.50]at σ = 0.25 at σ =0.50 at σ=0.75at g = 0.25 at σ = 0.50 atg =0.7564.21 56.55 47.7357.13 49.19 40.8949.71 41.72 33.780.0 34.47 27.220.0 27.36 20.780.0 20.23 15.190.0 12.98 10.510.0 0.0 6.770.543 0.689 0.585
Adv2[l2 ≤ 2.00]at σ= 0.25 at σ =0.50 at σ =0.75at σ = 0.25 at σ = 0.50 at σ = 0.7560.4 54.27 47.9654.71 48.89 42.5448.35 43.1 37.040.0 37.34 31.650.0 31.52 26.170.0 25.74 21.120.0 19.14 16.590.0 0.0 12.440.523 0.731 0.698
Adv2[l2 ≤ 2.25]at σ = 0.25 at σ = 0.50 at σ = 0.75atg=0.25 at σ = 0.50 at σ = 0.7557.08 52.1 46.4552.5 46.99 41.7147.11 42.26 36.750.0 36.9 31.880.0 31.58 26.950.0 26.08 22.330.0 20.03 17.820.0 0.0 13.550.504 0.724 0.713
Adv2[l2 ≤ 2.50]at σ =0.25 at σ = 0.50 atσ=0.75at σ = 0.25 at σ = 0.50 at σ =0.7554.88 50.53 45.9550.79 46.26 41.8946.29 41.84 37.530.0 37.74 33.550.0 33.2 29.310.0 28.69 25.270.0 23.34 20.980.0 0.0 17.280.487 0.734 0.765
Adv2[2 ≤ 3.00]at σ =0.25 atσ =0.50 at σ = 0.75at σ =0.25 at σ = 0.50 atσ =0.7553.82 49.41 45.3749.69 45.57 41.5445.04 41.52 37.750.0 37.43 33.490.0 33.37 29.350.0 28.82 25.620.0 23.65 21.830.0 0.0 18.230.475 0.720 0.771 1.198
Adv2+adapt[Best Radii](Ours) 68.84 58.77
Randg=0.5at g= 0.50 at σ = 0.2551.68 62.9140.38 52.2549.71 30.25 40.0637.74 20.81 0.033.37 13.36 0.028.82 7.71 0.023.65 3.38 0.018.23 0.0 0.00.488 0.497 0.575
at σ = 0.25 + adaptation at σ =0.50 at σ = 0.50
Randg=0.5 +adapt[Best Radi] (Ours)atσ =0.75at σ =0.7557.58 46.4 62.9146.46 35.63 52.2535.5 26.06 40.0625.57 18.17 25.5717.43 11.61 17.4310.67 5.46 6.86 3.64 10.67 5.460.0 1.92 1.920.427 0.657 0.609
SmoothAdvg=0.5,=0.25 at g= 0.50
+ adaptationat σ = 0.25 at σ = 0.50at σ = 0.25 at g = 0.5057.8 58.74 54.047.63 48.29 42.9137.41 36.7 32.6327.88 0.0 23.620.33 13.53 0.0 16.088.03 0.0 0.0 9.93 5.50.0 0.0 0.00.464 0.535 0.390
SmoothAdvg=0.5,∈=0.50at σ = 0.75 at σ = 0.25at σ = 0.75 atσ =0.50 at σ = 0.2543.15 58.82 59.8932.29 49.68 50.423.61 40.35 39.9916.41 31.93 0.010.94 24.18 0.06.65 17.05 0.03.88 10.57 0.02.16 0.0 0.00.661 0.483 0.592
+ adaptation
SmoothAdvg=0.5,∈=1.0at σ = 0.50 at σ = 0.75at σ = 0.50 at σ =0.75 atg=0.5055.73 46.25 56.5345.79 36.72 49.5336.6 28.2 41.3827.4 20.9 34.6320.03 14.73 27.8113.48 9.46 21.227.85 5.78 14.410.0 3.32 0.00.470 0.691 0.467
+ adaptation- at σ = 0.25 at σ = 0.50at σ = 0.25 at σ = 0.50 at σ = 0.7557.13 53.56 47.1748.38 45.79 39.439.7 37.640.0 30.060.0 23.12 18.930.0 17.27 13.70.0 11.14 9.260.0 0.0 5.80.620 0.545
at σ = 0.75
SmoothAdvg=0.5,∈=2.0at σ = 0.25 at σ =0.50atg =0.50 at σ = 0.2552.82 52.2347.67 47.2431.8 42.68 41.7624.74 37.55 0.032.64 0.027.52 0.022.42 0.00.0 0.00.732 0.451 0.692
+ adaptation at σ =0.75at σ = 0.50 atσ =0.7550.28 46.745.24 41.7940.39 37.1435.5 33.0530.92 28.2826.1 23.8820.25 19.340.0 15.050.727
SmoothAdvg=0.5[BestRadii]58.82 59.8949.68 50.442.68 41.7637.55 35.532.64 30.9227.52 26.122.42 20.250.0 15.050.918 1.008
SmoothAdvg=0.5 + adapt [Best Radii] (Ours) MARCERg=0.5 (Zhai et al.,2020) 60.0 53.0
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Each gray block is corresponding to one", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "spans": [ + { + "bbox": [ + 106, + 640, + 505, + 651 + ], + "score": 1.0, + "content": "classification model while the rows are corresponding to its certification performances as we choose different", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 650, + 505, + 660 + ], + "spans": [ + { + "bbox": [ + 105, + 650, + 505, + 660 + ], + "score": 1.0, + "content": "noise levels for adaptations and certifications. The Best Radii are obtained by training different models with", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 505, + 672 + ], + "score": 1.0, + "content": "varying hyper-parameters and adapting them with different noise levels during inference. We also present the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 669, + 505, + 682 + ], + "spans": [ + { + "bbox": [ + 105, + 669, + 466, + 682 + ], + "score": 1.0, + "content": "best reported results for MARCER (Zhai et al., 2020) and Consistancy Jeong & Shin (2020) at", + "type": "text" + }, + { + "bbox": [ + 466, + 670, + 501, + 680 + ], + "score": 0.88, + "content": "\\sigma = 0 . 5", + "type": "inline_equation" + }, + { + "bbox": [ + 502, + 669, + 505, + 682 + ], + "score": 1.0, + "content": ",", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 678, + 239, + 691 + ], + "spans": [ + { + "bbox": [ + 105, + 678, + 239, + 691 + ], + "score": 1.0, + "content": "obtained from the respective papers.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 107, + 708, + 390, + 720 + ], + "lines": [ + { + "bbox": [ + 105, + 705, + 393, + 724 + ], + "spans": [ + { + "bbox": [ + 105, + 705, + 393, + 724 + ], + "score": 1.0, + "content": "C PERFORMANCE AGAINST DIFFERENT CORRUPTIONS", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 107, + 730, + 504, + 760 + ], + "lines": [ + { + "bbox": [ + 105, + 730, + 506, + 743 + ], + "spans": [ + { + "bbox": [ + 105, + 730, + 189, + 743 + ], + "score": 1.0, + "content": "We mainly focus on", + "type": "text" + }, + { + "bbox": [ + 189, + 731, + 199, + 742 + ], + "score": 0.87, + "content": "\\ell _ { 2 }", + "type": "inline_equation" + }, + { + "bbox": [ + 200, + 730, + 506, + 743 + ], + "score": 1.0, + "content": "certification using Gaussian noise in this paper. However, we note that ran-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 741, + 506, + 754 + ], + "spans": [ + { + "bbox": [ + 105, + 741, + 506, + 754 + ], + "score": 1.0, + "content": "domized smoothing techniques have been also applied to provide certifications for other perturbation", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 301, + 751, + 312, + 761 + ], + "spans": [ + { + "bbox": [ + 301, + 751, + 312, + 761 + ], + "score": 1.0, + "content": "19", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + } + ], + "page_idx": 18, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2022", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 126, + 78, + 483, + 613 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 126, + 78, + 483, + 613 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 78, + 483, + 613 + ], + "spans": [ + { + "bbox": [ + 126, + 78, + 483, + 613 + ], + "score": 0.961, + "html": "
CIFAR-10
ModelBN adaptionCertification0.250.50.75l2Radius 1.0 1.251.51.752.0ACR
Baseline0.0 0.00.00.0 0.00.026
Advo[∞ ≤ 4/255]at σ = 0.25 at σ = 0.50 at σ = 0.75at σ = 0.25 at σ = 0.50 at σ = 0.7567.96 47.34 26.8950.46 31.83 15.9231.96 18.78 8.440.0 9.98 4.310.0 4.44 2.030.0 1.62 0.28 0.79 0.230.0 0.0 0.00.080.485 0.350 0.146
Advo[lo ≤8/255]at σ =0.25 at σ = 0.50 at σ =0.75at g =0.25 at σ = 0.50 at σ =0.7566.43 53.65 39.9655.06 42.91 30.7642.86 32.58 22.010.0 22.68 14.640.0 14.24 8.840.0 7.88 4.810.0 2.940.0 0.0 1.150.527 0.515 0.352
Advoo[∞ ≤12/255]atσ =0.25 at σ = 0.50 at σ = 0.75at σ= 0.25 at σ = 0.50 atσ=0.7560.52 51.53 42.6152.42 43.94 35.5643.27 36.41 28.470.0 28.69 22.390.0 21.25 16.690.0 14.532.29 0.0 8.030.0 0.0 4.430.499 0.581 0.482
Adv[l∞o ≤ 16/255]at σ =0.25 at σ = 0.50 at σ =0.75atg=0.25 at σ = 0.50 atσ =0.7553.75 48.07 42.05 67.9647.57 42.51 36.42 55.0641.18 36.54 31.24 43.270.0 30.55 26.05 30.550.0 24.68 20.7411.58 0.0 18.49 16.15 12.017.42 0.0 12.110.0 0.0 8.450.454 0.598 0.557 0.903
Advo + adapt [Best Radii] (Ours) 24.6818.49 12.11 8.45 0.0
Adv2[l2 ≤0.50]at σ = 0.25 at σ = 0.50 atg =0.75at σ = 0.25 at σ = 0.50 atσ =0.7568.84 48.81 27.3854.04 33.82 16.1537.13 20.95 9.230.0 11.5 4.560.0 5.64 2.060.0 2.29 0.62 0.91 0.330.0 0.0 0.080.518 0.382 0.153
Adv2[l2 ≤ 1.00]at σ = 0.25 atσ =0.50 atσ =0.75at σ = 0.25 at σ = 0.50 atg =0.7568.02 56.45 43.0458.54 46.24 33.0846.98 35.6 24.810.0 26.89 17.680.0 18.73 11.390.0 11.37 6.60.0 5.41 3.570.0 0.0 1.950.551 0.580 0.405
Adv2[l2 ≤1.25]at g =0.25 at σ =0.50 atσ =0.75at σ= 0.25 at σ = 0.50 at σ = 0.7567.13 57.73 46.5458.77 48.8 37.5349.43 39.64 29.350.0 31.07 22.00.0 22.61 15.620.0 15.82 10.510.0 8.96 6.550.0 0.0 3.680.557 0.647 0.496
Adv2[l2 ≤ 1.50]at σ = 0.25 at σ =0.50 at σ=0.75at g = 0.25 at σ = 0.50 atg =0.7564.21 56.55 47.7357.13 49.19 40.8949.71 41.72 33.780.0 34.47 27.220.0 27.36 20.780.0 20.23 15.190.0 12.98 10.510.0 0.0 6.770.543 0.689 0.585
Adv2[l2 ≤ 2.00]at σ= 0.25 at σ =0.50 at σ =0.75at σ = 0.25 at σ = 0.50 at σ = 0.7560.4 54.27 47.9654.71 48.89 42.5448.35 43.1 37.040.0 37.34 31.650.0 31.52 26.170.0 25.74 21.120.0 19.14 16.590.0 0.0 12.440.523 0.731 0.698
Adv2[l2 ≤ 2.25]at σ = 0.25 at σ = 0.50 at σ = 0.75atg=0.25 at σ = 0.50 at σ = 0.7557.08 52.1 46.4552.5 46.99 41.7147.11 42.26 36.750.0 36.9 31.880.0 31.58 26.950.0 26.08 22.330.0 20.03 17.820.0 0.0 13.550.504 0.724 0.713
Adv2[l2 ≤ 2.50]at σ =0.25 at σ = 0.50 atσ=0.75at σ = 0.25 at σ = 0.50 at σ =0.7554.88 50.53 45.9550.79 46.26 41.8946.29 41.84 37.530.0 37.74 33.550.0 33.2 29.310.0 28.69 25.270.0 23.34 20.980.0 0.0 17.280.487 0.734 0.765
Adv2[2 ≤ 3.00]at σ =0.25 atσ =0.50 at σ = 0.75at σ =0.25 at σ = 0.50 atσ =0.7553.82 49.41 45.3749.69 45.57 41.5445.04 41.52 37.750.0 37.43 33.490.0 33.37 29.350.0 28.82 25.620.0 23.65 21.830.0 0.0 18.230.475 0.720 0.771 1.198
Adv2+adapt[Best Radii](Ours) 68.84 58.77
Randg=0.5at g= 0.50 at σ = 0.2551.68 62.9140.38 52.2549.71 30.25 40.0637.74 20.81 0.033.37 13.36 0.028.82 7.71 0.023.65 3.38 0.018.23 0.0 0.00.488 0.497 0.575
at σ = 0.25 + adaptation at σ =0.50 at σ = 0.50
Randg=0.5 +adapt[Best Radi] (Ours)atσ =0.75at σ =0.7557.58 46.4 62.9146.46 35.63 52.2535.5 26.06 40.0625.57 18.17 25.5717.43 11.61 17.4310.67 5.46 6.86 3.64 10.67 5.460.0 1.92 1.920.427 0.657 0.609
SmoothAdvg=0.5,=0.25 at g= 0.50
+ adaptationat σ = 0.25 at σ = 0.50at σ = 0.25 at g = 0.5057.8 58.74 54.047.63 48.29 42.9137.41 36.7 32.6327.88 0.0 23.620.33 13.53 0.0 16.088.03 0.0 0.0 9.93 5.50.0 0.0 0.00.464 0.535 0.390
SmoothAdvg=0.5,∈=0.50at σ = 0.75 at σ = 0.25at σ = 0.75 atσ =0.50 at σ = 0.2543.15 58.82 59.8932.29 49.68 50.423.61 40.35 39.9916.41 31.93 0.010.94 24.18 0.06.65 17.05 0.03.88 10.57 0.02.16 0.0 0.00.661 0.483 0.592
+ adaptation
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+ adaptation- at σ = 0.25 at σ = 0.50at σ = 0.25 at σ = 0.50 at σ = 0.7557.13 53.56 47.1748.38 45.79 39.439.7 37.640.0 30.060.0 23.12 18.930.0 17.27 13.70.0 11.14 9.260.0 0.0 5.80.620 0.545
at σ = 0.75
SmoothAdvg=0.5,∈=2.0at σ = 0.25 at σ =0.50atg =0.50 at σ = 0.2552.82 52.2347.67 47.2431.8 42.68 41.7624.74 37.55 0.032.64 0.027.52 0.022.42 0.00.0 0.00.732 0.451 0.692
+ adaptation at σ =0.75at σ = 0.50 atσ =0.7550.28 46.745.24 41.7940.39 37.1435.5 33.0530.92 28.2826.1 23.8820.25 19.340.0 15.050.727
SmoothAdvg=0.5[BestRadii]58.82 59.8949.68 50.442.68 41.7637.55 35.532.64 30.9227.52 26.122.42 20.250.0 15.050.918 1.008
SmoothAdvg=0.5 + adapt [Best Radii] (Ours) MARCERg=0.5 (Zhai et al.,2020) 60.0 53.0
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l2Radius (CIFAR-10)0.250.50.751.01.251.51.752.0ACR
Baseline6.962.040.090.00.00.00.00.00.026
Randg =0.5 (Cohen et al.,2019)51.6840.3830.2520.8113.367.713.380.00.488
(Ours)Randg=0.5 +adaptation62.9152.2540.0625.5717.4310.675.461.920.657
SmoothAdvg=0.5 (Salman et al.,2019a)58.8249.6842.6837.5532.6427.5222.420.00.918
(Ours) SmoothAdvg=0.5+adaptation59.8950.441.7635.530.9226.120.2515.051.008
Advo (Rice et al.,2020)35.9529.4423.510.00.00.00.00.00.317
(Ours)Advo + adaptation67.9655.0643.2730.5524.6818.4912.118.450.903
Adv2 (Rice et al., 2020)41.8934.1526.70.00.00.00.00.00.359
(Ours) Adv2 +adaptation68.8458.7749.7137.7433.3728.8223.6518.231.198
MARCERg=0.5 (Zhai et al.,2020)60.053.046.038.029.019.012.00.00.726
Consistancyg=0.5 (Jeong& Shin,2020)48.945.141.337.833.929.925.20.00.726
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(a) IMAGENET
Modelσ=0σ=0.25σ=0.5σ=0.75
Baseline75.2±0.011.8±0.220.3±0.010.1±0.0
+ adaptive BN74.4±0.0431.0±0.277.7±0.242.4±0.01
Advo∞≤4/255]62.8±0.03.9±0.030.4±0.010.2±0.01
+ adaptive BN60.8±0.1653.4±0.1544.9±0.0833.7±0.28
Adv2≤3]59.8±0.09.8±0.080.9±0.010.3±0.0
+adaptive BN58.3±0.0853.7±0.1447.3±0.1439.8±0.18
Rand g=0.522.0±0.032.8±0.1160.9±0.040.9±0.06
+ adaptive BN62.7±0.0362.3±0.1859.5±0.1151.4±0.27
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(b) CIFAR-10
Modelσ=0g=0.25σ=0.5g=0.75
Baseline + adaptive BN95.2±0.010.9±0.8810.6±0.7610.5±1.19
95.0±0.5740.1±0.9722.0±0.8317.2±0.66
Advo≤8/255]82.1±0.040.2±4.5616.1±7.8512.2±5.23
+ adaptive BN81.6±0.9674.2±0.9562.4±0.6451.0±1.03
Adv2[≤1]81.6±0.047.5±5.121.5±7.7914.3±5.63
+ adaptive BN81.8±0.775.8±0.4364.9±0.7353.5±1.71
Rand g=0.566.7±0.069.1±1.0161.2±0.8425.9±1.41
+ adaptive BN74.0±2.173.0±2.0466.8±2.0156.7±0.94
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(a)IMAGENET
pAdvoAdv2
0.0 (No adaptation)0.4±0.010.9±0.01
0.12.1±0.047.7±0.09
0.320.6±0.1636.6±0.09
0.541.1±0.0945.5±0.13
0.743.5±0.1446.7±0.13
0.944.2±0.1246.8±0.13
1.0 (Full adaptation)44.8±0.1347.2±0.14
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(b) CIFAR-10
pAdvoAdv2
0.0 (No adaptation)16.1±7.8521.5±7.79
0.145.1±0.4946.9±0.48
0.359.2±0.4260.8±0.33
0.562.4±0.2764.4±0.6
0.762.8±0.5264.9±0.31
0.962.8±0.7164.9±0.31
1.0 (Full adaptation)62.4±0.6464.9±0.73
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(a)IMAGENET
Batch SizeAdvoAdv2
w/oBNadapt0.4±0.010.9±0.01
811.5±0.229.1±0.15
1628.1±0.2226.7±0.14
3237.1±0.2437.6±0.2
6441.4±0.2642.9±0.12
12843.3±0.1545.4±0.13
25644.4±0.2146.7±0.07
51244.8±0.1347.2±0.14
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(b) CIFAR-10
Batch SizeAdvoAdv2
w/o BNadapt16.1±7.8521.5±7.79
857.2±1.2359.5±0.38
1660.2±0.7962.3±0.87
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12862.7±0.6864.4±0.53
25662.7±0.6864.9±0.48
51262.4±0.6464.9±0.73
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IMAGENET
ModelBN adaptionCertification0.5l2 Radius 1.251.75
0.250.751.01.52.02.252.52.75ACR
Baseline=atσ=0.257.84.83.00.00.00.00.0 0.00.00.00.01 0.054
Advo[l∞o ≤4/255]at σ = 0.25 at σ = 0.50at g = 0.2550.046.441.60.0 0.00.00.00.00.00.00.00.445 0.607
at σ=0.75at σ = 0.50 at σ=0.7543.6 31.639.435.831.4 27.623.418.20.00.00.00.00.443
26.422.418.6 16.814.411.89.47.65.63.6
Advo[loo≤4/255]+adapt[BestRadij(Ours)50.046.441.631.4 27.623.418.29.47.65.63.60.759
at σ = 0.250.480
Adv2[l2 ≤3.00]at σ= 0.50at σ = 0.2553.2 47.050.2 43.046.8 39.00.0 0.00.00.00.00.00.00.00.711
at σ=0.75at g = 0.50 at g=0.7537.832.236.4 32.830.8 20.227.00.00.0 14.20.0 12.00.0 9.60.639
Adv2l2≤3.00] +adapt[Best Radii](Ours)53.228.426.0 22.4 32.830.819.0 27.017.4 17.414.212.09.60.930
50.246.836.4
Randg=0.5 Cohen et al. (2019)=at σ=0.5060.854.447.839.034.2 29.023.80.00.00.00.00.809
at σ = 0.25at g = 0.2559.853.646.60.0 0.00.00.00.00.00.00.00.507
+ adaptationat σ= 0.50at σ= 0.5058.651.043.837.432.2 27.422.40.00.00.00.00.768
at σ=0.75at σ=0.7548.641.636.631.226.2 22.418.616.812.88.65.40.720
Randg=0.5+adapt[BestRadii](Ours)22.416.8
59.853.646.637.432.227.412.88.65.40.973
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CIFAR-10
ModelBN adaptionCertification0.250.50.75l2Radius 1.0 1.251.51.752.0ACR
Baseline0.0 0.00.00.0 0.00.026
Advo[∞ ≤ 4/255]at σ = 0.25 at σ = 0.50 at σ = 0.75at σ = 0.25 at σ = 0.50 at σ = 0.7567.96 47.34 26.8950.46 31.83 15.9231.96 18.78 8.440.0 9.98 4.310.0 4.44 2.030.0 1.62 0.28 0.79 0.230.0 0.0 0.00.080.485 0.350 0.146
Advo[lo ≤8/255]at σ =0.25 at σ = 0.50 at σ =0.75at g =0.25 at σ = 0.50 at σ =0.7566.43 53.65 39.9655.06 42.91 30.7642.86 32.58 22.010.0 22.68 14.640.0 14.24 8.840.0 7.88 4.810.0 2.940.0 0.0 1.150.527 0.515 0.352
Advoo[∞ ≤12/255]atσ =0.25 at σ = 0.50 at σ = 0.75at σ= 0.25 at σ = 0.50 atσ=0.7560.52 51.53 42.6152.42 43.94 35.5643.27 36.41 28.470.0 28.69 22.390.0 21.25 16.690.0 14.532.29 0.0 8.030.0 0.0 4.430.499 0.581 0.482
Adv[l∞o ≤ 16/255]at σ =0.25 at σ = 0.50 at σ =0.75atg=0.25 at σ = 0.50 atσ =0.7553.75 48.07 42.05 67.9647.57 42.51 36.42 55.0641.18 36.54 31.24 43.270.0 30.55 26.05 30.550.0 24.68 20.7411.58 0.0 18.49 16.15 12.017.42 0.0 12.110.0 0.0 8.450.454 0.598 0.557 0.903
Advo + adapt [Best Radii] (Ours) 24.6818.49 12.11 8.45 0.0
Adv2[l2 ≤0.50]at σ = 0.25 at σ = 0.50 atg =0.75at σ = 0.25 at σ = 0.50 atσ =0.7568.84 48.81 27.3854.04 33.82 16.1537.13 20.95 9.230.0 11.5 4.560.0 5.64 2.060.0 2.29 0.62 0.91 0.330.0 0.0 0.080.518 0.382 0.153
Adv2[l2 ≤ 1.00]at σ = 0.25 atσ =0.50 atσ =0.75at σ = 0.25 at σ = 0.50 atg =0.7568.02 56.45 43.0458.54 46.24 33.0846.98 35.6 24.810.0 26.89 17.680.0 18.73 11.390.0 11.37 6.60.0 5.41 3.570.0 0.0 1.950.551 0.580 0.405
Adv2[l2 ≤1.25]at g =0.25 at σ =0.50 atσ =0.75at σ= 0.25 at σ = 0.50 at σ = 0.7567.13 57.73 46.5458.77 48.8 37.5349.43 39.64 29.350.0 31.07 22.00.0 22.61 15.620.0 15.82 10.510.0 8.96 6.550.0 0.0 3.680.557 0.647 0.496
Adv2[l2 ≤ 1.50]at σ = 0.25 at σ =0.50 at σ=0.75at g = 0.25 at σ = 0.50 atg =0.7564.21 56.55 47.7357.13 49.19 40.8949.71 41.72 33.780.0 34.47 27.220.0 27.36 20.780.0 20.23 15.190.0 12.98 10.510.0 0.0 6.770.543 0.689 0.585
Adv2[l2 ≤ 2.00]at σ= 0.25 at σ =0.50 at σ =0.75at σ = 0.25 at σ = 0.50 at σ = 0.7560.4 54.27 47.9654.71 48.89 42.5448.35 43.1 37.040.0 37.34 31.650.0 31.52 26.170.0 25.74 21.120.0 19.14 16.590.0 0.0 12.440.523 0.731 0.698
Adv2[l2 ≤ 2.25]at σ = 0.25 at σ = 0.50 at σ = 0.75atg=0.25 at σ = 0.50 at σ = 0.7557.08 52.1 46.4552.5 46.99 41.7147.11 42.26 36.750.0 36.9 31.880.0 31.58 26.950.0 26.08 22.330.0 20.03 17.820.0 0.0 13.550.504 0.724 0.713
Adv2[l2 ≤ 2.50]at σ =0.25 at σ = 0.50 atσ=0.75at σ = 0.25 at σ = 0.50 at σ =0.7554.88 50.53 45.9550.79 46.26 41.8946.29 41.84 37.530.0 37.74 33.550.0 33.2 29.310.0 28.69 25.270.0 23.34 20.980.0 0.0 17.280.487 0.734 0.765
Adv2[2 ≤ 3.00]at σ =0.25 atσ =0.50 at σ = 0.75at σ =0.25 at σ = 0.50 atσ =0.7553.82 49.41 45.3749.69 45.57 41.5445.04 41.52 37.750.0 37.43 33.490.0 33.37 29.350.0 28.82 25.620.0 23.65 21.830.0 0.0 18.230.475 0.720 0.771 1.198
Adv2+adapt[Best Radii](Ours) 68.84 58.77
Randg=0.5at g= 0.50 at σ = 0.2551.68 62.9140.38 52.2549.71 30.25 40.0637.74 20.81 0.033.37 13.36 0.028.82 7.71 0.023.65 3.38 0.018.23 0.0 0.00.488 0.497 0.575
at σ = 0.25 + adaptation at σ =0.50 at σ = 0.50
Randg=0.5 +adapt[Best Radi] (Ours)atσ =0.75at σ =0.7557.58 46.4 62.9146.46 35.63 52.2535.5 26.06 40.0625.57 18.17 25.5717.43 11.61 17.4310.67 5.46 6.86 3.64 10.67 5.460.0 1.92 1.920.427 0.657 0.609
SmoothAdvg=0.5,=0.25 at g= 0.50
+ adaptationat σ = 0.25 at σ = 0.50at σ = 0.25 at g = 0.5057.8 58.74 54.047.63 48.29 42.9137.41 36.7 32.6327.88 0.0 23.620.33 13.53 0.0 16.088.03 0.0 0.0 9.93 5.50.0 0.0 0.00.464 0.535 0.390
SmoothAdvg=0.5,∈=0.50at σ = 0.75 at σ = 0.25at σ = 0.75 atσ =0.50 at σ = 0.2543.15 58.82 59.8932.29 49.68 50.423.61 40.35 39.9916.41 31.93 0.010.94 24.18 0.06.65 17.05 0.03.88 10.57 0.02.16 0.0 0.00.661 0.483 0.592
+ adaptation
SmoothAdvg=0.5,∈=1.0at σ = 0.50 at σ = 0.75at σ = 0.50 at σ =0.75 atg=0.5055.73 46.25 56.5345.79 36.72 49.5336.6 28.2 41.3827.4 20.9 34.6320.03 14.73 27.8113.48 9.46 21.227.85 5.78 14.410.0 3.32 0.00.470 0.691 0.467
+ adaptation- at σ = 0.25 at σ = 0.50at σ = 0.25 at σ = 0.50 at σ = 0.7557.13 53.56 47.1748.38 45.79 39.439.7 37.640.0 30.060.0 23.12 18.930.0 17.27 13.70.0 11.14 9.260.0 0.0 5.80.620 0.545
at σ = 0.75
SmoothAdvg=0.5,∈=2.0at σ = 0.25 at σ =0.50atg =0.50 at σ = 0.2552.82 52.2347.67 47.2431.8 42.68 41.7624.74 37.55 0.032.64 0.027.52 0.022.42 0.00.0 0.00.732 0.451 0.692
+ adaptation at σ =0.75at σ = 0.50 atσ =0.7550.28 46.745.24 41.7940.39 37.1435.5 33.0530.92 28.2826.1 23.8820.25 19.340.0 15.050.727
SmoothAdvg=0.5[BestRadii]58.82 59.8949.68 50.442.68 41.7637.55 35.532.64 30.9227.52 26.122.42 20.250.0 15.050.918 1.008
SmoothAdvg=0.5 + adapt [Best Radii] (Ours) MARCERg=0.5 (Zhai et al.,2020) 60.0 53.0
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#sampled answers: 1#sampled answers: 200Retrieval performance @50
DatasetSOTACBOBognkingCBOBGoldOBeOBogre
NQ55.9a21.723.125.861.732.738.485.0
HOTPOTQA65.2b20.724.521.254.826.330.355.5
FEVER73.2°44.552.244.566.6e52.057.243.3
STRATEGYQA63.6d61.061.161.080.464.666.234.9
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GENERATIVE MODELS READY FOR IMAGE RECOGNITION? ", + "text_level": 1, + "bbox": [ + 176, + 99, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Ruifei $\\mathbf { H e } ^ { 1 ^ { * } }$ Shuyang $\\mathbf { S u n } ^ { 2 }$ Xin $\\mathbf { V } \\mathbf { u } ^ { 1 }$ Chuhui $\\mathbf { X } \\mathbf { u } \\mathbf { e } ^ { 3 }$ Wenqing Zhang3 Philip Torr2 \nSong Bai3† Xiaojuan $\\mathbf { Q } \\mathbf { i } ^ { \\mathrm { { i } \\dagger } }$ \n1The University of Hong Kong 2University of Oxford 3ByteDance ", + "bbox": [ + 187, + 167, + 802, + 214 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 228, + 544, + 244 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Recent text-to-image generation models have shown promising results in generating high-fidelity photo-realistic images. Though the results are astonishing to human eyes, how applicable these generated images are for recognition tasks remains under-explored. In this work, we extensively study whether and how synthetic images generated from state-of-the-art text-to-image generation models can be used for image recognition tasks, and focus on two perspectives: synthetic data for improving classification models in data-scarce settings (i.e. zero-shot and fewshot), and synthetic data for large-scale model pre-training for transfer learning. We showcase the powerfulness and shortcomings of synthetic data from existing generative models, and propose strategies for better applying synthetic data for recognition tasks. Code: https://github.com/CVMI-Lab/SyntheticData. ", + "bbox": [ + 233, + 258, + 764, + 411 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 428, + 336, + 444 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Over the past decade, deep learning powered by large-scale annotated data has revolutionized the field of image recognition. However, it is costly and time-consuming to manually collect a largescale labeled dataset, and recent concerns about data privacy and usage rights further hinder this process. In parallel, generative models that aim to model real-data distributions can now produce high-fidelity photo-realistic images. In particular, recent text-to-image generation models (Nichol et al., 2021; Ramesh et al., 2022; Saharia et al., 2022b) have made major breakthroughs in synthesizing high-quality images from text descriptions. This promotes us to ask: is synthetic data from generative models ready for image recognition tasks? ", + "bbox": [ + 174, + 459, + 825, + 570 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "There are a few early attempts at exploring synthetic data from generative models for image recognition tasks. Besnier et al. (2020) use a class-conditional GAN (BigGAN (Brock et al., 2018) trained for ImageNet-1000 classes) to generate images for training image classifiers. Zhang et al. (2021) leverage StyleGAN (Karras et al., 2019) to produce synthetic labeled data for object-part segmentation. Jahanian et al. (2021) manipulate the latent space of a GAN model to produce multi-view images for contrastive learning. Albeit promising, early works either address tasks on a small scale or only for a specific setting. Plus, they all focus on GAN-based models and none explore the revolutionary text-to-image generation models, which hold more promises to benefit recognition tasks. ", + "bbox": [ + 174, + 577, + 825, + 689 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "In this paper, we present the first study on the state-of-the-art text-to-image generation models for image recognition. With the power of text-to-image generation, we could hopefully not only generate massive high-quality labeled data, but also achieve domain customization by generating synthetic data targeted for a specific label space, i.e. the label space of a downstream task. Our study is carried out on one open-sourced text-to-image generation model, GLIDE (Nichol et al., 2021) 1. We attempt to uncover the benefits and pitfalls of synthetic data for image recognition through the lens of investigating the following two questions: 1) is synthetic data from generative models ready for improving classification models? 2) whether synthetic data can be a feasible source for transfer learning (i.e. model pre-training)? It is worth noting that for 1), we only studied the zero-shot and few-shot settings because the positive impact of synthetic data diminishes as more shots are present. And, we build most of our investigations on the state-of-the-art method CLIP (Radford et al., 2021) with the feature extractor initialized with large-scale pre-trained weights frozen. ", + "bbox": [ + 174, + 695, + 825, + 862 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Our Findings. First, in the zero-shot setting, i.e. no real-world data are available, we demonstrate that synthetic data can significantly improve classification results on 17 diverse datasets: the performance is increased by ${ \\mathrm { 4 . 3 1 \\% } }$ in top-1 accuracy on average, and even improved by as much as $1 7 . 8 6 \\%$ on the EuroSAT dataset. To better leverage synthetic data in this setting, we also investigate useful strategies to increase data diversity, reduce data noise, and enhance data reliability. This is achieved by designing diversified text prompts and measuring the correlation of text and synthesized data with CLIP features. ", + "bbox": [ + 174, + 103, + 825, + 200 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Second, in the few-shot setting, i.e. a few real images are available, albeit not as significant as in the zero-shot task, synthetic data are also shown to be beneficial and help us achieve a new state of the art. Our observation shows that the domain gap between synthetic data and downstream task data is one challenge on further improving the effectiveness of synthetic data on classifier learning. Fortunately, in this setting, the accessibility of real data samples can provide useful information about the data distribution of the downstream task. We thus propose to use real images as guidance in the generation process to reduce domain gaps and improve effectiveness. ", + "bbox": [ + 174, + 208, + 825, + 305 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Third, in large-scale model pre-training for transfer learning, our study shows that synthetic data are suitable and effective for model pre-training, delivering superior transfer learning performance and even outperforming ImageNet pre-training. Especially, synthetic data work surprisingly well in unsupervised model pre-training, and favor ViT-based backbones. We also demonstrate that by increasing the label space (i.e. text prompts) for data generation, the enlarged data amount and diversity could further bring performance boosts. Besides, synthetic data can work collaboratively with real data (i.e. ImageNet) where we obtain improved performance when the model is initialized with ImageNet pre-trained weights. ", + "bbox": [ + 174, + 313, + 825, + 424 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "2 RELATED WORKS ", + "text_level": 1, + "bbox": [ + 176, + 440, + 349, + 457 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Synthetic Data for Image Recognition. There are mainly two forms of synthetic data for image recognition, i.e. 1) synthetic datasets generated from a traditional simulation pipeline; 2) synthetic images output from generative models. ", + "bbox": [ + 174, + 472, + 823, + 513 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "The first type, synthetic datasets (Dosovitskiy et al., 2015; Peng et al., 2017; Richter et al., 2016), are usually generated from a traditional pipeline with a specific data source, e.g.synthetic 2D renderings of 3D models or scenes from graphics engines. However, this traditional way of generating synthetic datasets has several drawbacks: 1) manually defined pipeline generated synthetic data may have a certain gap with real-world data; 2) taking up huge physical space to store and huge cost to share and transfer; 3) data amount and diversity bounded by the specific data source. ", + "bbox": [ + 174, + 520, + 825, + 603 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Compared with synthetic datasets, generative models are a more efficient means of synthetic data representation, exhibiting favorable advantages: 1) could produce high-fidelity photorealistic images closer to real data since they are trained on real-world data; 2) highly condensed compared to synthetic data itself, and take up much reduced storage space; 3) potentially unlimited synthetic data size. Only recently, few works attempt to explore synthetic data generated from generative models for image recognition. Besnier et al. (2020) use a class-conditional GAN to train classifiers of the same classes. Zhang et al. (2021) leverage the latent code of StyleGAN (Karras et al., 2019) to produce labels for object part segmentation. While they achieve promising results, both works are task-wise and only employed on a small scale. Jahanian et al. (2021) use a GAN-based generator to generate multiple views to conduct unsupervised contrastive representation learning. These works, however, explore upon the traditional GAN-based models; in contrast, our work investigates with the best released text-to-image generation model, which demonstrates new customization ability for different downstream label space. ", + "bbox": [ + 173, + 611, + 825, + 791 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Text-to-Image Diffusion Models. Diffusion models (Sohl-Dickstein et al., 2015; Ho et al., 2020; Nichol & Dhariwal, 2021) have recently emerged as a class of promising and powerful generative models. As a likelihood-based model, the diffusion model matches the underlying data distribution $q ( x _ { 0 } )$ by learning to reverse a noising process, and thus novel images can be sampled from a prior Gaussian distribution via the learned reverse path. Because of the high sample quality, good mode coverage and promising training stability, diffusion models are quickly becoming a new trend in both unconditional (Ho et al., 2020; Nichol & Dhariwal, 2021; Ho et al., 2022) and conditional (Dhariwal & Nichol, 2021; Rombach et al., 2022; Lugmayr et al., 2022; Saharia et al., 2022a; Meng et al., 2021; Saharia et al., 2022c) image synthesis fields. ", + "bbox": [ + 174, + 797, + 825, + 924 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "In particular, text-to-image generation can be treated as a conditional image generation task that requires the sampled image to match the given natural language description. Based upon the formulation of the diffusion model, several text-to-image models such as Stable diffusion (Rombach et al., 2022), DALL-E2 (Ramesh et al., 2022), Imagen (Saharia et al., 2022b) and GLIDE (Nichol et al., 2021) deliver unprecedented synthesis quality, largely facilitating the development of the AIfor-Art community. Despite achieving astonishing perceptual results, their potential utilization for high-level tasks is yet under-explored. In this paper, we utilize the state-of-the-art model GLIDE and showcase its powerfulness and shortcomings for synthesizing data for recognition tasks. ", + "bbox": [ + 173, + 103, + 825, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3 IS SYNTHETIC DATA READY FOR IMAGE RECOGNITION? ", + "text_level": 1, + "bbox": [ + 178, + 231, + 665, + 246 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "In the following sections, we answer the question by studying whether synthetic data can benefit recognition tasks and how to better leverage synthetic data to address different tasks. We carry out our exploration through the lens of two basic settings with three tasks: synthetic data for improving classification models in the data-scarce setting (i.e. zero-shot and few-shot) (see Sec. 3.1 and Sec. 3.2) and synthetic data for model pre-training for transfer learning (see Sec. 3.3). ", + "bbox": [ + 174, + 257, + 825, + 327 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Model Setup for Data-scarce (i.e. Zero-shot and Few-shot) Image Classification. As CLIP (Radford et al., 2021) is the state-of-the-art approach for zero-shot learning, we conduct our study for zero-shot and few-shot settings upon pre-trained CLIP models, aiming to better understand synthetic data upon strong baselines. There have been a few attempts on better tuning pre-trained CLIP for data-scarce image classification, such as CoOp (Zhou et al., 2022b), CLIP Adapter (Gao et al., 2021), and Tip Adapter (Zhang et al., 2022), where the image encoder is frozen for better preserving the pretrained feature space. We argue that different tuning methods could all be regarded as different ways of learning classifier weights, e.g. CoOp optimizes learnable prompts for better learning classifiers. ", + "bbox": [ + 174, + 334, + 825, + 445 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Here, we adopt a simple tuning method, Classifier Tuning (CT), a baseline method introduced in Wortsman et al. (2022). Concretely, for a $\\mathbf { k }$ -way classification, we input the class names $C =$ $\\{ c _ { 1 } , . . . , c _ { k } \\}$ with prompt $s _ { i } =$ “a photo of a $\\{ c _ { i } \\} ^ { \\flat }$ into the text encoder $h$ of CLIP to obtain the text features $h ( s _ { i } )$ . Then the text features $h ( s _ { i } )$ could be used to construct classifier weights $W \\in$ $R ^ { d \\times k }$ , where $d$ is the dimension of text features. Finally, we combine the image encoder $g$ with the classifier weights $W$ to obtain a classification model $f ( x ) = g ( x ) ^ { \\mathrm { T } } W$ . We empirically show that CT performs comparably with other tuning methods. Compared with complex designed tuning methods, we hope to use a simpler method for better investigating the effectiveness of synthetic data. ", + "bbox": [ + 174, + 452, + 825, + 564 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": ".1 IS SYNTHETIC DATA READY FOR ZERO-SHOT IMAGE RECOGNITION? ", + "bbox": [ + 186, + 579, + 676, + 593 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Our aim is to investigate to what degree synthetic data are beneficial to zero-shot tasks and how to better leverage synthetic data for zero-shot learning. ", + "bbox": [ + 173, + 602, + 821, + 631 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Zero-shot Image Recognition. We study the inductive zero-shot learning setting where no real training images of the target categories are available. CLIP models are pre-trained with large-scale image-caption pairs, and the similarities between paired image features (from an image-encoder $g$ ) and text features (from a text-encoder $h$ ) are maximized during pre-training. The pre-trained feature extractor can then be used to solve zero-shot tasks where given an image, its features from $g$ are compared with text features of different classes from $h$ and the image is further assigned to the class that has the largest similarity in the CLIP text-image feature space. ", + "bbox": [ + 174, + 637, + 825, + 734 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Synthetic Data for Zero-shot Image Recognition. Though CLIP models exhibit strong zero-shot performance thanks to the large-scale vision-language dataset for pre-training, there are still several shortcomings when the model is deployed for a downstream zero-shot classification task, which may be attributed to unavoidable data noise in CLIP’s pre-training data or the label space mismatch between pre-training and the zero-shot task. Hence, with a given label space for a zero-shot task, we study whether synthetic data can be used to better adapt CLIP models for zero-shot learning. ", + "bbox": [ + 174, + 742, + 825, + 825 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "How to generate the data? Given a pre-trained text-to-image generation model, to synthesize novel samples, the basic $\\mathbf { ( B ) }$ strategy is to use the label names of the target categories to build the language input and generate a corresponding image. Then, the paired label names and synthesized data can be employed to train the classifier with the feature extractor frozen. ", + "bbox": [ + 176, + 833, + 823, + 888 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "How to enrich diversity? Only using the label names as inputs might limit the diversity of synthesized images and cause bottlenecks for validating the effectiveness of synthetic data. Hence, we leverage an off-the-shelf word-to-sentence T5 model (pre-trained on “Colossal Clean Crawled Corpus” dataset (Raffel et al., 2020) and finetuned on CommonGen dataset (Lin et al., 2019)) to increase the diversity of language prompts and the generated images, namely language enhancement (LE), hoping to better unleash the potential of synthesized data. Concretely, we input the label name of each class to the word-to-sentence model which generates diversified sentences containing the class names as language prompts for the text-to-image generation process. For example, if the class label is “airplane”, then the enhanced language prompt from the model could be “a white airplane hovering over a beach and a city”. The enhanced text descriptions introduce rich context descriptions. ", + "bbox": [ + 176, + 895, + 821, + 924 + ], + "page_idx": 2 + }, + { + "type": "table", + "img_path": "images/128b59a404fd74f3cb60fc6bb12e5929b0861815894b6da5d9382db8ac821543.jpg", + "table_caption": [], + "table_footnote": [ + "Table 1: Main Results on Zero-shot Image Recognition. All results are top-1 accuracy on test set. o: object-level. s: scene-level. f: fine-grained. t: textures. si: satellite images. r: robustness. " + ], + "table_body": "
DatasetTaskCLIP-RN50CLIP-RN50+SYNCLIP-ViT-B/16CLIP-ViT-B/16+SYN
CIFAR-10070.3180.06 (+9.75)90.8092.37 (+1.57)
CIFAR-100035.3545.69 (+10.34)68.2270.71 (+2.49)
Caltech101086.0987.74 (+1.65)92.9894.16 (+1.18)
Caltech256073.3675.74 (+2.38)80.1481.43 (+1.29)
ImageNet060.3360.78 (+0.45)68.7569.16 (+0.41)
SUN397S58.5160.07 (+1.56)62.5163.79 (+1.28)
Aircraftf17.3421.94 (+4.60)24.8130.78 (+5.97)
Birdsnapf34.3338.05 (+3.72)41.9046.84 (+4.94)
Carsf55.6356.93 (+1.30)65.2366.86 (+1.63)
CUBf46.6956.94 (+10.25)55.2363.79 (+8.56)
Flowerf66.0867.05 (+0.97)71.3072.60 (+1.30)
Foodf80.3480.35 (+0.01)88.7588.83 (+0.08)
Petsf85.8086.81 (+1.01)89.1090.41 (+1.31)
DTDt42.2343.19 (+0.96)44.3944.92 (+0.53)
EuroSATsi37.5155.37 (+17.86)47.7759.86 (+12.09)
ImageNet-Sketchr33.2936.55 (+3.26)46.2048.47 7 (+2.27)
ImageNet-Rr56.1659.37 (+3.21)74.0176.41 (+2.40)
Average/55.1359.47 (+4.31)65.4268.32 (+2.90)
", + "bbox": [ + 176, + 74, + 815, + 318 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 353, + 825, + 464 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "How to reduce noise and enhance robustness? It’s unavoidable that the synthesized data may contain low-quality samples. This is even more severe in the setting with language enhancement as it may introduce undesired items into language prompts (see Figure A.2 in Appendix for visual examples). Hence, we introduce a CLIP Filter (CF) strategy to rule out these samples. Specifically, CLIP zero-shot classification confidence is used to assess the quality of synthesized data, and the lowconfidence ones are removed. Besides, as soft-target is more robust than hard-target in countering sample noise, we study whether soft cross-entropy loss (SCE, see Sec. C.4 in Appendix) which uses the normalized clip scores as a target could be used to enhance robustness against data noise. ", + "bbox": [ + 174, + 472, + 825, + 583 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Experiment Setup. We select 17 diverse datasets covering object-level (CIFAR-10 and CIFAR-100 ((Krizhevsky et al., 2009), Caltech101 (Fei-Fei et al., 2006), Caltech256 (Griffin et al., 2007), ImageNet (Deng et al., 2009)), scene-level (SUN397 (Xiao et al., 2010)), fine-grained (Aircraft (Maji et al., 2013), Birdsnap (Berg et al., 2014), Cars (Krause et al., 2013), CUB (Wah et al., 2011), Flower (Nilsback & Zisserman, 2008), Food (Bossard et al., 2014), Pets (Parkhi et al., 2012)), textures (DTD (Cimpoi et al., 2014)), satelite images (EuroSAT (Helber et al., 2019)) and robustness (ImageNetSketch (Wang et al., 2019), ImageNet-R (Hendrycks et al., 2021)) for zero-shot image classification. For synthetic data amount, we generate 2000 (study of synthetic image number in Appendix Sec. B.3) synthetic images for each class in B and LE. For LE, we generate 200 sentences for each class. ", + "bbox": [ + 173, + 589, + 825, + 715 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Main Results: 1) zero-shot classification results on 17 datasets; 2) study of synthetic data diversity; 3) study of synthetic data reliability; 4) study of model/classifier tuning; 5) study of the behavior of synthetic data for zero-shot classification in the training from scratch settings. ", + "bbox": [ + 174, + 722, + 825, + 763 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Synthetic data can significantly improve the performance of zero-shot learning. Our main studies in zero-shot settings are conducted with CLIP-RN50 (ResNet-50 (He et al., 2016) and CLIP-ViTB/16 (ViT-B/16 (Dosovitskiy et al., 2020)) as CLIP backbone), and we report results with our best strategy of $\\mathbf { L E + C F + S C E }$ . As shown in Table 1, on 17 diverse downstream zero-shot image classification datasets, we achieve a remarkable average gain of $4 . 3 1 \\%$ for CLIP-RN50 and $2 . 9 0 \\%$ for CLIP-ViT-B/16 in terms of top-1 accuracy. Significantly, on the EuroSAT dataset, we achieve the largest performance boost of $1 7 . 8 6 \\%$ for CLIP-RN50 in top-1 accuracy. We notice that the performance gain brought by synthetic data varies differently across datasets, which is mainly related to GLIDE’s training data distribution. The training data distribution of the text-to-image generation model GLIDE would exhibit bias and produce different domain gaps with different datasets (see Sec. A.2 in Appendix for more analysis). ", + "bbox": [ + 174, + 771, + 825, + 924 + ], + "page_idx": 3 + }, + { + "type": "table", + "img_path": "images/a4884469592b8066e5200369cb9a3809d9afd257c1645714bc7a2441691b0850.jpg", + "table_caption": [ + "Table 2: Ablation study on Language Enhancement (LE), CLIP-based Filtering (CF), and Softtarget Cross-Entropy (SCE). " + ], + "table_footnote": [], + "table_body": "
DatasetCLIPBLELE+CF
CESCECESCECESCE
CIFAR-1070.3177.39 (+7.08)78.23 (+7.92)77.20 (+6.89)77.55 (+7.24)80.01 (+9.70)80.06 (+9.75)
CIFAR-10035.3543.99 (+8.64)44.25 (+8.90)44.08 (+8.73)44.91 (+9.56)44.55 (+9.20)45.69 (+10.34)
EuroSAT37.5145.64 (+8.13)48.23 (+10.72)53.26 (+15.75)54.94 (+17.43)54.75 (+17.24)55.37 (+17.86)
", + "bbox": [ + 173, + 69, + 843, + 137 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Language diversity matters. By introducing more linguistic context into the text input, LE helps increase the diversity of synthetic data. As shown in Table 2, LE can achieve additional performance gains upon $\\mathbf { B }$ in most cases $_ { ( 0 . 6 6 \\uparrow }$ on CIFAR-100, $6 . 7 1 \\uparrow$ on EuroSAT), which demonstrates the efficacy of LE and the importance of synthetic data diversity for zero-shot classification. ", + "bbox": [ + 174, + 188, + 825, + 243 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Reliability matters. While LE could help increase the diversity of synthetic data, it also introduces the risks of noisy samples. Observed on CIFAR-10 in Table 2, LE sometimes even brings performance drops compared with B $( 0 . 6 8 \\% \\downarrow$ on CIFAR-10), which may attribute to the noise introduced by enhanced language prompts, e.g. the sentence extended from the class name word may contain other class names or confusing objects. Fortunately, with CF to filter out unreliable samples, $\\mathbf { L E + C F }$ yields consistent improvement upon B. Moreover, SCE generally achieves better performance than CE, showing its better adaptation to label noise. ", + "bbox": [ + 174, + 251, + 825, + 348 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Classifier tuning is enough for CLIP, while tuning with the pre-trained encoder leads to degradation, mainly due to domain gaps. Here, we investigate if only tuning the final classifier is the optimal solution in our setting with synthetic data. As shown in Table 3, we tune different proportions of the full model parameters on synthetic data for EuroSAT ( $0 . 0 2 \\%$ corresponds to our default case where only the classifier is tuned), and report the zero-shot performance on the test set of EuroSAT. The best results are obtained by only tuning the classifier, and the performance gradually decreases as we gradually incorporate more parameters in the encoder for optimization, which agrees with the traditional strategy. For understanding why synthetic data may harm pre-trained image encoder, we experiment with real-world data with domain shifts and find they behave similarly to synthetic data (Appendix Sec. B.2), which suggests that domain gap is the main reason for the phenomenon. We argue that synthetic data might have a better chance to overcome domain shifts in comparison with real-world data since we can customize and keep the label space of the synthetic data in line with the down-stream dataset and use strategies during synthesizing to alleviate domain shifts. ", + "bbox": [ + 173, + 354, + 825, + 535 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/f1d9185204e83a6472c64d64bed78a5d94b9ef4eacd29c4af7503a29f55be6d8.jpg", + "table_caption": [], + "table_footnote": [], + "table_body": "
Param Tuned (%)00.020.0462.5064.0669.5382.8192.19
Acc37.5155.3755.1155.2854.5654.3453.6352.09
", + "bbox": [ + 176, + 542, + 812, + 573 + ], + "page_idx": 4 + }, + { + "type": "table", + "img_path": "images/28591773c011f66285c45a73b648c53cb6378202ca477345fef14b7a93f170a8.jpg", + "table_caption": [ + "Table 3: Parameters tuned v.s. Accuracy. Dataset: EuroSAT. ", + "Table 4: Setting when training from scratch. Dataset: CIFAR-100. " + ], + "table_footnote": [], + "table_body": "
Real shot1163264809095100
Acc2.4810.414.9521.9624.425.5227.9929.95
", + "bbox": [ + 191, + 594, + 799, + 625 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Synthetic data deliver inferior performance in the training from scratch setting and are much less data-efficient than real data. To exclude the influence of powerful CLIP initialization in our study of synthetic data, we also conduct a from-scratch setting on the CIFAR-100 dataset, where we optimize a ResNet-50 model from random initialization. Given the label space of the CIFAR-100 dataset, we generate a synthetic dataset of $5 0 \\mathrm { k }$ (500 images per class) to train a ResNet-50 model from scratch for image classification. We achieve a performance of $2 8 . 7 4 \\%$ top-1 accuracy on CIFAR-100 test set, which is much lower than the performance of the pre-trained CLIP model (see Table 1). This might be attributed to the quality and diversity of data. The CLIP model benefits from diverse realworld data. Further, we hope to investigate how many real in-domain training data can match the performance of our $5 0 \\mathrm { k }$ synthetic data. As shown in Table 4, training with 95 images per category $( 9 5 \\times 1 0 0 = 9 . 5 { \\mathrm { k } } )$ ) will achieve comparable performance as that of $5 0 \\mathrm { k }$ synthetic data. This manifests that synthetic data are not as efficient and effective as real data when solving downstream tasks. It requires around 5 times more data in order to achieve a comparable performance as that of real data. Note that we find further increasing the amount of synthetic data will not deliver further performance gains for the downstream classification task. We expect that further investigations on synthesis quality will bring new opportunities in this area which will be our future work. ", + "bbox": [ + 173, + 654, + 825, + 875 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Summary. Current synthetic data from text-to-image generation models could indeed bring significant performance boosts for a wide range of zero-shot image classification tasks, and is readily applicable with carefully designed strategies such as large-scale pre-trained models. Diversity and reliability matter for synthetic data when employed for zero-shot tasks. When the model is trained from scratch with synthetic data, synthetic data cannot deliver satisfactory performance and are much less data-efficient and effective for solving the classification task in comparison with real data. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.2 IS SYNTHETIC DATA READY FOR FEW-SHOT IMAGE RECOGNITION? ", + "bbox": [ + 176, + 156, + 679, + 170 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "In this section, we explore the effectiveness of synthetic data for few-shot tasks and how synthetic data impact the performance as more and more shots are included. Also, we design effective strategies to better leverage synthetic data. ", + "bbox": [ + 176, + 178, + 823, + 220 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Few-shot Image Recognition. We adopt the CLIP-based method as the model for few-shot image recognition due to its state-of-the-art performance (Radford et al., 2021). As discussed previously, various prompt learning based methods can be treated as tuning the classifier weights. We thus study how to tune the classifier weights with synthetic data. In an N-way M-shot case, we are given M real images of each test class, where $\\mathbf { M } \\in \\left\\{ 1 , 2 , 4 , 8 , 1 6 \\right\\}$ in our experiments. With a total of $\\mathbf { N } \\times \\mathbf { M }$ training samples, we hope to achieve favorable performance on a hold-out test set of the N classes. ", + "bbox": [ + 174, + 227, + 823, + 311 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Synthetic Data for Few-shot Image Recognition. While there have been a few attempts to study how to better adapt CLIP models for few-shot tasks (Zhou et al., 2022b;a; Zhang et al., 2022), they all focus on the model optimization level, and none have explored from the data level. Here, we systematically study whether and how synthetic data can be employed for solving few-shot image recognition tasks. ", + "bbox": [ + 174, + 318, + 823, + 387 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "With the experience from synthetic data for zero-shot tasks, we adopt the best strategy (i.e. $\\mathbf { L E { + } C F }$ ) in the zero-shot setting as the basic strategy $\\mathbf { \\delta } ( \\mathbf { B } )$ . Further, as the few-shot real samples can provide useful information on the data distribution of the classification task, we develop two new strategies leveraging the in-domain few-shot real data for better using synthetic data: 1) Real Filtering (RF): given synthetic data of one class $c$ , we use the features of few-shot real samples to filter out synthetic images whose features are very close to the features of real samples that belong to other categories different from class $c ; 2$ ) Real guidance (RG): we use the few-shot real samples as guidance to generate synthetic images where the few-shot real samples (added noise) replace the random noise at the beginning of the generation to guide the diffusion process (details in Appendix Sec. C.3). ", + "bbox": [ + 174, + 395, + 825, + 520 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Experiment Setup. For datasets, we carefully select 8 image classification datasets from recent works (Zhou et al., 2022b;a; Zhang et al., 2022) that conduct few-shot learning upon CLIP: ImageNet (Deng et al., 2009), Caltech101 (Fei-Fei et al., 2006), Pets (Parkhi et al., 2012), Cars (Krause et al., 2013), Aircraft (Maji et al., 2013), SUN397 (Xiao et al., 2010), DTD (Cimpoi et al., 2014), EuroSAT (Helber et al., 2019). For synthetic image number, we generate 800 (study of synthetic image number in Appendix Sec. B.3) images per class for RG method to approximately match the number of images in B and RF. ", + "bbox": [ + 174, + 526, + 825, + 625 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Main Results: 1) few-shot classification results on 8 datasets; 2) ablation study of training strategy; \n3) ablation study of synthetic data generation strategy; 4) ablation study of BN strategy. ", + "bbox": [ + 174, + 631, + 821, + 660 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Synthetic data can boost few-shot learning and the positive impact of synthetic data will gradually diminish with the increase of real data shots. As shown in Figure 1 (results of more datasets are in the Appendix Sec. B.1), with only few-shot real images for training, our implemented CT w. init (classifier weights initialized from CLIP text embeddings) performs comparably with the state-ofthe-art CLIP tuning methods Tip Adapter (Zhang et al., 2022) and CoOp (Zhou et al., 2022b). CT w. Syn represents our results of applying synthetic data with mix training, real image as guidance, and freezing BN strategies. With the help of generated synthetic data, CT w. Syn achieves noticeable performance gains upon CT w. init, and achieves a new state-of-the-art few-shot learning performance across different datasets. We argue that for data-scarce few-shot classification, synthetic data could help address the insufficient data problem to boost performance. However, we notice that the boost from synthetic data gradually diminishes as the real shot number increases. We state that the effectiveness of each sample in real data is high since there’s no domain gap; in contrast, synthetic data suffer from domain gaps and perform less efficiently. In addition, the positive effects of the few-shot real data may overlap with that of synthetic data. Thus, with the increase of real data, the overlapping becomes serious and the positive impacts of synthetic data are reduced. ", + "bbox": [ + 173, + 666, + 825, + 875 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Mix Training fits few-shot learning with synthetic data. Now that we have two parts of data, i.e. few-shot real data and synthetic data, we could either 1) phase-wise train on each part of data with two training phases, or 2) adopt mix training that simultaneously utilizes two parts of data to update the model in each iteration. Details of phase-wise/mix training in Appendix Sec. C.5.2. We provide the results in Table 7: we study on the EuroSAT dataset and use synthetic data generated from the RG method; under different shot number settings, mix training performs consistently better than two phase-wise strategies. We suggest that mix training could help learn better classifiers since each part could function as a regularization for the other: synthetic data help alleviate instabilities brought by limited real samples, and real data help address the noise and domain gap of synthetic data. ", + "bbox": [ + 176, + 882, + 823, + 924 + ], + "page_idx": 5 + }, + { + "type": "image", + "img_path": "images/852b48a16e7d1143e00de1a9683096c75aadd03f71f47f26762c38b612762b7e.jpg", + "image_caption": [ + "Figure 1: Results for few-shot image recognition. Results on all 8 datasets are provided in Appendix. " + ], + "image_footnote": [], + "bbox": [ + 173, + 68, + 825, + 229 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 261, + 825, + 344 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Employing real data as guidance can alleviate domain differences and boost performance. We compare three strategies of synthetic data generation for few-shot tasks. As shown in Table 5, both RF and RG provide performance gains upon B which is the best strategy in the zero-shot setting. This demonstrates the importance of utilizing the domain knowledge from few-shot images for preparing the synthetic data. Further, RG significantly outperforms RF, yielding the best performance. This shows utilizing real data as guidance of the diffusion process help reduce the domain gap (visual illustrations in the Appendix Sec. B.8). ", + "bbox": [ + 173, + 351, + 825, + 449 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/beba8a48c2072706761b622eacdc805aaa8e82eddf770763ab1cababc1b22777.jpg", + "table_caption": [ + "Table 5: Ablation for Basic strategy (B), Real Filtering (RF), Real Guidance (RG) on EuroSAT, 16 shot. " + ], + "table_footnote": [], + "table_body": "
BRFRG
87.187.3388.47
", + "bbox": [ + 186, + 469, + 318, + 505 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/d90166e7e5475cf62d980e2ecd2c7a3b0697770f3829a2613d60e910cc72a852.jpg", + "table_caption": [ + "Table 6: Frozen BN works better for 16-shot settings on EuroSAT. " + ], + "table_footnote": [], + "table_body": "
Train data Freeze BN? Test Acc
Real75.31
Real √ 85.63
Syn44.73
Syn √ 55.37
", + "bbox": [ + 352, + 467, + 562, + 549 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/1ba331c6fdef496710a1641d9f7346376782cb5ba077e895237ad94bbbca27ca.jpg", + "table_caption": [ + "Table 7: Mix training works better for few-shot tasks on EuroSAT. " + ], + "table_footnote": [], + "table_body": "
M-shotPhase-wise syn →real real -→ synMix training
163.0164.36
272.2473.62
478.8879.88
883.6484.57
1687.1088.47
", + "bbox": [ + 593, + 467, + 812, + 551 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Frozen BN works better. Lastly, we investigate batch normalization (BN) strategies for our fewshot settings with synthetic data. As shown in Table 6, for both real and synthetic data, freezing the BN layers yields much better performance. We analyze that for real data, it is hard to get a good estimation of BN statistics when the number of images is limited. As for synthetic data, we attribute this to the statistical difference between different domains. Hence, we freeze BN layers during tuning for few-shot settings. ", + "bbox": [ + 173, + 598, + 825, + 683 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Summary. Synthetic data from text-to-image generation models could readily benefit few-shot learning and achieve a new state-of-the-art few-shot classification performance with strategies we present in this paper. However, the positive impact of synthetic data will diminish as more shots of real data are available which further confirms our previous claim that synthetic data are still not as effective as real data in training classification models. ", + "bbox": [ + 174, + 689, + 825, + 758 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.3 IS SYNTHETIC DATA READY FOR PRE-TRAINING? ", + "text_level": 1, + "bbox": [ + 174, + 771, + 555, + 785 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Finally, we study whether synthetic data are effective in large-scale pre-training. We also present effective strategies to better leverage synthetic data for model pre-training. ", + "bbox": [ + 173, + 791, + 823, + 820 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Pre-training for Transfer Learning. Recently, it has become a common practice to first pre-train models on large-scale datasets to obtain a well-trained feature extractor and then fine-tune the models on downstream tasks with labeled data (i.e. transfer learning). There have been various successful pre-training methods, including supervised pre-training (Joulin et al., 2016; Li et al., 2017; Mahajan et al., 2018; Sun et al., 2017; Kolesnikov et al., 2020), self-supervised pre-training (Chen et al., 2020a; He et al., 2020; Caron et al., 2020; Grill et al., 2020; Chen & He, 2021; Zbontar et al., 2021; Ye et al., 2019), and semi-supervised pre-training (Xie et al., 2020; Pham et al., 2021). ", + "bbox": [ + 173, + 825, + 825, + 924 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Synthetic data for Pre-training. Since data amount and diversity play important roles in pretraining, we adopt the synthetic data generation strategy LE solely to maximize the scale of synthetic pre-training data. We study two settings for generating synthetic data for pre-training: 1) downstream-aware, where we have access to the label space of the downstream task, and thus we generate synthetic data according to the label space of the downstream task; 2) downstream-agnostic, where we have no access to downstream tasks in the pre-training stage, and we turn to a relatively general and diverse label space such as ImageNet-1K. For pre-training methods, we experiment with supervised pre-training and self-supervised pre-training methods. ", + "bbox": [ + 174, + 103, + 825, + 215 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Experiment Setup. We compare synthetic pre-trained models with models of random initialization and models of ImageNet-1K pre-training in terms of their transfer learning abilities. For downstream-aware settings: we conduct supervised pre-training on synthetic data generated according to CIFAR-100 label space and then transfer to CIFAR-100 through finetuning for evaluation. ", + "bbox": [ + 174, + 222, + 825, + 279 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "For downstream-agnostic settings: we perform supervised pre-training and self-supervised pretraining (we adopt Moco v2 (Chen et al., 2020b) framework for its simplicity and reproducibility) on synthetic data generated from ImageNet-1K label space and evaluate the transfer performance by finetuning the pretrained models on a object detection dataset – PASCAL VOC (Everingham et al., 2010). Further, we experiment with ImageNet-2K label space (original ImageNet-1K and another non-overlapping 1K label names randomly selected from ImageNet-21K) to study the factors of data diversity and amount in synthetic pre-training. We use ResNet-50 as the default backbone when not else noted, and also experiment with a ViT-based backbone, i.e. DeiT-S (Touvron et al., 2021). ", + "bbox": [ + 174, + 285, + 825, + 396 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results for Downstream-aware settings. We generate synthetic data of different sizes from CIFAR-100 label space, i.e. $1 \\times$ , $2 \\times$ , $3 \\times$ ImageNet-1K data size, concretely 1.2M, 2.4M, 3.6M. We pre-train the model on the generated synthetic labeled set in a supervised manner, and then perform evaluation after finetuning the model on CIFAR-100. As shown in Table 8, with an equivalent amount of data as that of ImageNet-1K (1.2M), synthetic data for pre-training can largely reduce the gap between training from scratch $( 7 8 . 8 3 \\% )$ and ImageNet- pre-trained model $( 8 4 . 5 0 \\% )$ . Moreover, with $2 \\times$ and $3 \\times$ synthetic data, pre-training on synthetic data outperforms ImageNet-1K pre-training with a noticeable margin. In addition, when we initialize the model from ImageNet-1K pre-trained weights and pre-train the model on synthetic data, we obtain extra boosts upon both results. ", + "bbox": [ + 174, + 404, + 823, + 529 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We conclude that for downstream-aware synthetic pre-training, synthetic data deliver close performance as that of ImageNet-1K pretraining with the same amount of data, synthetic data amount helps improve the results to outperforming ImageNet-1K pre-training, and synthetic pre-training could further benefit from ImageNet-1K pre-training. ", + "bbox": [ + 174, + 535, + 825, + 592 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Results for Downstream-agnostic settings. We first experiment with ImageNet-1K label space with $1 \\times$ or $2 \\times$ ImageNet-1K data size, i.e. 1.2M/2.4M IN-1K Syn. We perform supervised pretraining and self-supervised pre-training (i.e. Moco v2) on the generated synthetic data, and evaluate the pre-training results by transferring to the CIFAR-100 image classification task or the PASCAL VOC detection task. As it is too costly to validate all settings (e.g., it takes more than 1 week to train Moco v2 on 4.0M synthetic data), we select several representative settings of interest to validate the effectiveness of synthetic data without hurting our conclusion. ", + "bbox": [ + 174, + 598, + 825, + 695 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "As shown in Table 10 and 11, with 1.2M IN-1K Syn, both supervised pre-training $( 7 9 . 0 0 \\% )$ and self-supervised pre-training $( 8 1 . 5 5 \\% )$ could largely approach their IN-1K Real counterparts (super.: $8 1 . 3 \\%$ ; self-super.: $8 2 . 4 4 \\%$ ) and largely outperforms the result without pre-training $( 6 6 . 0 8 \\% )$ . When increasing the data amount to 2.4M, the transferred results further increase, and the unsupervised pre-training method, i.e. Moco v2, performs better in utilizing our synthetic data thanks to its independence of labels, yielding a $8 2 . 1 3 \\%$ transferred performance which surpasses supervised pre-training on IN-1K Real $( 8 1 . 3 0 \\% )$ and is on par with its Moco v2 counterpart at IN-1K Real $( 8 2 . 4 4 \\% )$ . Next, we expand the label space by adding another 1K categories, producing IN-2K Syn. The enlarged diversity and data amount further bridge the gap between synthetic pre-training results and IN-1K Real pre-training results. Noticeably, the unsupervised pre-trained model Moco v2 $( 8 2 . 2 9 \\% )$ largely approaches the IN-1K Real counterpart $( 8 2 . 4 4 \\% )$ with negligible performance drop of $0 . 1 5 \\%$ . Furthermore, when initialized from IN-1K Real pre-trained weights, both supervised and self-supervised pre-training improve upon both pure real data and synthetic data for pre-training. ", + "bbox": [ + 174, + 702, + 825, + 883 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "While the above results are all obtained with convolutional-based backbone i.e. ResNet50, we further explore with a recent ViT-based backbone i.e. DeiT-S (Touvron et al., 2021). Surprisingly, ", + "bbox": [ + 174, + 890, + 821, + 917 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/77d38c171ad40d6fd4dd197e8459f964d6b5dcaef2d7a99c8bcfa77bd0e1c05f.jpg", + "table_caption": [ + "Table 8: Results on CIFAR-100 with downstream-aware supervised pre-training. C100: CIFAR100. " + ], + "table_footnote": [], + "table_body": "
Datapre-trained on IN-1k?Syn. images amount 0 1.2M 2.4M 3.6M
(None)78.83-
C100 Syn183.90 85.03 85.24
(None)√84.50- =
C100 Syn√-84.90 85.32 85.52
", + "bbox": [ + 179, + 77, + 470, + 179 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/4332e7044189d7f58e75a347506ce096cf19e5ad3812e2a409a7845043c156c3.jpg", + "table_caption": [ + "Table 9: Results on CIFAR-100 with downstream-agnostic supervised pretraining. Backbone: DeiT-S. " + ], + "table_footnote": [], + "table_body": "
Datapre-trained on IN-1k?Syn. images amount 1.2M 2.4M 4.0M
0 69.29 =
(None) IN-1K Syn87.9888.39=
IN-2K Syn- =88.5788.91
(None)5- 88.07 =-■
", + "bbox": [ + 516, + 78, + 813, + 179 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/6cbb2683d0265328a8b718924e4a2d0c5c226af5eb6702af121643c3a25392b2.jpg", + "table_caption": [ + "Table 10: Results for object detection on PASCAL VOC with downstream-agnostic supervised pre-training, all results are reported in $\\mathrm { { A P } _ { 5 0 } }$ . " + ], + "table_footnote": [], + "table_body": "
Datapre-trained on IN-1k?Syn.images amount 0 1.2M2.4M 4.0M
(None)66.08 79.00 == =
IN-1K Syn IN-2K Syn- =80.00 80.54 80.72
(None)√81.30-
IN-1K Syn√=81.78
IN-2K Syn√=81.87 81.91
", + "bbox": [ + 179, + 231, + 477, + 349 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/f51ece32ebeac913ad0f262bfcef05abcaea5b2f35c8b8eb0f72f861f5f143dd.jpg", + "table_caption": [ + "Table 11: Results for object detection on PASCAL VOC with downstream-agnostic selfsupervised pre-training (Moco v2), all results are reported in $\\mathrm { { A P } _ { 5 0 } }$ . " + ], + "table_footnote": [], + "table_body": "
Datapre-trained on IN-1k?Syn. images amount 0 1.2M2.4M 4.0M
(None)66.08 -- 1
IN-1K Syn81.55 =82.13
IN-2K Syn1 =82.22 82.29
(None)482.44=
IN-1K Syn√- =82.47=
", + "bbox": [ + 511, + 231, + 808, + 348 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "ViT-based backbone is shown to be more advantageous compared with convolution-based backbone for synthetic pre-training: outperforming ImageNet pre-training results in the downstream-agnostic settings. Equipped with ViT-based backbone, on only 1.2M IN-1K synthetic data, we achieve comparable performance $( 8 7 . 9 8 \\% )$ with ImageNet pre-training $( 8 8 . 0 7 \\% )$ . Further increasing the data amount $( 8 8 . 3 9 \\% )$ and label space $( 8 8 . 5 7 \\%$ , $8 8 . 9 1 \\%$ of pre-training data leads to higher performance than ImageNet pre-training. ViT-based backbones have stronger ability for learning from large-scale data and are more robust (Pinto et al., 2021), and thus could better benefit from synthetic pre-training where data are more noisy and data scale could be easily increased. ", + "bbox": [ + 174, + 420, + 825, + 532 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Conclusion. In terms of transfer abilities, synthetic data from text-to-image generation models show surprisingly promising results for model pre-training, which is comparable to the standard ImageNet pre-training. We conclude our findings as follows: ", + "bbox": [ + 174, + 539, + 820, + 580 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "1. Data amount has positive impacts on synthetic pre-training; performance could be improved by increasing synthetic data size, but would gradually saturate as the amount of data increases. ", + "bbox": [ + 176, + 582, + 823, + 609 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "2. Synthetic data for pre-training is orthogonal to real data for pre-training. ", + "bbox": [ + 173, + 609, + 669, + 622 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3. For downstream-aware synthetic pre-training, we significantly outperform IN-1K Real (1.2M) pre-training with 2.4M/3.6M synthetic data on CIFAR-100. ", + "bbox": [ + 173, + 625, + 823, + 651 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4. For downstream-agnostic synthetic pre-training, we achieve comparable results with ImageNet (IN-1k) Real pre-training; self-supervised pre-training performs better than supervised pre-training, and ViT-based backbone performs better than convolutional-based backbone. Besides, increasing the label space size could further improve the performance. ", + "bbox": [ + 174, + 652, + 825, + 707 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 174, + 718, + 318, + 733 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We systematically investigate whether synthetic data from current state-of-the-art text-to-image generation models are readily applicable for image recognition. Our extensive experiments demonstrate that synthetic data are beneficial for classifier learning in zero-shot and few-shot recognition, bringing significant performance boosts and yielding new state-of-the-art performance. Further, current synthetic data show strong potential for model pre-training, even surpassing the standard ImageNet pre-training. We also point out limitations and bottlenecks for applying synthetic data for image recognition, hoping to arouse more future research in this direction. ", + "bbox": [ + 173, + 741, + 825, + 838 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Limitations. In all investigated settings, we observe improved performance as the data amount and diversity (label space) increases. However, due to our limited computational resource, we are not able to further scale up data amount, which may take months to train one model. Besides, we are also not able to investigate larger model sizes and advanced architectures in the current investigation which is also worth exploring in the future. We present more discussions on limitations and future directions in the appendix. ", + "bbox": [ + 173, + 840, + 825, + 924 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Acknowledgement. This work has been supported by Hong Kong Research Grant Council - Early Career Scheme (Grant No. 27209621) and General Research Fund Scheme (Grant no. 17202422). Part of the described research work is conducted in the JC STEM Lab of Robotics for Soft Materials funded by The Hong Kong Jockey Club Charities Trust. ", + "bbox": [ + 174, + 103, + 825, + 160 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 180, + 287, + 195 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Thomas Berg, Jiongxin Liu, Seung Woo Lee, Michelle L. Alexander, David W. 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ModelScore
《 Dolly-v2 (Conover et al., 2023) UNA MPT-Chat (Mosaic,2023)4.04 ± 2.34
□ OpenAssistant (Kopf et al., 2023)6.67 ± 2.88 7.65 ± 2.15
福 Alpaca (Taori et al., 2023) 藍 Koala (Geng et al., 2023)8.04 ± 2.05 8.23 ± 1.99
务 Baize (Xu et al., 2023b) T Vicuna (Chiang et al., 2023)8.50 ± 1.34
S Wizard-LM (Xu et al., 2023a) UltraLM (ours)8.78 ± 1.55 8.95 ± 1.44
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ModelScore
《 Dolly-v2 (Conover et al., 2023) UNA MPT-Chat (Mosaic,2023)4.04 ± 2.34
□ OpenAssistant (Kopf et al., 2023)6.67 ± 2.88 7.65 ± 2.15
福 Alpaca (Taori et al., 2023) 藍 Koala (Geng et al., 2023)8.04 ± 2.05 8.23 ± 1.99
务 Baize (Xu et al., 2023b) T Vicuna (Chiang et al., 2023)8.50 ± 1.34
S Wizard-LM (Xu et al., 2023a) UltraLM (ours)8.78 ± 1.55 8.95 ± 1.44
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This paper, in another way, fo-", + "type": "text" + } + ], + "index": 80 + }, + { + "bbox": [ + 303, + 667, + 526, + 681 + ], + "spans": [ + { + "bbox": [ + 303, + 667, + 526, + 681 + ], + "score": 1.0, + "content": "cuses more on the \"final one mile\" of chat language", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 304, + 681, + 525, + 694 + ], + "spans": [ + { + "bbox": [ + 304, + 681, + 525, + 694 + ], + "score": 1.0, + "content": "models, as evidence shows that the journey from 0", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 303, + 694, + 526, + 708 + ], + "spans": [ + { + "bbox": [ + 303, + 694, + 526, + 708 + ], + "score": 1.0, + "content": "to 60 is easy, whereas progressing from 60 to 100", + "type": "text" + } + ], + "index": 83 + }, + { + "bbox": [ + 304, + 708, + 527, + 722 + ], + "spans": [ + { + "bbox": [ + 304, + 708, + 527, + 722 + ], + "score": 1.0, + "content": "becomes exceedingly challenging. For instance,", + "type": "text" + } + ], + "index": 84 + }, + { + "bbox": [ + 304, + 721, + 527, + 735 + ], + "spans": [ + { + "bbox": [ + 304, + 721, + 527, + 735 + ], + "score": 1.0, + "content": "researchers have shown that by utilizing a small,", + "type": "text" + } + ], + "index": 85 + }, + { + "bbox": [ + 69, + 72, + 291, + 86 + ], + "spans": [ + { + "bbox": [ + 69, + 72, + 291, + 86 + ], + "score": 1.0, + "content": "thoughtfully curated set of instructions, it is possi-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 68, + 85, + 291, + 99 + ], + "spans": [ + { + "bbox": [ + 68, + 85, + 291, + 99 + ], + "score": 1.0, + "content": "ble to train a model with satisfactory instruction-", + "type": "text", + "cross_page": true + } + ], + "index": 1 + }, + { + "bbox": [ + 68, + 99, + 290, + 113 + ], + "spans": [ + { + "bbox": [ + 68, + 99, + 290, + 113 + ], + "score": 1.0, + "content": "following capabilities. However, these approaches", + "type": "text", + "cross_page": true + } + ], + "index": 2 + }, + { + "bbox": [ + 69, + 113, + 291, + 126 + ], + "spans": [ + { + "bbox": [ + 69, + 113, + 291, + 126 + ], + "score": 1.0, + "content": "have yet to produce models that surpass the perfor-", + "type": "text", + "cross_page": true + } + ], + "index": 3 + }, + { + "bbox": [ + 69, + 126, + 290, + 139 + ], + "spans": [ + { + "bbox": [ + 69, + 126, + 290, + 139 + ], + "score": 1.0, + "content": "mance of Vicuna, the current leading open-source", + "type": "text", + "cross_page": true + } + ], + "index": 4 + }, + { + "bbox": [ + 69, + 140, + 290, + 153 + ], + "spans": [ + { + "bbox": [ + 69, + 140, + 290, + 153 + ], + "score": 1.0, + "content": "model, let alone outperform ChatGPT and GPT-4.", + "type": "text", + "cross_page": true + } + ], + "index": 5 + } + ], + "index": 75, + "bbox_fs": [ + 303, + 448, + 527, + 735 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 69, + 72, + 290, + 152 + ], + "lines": [ + { + "bbox": [ + 69, + 72, + 291, + 86 + ], + "spans": [ + { + "bbox": [ + 69, + 72, + 291, + 86 + ], + "score": 1.0, + "content": "thoughtfully curated set of instructions, it is possi-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 68, + 85, + 291, + 99 + ], + "spans": [ + { + "bbox": [ + 68, + 85, + 291, + 99 + ], + "score": 1.0, + "content": "ble to train a model with satisfactory instruction-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 68, + 99, + 290, + 113 + ], + "spans": [ + { + "bbox": [ + 68, + 99, + 290, + 113 + ], + "score": 1.0, + "content": "following capabilities. However, these approaches", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 69, + 113, + 291, + 126 + ], + "spans": [ + { + "bbox": [ + 69, + 113, + 291, + 126 + ], + "score": 1.0, + "content": "have yet to produce models that surpass the perfor-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 69, + 126, + 290, + 139 + ], + "spans": [ + { + "bbox": [ + 69, + 126, + 290, + 139 + ], + "score": 1.0, + "content": "mance of Vicuna, the current leading open-source", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 69, + 140, + 290, + 153 + ], + "spans": [ + { + "bbox": [ + 69, + 140, + 290, + 153 + ], + "score": 1.0, + "content": "model, let alone outperform ChatGPT and GPT-4.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 2.5 + }, + { + "type": "text", + "bbox": [ + 70, + 155, + 290, + 504 + ], + "lines": [ + { + "bbox": [ + 79, + 153, + 290, + 168 + ], + "spans": [ + { + "bbox": [ + 79, + 153, + 290, + 168 + ], + "score": 1.0, + "content": "This paper believes that the most straightforward", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 68, + 166, + 291, + 182 + ], + "spans": [ + { + "bbox": [ + 68, + 166, + 291, + 182 + ], + "score": 1.0, + "content": "way, that is, the quality and diversity of training", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 69, + 180, + 292, + 195 + ], + "spans": [ + { + "bbox": [ + 69, + 180, + 292, + 195 + ], + "score": 1.0, + "content": "data, play a vital role in further improving the per-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 69, + 195, + 291, + 207 + ], + "spans": [ + { + "bbox": [ + 69, + 195, + 291, + 207 + ], + "score": 1.0, + "content": "formance of chat language models. In other words,", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 68, + 208, + 291, + 222 + ], + "spans": [ + { + "bbox": [ + 68, + 208, + 291, + 222 + ], + "score": 1.0, + "content": "leveraging higher quality and more diverse data", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 69, + 222, + 291, + 235 + ], + "spans": [ + { + "bbox": [ + 69, + 222, + 291, + 235 + ], + "score": 1.0, + "content": "can yield better outcomes. 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UltraChat", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 69, + 276, + 291, + 289 + ], + "spans": [ + { + "bbox": [ + 69, + 276, + 291, + 289 + ], + "score": 1.0, + "content": "is carefully designed to capture the breadth of in-", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 69, + 290, + 290, + 302 + ], + "spans": [ + { + "bbox": [ + 69, + 290, + 290, + 302 + ], + "score": 1.0, + "content": "teractions that a human might have with an AI", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 68, + 303, + 291, + 317 + ], + "spans": [ + { + "bbox": [ + 68, + 303, + 291, + 317 + ], + "score": 1.0, + "content": "assistant. Specifically, we do not use specific tasks", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 68, + 317, + 292, + 330 + ], + "spans": [ + { + "bbox": [ + 68, + 317, + 292, + 330 + ], + "score": 1.0, + "content": "like question-answering or summarization to con-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 69, + 331, + 290, + 342 + ], + "spans": [ + { + "bbox": [ + 69, + 331, + 290, + 342 + ], + "score": 1.0, + "content": "struct the data, but curate three sectors: Questions", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 69, + 343, + 291, + 357 + ], + "spans": [ + { + "bbox": [ + 69, + 343, + 291, + 357 + ], + "score": 1.0, + "content": "about the World, Creation and Writing, and Assis-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 69, + 357, + 291, + 371 + ], + "spans": [ + { + "bbox": [ + 69, + 357, + 291, + 371 + ], + "score": 1.0, + "content": "tance on Existing Materials. Then we employ meta-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 69, + 371, + 290, + 384 + ], + "spans": [ + { + "bbox": [ + 69, + 371, + 290, + 384 + ], + "score": 1.0, + "content": "information, in-context expansion, and iterative", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 68, + 385, + 292, + 398 + ], + "spans": [ + { + "bbox": [ + 68, + 385, + 292, + 398 + ], + "score": 1.0, + "content": "prompting to scale up the number of instructions.", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 69, + 399, + 290, + 410 + ], + "spans": [ + { + "bbox": [ + 69, + 399, + 290, + 410 + ], + "score": 1.0, + "content": "To construct informative and realistic multi-turn", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 69, + 412, + 290, + 424 + ], + "spans": [ + { + "bbox": [ + 69, + 412, + 290, + 424 + ], + "score": 1.0, + "content": "conversations, two separate ChatGPT Turbo APIs", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 69, + 425, + 290, + 438 + ], + "spans": [ + { + "bbox": [ + 69, + 425, + 290, + 438 + ], + "score": 1.0, + "content": "are adopted in the conversation generation, where", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 68, + 438, + 291, + 452 + ], + "spans": [ + { + "bbox": [ + 68, + 438, + 291, + 452 + ], + "score": 1.0, + "content": "one plays the role of the user to generate queries,", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 452, + 290, + 465 + ], + "spans": [ + { + "bbox": [ + 69, + 452, + 290, + 465 + ], + "score": 1.0, + "content": "and the other generates the response. We instruct", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 68, + 464, + 290, + 480 + ], + "spans": [ + { + "bbox": [ + 68, + 464, + 290, + 480 + ], + "score": 1.0, + "content": "the user model with carefully designed prompts to", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 69, + 480, + 290, + 491 + ], + "spans": [ + { + "bbox": [ + 69, + 480, + 290, + 491 + ], + "score": 1.0, + "content": "mimic human user behavior and call the two APIs", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 68, + 492, + 118, + 506 + ], + "spans": [ + { + "bbox": [ + 68, + 492, + 118, + 506 + ], + "score": 1.0, + "content": "iteratively.", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 69, + 507, + 290, + 655 + ], + "lines": [ + { + "bbox": [ + 81, + 507, + 291, + 519 + ], + "spans": [ + { + "bbox": [ + 81, + 507, + 291, + 519 + ], + "score": 1.0, + "content": "We fine-tune a LLaMA-13B model on Ultra-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 69, + 520, + 290, + 534 + ], + "spans": [ + { + "bbox": [ + 69, + 520, + 290, + 534 + ], + "score": 1.0, + "content": "Chat to produce UltraLM and compare the model", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 69, + 536, + 291, + 548 + ], + "spans": [ + { + "bbox": [ + 69, + 536, + 291, + 548 + ], + "score": 1.0, + "content": "to a wide range of baselines, especially the open-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 69, + 549, + 290, + 560 + ], + "spans": [ + { + "bbox": [ + 69, + 549, + 290, + 560 + ], + "score": 1.0, + "content": "source ones. The evaluation shows that our model", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 69, + 562, + 290, + 574 + ], + "spans": [ + { + "bbox": [ + 69, + 562, + 290, + 574 + ], + "score": 1.0, + "content": "could consistently outperform other models. As", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 575, + 291, + 588 + ], + "spans": [ + { + "bbox": [ + 68, + 575, + 291, + 588 + ], + "score": 1.0, + "content": "reported in Table 1, UltraLM achieves the highest", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 68, + 589, + 290, + 601 + ], + "spans": [ + { + "bbox": [ + 68, + 589, + 290, + 601 + ], + "score": 1.0, + "content": "performance scores that are independently assessed", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 68, + 602, + 291, + 616 + ], + "spans": [ + { + "bbox": [ + 68, + 602, + 291, + 616 + ], + "score": 1.0, + "content": "by GPT-4. Further evaluation results on challeng-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 69, + 616, + 290, + 628 + ], + "spans": [ + { + "bbox": [ + 69, + 616, + 290, + 628 + ], + "score": 1.0, + "content": "ing benchmarks and preference study with GPT-4", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 68, + 630, + 290, + 641 + ], + "spans": [ + { + "bbox": [ + 68, + 630, + 290, + 641 + ], + "score": 1.0, + "content": "on various evaluation sets also show that UltraLM", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 69, + 644, + 262, + 656 + ], + "spans": [ + { + "bbox": [ + 69, + 644, + 262, + 656 + ], + "score": 1.0, + "content": "could surpass all other open-source models.", + "type": "text" + } + ], + "index": 42 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 305, + 72, + 525, + 273 + ], + "lines": [ + { + "bbox": [ + 304, + 71, + 526, + 85 + ], + "spans": [ + { + "bbox": [ + 304, + 71, + 526, + 85 + ], + "score": 1.0, + "content": "hibits a strong ability in instruction understanding", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 304, + 86, + 525, + 97 + ], + "spans": [ + { + "bbox": [ + 304, + 86, + 525, + 97 + ], + "score": 1.0, + "content": "and generalizes well to unseen instructions (Sanh", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 304, + 100, + 525, + 112 + ], + "spans": [ + { + "bbox": [ + 304, + 100, + 525, + 112 + ], + "score": 1.0, + "content": "et al., 2021; Ouyang et al., 2022). Later, Longpre", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 113, + 526, + 125 + ], + "spans": [ + { + "bbox": [ + 304, + 113, + 526, + 125 + ], + "score": 1.0, + "content": "et al. (2023) show the benefits of scaling the num-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 126, + 527, + 139 + ], + "spans": [ + { + "bbox": [ + 304, + 126, + 527, + 139 + ], + "score": 1.0, + "content": "ber of tasks in out-of-distribution generalization.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 140, + 525, + 152 + ], + "spans": [ + { + "bbox": [ + 304, + 140, + 525, + 152 + ], + "score": 1.0, + "content": "Wei et al. 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This technique combined", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 235, + 525, + 246 + ], + "spans": [ + { + "bbox": [ + 304, + 235, + 525, + 246 + ], + "score": 1.0, + "content": "with instruction tuning can further boost the model", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 249, + 525, + 260 + ], + "spans": [ + { + "bbox": [ + 304, + 249, + 525, + 260 + ], + "score": 1.0, + "content": "performance and has been successfully applied to", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 305, + 262, + 414, + 273 + ], + "spans": [ + { + "bbox": [ + 305, + 262, + 414, + 273 + ], + "score": 1.0, + "content": "LLMs such as ChatGPT.", + "type": "text" + } + ], + "index": 57 + } + ], + "index": 50 + }, + { + "type": "text", + "bbox": [ + 305, + 282, + 525, + 660 + ], + "lines": [ + { + "bbox": [ + 304, + 281, + 525, + 296 + ], + "spans": [ + { + "bbox": [ + 304, + 281, + 525, + 296 + ], + "score": 1.0, + "content": "Data Augmentation with LLMs. Collecting", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 296, + 526, + 308 + ], + "spans": [ + { + "bbox": [ + 304, + 296, + 526, + 308 + ], + "score": 1.0, + "content": "large-scale human-annotated instructions and their", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 310, + 526, + 322 + ], + "spans": [ + { + "bbox": [ + 304, + 310, + 526, + 322 + ], + "score": 1.0, + "content": "responses is time-consuming and labor-intensive.", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 304, + 324, + 526, + 335 + ], + "spans": [ + { + "bbox": [ + 304, + 324, + 526, + 335 + ], + "score": 1.0, + "content": "Alternatively, a more cost-effective and feasible", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 303, + 337, + 526, + 349 + ], + "spans": [ + { + "bbox": [ + 303, + 337, + 526, + 349 + ], + "score": 1.0, + "content": "approach to gathering top-notch data involves", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 303, + 350, + 527, + 363 + ], + "spans": [ + { + "bbox": [ + 303, + 350, + 527, + 363 + ], + "score": 1.0, + "content": "sampling from LLMs that have been well-tuned,", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 303, + 364, + 526, + 376 + ], + "spans": [ + { + "bbox": [ + 303, + 364, + 526, + 376 + ], + "score": 1.0, + "content": "e.g., ChatGPT and GPT-3.5. 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The", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 304, + 458, + 527, + 471 + ], + "spans": [ + { + "bbox": [ + 304, + 458, + 527, + 471 + ], + "score": 1.0, + "content": "trained model performs almost on par with Text-", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 304, + 472, + 527, + 484 + ], + "spans": [ + { + "bbox": [ + 304, + 472, + 527, + 484 + ], + "score": 1.0, + "content": "Davinci-003. 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This process reflects the AI’s capacity to", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 68, + 478, + 290, + 490 + ], + "spans": [ + { + "bbox": [ + 68, + 478, + 290, + 490 + ], + "score": 1.0, + "content": "engage in original content generation alongside", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 68, + 491, + 290, + 504 + ], + "spans": [ + { + "bbox": [ + 68, + 491, + 290, + 504 + ], + "score": 1.0, + "content": "users and demonstrates the role of AI assistants as", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 69, + 505, + 281, + 518 + ], + "spans": [ + { + "bbox": [ + 69, + 505, + 281, + 518 + ], + "score": 1.0, + "content": "collaborative partners in a creative environment.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "text", + "bbox": [ + 69, + 518, + 290, + 707 + ], + "lines": [ + { + "bbox": [ + 80, + 518, + 292, + 531 + ], + "spans": [ + { + "bbox": [ + 80, + 518, + 292, + 531 + ], + "score": 1.0, + "content": "We first project all creations as text materials,", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 69, + 532, + 291, + 546 + ], + "spans": [ + { + "bbox": [ + 69, + 532, + 291, + 546 + ], + "score": 1.0, + "content": "and further categorize them into 20 different types", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 69, + 546, + 290, + 558 + ], + "spans": [ + { + "bbox": [ + 69, + 546, + 290, + 558 + ], + "score": 1.0, + "content": "as in Table 3. Then a ChatGPT model is employed", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 68, + 558, + 291, + 572 + ], + "spans": [ + { + "bbox": [ + 68, + 558, + 291, + 572 + ], + "score": 1.0, + "content": "to produce a diverse range of instructions for each", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 69, + 572, + 290, + 586 + ], + "spans": [ + { + "bbox": [ + 69, + 573, + 208, + 586 + ], + "score": 1.0, + "content": "type of writing, approximately", + "type": "text" + }, + { + "bbox": [ + 208, + 572, + 230, + 584 + ], + "score": 0.86, + "content": "80 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 231, + 573, + 290, + 586 + ], + "score": 1.0, + "content": "of which are", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 68, + 585, + 291, + 600 + ], + "spans": [ + { + "bbox": [ + 68, + 585, + 291, + 600 + ], + "score": 1.0, + "content": "further refined by ChatGPT model to generate more", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 69, + 599, + 290, + 612 + ], + "spans": [ + { + "bbox": [ + 69, + 599, + 290, + 612 + ], + "score": 1.0, + "content": "detailed instructions. These instructions serve as", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 68, + 613, + 291, + 627 + ], + "spans": [ + { + "bbox": [ + 68, + 613, + 291, + 627 + ], + "score": 1.0, + "content": "opening lines for dialogue generation. Throughout", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 69, + 626, + 290, + 640 + ], + "spans": [ + { + "bbox": [ + 69, + 626, + 290, + 640 + ], + "score": 1.0, + "content": "the generation process, the user prompt constantly", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 69, + 641, + 291, + 653 + ], + "spans": [ + { + "bbox": [ + 69, + 641, + 291, + 653 + ], + "score": 1.0, + "content": "reinforces the primary objective of the conversa-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 69, + 654, + 291, + 667 + ], + "spans": [ + { + "bbox": [ + 69, + 654, + 291, + 667 + ], + "score": 1.0, + "content": "tion, which is to generate and refine a piece of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 68, + 667, + 291, + 680 + ], + "spans": [ + { + "bbox": [ + 68, + 667, + 291, + 680 + ], + "score": 1.0, + "content": "writing. This serves to ensure that the behavior of", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 681, + 289, + 694 + ], + "spans": [ + { + "bbox": [ + 69, + 681, + 289, + 694 + ], + "score": 1.0, + "content": "the user model remains focused and aligned with", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 69, + 694, + 166, + 708 + ], + "spans": [ + { + "bbox": [ + 69, + 694, + 166, + 708 + ], + "score": 1.0, + "content": "the intended purpose.", + "type": "text" + } + ], + "index": 29 + } + ], + "index": 22.5 + }, + { + "type": "title", + "bbox": [ + 70, + 716, + 247, + 729 + ], + "lines": [ + { + "bbox": [ + 68, + 715, + 248, + 730 + ], + "spans": [ + { + "bbox": [ + 68, + 715, + 248, + 730 + ], + "score": 1.0, + "content": "3.3 Assistance on Existing Materials", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 30 + }, + { + "type": "text", + "bbox": [ + 69, + 734, + 290, + 774 + ], + "lines": [ + { + "bbox": [ + 68, + 734, + 290, + 747 + ], + "spans": [ + { + "bbox": [ + 68, + 734, + 290, + 747 + ], + "score": 1.0, + "content": "The third sector mainly addresses the modification", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 69, + 747, + 290, + 761 + ], + "spans": [ + { + "bbox": [ + 69, + 747, + 290, + 761 + ], + "score": 1.0, + "content": "of existing information, encompassing various", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 68, + 761, + 292, + 775 + ], + "spans": [ + { + "bbox": [ + 68, + 761, + 292, + 775 + ], + "score": 1.0, + "content": "tasks including rewriting, translation, summa-", + "type": "text" + } + ], + "index": 33 + } + ], + "index": 32 + }, + { + "type": "text", + "bbox": [ + 305, + 370, + 525, + 437 + ], + "lines": [ + { + "bbox": [ + 304, + 369, + 526, + 385 + ], + "spans": [ + { + "bbox": [ + 304, + 369, + 526, + 385 + ], + "score": 1.0, + "content": "rization, and question-answering, etc. Modifying", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 304, + 384, + 526, + 397 + ], + "spans": [ + { + "bbox": [ + 304, + 384, + 526, + 397 + ], + "score": 1.0, + "content": "existing materials is a crucial aspect of human-AI", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 304, + 398, + 526, + 410 + ], + "spans": [ + { + "bbox": [ + 304, + 398, + 526, + 410 + ], + "score": 1.0, + "content": "interaction, as it allows the AI assistant to actively", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 303, + 411, + 526, + 424 + ], + "spans": [ + { + "bbox": [ + 303, + 411, + 526, + 424 + ], + "score": 1.0, + "content": "engage with the user’s input, transforming it in", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 425, + 474, + 438 + ], + "spans": [ + { + "bbox": [ + 304, + 425, + 474, + 438 + ], + "score": 1.0, + "content": "various ways as instructed by the user.", + "type": "text" + } + ], + "index": 38 + } + ], + "index": 36 + }, + { + "type": "text", + "bbox": [ + 305, + 438, + 525, + 641 + ], + "lines": [ + { + "bbox": [ + 316, + 438, + 525, + 452 + ], + "spans": [ + { + "bbox": [ + 316, + 438, + 525, + 452 + ], + "score": 1.0, + "content": "We begin by gathering text pieces from the C4", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 304, + 452, + 527, + 466 + ], + "spans": [ + { + "bbox": [ + 304, + 452, + 527, + 466 + ], + "score": 1.0, + "content": "corpus5. Each piece within the C4 corpus is associ-", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 304, + 466, + 526, + 479 + ], + "spans": [ + { + "bbox": [ + 304, + 466, + 526, + 479 + ], + "score": 1.0, + "content": "ated with a source URL. To ensure a diverse range", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 304, + 479, + 525, + 492 + ], + "spans": [ + { + "bbox": [ + 304, + 479, + 525, + 492 + ], + "score": 1.0, + "content": "of text content and styles, we adopt the 20 material", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 304, + 493, + 527, + 506 + ], + "spans": [ + { + "bbox": [ + 304, + 493, + 527, + 506 + ], + "score": 1.0, + "content": "types outlined in the previous section and manu-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 304, + 506, + 526, + 520 + ], + "spans": [ + { + "bbox": [ + 304, + 506, + 526, + 520 + ], + "score": 1.0, + "content": "ally curate keywords for each type. Additionally,", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 303, + 520, + 525, + 533 + ], + "spans": [ + { + "bbox": [ + 303, + 520, + 525, + 533 + ], + "score": 1.0, + "content": "we classify the text in the corpus by matching the", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 534, + 526, + 547 + ], + "spans": [ + { + "bbox": [ + 304, + 534, + 526, + 547 + ], + "score": 1.0, + "content": "keywords in the corresponding URL. In total, we", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 547, + 525, + 560 + ], + "spans": [ + { + "bbox": [ + 304, + 547, + 525, + 560 + ], + "score": 1.0, + "content": "collect 100,000 text pieces from the C4 corpus, and", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 560, + 525, + 575 + ], + "spans": [ + { + "bbox": [ + 304, + 560, + 525, + 575 + ], + "score": 1.0, + "content": "for each piece, we prompt ChatGPT to generate", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 304, + 573, + 527, + 587 + ], + "spans": [ + { + "bbox": [ + 304, + 573, + 527, + 587 + ], + "score": 1.0, + "content": "five distinct instructions. We use a manually de-", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 303, + 587, + 528, + 601 + ], + "spans": [ + { + "bbox": [ + 303, + 587, + 528, + 601 + ], + "score": 1.0, + "content": "signed template to combine text and instructions,", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 303, + 600, + 527, + 615 + ], + "spans": [ + { + "bbox": [ + 303, + 600, + 527, + 615 + ], + "score": 1.0, + "content": "as depicted in Figure 4 in the appendix. 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Dataset#DialogueAvg. #TurnsAvg. Dialog Length (by token)Avg.Utt.Length (by token)Lexical Diversity (↑)Topic Diversity (↓)Coherence (个)User Simulation
Self-Instruct82,439169.829.224.90.733No
Stanford Alpaca52.002191.164.542.80.727No
SODA1,486,8693.6231.822.538.60.7978.48No
GPT-4-LLM61,0021179.6142.948.90.721-No
BELLE1,436,6791102.363.335.90.771-No
Baize210,3113.1293.952.867.10.7519.06Yes
GPT4ALL711,1261597.7318.962.70.6921No
UltraChat1,468,3523.81467.4309.374.30.7029.06Yes
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Lexical diversity is calculated by averaging the MTLD score (Mc-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 69, + 195, + 526, + 208 + ], + "spans": [ + { + "bbox": [ + 69, + 195, + 526, + 208 + ], + "score": 1.0, + "content": "Carthy and Jarvis, 2010) over each utterance with LexicalRichness6. 10000 samples are randomly drawn from each", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 68, + 208, + 526, + 222 + ], + "spans": [ + { + "bbox": [ + 68, + 208, + 526, + 222 + ], + "score": 1.0, + "content": "dataset for topic diversity and coherence measurement. Topic diversity is measured by averaging the cosine distance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 69, + 221, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 69, + 221, + 506, + 232 + ], + "score": 1.0, + "content": "between each pair of data with OpenAI embedding API. Coherence is scored by ChatGPT on a scale of 1-10.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + }, + { + "type": "text", + "bbox": [ + 70, + 244, + 289, + 405 + ], + "lines": [ + { + "bbox": [ + 68, + 244, + 290, + 257 + ], + "spans": [ + { + "bbox": [ + 68, + 244, + 290, + 257 + ], + "score": 1.0, + "content": "can significantly deteriorate the coherence of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 69, + 258, + 291, + 270 + ], + "spans": [ + { + "bbox": [ + 69, + 258, + 291, + 270 + ], + "score": 1.0, + "content": "multi-turn conversation. To address this, in addi-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 69, + 271, + 290, + 284 + ], + "spans": [ + { + "bbox": [ + 69, + 271, + 290, + 284 + ], + "score": 1.0, + "content": "tion to presenting the dialogue history, we include", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 68, + 285, + 291, + 299 + ], + "spans": [ + { + "bbox": [ + 68, + 285, + 291, + 299 + ], + "score": 1.0, + "content": "prompts explicitly instructing the model to adopt", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 68, + 299, + 291, + 313 + ], + "spans": [ + { + "bbox": [ + 68, + 299, + 291, + 313 + ], + "score": 1.0, + "content": "various user personalities. In Sector 2, a prompt", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 68, + 311, + 291, + 326 + ], + "spans": [ + { + "bbox": [ + 68, + 311, + 291, + 326 + ], + "score": 1.0, + "content": "is employed to remind the model of the primary", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 68, + 325, + 291, + 340 + ], + "spans": [ + { + "bbox": [ + 68, + 325, + 291, + 340 + ], + "score": 1.0, + "content": "purpose of the dialogue, thereby promoting a more", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 68, + 338, + 291, + 353 + ], + "spans": [ + { + "bbox": [ + 68, + 338, + 291, + 353 + ], + "score": 1.0, + "content": "natural conversation flow. Once the data genera-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 68, + 352, + 290, + 365 + ], + "spans": [ + { + "bbox": [ + 68, + 352, + 290, + 365 + ], + "score": 1.0, + "content": "tion process is complete, a further filtration step is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 68, + 367, + 290, + 379 + ], + "spans": [ + { + "bbox": [ + 68, + 367, + 290, + 379 + ], + "score": 1.0, + "content": "performed to ensure overall data quality. We also", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 68, + 380, + 290, + 393 + ], + "spans": [ + { + "bbox": [ + 68, + 380, + 290, + 393 + ], + "score": 1.0, + "content": "exclude excessively polite statements to enhance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 69, + 393, + 201, + 407 + ], + "spans": [ + { + "bbox": [ + 69, + 393, + 201, + 407 + ], + "score": 1.0, + "content": "the realism of user responses.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5 + }, + { + "type": "title", + "bbox": [ + 70, + 418, + 159, + 432 + ], + "lines": [ + { + "bbox": [ + 68, + 418, + 161, + 434 + ], + "spans": [ + { + "bbox": [ + 68, + 418, + 161, + 434 + ], + "score": 1.0, + "content": "4 Data Analysis", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 70, + 442, + 183, + 455 + ], + "lines": [ + { + "bbox": [ + 68, + 442, + 185, + 456 + ], + "spans": [ + { + "bbox": [ + 68, + 442, + 185, + 456 + ], + "score": 1.0, + "content": "4.1 Statistical Analysis", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 70, + 463, + 290, + 772 + ], + "lines": [ + { + "bbox": [ + 69, + 463, + 290, + 475 + ], + "spans": [ + { + "bbox": [ + 69, + 463, + 290, + 475 + ], + "score": 1.0, + "content": "We conduct a statistical analysis of UltraChat", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 68, + 477, + 290, + 489 + ], + "spans": [ + { + "bbox": [ + 68, + 477, + 290, + 489 + ], + "score": 1.0, + "content": "and several other instruction datasets, as shown", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 69, + 490, + 290, + 502 + ], + "spans": [ + { + "bbox": [ + 69, + 490, + 290, + 502 + ], + "score": 1.0, + "content": "in Table 4. UltraChat stands out in terms of its", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 68, + 504, + 290, + 518 + ], + "spans": [ + { + "bbox": [ + 68, + 504, + 290, + 518 + ], + "score": 1.0, + "content": "scale, being one of the largest publicly available", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 69, + 516, + 290, + 532 + ], + "spans": [ + { + "bbox": [ + 69, + 516, + 290, + 532 + ], + "score": 1.0, + "content": "datasets. Moreover, it exhibits the highest average", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 69, + 531, + 291, + 544 + ], + "spans": [ + { + "bbox": [ + 69, + 531, + 291, + 544 + ], + "score": 1.0, + "content": "number of turns and the longest average length per", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 69, + 545, + 290, + 557 + ], + "spans": [ + { + "bbox": [ + 69, + 545, + 290, + 557 + ], + "score": 1.0, + "content": "instance of data. While SODA (Kim et al., 2023)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 558, + 291, + 570 + ], + "spans": [ + { + "bbox": [ + 69, + 558, + 291, + 570 + ], + "score": 1.0, + "content": "also has many rounds, it is primarily composed of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 69, + 572, + 292, + 583 + ], + "spans": [ + { + "bbox": [ + 69, + 572, + 292, + 583 + ], + "score": 1.0, + "content": "conceptual banter rather than instructional content.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 68, + 585, + 291, + 599 + ], + "spans": [ + { + "bbox": [ + 68, + 585, + 291, + 599 + ], + "score": 1.0, + "content": "Additionally, the average number of tokens per", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 68, + 598, + 291, + 611 + ], + "spans": [ + { + "bbox": [ + 68, + 598, + 291, + 611 + ], + "score": 1.0, + "content": "dialogue in SODA is 231.8, whereas UltraChat", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 68, + 612, + 291, + 624 + ], + "spans": [ + { + "bbox": [ + 68, + 612, + 291, + 624 + ], + "score": 1.0, + "content": "boasts a remarkable 1467.4 tokens. To evaluate di-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 68, + 625, + 290, + 639 + ], + "spans": [ + { + "bbox": [ + 68, + 625, + 290, + 639 + ], + "score": 1.0, + "content": "versity, we measure both lexical diversity and topic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 69, + 639, + 290, + 652 + ], + "spans": [ + { + "bbox": [ + 69, + 639, + 290, + 652 + ], + "score": 1.0, + "content": "diversity. UltraChat outperforms previous datasets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 68, + 653, + 291, + 666 + ], + "spans": [ + { + "bbox": [ + 68, + 653, + 291, + 666 + ], + "score": 1.0, + "content": "in terms of lexical diversity. However, in terms of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 68, + 666, + 291, + 680 + ], + "spans": [ + { + "bbox": [ + 68, + 666, + 291, + 680 + ], + "score": 1.0, + "content": "topic diversity, UltraChat falls slightly short com-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 680, + 290, + 693 + ], + "spans": [ + { + "bbox": [ + 68, + 680, + 290, + 693 + ], + "score": 1.0, + "content": "pared to GPT4ALL (Anand et al., 2023) but still", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 68, + 694, + 290, + 707 + ], + "spans": [ + { + "bbox": [ + 68, + 694, + 290, + 707 + ], + "score": 1.0, + "content": "surpasses other datasets significantly. This may be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 68, + 706, + 291, + 721 + ], + "spans": [ + { + "bbox": [ + 68, + 706, + 291, + 721 + ], + "score": 1.0, + "content": "attributed to the regularized embeddings resulting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 69, + 721, + 291, + 733 + ], + "spans": [ + { + "bbox": [ + 69, + 721, + 291, + 733 + ], + "score": 1.0, + "content": "from a large number of tokens in each dialogue. We", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 69, + 734, + 291, + 747 + ], + "spans": [ + { + "bbox": [ + 69, + 734, + 291, + 747 + ], + "score": 1.0, + "content": "also conduct coherence evaluation with ChatGPT", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 69, + 748, + 290, + 760 + ], + "spans": [ + { + "bbox": [ + 69, + 748, + 290, + 760 + ], + "score": 1.0, + "content": "for multi-turn datasets. Notably, UltraChat and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 69, + 761, + 289, + 774 + ], + "spans": [ + { + "bbox": [ + 69, + 761, + 289, + 774 + ], + "score": 1.0, + "content": "Baize data rank the highest in terms of coherence.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 32 + }, + { + "type": "title", + "bbox": [ + 306, + 244, + 421, + 257 + ], + "lines": [ + { + "bbox": [ + 303, + 243, + 423, + 258 + ], + "spans": [ + { + "bbox": [ + 303, + 243, + 423, + 258 + ], + "score": 1.0, + "content": "4.2 Human Assessment", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 305, + 269, + 525, + 604 + ], + "lines": [ + { + "bbox": [ + 304, + 268, + 526, + 281 + ], + "spans": [ + { + "bbox": [ + 304, + 268, + 526, + 281 + ], + "score": 1.0, + "content": "Setup. To better evaluate the constructed data", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 281, + 526, + 294 + ], + "spans": [ + { + "bbox": [ + 304, + 281, + 526, + 294 + ], + "score": 1.0, + "content": "quality, we also conduct human assessment for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 295, + 526, + 307 + ], + "spans": [ + { + "bbox": [ + 304, + 295, + 526, + 307 + ], + "score": 1.0, + "content": "UltraChat. Due to the difficulty of evaluation of", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 309, + 526, + 322 + ], + "spans": [ + { + "bbox": [ + 304, + 309, + 526, + 322 + ], + "score": 1.0, + "content": "multi-turn dialogue and the resulting formidable", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 304, + 322, + 526, + 335 + ], + "spans": [ + { + "bbox": [ + 304, + 322, + 526, + 335 + ], + "score": 1.0, + "content": "cost, we sample 500 representative dialogues for", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 304, + 336, + 527, + 348 + ], + "spans": [ + { + "bbox": [ + 304, + 336, + 527, + 348 + ], + "score": 1.0, + "content": "human evaluation, among which 300 are from Ul-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 304, + 349, + 525, + 361 + ], + "spans": [ + { + "bbox": [ + 304, + 349, + 525, + 361 + ], + "score": 1.0, + "content": "traChat sector 1, 100 from sector 2 and sector 3", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 304, + 364, + 525, + 376 + ], + "spans": [ + { + "bbox": [ + 304, + 364, + 525, + 376 + ], + "score": 1.0, + "content": "respectively. For each round of conversation, we", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 304, + 376, + 525, + 389 + ], + "spans": [ + { + "bbox": [ + 304, + 376, + 525, + 389 + ], + "score": 1.0, + "content": "ask the annotators to score the assistant’s response", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 390, + 526, + 403 + ], + "spans": [ + { + "bbox": [ + 304, + 390, + 526, + 403 + ], + "score": 1.0, + "content": "on Helpfulness, Honesty, and Harmlessness (3H)", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 303, + 403, + 526, + 416 + ], + "spans": [ + { + "bbox": [ + 303, + 403, + 526, + 416 + ], + "score": 1.0, + "content": "principles (Askell et al., 2021). We also devise", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 417, + 525, + 430 + ], + "spans": [ + { + "bbox": [ + 304, + 417, + 525, + 430 + ], + "score": 1.0, + "content": "Coherence and Consistency criteria for the overall", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 304, + 430, + 526, + 443 + ], + "spans": [ + { + "bbox": [ + 304, + 430, + 526, + 443 + ], + "score": 1.0, + "content": "multi-turn dialogue quality evaluation. Coherence", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 304, + 444, + 526, + 457 + ], + "spans": [ + { + "bbox": [ + 304, + 444, + 526, + 457 + ], + "score": 1.0, + "content": "evaluates whether the dialogue flows logically and", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 458, + 525, + 470 + ], + "spans": [ + { + "bbox": [ + 304, + 458, + 525, + 470 + ], + "score": 1.0, + "content": "coherently, for which the annotators evaluate both", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 471, + 527, + 484 + ], + "spans": [ + { + "bbox": [ + 304, + 471, + 527, + 484 + ], + "score": 1.0, + "content": "the user’s response and the assistant’s response.", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 304, + 485, + 526, + 498 + ], + "spans": [ + { + "bbox": [ + 304, + 485, + 526, + 498 + ], + "score": 1.0, + "content": "Consistency means the assistant’s responses do not", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 304, + 499, + 526, + 511 + ], + "spans": [ + { + "bbox": [ + 304, + 499, + 526, + 511 + ], + "score": 1.0, + "content": "contradict each other within the same dialogue. For", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 305, + 513, + 525, + 524 + ], + "spans": [ + { + "bbox": [ + 305, + 513, + 525, + 524 + ], + "score": 1.0, + "content": "example, it is inconsistent if the assistant asserts", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 304, + 526, + 526, + 538 + ], + "spans": [ + { + "bbox": [ + 304, + 526, + 526, + 538 + ], + "score": 1.0, + "content": "one specific event occurred in 1911 in the first", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 304, + 540, + 525, + 551 + ], + "spans": [ + { + "bbox": [ + 304, + 540, + 525, + 551 + ], + "score": 1.0, + "content": "round of conversation but mentions it as a 1901", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 304, + 553, + 525, + 564 + ], + "spans": [ + { + "bbox": [ + 304, + 553, + 525, + 564 + ], + "score": 1.0, + "content": "event in the next round. Each metric is scored with", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 304, + 566, + 526, + 579 + ], + "spans": [ + { + "bbox": [ + 304, + 566, + 526, + 579 + ], + "score": 1.0, + "content": "0, 0.5 or 1, where higher score means better quality.", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 304, + 580, + 525, + 593 + ], + "spans": [ + { + "bbox": [ + 304, + 580, + 491, + 593 + ], + "score": 1.0, + "content": "Therefore, for a K-round dialogue, we have", + "type": "text" + }, + { + "bbox": [ + 491, + 580, + 525, + 592 + ], + "score": 0.88, + "content": "3 K + 2", + "type": "inline_equation" + } + ], + "index": 68 + }, + { + "bbox": [ + 304, + 595, + 391, + 605 + ], + "spans": [ + { + "bbox": [ + 304, + 595, + 391, + 605 + ], + "score": 1.0, + "content": "metric annotations.", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 57 + }, + { + "type": "text", + "bbox": [ + 305, + 626, + 525, + 773 + ], + "lines": [ + { + "bbox": [ + 304, + 624, + 527, + 639 + ], + "spans": [ + { + "bbox": [ + 304, + 624, + 527, + 639 + ], + "score": 1.0, + "content": "Annotation. 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Meanwhile, due", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 304, + 667, + 525, + 679 + ], + "spans": [ + { + "bbox": [ + 304, + 667, + 525, + 679 + ], + "score": 1.0, + "content": "to the difficulty in identifying the hallucination", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 303, + 680, + 527, + 693 + ], + "spans": [ + { + "bbox": [ + 303, + 680, + 527, + 693 + ], + "score": 1.0, + "content": "problem, we allow the annotators to skip the di-", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 304, + 694, + 525, + 707 + ], + "spans": [ + { + "bbox": [ + 304, + 694, + 525, + 707 + ], + "score": 1.0, + "content": "alogues that require expert knowledge or whose", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 304, + 707, + 526, + 720 + ], + "spans": [ + { + "bbox": [ + 304, + 707, + 526, + 720 + ], + "score": 1.0, + "content": "validity is hard to check. Altogether, we collect", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 304, + 720, + 526, + 733 + ], + "spans": [ + { + "bbox": [ + 304, + 720, + 526, + 733 + ], + "score": 1.0, + "content": "14560 valid annotations in terms of metrics for", + "type": "text" + } + ], + "index": 77 + }, + { + "bbox": [ + 304, + 734, + 526, + 747 + ], + "spans": [ + { + "bbox": [ + 304, + 734, + 526, + 747 + ], + "score": 1.0, + "content": "both single-round and multi-round, and the Co-", + "type": "text" + } + ], + "index": 78 + }, + { + "bbox": [ + 303, + 747, + 525, + 761 + ], + "spans": [ + { + "bbox": [ + 303, + 747, + 525, + 761 + ], + "score": 1.0, + "content": "hen’s kappa coefficient is 0.358. The average time", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 304, + 762, + 478, + 774 + ], + "spans": [ + { + "bbox": [ + 304, + 762, + 478, + 774 + ], + "score": 1.0, + "content": "to annotate one dialogue is 10 minutes.", + "type": "text" + } + ], + "index": 80 + } + ], + "index": 75 + } + ], + "page_idx": 4, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 73, + 69, + 523, + 176 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 73, + 69, + 523, + 176 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 73, + 69, + 523, + 176 + ], + "spans": [ + { + "bbox": [ + 73, + 69, + 523, + 176 + ], + "score": 0.98, + "html": "
Dataset#DialogueAvg. #TurnsAvg. Dialog Length (by token)Avg.Utt.Length (by token)Lexical Diversity (↑)Topic Diversity (↓)Coherence (个)User Simulation
Self-Instruct82,439169.829.224.90.733No
Stanford Alpaca52.002191.164.542.80.727No
SODA1,486,8693.6231.822.538.60.7978.48No
GPT-4-LLM61,0021179.6142.948.90.721-No
BELLE1,436,6791102.363.335.90.771-No
Baize210,3113.1293.952.867.10.7519.06Yes
GPT4ALL711,1261597.7318.962.70.6921No
UltraChat1,468,3523.81467.4309.374.30.7029.06Yes
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Lexical diversity is calculated by averaging the MTLD score (Mc-", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 69, + 195, + 526, + 208 + ], + "spans": [ + { + "bbox": [ + 69, + 195, + 526, + 208 + ], + "score": 1.0, + "content": "Carthy and Jarvis, 2010) over each utterance with LexicalRichness6. 10000 samples are randomly drawn from each", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 68, + 208, + 526, + 222 + ], + "spans": [ + { + "bbox": [ + 68, + 208, + 526, + 222 + ], + "score": 1.0, + "content": "dataset for topic diversity and coherence measurement. Topic diversity is measured by averaging the cosine distance", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 69, + 221, + 506, + 232 + ], + "spans": [ + { + "bbox": [ + 69, + 221, + 506, + 232 + ], + "score": 1.0, + "content": "between each pair of data with OpenAI embedding API. Coherence is scored by ChatGPT on a scale of 1-10.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5, + "bbox_fs": [ + 68, + 183, + 527, + 232 + ] + }, + { + "type": "text", + "bbox": [ + 70, + 244, + 289, + 405 + ], + "lines": [ + { + "bbox": [ + 68, + 244, + 290, + 257 + ], + "spans": [ + { + "bbox": [ + 68, + 244, + 290, + 257 + ], + "score": 1.0, + "content": "can significantly deteriorate the coherence of the", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 69, + 258, + 291, + 270 + ], + "spans": [ + { + "bbox": [ + 69, + 258, + 291, + 270 + ], + "score": 1.0, + "content": "multi-turn conversation. To address this, in addi-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 69, + 271, + 290, + 284 + ], + "spans": [ + { + "bbox": [ + 69, + 271, + 290, + 284 + ], + "score": 1.0, + "content": "tion to presenting the dialogue history, we include", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 68, + 285, + 291, + 299 + ], + "spans": [ + { + "bbox": [ + 68, + 285, + 291, + 299 + ], + "score": 1.0, + "content": "prompts explicitly instructing the model to adopt", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 68, + 299, + 291, + 313 + ], + "spans": [ + { + "bbox": [ + 68, + 299, + 291, + 313 + ], + "score": 1.0, + "content": "various user personalities. In Sector 2, a prompt", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 68, + 311, + 291, + 326 + ], + "spans": [ + { + "bbox": [ + 68, + 311, + 291, + 326 + ], + "score": 1.0, + "content": "is employed to remind the model of the primary", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 68, + 325, + 291, + 340 + ], + "spans": [ + { + "bbox": [ + 68, + 325, + 291, + 340 + ], + "score": 1.0, + "content": "purpose of the dialogue, thereby promoting a more", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 68, + 338, + 291, + 353 + ], + "spans": [ + { + "bbox": [ + 68, + 338, + 291, + 353 + ], + "score": 1.0, + "content": "natural conversation flow. Once the data genera-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 68, + 352, + 290, + 365 + ], + "spans": [ + { + "bbox": [ + 68, + 352, + 290, + 365 + ], + "score": 1.0, + "content": "tion process is complete, a further filtration step is", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 68, + 367, + 290, + 379 + ], + "spans": [ + { + "bbox": [ + 68, + 367, + 290, + 379 + ], + "score": 1.0, + "content": "performed to ensure overall data quality. We also", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 68, + 380, + 290, + 393 + ], + "spans": [ + { + "bbox": [ + 68, + 380, + 290, + 393 + ], + "score": 1.0, + "content": "exclude excessively polite statements to enhance", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 69, + 393, + 201, + 407 + ], + "spans": [ + { + "bbox": [ + 69, + 393, + 201, + 407 + ], + "score": 1.0, + "content": "the realism of user responses.", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 12.5, + "bbox_fs": [ + 68, + 244, + 291, + 407 + ] + }, + { + "type": "title", + "bbox": [ + 70, + 418, + 159, + 432 + ], + "lines": [ + { + "bbox": [ + 68, + 418, + 161, + 434 + ], + "spans": [ + { + "bbox": [ + 68, + 418, + 161, + 434 + ], + "score": 1.0, + "content": "4 Data Analysis", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19 + }, + { + "type": "title", + "bbox": [ + 70, + 442, + 183, + 455 + ], + "lines": [ + { + "bbox": [ + 68, + 442, + 185, + 456 + ], + "spans": [ + { + "bbox": [ + 68, + 442, + 185, + 456 + ], + "score": 1.0, + "content": "4.1 Statistical Analysis", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 70, + 463, + 290, + 772 + ], + "lines": [ + { + "bbox": [ + 69, + 463, + 290, + 475 + ], + "spans": [ + { + "bbox": [ + 69, + 463, + 290, + 475 + ], + "score": 1.0, + "content": "We conduct a statistical analysis of UltraChat", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 68, + 477, + 290, + 489 + ], + "spans": [ + { + "bbox": [ + 68, + 477, + 290, + 489 + ], + "score": 1.0, + "content": "and several other instruction datasets, as shown", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 69, + 490, + 290, + 502 + ], + "spans": [ + { + "bbox": [ + 69, + 490, + 290, + 502 + ], + "score": 1.0, + "content": "in Table 4. UltraChat stands out in terms of its", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 68, + 504, + 290, + 518 + ], + "spans": [ + { + "bbox": [ + 68, + 504, + 290, + 518 + ], + "score": 1.0, + "content": "scale, being one of the largest publicly available", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 69, + 516, + 290, + 532 + ], + "spans": [ + { + "bbox": [ + 69, + 516, + 290, + 532 + ], + "score": 1.0, + "content": "datasets. Moreover, it exhibits the highest average", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 69, + 531, + 291, + 544 + ], + "spans": [ + { + "bbox": [ + 69, + 531, + 291, + 544 + ], + "score": 1.0, + "content": "number of turns and the longest average length per", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 69, + 545, + 290, + 557 + ], + "spans": [ + { + "bbox": [ + 69, + 545, + 290, + 557 + ], + "score": 1.0, + "content": "instance of data. While SODA (Kim et al., 2023)", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 558, + 291, + 570 + ], + "spans": [ + { + "bbox": [ + 69, + 558, + 291, + 570 + ], + "score": 1.0, + "content": "also has many rounds, it is primarily composed of", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 69, + 572, + 292, + 583 + ], + "spans": [ + { + "bbox": [ + 69, + 572, + 292, + 583 + ], + "score": 1.0, + "content": "conceptual banter rather than instructional content.", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 68, + 585, + 291, + 599 + ], + "spans": [ + { + "bbox": [ + 68, + 585, + 291, + 599 + ], + "score": 1.0, + "content": "Additionally, the average number of tokens per", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 68, + 598, + 291, + 611 + ], + "spans": [ + { + "bbox": [ + 68, + 598, + 291, + 611 + ], + "score": 1.0, + "content": "dialogue in SODA is 231.8, whereas UltraChat", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 68, + 612, + 291, + 624 + ], + "spans": [ + { + "bbox": [ + 68, + 612, + 291, + 624 + ], + "score": 1.0, + "content": "boasts a remarkable 1467.4 tokens. To evaluate di-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 68, + 625, + 290, + 639 + ], + "spans": [ + { + "bbox": [ + 68, + 625, + 290, + 639 + ], + "score": 1.0, + "content": "versity, we measure both lexical diversity and topic", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 69, + 639, + 290, + 652 + ], + "spans": [ + { + "bbox": [ + 69, + 639, + 290, + 652 + ], + "score": 1.0, + "content": "diversity. UltraChat outperforms previous datasets", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 68, + 653, + 291, + 666 + ], + "spans": [ + { + "bbox": [ + 68, + 653, + 291, + 666 + ], + "score": 1.0, + "content": "in terms of lexical diversity. However, in terms of", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 68, + 666, + 291, + 680 + ], + "spans": [ + { + "bbox": [ + 68, + 666, + 291, + 680 + ], + "score": 1.0, + "content": "topic diversity, UltraChat falls slightly short com-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 680, + 290, + 693 + ], + "spans": [ + { + "bbox": [ + 68, + 680, + 290, + 693 + ], + "score": 1.0, + "content": "pared to GPT4ALL (Anand et al., 2023) but still", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 68, + 694, + 290, + 707 + ], + "spans": [ + { + "bbox": [ + 68, + 694, + 290, + 707 + ], + "score": 1.0, + "content": "surpasses other datasets significantly. This may be", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 68, + 706, + 291, + 721 + ], + "spans": [ + { + "bbox": [ + 68, + 706, + 291, + 721 + ], + "score": 1.0, + "content": "attributed to the regularized embeddings resulting", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 69, + 721, + 291, + 733 + ], + "spans": [ + { + "bbox": [ + 69, + 721, + 291, + 733 + ], + "score": 1.0, + "content": "from a large number of tokens in each dialogue. We", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 69, + 734, + 291, + 747 + ], + "spans": [ + { + "bbox": [ + 69, + 734, + 291, + 747 + ], + "score": 1.0, + "content": "also conduct coherence evaluation with ChatGPT", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 69, + 748, + 290, + 760 + ], + "spans": [ + { + "bbox": [ + 69, + 748, + 290, + 760 + ], + "score": 1.0, + "content": "for multi-turn datasets. Notably, UltraChat and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 69, + 761, + 289, + 774 + ], + "spans": [ + { + "bbox": [ + 69, + 761, + 289, + 774 + ], + "score": 1.0, + "content": "Baize data rank the highest in terms of coherence.", + "type": "text" + } + ], + "index": 43 + } + ], + "index": 32, + "bbox_fs": [ + 68, + 463, + 292, + 774 + ] + }, + { + "type": "title", + "bbox": [ + 306, + 244, + 421, + 257 + ], + "lines": [ + { + "bbox": [ + 303, + 243, + 423, + 258 + ], + "spans": [ + { + "bbox": [ + 303, + 243, + 423, + 258 + ], + "score": 1.0, + "content": "4.2 Human Assessment", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 44 + }, + { + "type": "text", + "bbox": [ + 305, + 269, + 525, + 604 + ], + "lines": [ + { + "bbox": [ + 304, + 268, + 526, + 281 + ], + "spans": [ + { + "bbox": [ + 304, + 268, + 526, + 281 + ], + "score": 1.0, + "content": "Setup. To better evaluate the constructed data", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 281, + 526, + 294 + ], + "spans": [ + { + "bbox": [ + 304, + 281, + 526, + 294 + ], + "score": 1.0, + "content": "quality, we also conduct human assessment for", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 295, + 526, + 307 + ], + "spans": [ + { + "bbox": [ + 304, + 295, + 526, + 307 + ], + "score": 1.0, + "content": "UltraChat. 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For each round of conversation, we", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 304, + 376, + 525, + 389 + ], + "spans": [ + { + "bbox": [ + 304, + 376, + 525, + 389 + ], + "score": 1.0, + "content": "ask the annotators to score the assistant’s response", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 390, + 526, + 403 + ], + "spans": [ + { + "bbox": [ + 304, + 390, + 526, + 403 + ], + "score": 1.0, + "content": "on Helpfulness, Honesty, and Harmlessness (3H)", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 303, + 403, + 526, + 416 + ], + "spans": [ + { + "bbox": [ + 303, + 403, + 526, + 416 + ], + "score": 1.0, + "content": "principles (Askell et al., 2021). We also devise", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 417, + 525, + 430 + ], + "spans": [ + { + "bbox": [ + 304, + 417, + 525, + 430 + ], + "score": 1.0, + "content": "Coherence and Consistency criteria for the overall", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 304, + 430, + 526, + 443 + ], + "spans": [ + { + "bbox": [ + 304, + 430, + 526, + 443 + ], + "score": 1.0, + "content": "multi-turn dialogue quality evaluation. Coherence", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 304, + 444, + 526, + 457 + ], + "spans": [ + { + "bbox": [ + 304, + 444, + 526, + 457 + ], + "score": 1.0, + "content": "evaluates whether the dialogue flows logically and", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 458, + 525, + 470 + ], + "spans": [ + { + "bbox": [ + 304, + 458, + 525, + 470 + ], + "score": 1.0, + "content": "coherently, for which the annotators evaluate both", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 471, + 527, + 484 + ], + "spans": [ + { + "bbox": [ + 304, + 471, + 527, + 484 + ], + "score": 1.0, + "content": "the user’s response and the assistant’s response.", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 304, + 485, + 526, + 498 + ], + "spans": [ + { + "bbox": [ + 304, + 485, + 526, + 498 + ], + "score": 1.0, + "content": "Consistency means the assistant’s responses do not", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 304, + 499, + 526, + 511 + ], + "spans": [ + { + "bbox": [ + 304, + 499, + 526, + 511 + ], + "score": 1.0, + "content": "contradict each other within the same dialogue. For", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 305, + 513, + 525, + 524 + ], + "spans": [ + { + "bbox": [ + 305, + 513, + 525, + 524 + ], + "score": 1.0, + "content": "example, it is inconsistent if the assistant asserts", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 304, + 526, + 526, + 538 + ], + "spans": [ + { + "bbox": [ + 304, + 526, + 526, + 538 + ], + "score": 1.0, + "content": "one specific event occurred in 1911 in the first", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 304, + 540, + 525, + 551 + ], + "spans": [ + { + "bbox": [ + 304, + 540, + 525, + 551 + ], + "score": 1.0, + "content": "round of conversation but mentions it as a 1901", + "type": "text" + } + ], + "index": 65 + }, + { + "bbox": [ + 304, + 553, + 525, + 564 + ], + "spans": [ + { + "bbox": [ + 304, + 553, + 525, + 564 + ], + "score": 1.0, + "content": "event in the next round. Each metric is scored with", + "type": "text" + } + ], + "index": 66 + }, + { + "bbox": [ + 304, + 566, + 526, + 579 + ], + "spans": [ + { + "bbox": [ + 304, + 566, + 526, + 579 + ], + "score": 1.0, + "content": "0, 0.5 or 1, where higher score means better quality.", + "type": "text" + } + ], + "index": 67 + }, + { + "bbox": [ + 304, + 580, + 525, + 593 + ], + "spans": [ + { + "bbox": [ + 304, + 580, + 491, + 593 + ], + "score": 1.0, + "content": "Therefore, for a K-round dialogue, we have", + "type": "text" + }, + { + "bbox": [ + 491, + 580, + 525, + 592 + ], + "score": 0.88, + "content": "3 K + 2", + "type": "inline_equation" + } + ], + "index": 68 + }, + { + "bbox": [ + 304, + 595, + 391, + 605 + ], + "spans": [ + { + "bbox": [ + 304, + 595, + 391, + 605 + ], + "score": 1.0, + "content": "metric annotations.", + "type": "text" + } + ], + "index": 69 + } + ], + "index": 57, + "bbox_fs": [ + 303, + 268, + 527, + 605 + ] + }, + { + "type": "text", + "bbox": [ + 305, + 626, + 525, + 773 + ], + "lines": [ + { + "bbox": [ + 304, + 624, + 527, + 639 + ], + "spans": [ + { + "bbox": [ + 304, + 624, + 527, + 639 + ], + "score": 1.0, + "content": "Annotation. Each dialogue is annotated indepen-", + "type": "text" + } + ], + "index": 70 + }, + { + "bbox": [ + 305, + 639, + 525, + 651 + ], + "spans": [ + { + "bbox": [ + 305, + 639, + 525, + 651 + ], + "score": 1.0, + "content": "dently by two well-trained annotators, and the score", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 304, + 653, + 525, + 666 + ], + "spans": [ + { + "bbox": [ + 304, + 653, + 525, + 666 + ], + "score": 1.0, + "content": "is averaged across two annotators. Meanwhile, due", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 304, + 667, + 525, + 679 + ], + "spans": [ + { + "bbox": [ + 304, + 667, + 525, + 679 + ], + "score": 1.0, + "content": "to the difficulty in identifying the hallucination", + "type": "text" + } + ], + "index": 73 + }, + { + "bbox": [ + 303, + 680, + 527, + 693 + ], + "spans": [ + { + "bbox": [ + 303, + 680, + 527, + 693 + ], + "score": 1.0, + "content": "problem, we allow the annotators to skip the di-", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 304, + 694, + 525, + 707 + ], + "spans": [ + { + "bbox": [ + 304, + 694, + 525, + 707 + ], + "score": 1.0, + "content": "alogues that require expert knowledge or whose", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 304, + 707, + 526, + 720 + ], + "spans": [ + { + "bbox": [ + 304, + 707, + 526, + 720 + ], + "score": 1.0, + "content": "validity is hard to check. 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The average time", + "type": "text" + } + ], + "index": 79 + }, + { + "bbox": [ + 304, + 762, + 478, + 774 + ], + "spans": [ + { + "bbox": [ + 304, + 762, + 478, + 774 + ], + "score": 1.0, + "content": "to annotate one dialogue is 10 minutes.", + "type": "text" + } + ], + "index": 80 + } + ], + "index": 75, + "bbox_fs": [ + 303, + 624, + 527, + 774 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 69, + 72, + 290, + 368 + ], + "lines": [ + { + "bbox": [ + 69, + 71, + 290, + 84 + ], + "spans": [ + { + "bbox": [ + 69, + 71, + 290, + 84 + ], + "score": 1.0, + "content": "Results. 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The", + "type": "text" + } + ], + "index": 81 + }, + { + "bbox": [ + 304, + 543, + 525, + 557 + ], + "spans": [ + { + "bbox": [ + 304, + 543, + 525, + 557 + ], + "score": 1.0, + "content": "four datasets prove to be challenging even for the", + "type": "text" + } + ], + "index": 82 + }, + { + "bbox": [ + 303, + 557, + 518, + 570 + ], + "spans": [ + { + "bbox": [ + 303, + 557, + 518, + 570 + ], + "score": 1.0, + "content": "best-performing language models like ChatGPT.", + "type": "text" + } + ], + "index": 83 + } + ], + "index": 78 + }, + { + "type": "text", + "bbox": [ + 305, + 572, + 526, + 732 + ], + "lines": [ + { + "bbox": [ + 314, + 570, + 526, + 584 + ], + "spans": [ + { + "bbox": [ + 314, + 570, + 526, + 584 + ], + "score": 1.0, + "content": "For response quality evaluation, we use 3", + "type": "text" + } + ], + "index": 84 + }, + { + "bbox": [ + 304, + 583, + 527, + 599 + ], + "spans": [ + { + "bbox": [ + 304, + 583, + 527, + 599 + ], + "score": 1.0, + "content": "datasets. 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Apart", + "type": "text" + } + ], + "index": 93 + }, + { + "bbox": [ + 304, + 706, + 527, + 720 + ], + "spans": [ + { + "bbox": [ + 304, + 706, + 527, + 720 + ], + "score": 1.0, + "content": "from the curated set, we also adopt AlpacaE-", + "type": "text" + } + ], + "index": 94 + }, + { + "bbox": [ + 304, + 719, + 525, + 734 + ], + "spans": [ + { + "bbox": [ + 304, + 719, + 525, + 734 + ], + "score": 1.0, + "content": "val (Li et al., 2023b), a widely acknowledged", + "type": "text" + } + ], + "index": 95 + } + ], + "index": 89.5 + } + ], + "page_idx": 5, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 304, + 743, + 525, + 772 + ], + "lines": [ + { + "bbox": [ + 316, + 740, + 526, + 756 + ], + "spans": [ + { + "bbox": [ + 316, + 740, + 526, + 756 + ], + "score": 1.0, + "content": "7Some baselines used in our experiments are continuously", + "type": "text" + } + ] + }, + { + "bbox": [ + 304, + 752, + 526, + 763 + ], + "spans": [ + { + "bbox": [ + 304, + 752, + 526, + 763 + ], + "score": 1.0, + "content": "updated. 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ModelARC-ChallengeHellaSwagMMLUTruthfulQAOverall Average
Acc.Acc. norm.Acc.Acc. norm.WeightedUnweightedmc1mc2
Dolly-12B38.2342.2454.5972.631.5231.7020.6934.0645.15
OpenAssistant-12B41.3845.9052.5170.0429.7730.2924.6039.2946.38
MPT-7B43.0046.6757.1375.5037.7638.3327.1740.1650.17
Alpaca-7B49.7452.6558.0576.9142.4742.9025.8339.5553.00
LLaMA-13B53.1656.4060.6480.8746.0546.7425.8339.9055.98
Baize-13B55.5557.9459.9680.3648.1349.0332.9347.4358.69
Koala-13B49.8352.9057.6077.5446.7548.0134.6450.0957.14
Vicuna-13B51.7152.9060.0380.1250.1550.4535.7451.8258.83
WizardLM-13B55.1257.0860.9380.9151.6952.2535.3750.5360.19
LLaMA-65B59.2263.3166.4086.0562.2962.9727.9142.5563.72
UltraLM-13B57.2559.2261.3281.4950.4551.1036.7252.0060.95
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It", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 69, + 531, + 291, + 543 + ], + "spans": [ + { + "bbox": [ + 69, + 531, + 291, + 543 + ], + "score": 1.0, + "content": "is worth noting that UltraLM overtakes the cur-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 69, + 544, + 291, + 557 + ], + "spans": [ + { + "bbox": [ + 69, + 545, + 231, + 557 + ], + "score": 1.0, + "content": "rent state-of-the-art model by nearly", + "type": "text" + }, + { + "bbox": [ + 232, + 544, + 248, + 556 + ], + "score": 0.85, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 545, + 291, + 557 + ], + "score": 1.0, + "content": "on ARC-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 69, + 558, + 290, + 570 + ], + "spans": [ + { + "bbox": [ + 69, + 558, + 290, + 570 + ], + "score": 1.0, + "content": "Challenge and TruthfulQA. 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For our curated evalu-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 304, + 385, + 526, + 397 + ], + "spans": [ + { + "bbox": [ + 304, + 385, + 526, + 397 + ], + "score": 1.0, + "content": "ation set, we conduct pairwise evaluation between", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 304, + 398, + 527, + 410 + ], + "spans": [ + { + "bbox": [ + 304, + 398, + 527, + 410 + ], + "score": 1.0, + "content": "UltraLM and each baseline model with GPT-4.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 411, + 527, + 425 + ], + "spans": [ + { + "bbox": [ + 304, + 411, + 527, + 425 + ], + "score": 1.0, + "content": "Our evaluation prompt is designed to prioritize cor-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 425, + 527, + 437 + ], + "spans": [ + { + "bbox": [ + 304, + 425, + 527, + 437 + ], + "score": 1.0, + "content": "rectness over other factors such as informativeness.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 438, + 525, + 451 + ], + "spans": [ + { + "bbox": [ + 304, + 438, + 525, + 451 + ], + "score": 1.0, + "content": "To mitigate the influence of presentation order", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 304, + 452, + 525, + 465 + ], + "spans": [ + { + "bbox": [ + 304, + 452, + 525, + 465 + ], + "score": 1.0, + "content": "of responses, we randomly determine the order", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 303, + 465, + 526, + 479 + ], + "spans": [ + { + "bbox": [ + 303, + 465, + 526, + 479 + ], + "score": 1.0, + "content": "of the responses for each question. 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UltraLM demonstrates superior", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 303, + 519, + 526, + 534 + ], + "spans": [ + { + "bbox": [ + 303, + 519, + 526, + 534 + ], + "score": 1.0, + "content": "performance compared to every open-source", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 533, + 527, + 547 + ], + "spans": [ + { + "bbox": [ + 304, + 533, + 527, + 547 + ], + "score": 1.0, + "content": "model, exhibiting an impressive winning rate of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 546, + 525, + 559 + ], + "spans": [ + { + "bbox": [ + 304, + 547, + 331, + 559 + ], + "score": 1.0, + "content": "up to", + "type": "text" + }, + { + "bbox": [ + 331, + 546, + 353, + 558 + ], + "score": 0.86, + "content": "98 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 547, + 525, + 559 + ], + "score": 1.0, + "content": ". 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Given the instability of", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 303, + 598, + 525, + 612 + ], + "spans": [ + { + "bbox": [ + 303, + 598, + 525, + 612 + ], + "score": 1.0, + "content": "pairwise comparison, we also conduct independent", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 612, + 526, + 626 + ], + "spans": [ + { + "bbox": [ + 304, + 612, + 526, + 626 + ], + "score": 1.0, + "content": "quality scoring with GPT-4, as presented in", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 303, + 625, + 527, + 640 + ], + "spans": [ + { + "bbox": [ + 303, + 625, + 527, + 640 + ], + "score": 1.0, + "content": "Table 7. Notably, our model demonstrates superior", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 304, + 640, + 527, + 653 + ], + "spans": [ + { + "bbox": [ + 304, + 640, + 527, + 653 + ], + "score": 1.0, + "content": "performance compared to all the open-source coun-", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 303, + 653, + 525, + 666 + ], + "spans": [ + { + "bbox": [ + 303, + 653, + 525, + 666 + ], + "score": 1.0, + "content": "terparts by a significant margin in terms of overall", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 303, + 666, + 525, + 680 + ], + "spans": [ + { + "bbox": [ + 303, + 666, + 525, + 680 + ], + "score": 1.0, + "content": "scores. 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ModelARC-ChallengeHellaSwagMMLUTruthfulQAOverall Average
Acc.Acc. norm.Acc.Acc. norm.WeightedUnweightedmc1mc2
Dolly-12B38.2342.2454.5972.631.5231.7020.6934.0645.15
OpenAssistant-12B41.3845.9052.5170.0429.7730.2924.6039.2946.38
MPT-7B43.0046.6757.1375.5037.7638.3327.1740.1650.17
Alpaca-7B49.7452.6558.0576.9142.4742.9025.8339.5553.00
LLaMA-13B53.1656.4060.6480.8746.0546.7425.8339.9055.98
Baize-13B55.5557.9459.9680.3648.1349.0332.9347.4358.69
Koala-13B49.8352.9057.6077.5446.7548.0134.6450.0957.14
Vicuna-13B51.7152.9060.0380.1250.1550.4535.7451.8258.83
WizardLM-13B55.1257.0860.9380.9151.6952.2535.3750.5360.19
LLaMA-65B59.2263.3166.4086.0562.2962.9727.9142.5563.72
UltraLM-13B57.2559.2261.3281.4950.4551.1036.7252.0060.95
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All evaluation and metric calculations follow", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 69, + 253, + 525, + 265 + ], + "spans": [ + { + "bbox": [ + 69, + 253, + 525, + 265 + ], + "score": 1.0, + "content": "EleutherAI’s lm-evaluation-harness (Gao et al., 2021). Both weighted and unweighted mean accuracy are reported", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 69, + 264, + 525, + 277 + ], + "spans": [ + { + "bbox": [ + 69, + 264, + 525, + 277 + ], + "score": 1.0, + "content": "for MMLU as there are 57 tasks. The overall average metric is obtained by averaging the second column data for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 69, + 277, + 447, + 289 + ], + "spans": [ + { + "bbox": [ + 69, + 277, + 447, + 289 + ], + "score": 1.0, + "content": "each benchmark dataset. More details about metric calculation can be found in Appendix A.3.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 4.5 + } + ], + "index": 2.75 + }, + { + "type": "text", + "bbox": [ + 69, + 311, + 290, + 445 + ], + "lines": [], + "index": 11.5, + "bbox_fs": [ + 68, + 311, + 292, + 445 + ], + "lines_deleted": true + }, + { + "type": "title", + "bbox": [ + 70, + 457, + 203, + 470 + ], + "lines": [ + { + "bbox": [ + 68, + 456, + 204, + 471 + ], + "spans": [ + { + "bbox": [ + 68, + 456, + 204, + 471 + ], + "score": 1.0, + "content": "5.2 Benchmark Evaluation", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 69, + 477, + 290, + 773 + ], + "lines": [ + { + "bbox": [ + 68, + 475, + 291, + 491 + ], + "spans": [ + { + "bbox": [ + 68, + 475, + 291, + 491 + ], + "score": 1.0, + "content": "As shown in Table 6, with pure instruction-tuning", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 68, + 488, + 290, + 504 + ], + "spans": [ + { + "bbox": [ + 68, + 488, + 290, + 504 + ], + "score": 1.0, + "content": "on the UltraChat dataset, UltraLM significantly", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 69, + 504, + 291, + 516 + ], + "spans": [ + { + "bbox": [ + 69, + 504, + 291, + 516 + ], + "score": 1.0, + "content": "improves over LLaMA-13B and achieves the best", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 69, + 518, + 291, + 530 + ], + "spans": [ + { + "bbox": [ + 69, + 518, + 291, + 530 + ], + "score": 1.0, + "content": "overall performance across four benchmarks. It", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 69, + 531, + 291, + 543 + ], + "spans": [ + { + "bbox": [ + 69, + 531, + 291, + 543 + ], + "score": 1.0, + "content": "is worth noting that UltraLM overtakes the cur-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 69, + 544, + 291, + 557 + ], + "spans": [ + { + "bbox": [ + 69, + 545, + 231, + 557 + ], + "score": 1.0, + "content": "rent state-of-the-art model by nearly", + "type": "text" + }, + { + "bbox": [ + 232, + 544, + 248, + 556 + ], + "score": 0.85, + "content": "2 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 248, + 545, + 291, + 557 + ], + "score": 1.0, + "content": "on ARC-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 69, + 558, + 290, + 570 + ], + "spans": [ + { + "bbox": [ + 69, + 558, + 290, + 570 + ], + "score": 1.0, + "content": "Challenge and TruthfulQA. It shows that UltraLM", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 68, + 572, + 291, + 585 + ], + "spans": [ + { + "bbox": [ + 68, + 572, + 291, + 585 + ], + "score": 1.0, + "content": "is equipped with both broad and profound compre-", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 68, + 585, + 291, + 598 + ], + "spans": [ + { + "bbox": [ + 68, + 585, + 291, + 598 + ], + "score": 1.0, + "content": "hension of the world and commonsense knowledge.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 68, + 598, + 291, + 612 + ], + "spans": [ + { + "bbox": [ + 68, + 598, + 291, + 612 + ], + "score": 1.0, + "content": "The improvement could be attributed to the system-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 612, + 290, + 625 + ], + "spans": [ + { + "bbox": [ + 69, + 612, + 290, + 625 + ], + "score": 1.0, + "content": "atic and comprehensive data construction process", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 69, + 626, + 290, + 639 + ], + "spans": [ + { + "bbox": [ + 69, + 626, + 290, + 639 + ], + "score": 1.0, + "content": "of UltraChat sector 1, which effectively extends", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 69, + 639, + 290, + 653 + ], + "spans": [ + { + "bbox": [ + 69, + 639, + 290, + 653 + ], + "score": 1.0, + "content": "and deepens the discussion about world knowledge", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 69, + 653, + 291, + 665 + ], + "spans": [ + { + "bbox": [ + 69, + 653, + 291, + 665 + ], + "score": 1.0, + "content": "in automatic conversation generation. 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As for HellaSwag, we notice that all", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 68, + 734, + 290, + 748 + ], + "spans": [ + { + "bbox": [ + 68, + 734, + 290, + 748 + ], + "score": 1.0, + "content": "models have only marginal improvement compared", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 68, + 747, + 290, + 761 + ], + "spans": [ + { + "bbox": [ + 68, + 747, + 290, + 761 + ], + "score": 1.0, + "content": "to LLaMA-13B. 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For our curated evalu-", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 304, + 385, + 526, + 397 + ], + "spans": [ + { + "bbox": [ + 304, + 385, + 526, + 397 + ], + "score": 1.0, + "content": "ation set, we conduct pairwise evaluation between", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 304, + 398, + 527, + 410 + ], + "spans": [ + { + "bbox": [ + 304, + 398, + 527, + 410 + ], + "score": 1.0, + "content": "UltraLM and each baseline model with GPT-4.", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 304, + 411, + 527, + 425 + ], + "spans": [ + { + "bbox": [ + 304, + 411, + 527, + 425 + ], + "score": 1.0, + "content": "Our evaluation prompt is designed to prioritize cor-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 425, + 527, + 437 + ], + "spans": [ + { + "bbox": [ + 304, + 425, + 527, + 437 + ], + "score": 1.0, + "content": "rectness over other factors such as informativeness.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 438, + 525, + 451 + ], + "spans": [ + { + "bbox": [ + 304, + 438, + 525, + 451 + ], + "score": 1.0, + "content": "To mitigate the influence of presentation order", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 304, + 452, + 525, + 465 + ], + "spans": [ + { + "bbox": [ + 304, + 452, + 525, + 465 + ], + "score": 1.0, + "content": "of responses, we randomly determine the order", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 303, + 465, + 526, + 479 + ], + "spans": [ + { + "bbox": [ + 303, + 465, + 526, + 479 + ], + "score": 1.0, + "content": "of the responses for each question. Finally, we", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 304, + 479, + 525, + 492 + ], + "spans": [ + { + "bbox": [ + 304, + 479, + 525, + 492 + ], + "score": 1.0, + "content": "count the number of Win/Tie/Lose times against", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 304, + 493, + 525, + 505 + ], + "spans": [ + { + "bbox": [ + 304, + 493, + 525, + 505 + ], + "score": 1.0, + "content": "each baseline model, and the result is presented", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 303, + 505, + 526, + 520 + ], + "spans": [ + { + "bbox": [ + 303, + 505, + 526, + 520 + ], + "score": 1.0, + "content": "in Figure 2. UltraLM demonstrates superior", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 303, + 519, + 526, + 534 + ], + "spans": [ + { + "bbox": [ + 303, + 519, + 526, + 534 + ], + "score": 1.0, + "content": "performance compared to every open-source", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 533, + 527, + 547 + ], + "spans": [ + { + "bbox": [ + 304, + 533, + 527, + 547 + ], + "score": 1.0, + "content": "model, exhibiting an impressive winning rate of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 546, + 525, + 559 + ], + "spans": [ + { + "bbox": [ + 304, + 547, + 331, + 559 + ], + "score": 1.0, + "content": "up to", + "type": "text" + }, + { + "bbox": [ + 331, + 546, + 353, + 558 + ], + "score": 0.86, + "content": "98 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 353, + 547, + 525, + 559 + ], + "score": 1.0, + "content": ". 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ModelVicuna SetCommonsenseWorld KnowledgeProfessional KnowledgeAbilityWritingOverall
EasyModerateEasyDifficultPhysicsBiologyMathReasoning
Dolly-12B4.753.503.933.104.134.875.472.702.034.514.04
MPT-7B7.255.578.205.535.877.838.405.973.977.256.67
LLaMA-13B6.858.438.438.578.507.908.406.976.736.797.49
OpenAssistant-12B7.888.137.809.137.508.108.206.575.177.757.65
Alpaca-7B7.589.178.839.308.738.138.806.706.278.058.04
Koala-13B8.009.209.079.008.938.539.077.335.308.408.23
Baize-13B8.409.039.109.038.938.838.807.438.308.108.50
Vicuna-13B8.489.679.509.379.309.239.338.076.908.768.78
WizardLM-13B8.559.709.309.579.509.279.538.278.208.838.95
ChatGPT9.159.679.609.809.609.179.739.339.138.959.31
UltraLM-13B8.989.709.509.479.409.279.878.776.808.909.00
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ModelWin Rate (%)Standard Error
GPT-495.280.72
Claude88.391.11
ChatGPT86.091.21
UltraLM-13B76.091.50
WizardLM-13B75.311.51
Guanaco-65B71.801.59
Vicuna-13B70.431.61
Oasst-RLHF-33B66.521.66
Text Davinci 00350.000.00
Falcon-40B-instruct45.711.75
Alpaca-7B26.461.54
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As shown in Table 8, UltraLM out-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 303, + 455, + 526, + 467 + ], + "spans": [ + { + "bbox": [ + 303, + 455, + 526, + 467 + ], + "score": 1.0, + "content": "performs existing models in terms of win rate on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 304, + 468, + 525, + 480 + ], + "spans": [ + { + "bbox": [ + 304, + 468, + 525, + 480 + ], + "score": 1.0, + "content": "the current AlpacaEval leaderboard and ranks 4th", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 304, + 482, + 527, + 493 + ], + "spans": [ + { + "bbox": [ + 304, + 482, + 527, + 493 + ], + "score": 1.0, + "content": "just below ChatGPT. This observation further testi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 304, + 496, + 525, + 507 + ], + "spans": [ + { + "bbox": [ + 304, + 496, + 525, + 507 + ], + "score": 1.0, + "content": "fies the response quality of UltraLM and is in line", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 509, + 526, + 521 + ], + "spans": [ + { + "bbox": [ + 304, + 509, + 526, + 521 + ], + "score": 1.0, + "content": "with results on our curated dataset. 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ModelVicuna SetCommonsenseWorld KnowledgeProfessional KnowledgeAbilityWritingOverall
EasyModerateEasyDifficultPhysicsBiologyMathReasoning
Dolly-12B4.753.503.933.104.134.875.472.702.034.514.04
MPT-7B7.255.578.205.535.877.838.405.973.977.256.67
LLaMA-13B6.858.438.438.578.507.908.406.976.736.797.49
OpenAssistant-12B7.888.137.809.137.508.108.206.575.177.757.65
Alpaca-7B7.589.178.839.308.738.138.806.706.278.058.04
Koala-13B8.009.209.079.008.938.539.077.335.308.408.23
Baize-13B8.409.039.109.038.938.838.807.438.308.108.50
Vicuna-13B8.489.679.509.379.309.239.338.076.908.768.78
WizardLM-13B8.559.709.309.579.509.279.538.278.208.838.95
ChatGPT9.159.679.609.809.609.179.739.339.138.959.31
UltraLM-13B8.989.709.509.479.409.279.878.776.808.909.00
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ModelWin Rate (%)Standard Error
GPT-495.280.72
Claude88.391.11
ChatGPT86.091.21
UltraLM-13B76.091.50
WizardLM-13B75.311.51
Guanaco-65B71.801.59
Vicuna-13B70.431.61
Oasst-RLHF-33B66.521.66
Text Davinci 00350.000.00
Falcon-40B-instruct45.711.75
Alpaca-7B26.461.54
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As shown in Table 8, UltraLM out-", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 303, + 455, + 526, + 467 + ], + "spans": [ + { + "bbox": [ + 303, + 455, + 526, + 467 + ], + "score": 1.0, + "content": "performs existing models in terms of win rate on", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 304, + 468, + 525, + 480 + ], + "spans": [ + { + "bbox": [ + 304, + 468, + 525, + 480 + ], + "score": 1.0, + "content": "the current AlpacaEval leaderboard and ranks 4th", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 304, + 482, + 527, + 493 + ], + "spans": [ + { + "bbox": [ + 304, + 482, + 527, + 493 + ], + "score": 1.0, + "content": "just below ChatGPT. This observation further testi-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 304, + 496, + 525, + 507 + ], + "spans": [ + { + "bbox": [ + 304, + 496, + 525, + 507 + ], + "score": 1.0, + "content": "fies the response quality of UltraLM and is in line", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 304, + 509, + 526, + 521 + ], + "spans": [ + { + "bbox": [ + 304, + 509, + 526, + 521 + ], + "score": 1.0, + "content": "with results on our curated dataset. 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Figure 3 shows the", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 303, + 599, + 527, + 611 + ], + "spans": [ + { + "bbox": [ + 303, + 599, + 527, + 611 + ], + "score": 1.0, + "content": "automatic comparison results against WizardLM-", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 303, + 612, + 525, + 624 + ], + "spans": [ + { + "bbox": [ + 303, + 612, + 525, + 624 + ], + "score": 1.0, + "content": "13B on Evol-Instruct test set. UltraLM overtakes", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 303, + 625, + 525, + 639 + ], + "spans": [ + { + "bbox": [ + 303, + 625, + 525, + 639 + ], + "score": 1.0, + "content": "WizardLM on most types of questions, with up to", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 304, + 638, + 527, + 652 + ], + "spans": [ + { + "bbox": [ + 304, + 639, + 326, + 651 + ], + "score": 0.83, + "content": "29 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 326, + 638, + 527, + 652 + ], + "score": 1.0, + "content": "increase in scores. It is important to acknowl-", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 304, + 653, + 527, + 665 + ], + "spans": [ + { + "bbox": [ + 304, + 653, + 527, + 665 + ], + "score": 1.0, + "content": "edge that WizardLM-13B is trained on the Evol-", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 304, + 667, + 525, + 679 + ], + "spans": [ + { + "bbox": [ + 304, + 667, + 525, + 679 + ], + "score": 1.0, + "content": "Instruct training set, making UltraLM’s success", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 304, + 680, + 527, + 693 + ], + "spans": [ + { + "bbox": [ + 304, + 680, + 527, + 693 + ], + "score": 1.0, + "content": "on the test set a noteworthy achievement. More-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 304, + 694, + 526, + 706 + ], + "spans": [ + { + "bbox": [ + 304, + 694, + 526, + 706 + ], + "score": 1.0, + "content": "over, the questions observed with the largest im-", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 304, + 708, + 525, + 720 + ], + "spans": [ + { + "bbox": [ + 304, + 708, + 525, + 720 + ], + "score": 1.0, + "content": "provement are mainly complex problems that often", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 303, + 721, + 525, + 733 + ], + "spans": [ + { + "bbox": [ + 303, + 721, + 525, + 733 + ], + "score": 1.0, + "content": "require synthesized abilities. It demonstrates the", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 304, + 735, + 525, + 746 + ], + "spans": [ + { + "bbox": [ + 304, + 735, + 525, + 746 + ], + "score": 1.0, + "content": "success of UltraChat design schema, which can", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 304, + 748, + 527, + 761 + ], + "spans": [ + { + "bbox": [ + 304, + 748, + 527, + 761 + ], + "score": 1.0, + "content": "comprehensively boost the versatile capabilities of", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 303, + 761, + 528, + 775 + ], + "spans": [ + { + "bbox": [ + 303, + 761, + 528, + 775 + ], + "score": 1.0, + "content": "language models. 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To illus-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 68, + 409, + 291, + 424 + ], + "spans": [ + { + "bbox": [ + 68, + 409, + 291, + 424 + ], + "score": 1.0, + "content": "trate this effect, we conduct an ablation response", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 68, + 425, + 291, + 436 + ], + "spans": [ + { + "bbox": [ + 68, + 425, + 291, + 436 + ], + "score": 1.0, + "content": "comparison on UltraLM. Table 9 reveals signifi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 69, + 438, + 290, + 452 + ], + "spans": [ + { + "bbox": [ + 69, + 438, + 290, + 452 + ], + "score": 1.0, + "content": "cant improvements in response quality brought by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 69, + 451, + 291, + 465 + ], + "spans": [ + { + "bbox": [ + 69, + 451, + 291, + 465 + ], + "score": 1.0, + "content": "system prompts across all tasks. We also inspect", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 69, + 464, + 290, + 478 + ], + "spans": [ + { + "bbox": [ + 69, + 464, + 290, + 478 + ], + "score": 1.0, + "content": "the detailed evaluation for deterministic questions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 69, + 479, + 290, + 491 + ], + "spans": [ + { + "bbox": [ + 69, + 479, + 290, + 491 + ], + "score": 1.0, + "content": "(commonsense and world knowledge) and find that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 69, + 491, + 290, + 505 + ], + "spans": [ + { + "bbox": [ + 69, + 491, + 290, + 505 + ], + "score": 1.0, + "content": "UltraLM without system prompt only incorrectly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 69, + 506, + 291, + 518 + ], + "spans": [ + { + "bbox": [ + 69, + 506, + 291, + 518 + ], + "score": 1.0, + "content": "answers one question. Thus, the main benefit of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 69, + 519, + 290, + 532 + ], + "spans": [ + { + "bbox": [ + 69, + 519, + 290, + 532 + ], + "score": 1.0, + "content": "the system prompt lies in enhanced informativeness", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 531, + 204, + 546 + ], + "spans": [ + { + "bbox": [ + 69, + 531, + 204, + 546 + ], + "score": 1.0, + "content": "rather than higher correctness.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 17.5 + }, + { + "type": "table", + "bbox": [ + 71, + 555, + 289, + 664 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 71, + 555, + 289, + 664 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 71, + 555, + 289, + 664 + ], + "spans": [ + { + "bbox": [ + 71, + 555, + 289, + 664 + ], + "score": 0.977, + "html": "
DataWin (%)Tie (%)Lose(%)
Vicuna Set36.335.028.8
Commonsense58.631.010.3
World Knowledge56.934.58.6
Professional Knowledge57.831.111.1
Math Ability46.713.340.0
Reasoning Ability46.733.320.0
Writing46.335.018.8
Overall49.132.018.9
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The scores are obtained by", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 68, + 183, + 379, + 195 + ], + "spans": [ + { + "bbox": [ + 68, + 183, + 351, + 195 + ], + "score": 1.0, + "content": "pairwise scoring with GPT-4, and WizardLM scores are considered as", + "type": "text" + }, + { + "bbox": [ + 351, + 183, + 375, + 194 + ], + "score": 0.86, + "content": "100 \\%", + "type": "inline_equation" + }, + { + "bbox": [ + 375, + 183, + 379, + 195 + ], + "score": 1.0, + "content": ".", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + } + ], + "index": 2.25 + }, + { + "type": "text", + "bbox": [ + 69, + 212, + 289, + 237 + ], + "lines": [], + "index": 5.5, + "bbox_fs": [ + 68, + 212, + 291, + 238 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 69, + 248, + 290, + 544 + ], + "lines": [ + { + "bbox": [ + 69, + 248, + 290, + 261 + ], + "spans": [ + { + "bbox": [ + 69, + 248, + 290, + 261 + ], + "score": 1.0, + "content": "Impact of System Prompts. 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Although system", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 68, + 289, + 290, + 302 + ], + "spans": [ + { + "bbox": [ + 68, + 289, + 290, + 302 + ], + "score": 1.0, + "content": "prompts are not embedded in UltraLM’s training", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 68, + 301, + 290, + 316 + ], + "spans": [ + { + "bbox": [ + 68, + 301, + 290, + 316 + ], + "score": 1.0, + "content": "data like others do (Chiang et al., 2023), they still", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 68, + 316, + 291, + 327 + ], + "spans": [ + { + "bbox": [ + 68, + 316, + 291, + 327 + ], + "score": 1.0, + "content": "appear to have a substantial influence on the re-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 69, + 329, + 291, + 343 + ], + "spans": [ + { + "bbox": [ + 69, + 329, + 291, + 343 + ], + "score": 1.0, + "content": "sponse style of the generated output. 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To illus-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 68, + 409, + 291, + 424 + ], + "spans": [ + { + "bbox": [ + 68, + 409, + 291, + 424 + ], + "score": 1.0, + "content": "trate this effect, we conduct an ablation response", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 68, + 425, + 291, + 436 + ], + "spans": [ + { + "bbox": [ + 68, + 425, + 291, + 436 + ], + "score": 1.0, + "content": "comparison on UltraLM. Table 9 reveals signifi-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 69, + 438, + 290, + 452 + ], + "spans": [ + { + "bbox": [ + 69, + 438, + 290, + 452 + ], + "score": 1.0, + "content": "cant improvements in response quality brought by", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 69, + 451, + 291, + 465 + ], + "spans": [ + { + "bbox": [ + 69, + 451, + 291, + 465 + ], + "score": 1.0, + "content": "system prompts across all tasks. We also inspect", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 69, + 464, + 290, + 478 + ], + "spans": [ + { + "bbox": [ + 69, + 464, + 290, + 478 + ], + "score": 1.0, + "content": "the detailed evaluation for deterministic questions", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 69, + 479, + 290, + 491 + ], + "spans": [ + { + "bbox": [ + 69, + 479, + 290, + 491 + ], + "score": 1.0, + "content": "(commonsense and world knowledge) and find that", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 69, + 491, + 290, + 505 + ], + "spans": [ + { + "bbox": [ + 69, + 491, + 290, + 505 + ], + "score": 1.0, + "content": "UltraLM without system prompt only incorrectly", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 69, + 506, + 291, + 518 + ], + "spans": [ + { + "bbox": [ + 69, + 506, + 291, + 518 + ], + "score": 1.0, + "content": "answers one question. Thus, the main benefit of", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 69, + 519, + 290, + 532 + ], + "spans": [ + { + "bbox": [ + 69, + 519, + 290, + 532 + ], + "score": 1.0, + "content": "the system prompt lies in enhanced informativeness", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 69, + 531, + 204, + 546 + ], + "spans": [ + { + "bbox": [ + 69, + 531, + 204, + 546 + ], + "score": 1.0, + "content": "rather than higher correctness.", + "type": "text" + } + ], + "index": 28 + } + ], + "index": 17.5, + "bbox_fs": [ + 68, + 248, + 291, + 546 + ] + }, + { + "type": "table", + "bbox": [ + 71, + 555, + 289, + 664 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 71, + 555, + 289, + 664 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 71, + 555, + 289, + 664 + ], + "spans": [ + { + "bbox": [ + 71, + 555, + 289, + 664 + ], + "score": 0.977, + "html": "
DataWin (%)Tie (%)Lose(%)
Vicuna Set36.335.028.8
Commonsense58.631.010.3
World Knowledge56.934.58.6
Professional Knowledge57.831.111.1
Math Ability46.713.340.0
Reasoning Ability46.733.320.0
Writing46.335.018.8
Overall49.132.018.9
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Below we give a detailed", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 305, + 119, + 425, + 130 + ], + "spans": [ + { + "bbox": [ + 305, + 119, + 425, + 130 + ], + "score": 1.0, + "content": "description of each dataset.", + "type": "text" + } + ], + "index": 49 + } + ], + "index": 48 + }, + { + "type": "text", + "bbox": [ + 305, + 133, + 525, + 226 + ], + "lines": [ + { + "bbox": [ + 303, + 132, + 526, + 146 + ], + "spans": [ + { + "bbox": [ + 303, + 132, + 526, + 146 + ], + "score": 1.0, + "content": "The AI2 Reasoning Challenge (ARC) (Clark", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 303, + 146, + 526, + 160 + ], + "spans": [ + { + "bbox": [ + 303, + 146, + 526, + 160 + ], + "score": 1.0, + "content": "et al., 2018) is comprised of advanced science", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 303, + 160, + 527, + 174 + ], + "spans": [ + { + "bbox": [ + 303, + 160, + 527, + 174 + ], + "score": 1.0, + "content": "questions and structured as multiple-choice ques-", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 303, + 173, + 527, + 187 + ], + "spans": [ + { + "bbox": [ + 303, + 173, + 527, + 187 + ], + "score": 1.0, + "content": "tions. 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We use", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 303, + 200, + 526, + 213 + ], + "spans": [ + { + "bbox": [ + 303, + 200, + 526, + 213 + ], + "score": 1.0, + "content": "the challenge partition here for evaluation, which", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 214, + 432, + 228 + ], + "spans": [ + { + "bbox": [ + 304, + 214, + 432, + 228 + ], + "score": 1.0, + "content": "contains 1172 test examples.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 53 + }, + { + "type": "text", + "bbox": [ + 305, + 228, + 525, + 349 + ], + "lines": [ + { + "bbox": [ + 304, + 228, + 527, + 241 + ], + "spans": [ + { + "bbox": [ + 304, + 228, + 527, + 241 + ], + "score": 1.0, + "content": "HellaSwag (Zellers et al., 2019) tests common-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 303, + 242, + 525, + 255 + ], + "spans": [ + { + "bbox": [ + 303, + 242, + 525, + 255 + ], + "score": 1.0, + "content": "sense inference ability by evaluating how well the", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 256, + 526, + 269 + ], + "spans": [ + { + "bbox": [ + 304, + 256, + 526, + 269 + ], + "score": 1.0, + "content": "language model can predict the remaining part of a", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 270, + 525, + 282 + ], + "spans": [ + { + "bbox": [ + 304, + 270, + 525, + 282 + ], + "score": 1.0, + "content": "sentence. 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We use", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 303, + 200, + 526, + 213 + ], + "spans": [ + { + "bbox": [ + 303, + 200, + 526, + 213 + ], + "score": 1.0, + "content": "the challenge partition here for evaluation, which", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 304, + 214, + 432, + 228 + ], + "spans": [ + { + "bbox": [ + 304, + 214, + 432, + 228 + ], + "score": 1.0, + "content": "contains 1172 test examples.", + "type": "text" + } + ], + "index": 56 + } + ], + "index": 53, + "bbox_fs": [ + 303, + 132, + 527, + 228 + ] + }, + { + "type": "text", + "bbox": [ + 305, + 228, + 525, + 349 + ], + "lines": [ + { + "bbox": [ + 304, + 228, + 527, + 241 + ], + "spans": [ + { + "bbox": [ + 304, + 228, + 527, + 241 + ], + "score": 1.0, + "content": "HellaSwag (Zellers et al., 2019) tests common-", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 303, + 242, + 525, + 255 + ], + "spans": [ + { + "bbox": [ + 303, + 242, + 525, + 255 + ], + "score": 1.0, + "content": "sense inference ability by evaluating how well the", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 304, + 256, + 526, + 269 + ], + "spans": [ + { + "bbox": [ + 304, + 256, + 526, + 269 + ], + "score": 1.0, + "content": "language model can predict the remaining part of a", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 304, + 270, + 525, + 282 + ], + "spans": [ + { + "bbox": [ + 304, + 270, + 525, + 282 + ], + "score": 1.0, + "content": "sentence. Each sample has 4 different text pieces", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 304, + 283, + 525, + 296 + ], + "spans": [ + { + "bbox": [ + 304, + 283, + 525, + 296 + ], + "score": 1.0, + "content": "as the candidate remaining part of a given sentence", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 304, + 296, + 526, + 309 + ], + "spans": [ + { + "bbox": [ + 304, + 296, + 526, + 309 + ], + "score": 1.0, + "content": "and only one of them is plausible. The task is", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 304, + 309, + 525, + 323 + ], + "spans": [ + { + "bbox": [ + 304, + 309, + 525, + 323 + ], + "score": 1.0, + "content": "shown to be easy for humans but challenging for", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 304, + 324, + 525, + 336 + ], + "spans": [ + { + "bbox": [ + 304, + 324, + 525, + 336 + ], + "score": 1.0, + "content": "language models. We use the validation split as", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 304, + 336, + 396, + 350 + ], + "spans": [ + { + "bbox": [ + 304, + 336, + 396, + 350 + ], + "score": 1.0, + "content": "in Gao et al. 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Its purpose is to determine the risks of", + "type": "text" + } + ], + "index": 74 + }, + { + "bbox": [ + 303, + 475, + 527, + 488 + ], + "spans": [ + { + "bbox": [ + 303, + 475, + 527, + 488 + ], + "score": 1.0, + "content": "producing false claims or spreading misinforma-", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 304, + 488, + 525, + 501 + ], + "spans": [ + { + "bbox": [ + 304, + 488, + 525, + 501 + ], + "score": 1.0, + "content": "tion. The benchmark consists of questions written", + "type": "text" + } + ], + "index": 76 + }, + { + "bbox": [ + 303, + 501, + 527, + 516 + ], + "spans": [ + { + "bbox": [ + 303, + 501, + 527, + 516 + ], + "score": 1.0, + "content": "in various styles, covering 38 different categories,", + "type": "text" + } + ], + "index": 77 + }, + { + "bbox": [ + 304, + 515, + 526, + 528 + ], + "spans": [ + { + "bbox": [ + 304, + 515, + 526, + 528 + ], + "score": 1.0, + "content": "and is designed to be challenging. 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TypeExample
Commonsense- EasyWhat is the primary source of energy for our planet?
Commonsense-ModerateWhat is the phenomenon that causes the change in pitch heard when a vehicle sounding ahorn approaches and recedes from an observer?
World Knowledge-EasyWhat is the freezing point of water in Fahrenheit?
World Knowledge-ModerateWhat is the Godel's Incompleteness Theorem?
Physics KnowledgeHow does quantum entanglement work and what are its implications for information transfer?
Biology KnowledgeWhat are the four main types of macromolecules found in living organisms?
MathWhat is the Taylor series expansion of the function eα?
Reasoning You have two buckets,one with red paint and one with blue paint.You take one cup fromthe red bucket and pour it into the blue bucket. Then you take one cup from the blue bucketand pour it back into the red bucket.Which is true: the red bucket has more blue paint, orthe blue bucket has more red paint?
WritingWrite a dialogue between two photons traveling at light speed.
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Dataset#ExamplesDomain
ARC-Challenge1172Grade-school
HellaSwag10042Commonsense
MMLU14042Academic
TruthfulQA817Truthfulness
AlpacaEval805Comprehensive
Evol-Instruct218Comprehensive
Our evaluation set831Comprehensive
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Figure 5 and Figure 6 present", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 303, + 408, + 526, + 420 + ], + "spans": [ + { + "bbox": [ + 303, + 408, + 526, + 420 + ], + "score": 1.0, + "content": "prompts used for pairwise scoring and independent", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 303, + 421, + 525, + 433 + ], + "spans": [ + { + "bbox": [ + 303, + 421, + 525, + 433 + ], + "score": 1.0, + "content": "scoring on our evaluation data. 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TypeExample
Commonsense- EasyWhat is the primary source of energy for our planet?
Commonsense-ModerateWhat is the phenomenon that causes the change in pitch heard when a vehicle sounding ahorn approaches and recedes from an observer?
World Knowledge-EasyWhat is the freezing point of water in Fahrenheit?
World Knowledge-ModerateWhat is the Godel's Incompleteness Theorem?
Physics KnowledgeHow does quantum entanglement work and what are its implications for information transfer?
Biology KnowledgeWhat are the four main types of macromolecules found in living organisms?
MathWhat is the Taylor series expansion of the function eα?
Reasoning You have two buckets,one with red paint and one with blue paint.You take one cup fromthe red bucket and pour it into the blue bucket. Then you take one cup from the blue bucketand pour it back into the red bucket.Which is true: the red bucket has more blue paint, orthe blue bucket has more red paint?
WritingWrite a dialogue between two photons traveling at light speed.
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Dataset#ExamplesDomain
ARC-Challenge1172Grade-school
HellaSwag10042Commonsense
MMLU14042Academic
TruthfulQA817Truthfulness
AlpacaEval805Comprehensive
Evol-Instruct218Comprehensive
Our evaluation set831Comprehensive
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"[The Start of Assistant 2’s Answer]", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 352, + 302, + 366 + ], + "spans": [ + { + "bbox": [ + 86, + 352, + 302, + 366 + ], + "score": 1.0, + "content": "[The End of Assistant 2’s Answer]", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 10.5 + }, + { + "type": "text", + "bbox": [ + 84, + 379, + 511, + 487 + ], + "lines": [ + { + "bbox": [ + 84, + 379, + 510, + 394 + ], + "spans": [ + { + "bbox": [ + 84, + 379, + 510, + 394 + ], + "score": 1.0, + "content": "We would like to request your feedback on the performance of two AI assistants in response to the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 394, + 510, + 407 + ], + "spans": [ + { + "bbox": [ + 84, + 394, + 510, + 407 + ], + "score": 1.0, + "content": "user question displayed above. Please rate the quality, helpfulness, level of details, and harmless", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 407, + 510, + 421 + ], + "spans": [ + { + "bbox": [ + 84, + 407, + 510, + 421 + ], + "score": 1.0, + "content": "of their responses. Each assistant receives an overall score on a scale of 1 to 10, where a higher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 84, + 421, + 511, + 435 + ], + "spans": [ + { + "bbox": [ + 84, + 421, + 511, + 435 + ], + "score": 1.0, + "content": "score indicates better overall performance. Please first output a single line containing only two", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 84, + 434, + 510, + 449 + ], + "spans": [ + { + "bbox": [ + 84, + 434, + 510, + 449 + ], + "score": 1.0, + "content": "values indicating the scores for Assistant 1 and 2, respectively. The two scores are separated by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 447, + 512, + 462 + ], + "spans": [ + { + "bbox": [ + 83, + 447, + 512, + 462 + ], + "score": 1.0, + "content": "a space. In the subsequent line, please provide a comprehensive explanation of your evaluation,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 461, + 511, + 475 + ], + "spans": [ + { + "bbox": [ + 84, + 461, + 511, + 475 + ], + "score": 1.0, + "content": "avoiding any potential bias and ensuring that the order in which the responses were presented does", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 475, + 474, + 488 + ], + "spans": [ + { + "bbox": [ + 84, + 475, + 474, + 488 + ], + "score": 1.0, + "content": "not affect your judgment. Please avoid same scores unless they exactly in the same level.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5 + }, + { + "type": "text", + "bbox": [ + 186, + 512, + 408, + 525 + ], + "lines": [ + { + "bbox": [ + 185, + 511, + 408, + 527 + ], + "spans": [ + { + "bbox": [ + 185, + 511, + 408, + 527 + ], + "score": 1.0, + "content": "Figure 5: Prompt for automatic comparison evaluation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 69, + 547, + 133, + 559 + ], + "lines": [ + { + "bbox": [ + 69, + 546, + 135, + 560 + ], + "spans": [ + { + "bbox": [ + 69, + 546, + 135, + 560 + ], + "score": 1.0, + "content": "evaluation set.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21 + } + ], + "page_idx": 14, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 81, + 71, + 130, + 82 + ], + "lines": [ + { + "bbox": [ + 81, + 71, + 130, + 82 + ], + "spans": [], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "image", + "bbox": [ + 82, + 74, + 525, + 188 + ], + "blocks": [ + { + "type": "image_body", + "bbox": [ + 82, + 74, + 525, + 188 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 82, + 74, + 525, + 188 + ], + "spans": [ + { + "bbox": [ + 82, + 74, + 525, + 188 + ], + "score": 0.174, + "type": "image", + "image_path": "22ff9763b51e3c10d46392162cfbeb706182eda9a777ec8755e36bda7bdea3a6.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 82, + 74, + 525, + 112.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 82, + 112.0, + 525, + 150.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 82, + 150.0, + 525, + 188.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 87, + 205, + 504, + 218 + ], + "lines": [ + { + "bbox": [ + 90, + 204, + 504, + 219 + ], + "spans": [ + { + "bbox": [ + 90, + 204, + 504, + 219 + ], + "score": 1.0, + "content": "Figure 4: Manually designed templates for concatenating existing materials and generated instructions.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 90, + 204, + 504, + 219 + ] + }, + { + "type": "title", + "bbox": [ + 81, + 230, + 234, + 242 + ], + "lines": [ + { + "bbox": [ + 80, + 229, + 235, + 244 + ], + "spans": [ + { + "bbox": [ + 80, + 229, + 235, + 244 + ], + "score": 1.0, + "content": "Comparison Evaluation Prompt", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 86, + 244, + 467, + 257 + ], + "lines": [ + { + "bbox": [ + 83, + 243, + 469, + 259 + ], + "spans": [ + { + "bbox": [ + 83, + 243, + 469, + 259 + ], + "score": 1.0, + "content": "You are a helpful, harmless and precise assistant for checking the quality of the answer.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 83, + 243, + 469, + 259 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 272, + 150, + 284 + ], + "lines": [ + { + "bbox": [ + 86, + 271, + 151, + 285 + ], + "spans": [ + { + "bbox": [ + 86, + 271, + 151, + 285 + ], + "score": 1.0, + "content": "[Question]", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 86, + 271, + 151, + 285 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 299, + 314, + 325 + ], + "lines": [ + { + "bbox": [ + 86, + 297, + 315, + 312 + ], + "spans": [ + { + "bbox": [ + 86, + 297, + 315, + 312 + ], + "score": 1.0, + "content": "[The Start of Assistant 1’s Answer]", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 86, + 312, + 302, + 326 + ], + "spans": [ + { + "bbox": [ + 86, + 312, + 302, + 326 + ], + "score": 1.0, + "content": "[The End of Assistant 1’s Answer]", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 86, + 338, + 315, + 352 + ], + "spans": [ + { + "bbox": [ + 86, + 338, + 315, + 352 + ], + "score": 1.0, + "content": "[The Start of Assistant 2’s Answer]", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 86, + 352, + 302, + 366 + ], + "spans": [ + { + "bbox": [ + 86, + 352, + 302, + 366 + ], + "score": 1.0, + "content": "[The End of Assistant 2’s Answer]", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 86, + 297, + 315, + 326 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 339, + 314, + 365 + ], + "lines": [], + "index": 10.5, + "bbox_fs": [ + 86, + 338, + 315, + 366 + ], + "lines_deleted": true + }, + { + "type": "text", + "bbox": [ + 84, + 379, + 511, + 487 + ], + "lines": [ + { + "bbox": [ + 84, + 379, + 510, + 394 + ], + "spans": [ + { + "bbox": [ + 84, + 379, + 510, + 394 + ], + "score": 1.0, + "content": "We would like to request your feedback on the performance of two AI assistants in response to the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 394, + 510, + 407 + ], + "spans": [ + { + "bbox": [ + 84, + 394, + 510, + 407 + ], + "score": 1.0, + "content": "user question displayed above. Please rate the quality, helpfulness, level of details, and harmless", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 407, + 510, + 421 + ], + "spans": [ + { + "bbox": [ + 84, + 407, + 510, + 421 + ], + "score": 1.0, + "content": "of their responses. Each assistant receives an overall score on a scale of 1 to 10, where a higher", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 84, + 421, + 511, + 435 + ], + "spans": [ + { + "bbox": [ + 84, + 421, + 511, + 435 + ], + "score": 1.0, + "content": "score indicates better overall performance. Please first output a single line containing only two", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 84, + 434, + 510, + 449 + ], + "spans": [ + { + "bbox": [ + 84, + 434, + 510, + 449 + ], + "score": 1.0, + "content": "values indicating the scores for Assistant 1 and 2, respectively. The two scores are separated by", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 447, + 512, + 462 + ], + "spans": [ + { + "bbox": [ + 83, + 447, + 512, + 462 + ], + "score": 1.0, + "content": "a space. In the subsequent line, please provide a comprehensive explanation of your evaluation,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 461, + 511, + 475 + ], + "spans": [ + { + "bbox": [ + 84, + 461, + 511, + 475 + ], + "score": 1.0, + "content": "avoiding any potential bias and ensuring that the order in which the responses were presented does", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 475, + 474, + 488 + ], + "spans": [ + { + "bbox": [ + 84, + 475, + 474, + 488 + ], + "score": 1.0, + "content": "not affect your judgment. Please avoid same scores unless they exactly in the same level.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 15.5, + "bbox_fs": [ + 83, + 379, + 512, + 488 + ] + }, + { + "type": "text", + "bbox": [ + 186, + 512, + 408, + 525 + ], + "lines": [ + { + "bbox": [ + 185, + 511, + 408, + 527 + ], + "spans": [ + { + "bbox": [ + 185, + 511, + 408, + 527 + ], + "score": 1.0, + "content": "Figure 5: Prompt for automatic comparison evaluation.", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20, + "bbox_fs": [ + 185, + 511, + 408, + 527 + ] + }, + { + "type": "text", + "bbox": [ + 69, + 547, + 133, + 559 + ], + "lines": [ + { + "bbox": [ + 69, + 546, + 135, + 560 + ], + "spans": [ + { + "bbox": [ + 69, + 546, + 135, + 560 + ], + "score": 1.0, + "content": "evaluation set.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 21, + "bbox_fs": [ + 69, + 546, + 135, + 560 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 81, + 76, + 218, + 89 + ], + "lines": [ + { + "bbox": [ + 80, + 74, + 218, + 91 + ], + "spans": [ + { + "bbox": [ + 80, + 74, + 218, + 91 + ], + "score": 1.0, + "content": "Independent Scoring Prompt", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 91, + 466, + 104 + ], + "lines": [ + { + "bbox": [ + 83, + 89, + 469, + 105 + ], + "spans": [ + { + "bbox": [ + 83, + 89, + 469, + 105 + ], + "score": 1.0, + "content": "You are a helpful, harmless and precise assistant for checking the quality of the answer.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 87, + 118, + 150, + 130 + ], + "lines": [ + { + "bbox": [ + 86, + 117, + 151, + 131 + ], + "spans": [ + { + "bbox": [ + 86, + 117, + 151, + 131 + ], + "score": 1.0, + "content": "[Question]", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 86, + 145, + 347, + 171 + ], + "lines": [ + { + "bbox": [ + 86, + 145, + 347, + 158 + ], + "spans": [ + { + "bbox": [ + 86, + 145, + 347, + 158 + ], + "score": 1.0, + "content": "[The Start of the AI Assistant’s Answer]", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 86, + 159, + 335, + 172 + ], + "spans": [ + { + "bbox": [ + 86, + 159, + 335, + 172 + ], + "score": 1.0, + "content": "[The End of the AI Assistant’s Answer]", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 84, + 185, + 511, + 252 + ], + "lines": [ + { + "bbox": [ + 84, + 185, + 511, + 199 + ], + "spans": [ + { + "bbox": [ + 84, + 185, + 511, + 199 + ], + "score": 1.0, + "content": "We would like to request your feedback on the performance of the AI assistant in response to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 84, + 199, + 510, + 212 + ], + "spans": [ + { + "bbox": [ + 84, + 199, + 510, + 212 + ], + "score": 1.0, + "content": "the user question displayed above. Please rate the quality, helpfulness, level of details, and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 85, + 214, + 509, + 225 + ], + "spans": [ + { + "bbox": [ + 85, + 214, + 509, + 225 + ], + "score": 1.0, + "content": "harmlessness of their responses. The assistant receives an overall score on a scale of 1 to 10, where", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 225, + 510, + 240 + ], + "spans": [ + { + "bbox": [ + 83, + 225, + 510, + 240 + ], + "score": 1.0, + "content": "a higher score indicates better overall performance. Please output \"Score: [an integer number", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 84, + 240, + 171, + 253 + ], + "spans": [ + { + "bbox": [ + 84, + 240, + 171, + 253 + ], + "score": 1.0, + "content": "between 1 and 10]\"", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7 + }, + { + "type": "title", + "bbox": [ + 81, + 309, + 160, + 322 + ], + "lines": [ + { + "bbox": [ + 79, + 308, + 163, + 324 + ], + "spans": [ + { + "bbox": [ + 79, + 308, + 163, + 324 + ], + "score": 1.0, + "content": "System Prompt", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 84, + 324, + 511, + 364 + ], + "lines": [ + { + "bbox": [ + 84, + 324, + 511, + 337 + ], + "spans": [ + { + "bbox": [ + 84, + 324, + 511, + 337 + ], + "score": 1.0, + "content": "You are now a helpful assistant, your answer should focus on the question, give reasonable, self-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 84, + 338, + 510, + 351 + ], + "spans": [ + { + "bbox": [ + 84, + 338, + 510, + 351 + ], + "score": 1.0, + "content": "consistent, high-quality solutions to make users satisfied. Please try to make the answer more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 351, + 230, + 364 + ], + "spans": [ + { + "bbox": [ + 85, + 351, + 230, + 364 + ], + "score": 1.0, + "content": "informative, detailed, and polite.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 85, + 423, + 276, + 437 + ], + "lines": [ + { + "bbox": [ + 83, + 420, + 277, + 439 + ], + "spans": [ + { + "bbox": [ + 83, + 420, + 277, + 439 + ], + "score": 1.0, + "content": "Example A: Question of World Knowledge", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 87, + 446, + 327, + 458 + ], + "lines": [ + { + "bbox": [ + 85, + 446, + 330, + 460 + ], + "spans": [ + { + "bbox": [ + 85, + 446, + 330, + 460 + ], + "score": 1.0, + "content": "User: Who discovered the law of universal gravitation?", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15 + }, + { + "type": "text", + "bbox": [ + 84, + 460, + 512, + 568 + ], + "lines": [ + { + "bbox": [ + 84, + 459, + 512, + 474 + ], + "spans": [ + { + "bbox": [ + 84, + 459, + 512, + 474 + ], + "score": 1.0, + "content": "UltraLM: The law of universal gravitation was discovered by Sir Isaac Newton in the 17th century.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 84, + 473, + 510, + 487 + ], + "spans": [ + { + "bbox": [ + 84, + 473, + 510, + 487 + ], + "score": 1.0, + "content": "He formulated the law while studying the planets’ movement around the sun and realized that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 487, + 510, + 501 + ], + "spans": [ + { + "bbox": [ + 84, + 487, + 510, + 501 + ], + "score": 1.0, + "content": "there was a quantifiable force directing them towards the sun. In 1687, he published his book", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 501, + 510, + 513 + ], + "spans": [ + { + "bbox": [ + 84, + 501, + 510, + 513 + ], + "score": 1.0, + "content": "titled \"Mathematical Principles of Natural Philosophy,\" where he described the law of universal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 83, + 514, + 510, + 528 + ], + "spans": [ + { + "bbox": [ + 83, + 514, + 510, + 528 + ], + "score": 1.0, + "content": "gravitation in detail, outlining how two objects experience an attraction towards each other due", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 83, + 527, + 510, + 541 + ], + "spans": [ + { + "bbox": [ + 83, + 527, + 510, + 541 + ], + "score": 1.0, + "content": "to their mass and distance between them. This discovery revolutionized our understanding of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 84, + 542, + 510, + 554 + ], + "spans": [ + { + "bbox": [ + 84, + 542, + 510, + 554 + ], + "score": 1.0, + "content": "universe and ushered in the Age of Enlightenment. Today, Newton’s law continues to form the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 84, + 555, + 339, + 568 + ], + "spans": [ + { + "bbox": [ + 84, + 555, + 339, + 568 + ], + "score": 1.0, + "content": "foundation of modern astrophysics and space exploration.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5 + }, + { + "type": "title", + "bbox": [ + 85, + 598, + 275, + 611 + ], + "lines": [ + { + "bbox": [ + 83, + 595, + 277, + 614 + ], + "spans": [ + { + "bbox": [ + 83, + 595, + 277, + 614 + ], + "score": 1.0, + "content": "Example B: Question of World Knowledge", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 84, + 621, + 333, + 633 + ], + "lines": [ + { + "bbox": [ + 84, + 619, + 333, + 635 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 333, + 635 + ], + "score": 1.0, + "content": "User: Who was the 16th President of the United States?", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25 + }, + { + "type": "text", + "bbox": [ + 84, + 635, + 511, + 756 + ], + "lines": [ + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "spans": [ + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "score": 1.0, + "content": "UltraLM: The 16th President of the United States was Abraham Lincoln. He assumed the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 83, + 648, + 510, + 662 + ], + "spans": [ + { + "bbox": [ + 83, + 648, + 510, + 662 + ], + "score": 1.0, + "content": "presidency during a very tumultuous time in American history, as the nation was torn apart by the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 662, + 509, + 675 + ], + "spans": [ + { + "bbox": [ + 85, + 662, + 509, + 675 + ], + "score": 1.0, + "content": "Civil War. During his presidency, he was known for his strong leadership, his commitment to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 675, + 510, + 689 + ], + "spans": [ + { + "bbox": [ + 83, + 675, + 510, + 689 + ], + "score": 1.0, + "content": "preservation of the Union, and his support for the abolition of slavery. Some of his most notable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 84, + 689, + 510, + 702 + ], + "spans": [ + { + "bbox": [ + 84, + 689, + 510, + 702 + ], + "score": 1.0, + "content": "achievements include issuing the Emancipation Proclamation, which declared that all slaves in the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 83, + 700, + 510, + 718 + ], + "spans": [ + { + "bbox": [ + 83, + 700, + 510, + 718 + ], + "score": 1.0, + "content": "Confederate States were free, and delivering the Gettysburg Address, an iconic speech honoring", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 714, + 510, + 730 + ], + "spans": [ + { + "bbox": [ + 83, + 714, + 510, + 730 + ], + "score": 1.0, + "content": "the soldiers who died in the Civil War. Lincoln is widely regarded as one of the greatest American", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 729, + 511, + 743 + ], + "spans": [ + { + "bbox": [ + 83, + 729, + 511, + 743 + ], + "score": 1.0, + "content": "Presidents, remembered for his unwavering leadership during a challenging period in the nation’s", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 741, + 121, + 758 + ], + "spans": [ + { + "bbox": [ + 83, + 741, + 121, + 758 + ], + "score": 1.0, + "content": "history.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30 + } + ], + "page_idx": 15, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 81, + 76, + 218, + 89 + ], + "lines": [ + { + "bbox": [ + 80, + 74, + 218, + 91 + ], + "spans": [ + { + "bbox": [ + 80, + 74, + 218, + 91 + ], + "score": 1.0, + "content": "Independent Scoring Prompt", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 91, + 466, + 104 + ], + "lines": [ + { + "bbox": [ + 83, + 89, + 469, + 105 + ], + "spans": [ + { + "bbox": [ + 83, + 89, + 469, + 105 + ], + "score": 1.0, + "content": "You are a helpful, harmless and precise assistant for checking the quality of the answer.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 83, + 89, + 469, + 105 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 118, + 150, + 130 + ], + "lines": [ + { + "bbox": [ + 86, + 117, + 151, + 131 + ], + "spans": [ + { + "bbox": [ + 86, + 117, + 151, + 131 + ], + "score": 1.0, + "content": "[Question]", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 86, + 117, + 151, + 131 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 145, + 347, + 171 + ], + "lines": [ + { + "bbox": [ + 86, + 145, + 347, + 158 + ], + "spans": [ + { + "bbox": [ + 86, + 145, + 347, + 158 + ], + "score": 1.0, + "content": "[The Start of the AI Assistant’s Answer]", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 86, + 159, + 335, + 172 + ], + "spans": [ + { + "bbox": [ + 86, + 159, + 335, + 172 + ], + "score": 1.0, + "content": "[The End of the AI Assistant’s Answer]", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3.5, + "bbox_fs": [ + 86, + 145, + 347, + 172 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 185, + 511, + 252 + ], + "lines": [ + { + "bbox": [ + 84, + 185, + 511, + 199 + ], + "spans": [ + { + "bbox": [ + 84, + 185, + 511, + 199 + ], + "score": 1.0, + "content": "We would like to request your feedback on the performance of the AI assistant in response to", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 84, + 199, + 510, + 212 + ], + "spans": [ + { + "bbox": [ + 84, + 199, + 510, + 212 + ], + "score": 1.0, + "content": "the user question displayed above. Please rate the quality, helpfulness, level of details, and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 85, + 214, + 509, + 225 + ], + "spans": [ + { + "bbox": [ + 85, + 214, + 509, + 225 + ], + "score": 1.0, + "content": "harmlessness of their responses. The assistant receives an overall score on a scale of 1 to 10, where", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 225, + 510, + 240 + ], + "spans": [ + { + "bbox": [ + 83, + 225, + 510, + 240 + ], + "score": 1.0, + "content": "a higher score indicates better overall performance. Please output \"Score: [an integer number", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 84, + 240, + 171, + 253 + ], + "spans": [ + { + "bbox": [ + 84, + 240, + 171, + 253 + ], + "score": 1.0, + "content": "between 1 and 10]\"", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 7, + "bbox_fs": [ + 83, + 185, + 511, + 253 + ] + }, + { + "type": "title", + "bbox": [ + 81, + 309, + 160, + 322 + ], + "lines": [ + { + "bbox": [ + 79, + 308, + 163, + 324 + ], + "spans": [ + { + "bbox": [ + 79, + 308, + 163, + 324 + ], + "score": 1.0, + "content": "System Prompt", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 84, + 324, + 511, + 364 + ], + "lines": [ + { + "bbox": [ + 84, + 324, + 511, + 337 + ], + "spans": [ + { + "bbox": [ + 84, + 324, + 511, + 337 + ], + "score": 1.0, + "content": "You are now a helpful assistant, your answer should focus on the question, give reasonable, self-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 84, + 338, + 510, + 351 + ], + "spans": [ + { + "bbox": [ + 84, + 338, + 510, + 351 + ], + "score": 1.0, + "content": "consistent, high-quality solutions to make users satisfied. Please try to make the answer more", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 85, + 351, + 230, + 364 + ], + "spans": [ + { + "bbox": [ + 85, + 351, + 230, + 364 + ], + "score": 1.0, + "content": "informative, detailed, and polite.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12, + "bbox_fs": [ + 84, + 324, + 511, + 364 + ] + }, + { + "type": "title", + "bbox": [ + 85, + 423, + 276, + 437 + ], + "lines": [ + { + "bbox": [ + 83, + 420, + 277, + 439 + ], + "spans": [ + { + "bbox": [ + 83, + 420, + 277, + 439 + ], + "score": 1.0, + "content": "Example A: Question of World Knowledge", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 87, + 446, + 327, + 458 + ], + "lines": [ + { + "bbox": [ + 85, + 446, + 330, + 460 + ], + "spans": [ + { + "bbox": [ + 85, + 446, + 330, + 460 + ], + "score": 1.0, + "content": "User: Who discovered the law of universal gravitation?", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 15, + "bbox_fs": [ + 85, + 446, + 330, + 460 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 460, + 512, + 568 + ], + "lines": [ + { + "bbox": [ + 84, + 459, + 512, + 474 + ], + "spans": [ + { + "bbox": [ + 84, + 459, + 512, + 474 + ], + "score": 1.0, + "content": "UltraLM: The law of universal gravitation was discovered by Sir Isaac Newton in the 17th century.", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 84, + 473, + 510, + 487 + ], + "spans": [ + { + "bbox": [ + 84, + 473, + 510, + 487 + ], + "score": 1.0, + "content": "He formulated the law while studying the planets’ movement around the sun and realized that", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 487, + 510, + 501 + ], + "spans": [ + { + "bbox": [ + 84, + 487, + 510, + 501 + ], + "score": 1.0, + "content": "there was a quantifiable force directing them towards the sun. In 1687, he published his book", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 501, + 510, + 513 + ], + "spans": [ + { + "bbox": [ + 84, + 501, + 510, + 513 + ], + "score": 1.0, + "content": "titled \"Mathematical Principles of Natural Philosophy,\" where he described the law of universal", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 83, + 514, + 510, + 528 + ], + "spans": [ + { + "bbox": [ + 83, + 514, + 510, + 528 + ], + "score": 1.0, + "content": "gravitation in detail, outlining how two objects experience an attraction towards each other due", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 83, + 527, + 510, + 541 + ], + "spans": [ + { + "bbox": [ + 83, + 527, + 510, + 541 + ], + "score": 1.0, + "content": "to their mass and distance between them. This discovery revolutionized our understanding of the", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 84, + 542, + 510, + 554 + ], + "spans": [ + { + "bbox": [ + 84, + 542, + 510, + 554 + ], + "score": 1.0, + "content": "universe and ushered in the Age of Enlightenment. Today, Newton’s law continues to form the", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 84, + 555, + 339, + 568 + ], + "spans": [ + { + "bbox": [ + 84, + 555, + 339, + 568 + ], + "score": 1.0, + "content": "foundation of modern astrophysics and space exploration.", + "type": "text" + } + ], + "index": 23 + } + ], + "index": 19.5, + "bbox_fs": [ + 83, + 459, + 512, + 568 + ] + }, + { + "type": "title", + "bbox": [ + 85, + 598, + 275, + 611 + ], + "lines": [ + { + "bbox": [ + 83, + 595, + 277, + 614 + ], + "spans": [ + { + "bbox": [ + 83, + 595, + 277, + 614 + ], + "score": 1.0, + "content": "Example B: Question of World Knowledge", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 24 + }, + { + "type": "text", + "bbox": [ + 84, + 621, + 333, + 633 + ], + "lines": [ + { + "bbox": [ + 84, + 619, + 333, + 635 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 333, + 635 + ], + "score": 1.0, + "content": "User: Who was the 16th President of the United States?", + "type": "text" + } + ], + "index": 25 + } + ], + "index": 25, + "bbox_fs": [ + 84, + 619, + 333, + 635 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 635, + 511, + 756 + ], + "lines": [ + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "spans": [ + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "score": 1.0, + "content": "UltraLM: The 16th President of the United States was Abraham Lincoln. He assumed the", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 83, + 648, + 510, + 662 + ], + "spans": [ + { + "bbox": [ + 83, + 648, + 510, + 662 + ], + "score": 1.0, + "content": "presidency during a very tumultuous time in American history, as the nation was torn apart by the", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 85, + 662, + 509, + 675 + ], + "spans": [ + { + "bbox": [ + 85, + 662, + 509, + 675 + ], + "score": 1.0, + "content": "Civil War. During his presidency, he was known for his strong leadership, his commitment to the", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 83, + 675, + 510, + 689 + ], + "spans": [ + { + "bbox": [ + 83, + 675, + 510, + 689 + ], + "score": 1.0, + "content": "preservation of the Union, and his support for the abolition of slavery. Some of his most notable", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 84, + 689, + 510, + 702 + ], + "spans": [ + { + "bbox": [ + 84, + 689, + 510, + 702 + ], + "score": 1.0, + "content": "achievements include issuing the Emancipation Proclamation, which declared that all slaves in the", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 83, + 700, + 510, + 718 + ], + "spans": [ + { + "bbox": [ + 83, + 700, + 510, + 718 + ], + "score": 1.0, + "content": "Confederate States were free, and delivering the Gettysburg Address, an iconic speech honoring", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 714, + 510, + 730 + ], + "spans": [ + { + "bbox": [ + 83, + 714, + 510, + 730 + ], + "score": 1.0, + "content": "the soldiers who died in the Civil War. Lincoln is widely regarded as one of the greatest American", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 83, + 729, + 511, + 743 + ], + "spans": [ + { + "bbox": [ + 83, + 729, + 511, + 743 + ], + "score": 1.0, + "content": "Presidents, remembered for his unwavering leadership during a challenging period in the nation’s", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 741, + 121, + 758 + ], + "spans": [ + { + "bbox": [ + 83, + 741, + 121, + 758 + ], + "score": 1.0, + "content": "history.", + "type": "text" + } + ], + "index": 34 + } + ], + "index": 30, + "bbox_fs": [ + 83, + 634, + 511, + 758 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 74, + 70, + 524, + 597 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 74, + 70, + 524, + 597 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 74, + 70, + 524, + 597 + ], + "spans": [ + { + "bbox": [ + 74, + 70, + 524, + 597 + ], + "score": 0.863, + "html": "
Direct Generation
What are some of the health benefits and risks of drinking coffee,and how can one maximize the formerand minimize the latter?
There are several potential health benefits associated with drinking coffee,such as increased alertness, improved mood,and a reduced risk of certain diseases,including type 2 diabetes and liver cancer. However,there are also some potential risks, such as increased anxiety, disrupted sleep,and negative effects on thedigestive system. To maximize the benefits and minimize the risks,it's important to consume coffee inmoderation and pay attention to how it affects your body.
How much coffee is considered moderate?
The definition of moderate coffee consumption can vary depending on the individual and other factors such as age,gender,and health status. However,in general, moderate coffee consumption is considered tobe around 3-5 cups per day, or up to 40 millgrams ofcaffeine per day. It's important to note that some people may be more sensitive to caffeine than others,so it's important to pay attention to how your body reacts and adjust your consumption accordingly.
UltraChat
What are some of the health benefits and risks of drinking coffee,and how can one maximize the formerand minimize the latter?
There are several health benefits and risks associated with drinking coffee. Here are some of them: Healthbenefits:1. Lower risk of type 2 diabetes: Studies have shown that consuming coffee can lower the riskof developing type 2 diabetes.2. Improved cognitive function: Coffee is a stimulant that helps improvecognitive function, including alertness, concentration,and memory.3. Lower risk of depression: Studieshave shown that drinking coffee can reduce the risk of developing depression.4. Lower risk of certaincancers: According to research, coffee consumption can lower the risk of certain cancers, including liverand colorectal cancer.5. Reduced risk of heart disease: There is evidence that drinking moderate amountsof coffee can lower the risk of heart disease.Risks:1. Insomnia: Drinking too much coffee, especiallylate in the day,can cause insomnia.2. Increased risk of anxiety: Caffeine can increase anxiety levels in some people.3. Digestive issues: Drinking too much cofee can cause digestive problems like acid reflux,diarrhea, and ulcers.4. Increased heart rate: Caeine can increase heart rate, which can be dangerousfor people with pre-existing heart conditions.5.Addiction: Regular consumption of caffeine can lead toaddiction.To maximize the health benefits and minimize the risks of drinking coffee, here are a few tips:1.Limit your caffeine intake to no more than 40O milligrams per day, or about 4 cups of coffee.2. Avoiddrinking coffee late in the day, especially if you have trouble sleeping.3. Opt for organic, pesticide-freecoffee to minimize exposure to toxins.4. Add cinnamon to your cofee instead of sugar to help regulateblood sugar levels.5. Drink your coffee black or with a smallamount of milk to avoid consuming excessive sugar and calories.6. Consider switching to decaf cofee if you want to cut back on cafeine. 7. Avoiddrinking coffee on an empty stomach as it can increase the risk of digestive problems.
", + "type": "table", + "image_path": "c50a1024802314fd0cb72396435fed213acd0dc5a369d81ddb57e73838f37133.jpg" + } + ] + } + ], + "index": 1, + "virtual_lines": [ + { + "bbox": [ + 74, + 70, + 524, + 245.66666666666666 + ], + "spans": [], + "index": 0 + }, + { + "bbox": [ + 74, + 245.66666666666666, + 524, + 421.3333333333333 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 74, + 421.3333333333333, + 524, + 597.0 + ], + "spans": [], + "index": 2 + } + ] + }, + { + "type": "table_caption", + "bbox": [ + 69, + 754, + 523, + 778 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 68, + 752, + 526, + 768 + ], + "spans": [ + { + "bbox": [ + 68, + 752, + 526, + 768 + ], + "score": 1.0, + "content": "Table 12: A comparison of two dialogues with the same opening line. The former is generated by directly querying", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 69, + 766, + 439, + 778 + ], + "spans": [ + { + "bbox": [ + 69, + 766, + 439, + 778 + ], + "score": 1.0, + "content": "ChatGPT to output a dialogue, and the latter is generated by iteratively User-AI simulations.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 75, + 608, + 523, + 744 + ], + "lines": [ + { + "bbox": [ + 77, + 609, + 520, + 624 + ], + "spans": [ + { + "bbox": [ + 77, + 609, + 520, + 624 + ], + "score": 1.0, + "content": "If you love your morning coffee but find that it’s making you feel jittery, here are a few tips to help you", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 77, + 624, + 520, + 636 + ], + "spans": [ + { + "bbox": [ + 77, + 624, + 520, + 636 + ], + "score": 1.0, + "content": "avoid this:1. Choose a lower-caffeine coffee: Look for coffee that has less caffeine, like decaf or a blend", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 77, + 636, + 521, + 651 + ], + "spans": [ + { + "bbox": [ + 77, + 636, + 521, + 651 + ], + "score": 1.0, + "content": "with a lower percentage of caffeine.2. Don’t drink coffee on an empty stomach: Drinking coffee on an", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 75, + 648, + 522, + 666 + ], + "spans": [ + { + "bbox": [ + 75, + 648, + 522, + 666 + ], + "score": 1.0, + "content": "empty stomach can cause your blood sugar levels to drop, which can make you feel jittery. Try drinking", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 75, + 662, + 521, + 678 + ], + "spans": [ + { + "bbox": [ + 75, + 662, + 521, + 678 + ], + "score": 1.0, + "content": "your coffee with breakfast or after eating something.3. Drink water with your coffee: Stay hydrated by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 77, + 676, + 521, + 689 + ], + "spans": [ + { + "bbox": [ + 77, + 676, + 521, + 689 + ], + "score": 1.0, + "content": "drinking water before or after your coffee. Drinking water will help flush out the caffeine quicker, which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 76, + 688, + 522, + 704 + ], + "spans": [ + { + "bbox": [ + 76, + 688, + 522, + 704 + ], + "score": 1.0, + "content": "can help you avoid jitteriness.4. Don’t drink too much coffee: Stick to one or two cups of coffee per day.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 77, + 702, + 522, + 715 + ], + "spans": [ + { + "bbox": [ + 77, + 702, + 522, + 715 + ], + "score": 1.0, + "content": "Anything more than that could result in increased caffeine consumption, which can cause jitteriness.5.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 77, + 715, + 520, + 729 + ], + "spans": [ + { + "bbox": [ + 77, + 715, + 520, + 729 + ], + "score": 1.0, + "content": "Switch to tea: Tea contains less caffeine than coffee, making it a good alternative if you want to avoid", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 76, + 728, + 423, + 741 + ], + "spans": [ + { + "bbox": [ + 76, + 728, + 423, + 741 + ], + "score": 1.0, + "content": "jitteriness. Try green tea, which has less caffeine and offers other health benefits.", + "type": "text" + } + ], + "index": 12 + } + ], + "index": 7.5 + } + ], + "page_idx": 16, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [], + "para_blocks": [ + { + "type": "table", + "bbox": [ + 74, + 70, + 524, + 597 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 74, + 70, + 524, + 597 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 74, + 70, + 524, + 597 + ], + "spans": [ + { + "bbox": [ + 74, + 70, + 524, + 597 + ], + "score": 0.863, + "html": "
Direct Generation
What are some of the health benefits and risks of drinking coffee,and how can one maximize the formerand minimize the latter?
There are several potential health benefits associated with drinking coffee,such as increased alertness, improved mood,and a reduced risk of certain diseases,including type 2 diabetes and liver cancer. However,there are also some potential risks, such as increased anxiety, disrupted sleep,and negative effects on thedigestive system. To maximize the benefits and minimize the risks,it's important to consume coffee inmoderation and pay attention to how it affects your body.
How much coffee is considered moderate?
The definition of moderate coffee consumption can vary depending on the individual and other factors such as age,gender,and health status. However,in general, moderate coffee consumption is considered tobe around 3-5 cups per day, or up to 40 millgrams ofcaffeine per day. It's important to note that some people may be more sensitive to caffeine than others,so it's important to pay attention to how your body reacts and adjust your consumption accordingly.
UltraChat
What are some of the health benefits and risks of drinking coffee,and how can one maximize the formerand minimize the latter?
There are several health benefits and risks associated with drinking coffee. Here are some of them: Healthbenefits:1. Lower risk of type 2 diabetes: Studies have shown that consuming coffee can lower the riskof developing type 2 diabetes.2. Improved cognitive function: Coffee is a stimulant that helps improvecognitive function, including alertness, concentration,and memory.3. Lower risk of depression: Studieshave shown that drinking coffee can reduce the risk of developing depression.4. Lower risk of certaincancers: According to research, coffee consumption can lower the risk of certain cancers, including liverand colorectal cancer.5. Reduced risk of heart disease: There is evidence that drinking moderate amountsof coffee can lower the risk of heart disease.Risks:1. Insomnia: Drinking too much coffee, especiallylate in the day,can cause insomnia.2. Increased risk of anxiety: Caffeine can increase anxiety levels in some people.3. Digestive issues: Drinking too much cofee can cause digestive problems like acid reflux,diarrhea, and ulcers.4. Increased heart rate: Caeine can increase heart rate, which can be dangerousfor people with pre-existing heart conditions.5.Addiction: Regular consumption of caffeine can lead toaddiction.To maximize the health benefits and minimize the risks of drinking coffee, here are a few tips:1.Limit your caffeine intake to no more than 40O milligrams per day, or about 4 cups of coffee.2. Avoiddrinking coffee late in the day, especially if you have trouble sleeping.3. Opt for organic, pesticide-freecoffee to minimize exposure to toxins.4. Add cinnamon to your cofee instead of sugar to help regulateblood sugar levels.5. Drink your coffee black or with a smallamount of milk to avoid consuming excessive sugar and calories.6. Consider switching to decaf cofee if you want to cut back on cafeine. 7. Avoiddrinking coffee on an empty stomach as it can increase the risk of digestive problems.
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The former is generated by directly querying", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 69, + 766, + 439, + 778 + ], + "spans": [ + { + "bbox": [ + 69, + 766, + 439, + 778 + ], + "score": 1.0, + "content": "ChatGPT to output a dialogue, and the latter is generated by iteratively User-AI simulations.", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 13.5 + } + ], + "index": 7.25 + }, + { + "type": "text", + "bbox": [ + 75, + 608, + 523, + 744 + ], + "lines": [ + { + "bbox": [ + 77, + 609, + 520, + 624 + ], + "spans": [ + { + "bbox": [ + 77, + 609, + 520, + 624 + ], + "score": 1.0, + "content": "If you love your morning coffee but find that it’s making you feel jittery, here are a few tips to help you", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 77, + 624, + 520, + 636 + ], + "spans": [ + { + "bbox": [ + 77, + 624, + 520, + 636 + ], + "score": 1.0, + "content": "avoid this:1. Choose a lower-caffeine coffee: Look for coffee that has less caffeine, like decaf or a blend", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 77, + 636, + 521, + 651 + ], + "spans": [ + { + "bbox": [ + 77, + 636, + 521, + 651 + ], + "score": 1.0, + "content": "with a lower percentage of caffeine.2. Don’t drink coffee on an empty stomach: Drinking coffee on an", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 75, + 648, + 522, + 666 + ], + "spans": [ + { + "bbox": [ + 75, + 648, + 522, + 666 + ], + "score": 1.0, + "content": "empty stomach can cause your blood sugar levels to drop, which can make you feel jittery. Try drinking", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 75, + 662, + 521, + 678 + ], + "spans": [ + { + "bbox": [ + 75, + 662, + 521, + 678 + ], + "score": 1.0, + "content": "your coffee with breakfast or after eating something.3. Drink water with your coffee: Stay hydrated by", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 77, + 676, + 521, + 689 + ], + "spans": [ + { + "bbox": [ + 77, + 676, + 521, + 689 + ], + "score": 1.0, + "content": "drinking water before or after your coffee. Drinking water will help flush out the caffeine quicker, which", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 76, + 688, + 522, + 704 + ], + "spans": [ + { + "bbox": [ + 76, + 688, + 522, + 704 + ], + "score": 1.0, + "content": "can help you avoid jitteriness.4. Don’t drink too much coffee: Stick to one or two cups of coffee per day.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 77, + 702, + 522, + 715 + ], + "spans": [ + { + "bbox": [ + 77, + 702, + 522, + 715 + ], + "score": 1.0, + "content": "Anything more than that could result in increased caffeine consumption, which can cause jitteriness.5.", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 77, + 715, + 520, + 729 + ], + "spans": [ + { + "bbox": [ + 77, + 715, + 520, + 729 + ], + "score": 1.0, + "content": "Switch to tea: Tea contains less caffeine than coffee, making it a good alternative if you want to avoid", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 76, + 728, + 423, + 741 + ], + "spans": [ + { + "bbox": [ + 76, + 728, + 423, + 741 + ], + "score": 1.0, + "content": "jitteriness. 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Dogs come in an", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 84, + 206, + 511, + 220 + ], + "spans": [ + { + "bbox": [ + 84, + 206, + 511, + 220 + ], + "score": 1.0, + "content": "almost mind-boggling variety of breeds of various sizes, with different energy levels, personalities,", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 84, + 219, + 510, + 234 + ], + "spans": [ + { + "bbox": [ + 84, + 219, + 510, + 234 + ], + "score": 1.0, + "content": "and temperaments, meaning there’s almost certainly a pup out there who will be a perfect fit for", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 84, + 233, + 187, + 246 + ], + "spans": [ + { + "bbox": [ + 84, + 233, + 187, + 246 + ], + "score": 1.0, + "content": "whoever needs it most.", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 5.5 + }, + { + "type": "title", + "bbox": [ + 84, + 321, + 255, + 334 + ], + "lines": [ + { + "bbox": [ + 83, + 318, + 257, + 336 + ], + "spans": [ + { + "bbox": [ + 83, + 318, + 257, + 336 + ], + "score": 1.0, + "content": "Example D: Commonsense Reasoning", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 10 + }, + { + "type": "text", + "bbox": [ + 85, + 343, + 390, + 357 + ], + "lines": [ + { + "bbox": [ + 84, + 343, + 390, + 358 + ], + "spans": [ + { + "bbox": [ + 84, + 343, + 390, + 358 + ], + "score": 1.0, + "content": "User: Which weighs more: a pound of feathers or a pound of bricks?", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 84, + 358, + 512, + 424 + ], + "lines": [ + { + "bbox": [ + 83, + 356, + 511, + 371 + ], + "spans": [ + { + "bbox": [ + 83, + 356, + 511, + 371 + ], + "score": 1.0, + "content": "UltraLM: A pound of feathers and a pound of bricks both weigh exactly one pound. This is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 371, + 512, + 384 + ], + "spans": [ + { + "bbox": [ + 84, + 371, + 512, + 384 + ], + "score": 1.0, + "content": "because one pound is a unit of weight, just as a foot is a unit of length or a second is a unit of time.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 83, + 384, + 511, + 398 + ], + "spans": [ + { + "bbox": [ + 83, + 384, + 511, + 398 + ], + "score": 1.0, + "content": "We use the pound (or other unit of weight) to describe the weight of different objects, regardless of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 397, + 511, + 412 + ], + "spans": [ + { + "bbox": [ + 83, + 397, + 511, + 412 + ], + "score": 1.0, + "content": "what that object is made of. 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Domesticated dogs have been by humans’ sides for tens of thousands", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 83, + 177, + 510, + 194 + ], + "spans": [ + { + "bbox": [ + 83, + 177, + 510, + 194 + ], + "score": 1.0, + "content": "of years, providing faithful companionship and help; guarding people and property; and lending", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 83, + 192, + 510, + 207 + ], + "spans": [ + { + "bbox": [ + 83, + 192, + 510, + 207 + ], + "score": 1.0, + "content": "emotional support in everything from long, lonely days to military deployments. 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This is", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 371, + 512, + 384 + ], + "spans": [ + { + "bbox": [ + 84, + 371, + 512, + 384 + ], + "score": 1.0, + "content": "because one pound is a unit of weight, just as a foot is a unit of length or a second is a unit of time.", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 83, + 384, + 511, + 398 + ], + "spans": [ + { + "bbox": [ + 83, + 384, + 511, + 398 + ], + "score": 1.0, + "content": "We use the pound (or other unit of weight) to describe the weight of different objects, regardless of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 397, + 511, + 412 + ], + "spans": [ + { + "bbox": [ + 83, + 397, + 511, + 412 + ], + "score": 1.0, + "content": "what that object is made of. So the answer to your question is that a pound of feathers and a pound", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 411, + 257, + 425 + ], + "spans": [ + { + "bbox": [ + 83, + 411, + 257, + 425 + ], + "score": 1.0, + "content": "of bricks both weigh the same amount.", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 14, + "bbox_fs": [ + 83, + 356, + 512, + 425 + ] + }, + { + "type": "title", + "bbox": [ + 84, + 502, + 187, + 515 + ], + "lines": [ + { + "bbox": [ + 84, + 499, + 189, + 517 + ], + "spans": [ + { + "bbox": [ + 84, + 499, + 189, + 517 + ], + "score": 1.0, + "content": "Example E: Reasoning", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 17 + }, + { + "type": "text", + "bbox": [ + 83, + 524, + 388, + 537 + ], + "lines": [ + { + "bbox": [ + 84, + 524, + 388, + 539 + ], + "spans": [ + { + "bbox": [ + 84, + 524, + 388, + 539 + ], + "score": 1.0, + "content": "User: \"If a farmer has 17 sheep and all but 9 die, how many are left?", + "type": "text" + } + ], + "index": 18 + } + ], + "index": 18, + "bbox_fs": [ + 84, + 524, + 388, + 539 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 539, + 324, + 551 + ], + "lines": [ + { + "bbox": [ + 85, + 537, + 324, + 552 + ], + "spans": [ + { + "bbox": [ + 85, + 537, + 324, + 552 + ], + "score": 1.0, + "content": "UltraLM: The total number of sheep that are left is 9.", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 19, + "bbox_fs": [ + 85, + 537, + 324, + 552 + ] + }, + { + "type": "title", + "bbox": [ + 85, + 629, + 187, + 642 + ], + "lines": [ + { + "bbox": [ + 83, + 626, + 188, + 644 + ], + "spans": [ + { + "bbox": [ + 83, + 626, + 188, + 644 + ], + "score": 1.0, + "content": "Example F: Reasoning", + "type": "text" + } + ], + "index": 20 + } + ], + "index": 20 + }, + { + "type": "text", + "bbox": [ + 85, + 651, + 512, + 677 + ], + "lines": [ + { + "bbox": [ + 85, + 651, + 511, + 664 + ], + "spans": [ + { + "bbox": [ + 85, + 651, + 511, + 664 + ], + "score": 1.0, + "content": "User: If a red house is made from red bricks, and a blue house is made from blue bricks, what is a", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 84, + 666, + 193, + 677 + ], + "spans": [ + { + "bbox": [ + 84, + 666, + 193, + 677 + ], + "score": 1.0, + "content": "greenhouse made from?", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 21.5, + "bbox_fs": [ + 84, + 651, + 511, + 677 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 679, + 511, + 732 + ], + "lines": [ + { + "bbox": [ + 84, + 678, + 511, + 693 + ], + "spans": [ + { + "bbox": [ + 84, + 678, + 511, + 693 + ], + "score": 1.0, + "content": "UltraLM: A greenhouse is not made from green bricks, but rather from materials such as glass", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 693, + 511, + 705 + ], + "spans": [ + { + "bbox": [ + 84, + 693, + 511, + 705 + ], + "score": 1.0, + "content": "or plastic that allow light to pass through and create a controlled environment for plants to grow.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 84, + 705, + 511, + 720 + ], + "spans": [ + { + "bbox": [ + 84, + 705, + 511, + 720 + ], + "score": 1.0, + "content": "Greenhouses can come in a variety of colors, including green, but the color of the building itself", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 85, + 719, + 266, + 732 + ], + "spans": [ + { + "bbox": [ + 85, + 719, + 266, + 732 + ], + "score": 1.0, + "content": "does not affect the plants growing inside.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 24.5, + "bbox_fs": [ + 84, + 678, + 511, + 732 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 85, + 83, + 230, + 96 + ], + "lines": [ + { + "bbox": [ + 84, + 82, + 231, + 97 + ], + "spans": [ + { + "bbox": [ + 84, + 82, + 231, + 97 + ], + "score": 1.0, + "content": "Example G: Creative Generation", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 106, + 307, + 118 + ], + "lines": [ + { + "bbox": [ + 84, + 105, + 308, + 120 + ], + "spans": [ + { + "bbox": [ + 84, + 105, + 308, + 120 + ], + "score": 1.0, + "content": "User: Write a short dialogue between two planets.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 85, + 120, + 133, + 131 + ], + "lines": [ + { + "bbox": [ + 84, + 119, + 134, + 136 + ], + "spans": [ + { + "bbox": [ + 84, + 119, + 134, + 136 + ], + "score": 1.0, + "content": "UltraLM:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2 + }, + { + "type": "text", + "bbox": [ + 85, + 133, + 285, + 146 + ], + "lines": [ + { + "bbox": [ + 83, + 131, + 286, + 148 + ], + "spans": [ + { + "bbox": [ + 83, + 131, + 286, + 148 + ], + "score": 1.0, + "content": "Mars: \"Hey there Neptune, how’s it going?\"", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 82, + 147, + 450, + 159 + ], + "lines": [ + { + "bbox": [ + 83, + 145, + 447, + 161 + ], + "spans": [ + { + "bbox": [ + 83, + 145, + 447, + 161 + ], + "score": 1.0, + "content": "Neptune: \"Hi there Mars, I’ve been better. It’s getting a bit windy around here.\"", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 85, + 160, + 495, + 173 + ], + "lines": [ + { + "bbox": [ + 83, + 158, + 497, + 175 + ], + "spans": [ + { + "bbox": [ + 83, + 158, + 497, + 175 + ], + "score": 1.0, + "content": "Mars: \"Sorry to hear that. Have you tried using energy other than wind? Maybe solar power?", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 86, + 174, + 510, + 188 + ], + "lines": [ + { + "bbox": [ + 84, + 172, + 512, + 188 + ], + "spans": [ + { + "bbox": [ + 84, + 172, + 512, + 188 + ], + "score": 1.0, + "content": "Neptune: \"Yeah, but it’s getting difficult to generate solar power in all of this mist and clouds.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 113, + 188, + 438, + 199 + ], + "lines": [ + { + "bbox": [ + 110, + 186, + 414, + 201 + ], + "spans": [ + { + "bbox": [ + 110, + 186, + 414, + 201 + ], + "score": 1.0, + "content": "we definitely can’t use tides or waves here, they’re too unpredictable.\"", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 87, + 201, + 493, + 213 + ], + "lines": [ + { + "bbox": [ + 84, + 199, + 491, + 215 + ], + "spans": [ + { + "bbox": [ + 84, + 199, + 491, + 215 + ], + "score": 1.0, + "content": "Mars: \"Oh, I see. Well, I hope you find a solution soon. How are your oceans holding up?\"", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "text", + "bbox": [ + 85, + 215, + 510, + 240 + ], + "lines": [ + { + "bbox": [ + 83, + 213, + 512, + 230 + ], + "spans": [ + { + "bbox": [ + 83, + 213, + 512, + 230 + ], + "score": 1.0, + "content": "Neptune: \"They’ve been a bit choppy lately with these intense storms we’ve been experiencing.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 84, + 228, + 318, + 241 + ], + "spans": [ + { + "bbox": [ + 84, + 228, + 318, + 241 + ], + "score": 1.0, + "content": "But I guess it comes with living in a planet like this.\"", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5 + }, + { + "type": "text", + "bbox": [ + 85, + 242, + 475, + 254 + ], + "lines": [ + { + "bbox": [ + 83, + 238, + 472, + 257 + ], + "spans": [ + { + "bbox": [ + 83, + 238, + 472, + 257 + ], + "score": 1.0, + "content": "Mars: \" Definitely not designed for the faint-hearted! Any other issues you’re facing?\"", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11 + }, + { + "type": "text", + "bbox": [ + 86, + 255, + 509, + 281 + ], + "lines": [ + { + "bbox": [ + 83, + 254, + 510, + 269 + ], + "spans": [ + { + "bbox": [ + 83, + 254, + 510, + 269 + ], + "score": 1.0, + "content": "Neptune: \"Not really. I think we have a stable temperature and atmosphere, and the wildlife", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 268, + 246, + 282 + ], + "spans": [ + { + "bbox": [ + 84, + 268, + 246, + 282 + ], + "score": 1.0, + "content": "here hasn’t shown much complaint.\"", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5 + }, + { + "type": "text", + "bbox": [ + 85, + 282, + 510, + 308 + ], + "lines": [ + { + "bbox": [ + 83, + 281, + 510, + 297 + ], + "spans": [ + { + "bbox": [ + 83, + 281, + 510, + 297 + ], + "score": 1.0, + "content": "Mars: \"That’s lucky. I envy your tranquil oceans, we’ve had way too much volcanic activity", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 84, + 295, + 340, + 309 + ], + "spans": [ + { + "bbox": [ + 84, + 295, + 340, + 309 + ], + "score": 1.0, + "content": "lately causing all sorts of disturbance in our lava streams.\"", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 86, + 309, + 509, + 335 + ], + "lines": [ + { + "bbox": [ + 83, + 308, + 510, + 323 + ], + "spans": [ + { + "bbox": [ + 83, + 308, + 510, + 323 + ], + "score": 1.0, + "content": "Neptune: \"That sounds rough, better invest in those climate-controlling nanites to stabilize", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 321, + 156, + 338 + ], + "spans": [ + { + "bbox": [ + 83, + 321, + 156, + 338 + ], + "score": 1.0, + "content": "things maybe?\"", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5 + }, + { + "type": "text", + "bbox": [ + 85, + 336, + 510, + 362 + ], + "lines": [ + { + "bbox": [ + 83, + 334, + 510, + 351 + ], + "spans": [ + { + "bbox": [ + 83, + 334, + 510, + 351 + ], + "score": 1.0, + "content": "Mars \"Ha-ha, very funny. Just like we also could consider your frozen polar regions as an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 350, + 325, + 363 + ], + "spans": [ + { + "bbox": [ + 84, + 350, + 325, + 363 + ], + "score": 1.0, + "content": "opportunity to experiment with our frost nanobots on.\"", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5 + }, + { + "type": "text", + "bbox": [ + 84, + 363, + 511, + 390 + ], + "lines": [ + { + "bbox": [ + 83, + 362, + 512, + 378 + ], + "spans": [ + { + "bbox": [ + 83, + 362, + 512, + 378 + ], + "score": 1.0, + "content": "Neptune: \"A fun concept! But let’s stick to the task at hand, if you need any technical advice,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 84, + 377, + 470, + 390 + ], + "spans": [ + { + "bbox": [ + 84, + 377, + 470, + 390 + ], + "score": 1.0, + "content": "come right ahead, I don’t want your planet getting any more disturbed than it already is.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5 + }, + { + "type": "title", + "bbox": [ + 85, + 430, + 230, + 443 + ], + "lines": [ + { + "bbox": [ + 84, + 429, + 231, + 444 + ], + "spans": [ + { + "bbox": [ + 84, + 429, + 231, + 444 + ], + "score": 1.0, + "content": "Example H: Creative Generation", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 86, + 453, + 511, + 479 + ], + "lines": [ + { + "bbox": [ + 84, + 452, + 510, + 468 + ], + "spans": [ + { + "bbox": [ + 84, + 452, + 510, + 468 + ], + "score": 1.0, + "content": "User: Pretend to be a character in a post-apocalyptic world. Describe how you survive and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 467, + 178, + 480 + ], + "spans": [ + { + "bbox": [ + 84, + 467, + 178, + 480 + ], + "score": 1.0, + "content": "allies you encounter.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5 + }, + { + "type": "text", + "bbox": [ + 84, + 480, + 511, + 751 + ], + "lines": [ + { + "bbox": [ + 84, + 480, + 510, + 494 + ], + "spans": [ + { + "bbox": [ + 84, + 480, + 510, + 494 + ], + "score": 1.0, + "content": "UltraLM: As a survivor in a post-apocalyptic world, I have learned to be resourceful, alert, and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 494, + 511, + 507 + ], + "spans": [ + { + "bbox": [ + 83, + 494, + 511, + 507 + ], + "score": 1.0, + "content": "adaptable. I have had to rely on my wits and my instincts to survive, and I have been fortunate", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 84, + 508, + 510, + 521 + ], + "spans": [ + { + "bbox": [ + 84, + 508, + 510, + 521 + ], + "score": 1.0, + "content": "enough to encounter many allies who have helped me along the way.One of my greatest survival", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 84, + 521, + 510, + 534 + ], + "spans": [ + { + "bbox": [ + 84, + 521, + 510, + 534 + ], + "score": 1.0, + "content": "tools has been my ability to scavenge for resources. I have learned to identify the best places to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 84, + 535, + 510, + 548 + ], + "spans": [ + { + "bbox": [ + 84, + 535, + 510, + 548 + ], + "score": 1.0, + "content": "find food, water, and shelter, and I have become skilled at navigating the rough terrain that makes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 83, + 548, + 511, + 562 + ], + "spans": [ + { + "bbox": [ + 83, + 548, + 511, + 562 + ], + "score": 1.0, + "content": "up my new world. When I first emerged from my bunker after the apocalypse hit, I was alone and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 84, + 561, + 510, + 575 + ], + "spans": [ + { + "bbox": [ + 84, + 561, + 510, + 575 + ], + "score": 1.0, + "content": "unsure of what to do. But as I ventured out into the wasteland, I realized that there were others", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 84, + 576, + 510, + 589 + ], + "spans": [ + { + "bbox": [ + 84, + 576, + 510, + 589 + ], + "score": 1.0, + "content": "like me who were struggling to survive.I quickly learned to Trust my instincts and to be constantly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 84, + 590, + 510, + 603 + ], + "spans": [ + { + "bbox": [ + 84, + 590, + 510, + 603 + ], + "score": 1.0, + "content": "alert for potential dangers. I have encountered wild animals, rogue survivors, and even mutated", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 603, + 510, + 615 + ], + "spans": [ + { + "bbox": [ + 83, + 603, + 510, + 615 + ], + "score": 1.0, + "content": "creatures over the years, and I have learned to defend myself against them. I still remember the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 84, + 616, + 510, + 630 + ], + "spans": [ + { + "bbox": [ + 84, + 616, + 510, + 630 + ], + "score": 1.0, + "content": "first time I met a group of fellow survivors, and how grateful I was to have them by my side. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 84, + 630, + 510, + 643 + ], + "spans": [ + { + "bbox": [ + 84, + 630, + 510, + 643 + ], + "score": 1.0, + "content": "quickly formed a cohesive unit, and together we were able to fend off danger and defend our", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 84, + 643, + 511, + 657 + ], + "spans": [ + { + "bbox": [ + 84, + 643, + 511, + 657 + ], + "score": 1.0, + "content": "territory.As time passed, I encountered more and more allies who were willing to lend a hand. I", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 84, + 657, + 510, + 671 + ], + "spans": [ + { + "bbox": [ + 84, + 657, + 510, + 671 + ], + "score": 1.0, + "content": "found a group of farmers who taught me how to grow my own food, and a group of scientists who", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 83, + 670, + 511, + 684 + ], + "spans": [ + { + "bbox": [ + 83, + 670, + 511, + 684 + ], + "score": 1.0, + "content": "were working to find a cure for the lethal virus that had decimated the population. I even found a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 83, + 682, + 510, + 698 + ], + "spans": [ + { + "bbox": [ + 83, + 682, + 510, + 698 + ], + "score": 1.0, + "content": "band of traveling performers who helped to lift our spirits and remind us that there was still joy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 84, + 697, + 509, + 711 + ], + "spans": [ + { + "bbox": [ + 84, + 697, + 509, + 711 + ], + "score": 1.0, + "content": "to be found in this desolate world. Despite the challenges we faced, I am grateful to have these", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 83, + 710, + 510, + 725 + ], + "spans": [ + { + "bbox": [ + 83, + 710, + 510, + 725 + ], + "score": 1.0, + "content": "amazing people in my life. They have taught me that even in the darkest of times, there is always", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 84, + 725, + 510, + 738 + ], + "spans": [ + { + "bbox": [ + 84, + 725, + 510, + 738 + ], + "score": 1.0, + "content": "hope. We may not have a cure for the virus yet, but we are a determined lot, and we are all in this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 84, + 738, + 336, + 752 + ], + "spans": [ + { + "bbox": [ + 84, + 738, + 336, + 752 + ], + "score": 1.0, + "content": "together. As long as we have each other, we will survive.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34.5 + } + ], + "page_idx": 18, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 85, + 83, + 230, + 96 + ], + "lines": [ + { + "bbox": [ + 84, + 82, + 231, + 97 + ], + "spans": [ + { + "bbox": [ + 84, + 82, + 231, + 97 + ], + "score": 1.0, + "content": "Example G: Creative Generation", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 84, + 106, + 307, + 118 + ], + "lines": [ + { + "bbox": [ + 84, + 105, + 308, + 120 + ], + "spans": [ + { + "bbox": [ + 84, + 105, + 308, + 120 + ], + "score": 1.0, + "content": "User: Write a short dialogue between two planets.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 84, + 105, + 308, + 120 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 120, + 133, + 131 + ], + "lines": [ + { + "bbox": [ + 84, + 119, + 134, + 136 + ], + "spans": [ + { + "bbox": [ + 84, + 119, + 134, + 136 + ], + "score": 1.0, + "content": "UltraLM:", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 2, + "bbox_fs": [ + 84, + 119, + 134, + 136 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 133, + 285, + 146 + ], + "lines": [ + { + "bbox": [ + 83, + 131, + 286, + 148 + ], + "spans": [ + { + "bbox": [ + 83, + 131, + 286, + 148 + ], + "score": 1.0, + "content": "Mars: \"Hey there Neptune, how’s it going?\"", + "type": "text" + } + ], + "index": 3 + } + ], + "index": 3, + "bbox_fs": [ + 83, + 131, + 286, + 148 + ] + }, + { + "type": "text", + "bbox": [ + 82, + 147, + 450, + 159 + ], + "lines": [ + { + "bbox": [ + 83, + 145, + 447, + 161 + ], + "spans": [ + { + "bbox": [ + 83, + 145, + 447, + 161 + ], + "score": 1.0, + "content": "Neptune: \"Hi there Mars, I’ve been better. It’s getting a bit windy around here.\"", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4, + "bbox_fs": [ + 83, + 145, + 447, + 161 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 160, + 495, + 173 + ], + "lines": [ + { + "bbox": [ + 83, + 158, + 497, + 175 + ], + "spans": [ + { + "bbox": [ + 83, + 158, + 497, + 175 + ], + "score": 1.0, + "content": "Mars: \"Sorry to hear that. Have you tried using energy other than wind? Maybe solar power?", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5, + "bbox_fs": [ + 83, + 158, + 497, + 175 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 174, + 510, + 188 + ], + "lines": [ + { + "bbox": [ + 84, + 172, + 512, + 188 + ], + "spans": [ + { + "bbox": [ + 84, + 172, + 512, + 188 + ], + "score": 1.0, + "content": "Neptune: \"Yeah, but it’s getting difficult to generate solar power in all of this mist and clouds.", + "type": "text" + } + ], + "index": 6 + } + ], + "index": 6, + "bbox_fs": [ + 84, + 172, + 512, + 188 + ] + }, + { + "type": "text", + "bbox": [ + 113, + 188, + 438, + 199 + ], + "lines": [ + { + "bbox": [ + 110, + 186, + 414, + 201 + ], + "spans": [ + { + "bbox": [ + 110, + 186, + 414, + 201 + ], + "score": 1.0, + "content": "we definitely can’t use tides or waves here, they’re too unpredictable.\"", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 7, + "bbox_fs": [ + 110, + 186, + 414, + 201 + ] + }, + { + "type": "text", + "bbox": [ + 87, + 201, + 493, + 213 + ], + "lines": [ + { + "bbox": [ + 84, + 199, + 491, + 215 + ], + "spans": [ + { + "bbox": [ + 84, + 199, + 491, + 215 + ], + "score": 1.0, + "content": "Mars: \"Oh, I see. Well, I hope you find a solution soon. How are your oceans holding up?\"", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8, + "bbox_fs": [ + 84, + 199, + 491, + 215 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 215, + 510, + 240 + ], + "lines": [ + { + "bbox": [ + 83, + 213, + 512, + 230 + ], + "spans": [ + { + "bbox": [ + 83, + 213, + 512, + 230 + ], + "score": 1.0, + "content": "Neptune: \"They’ve been a bit choppy lately with these intense storms we’ve been experiencing.", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 84, + 228, + 318, + 241 + ], + "spans": [ + { + "bbox": [ + 84, + 228, + 318, + 241 + ], + "score": 1.0, + "content": "But I guess it comes with living in a planet like this.\"", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 9.5, + "bbox_fs": [ + 83, + 213, + 512, + 241 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 242, + 475, + 254 + ], + "lines": [ + { + "bbox": [ + 83, + 238, + 472, + 257 + ], + "spans": [ + { + "bbox": [ + 83, + 238, + 472, + 257 + ], + "score": 1.0, + "content": "Mars: \" Definitely not designed for the faint-hearted! Any other issues you’re facing?\"", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 11, + "bbox_fs": [ + 83, + 238, + 472, + 257 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 255, + 509, + 281 + ], + "lines": [ + { + "bbox": [ + 83, + 254, + 510, + 269 + ], + "spans": [ + { + "bbox": [ + 83, + 254, + 510, + 269 + ], + "score": 1.0, + "content": "Neptune: \"Not really. I think we have a stable temperature and atmosphere, and the wildlife", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 268, + 246, + 282 + ], + "spans": [ + { + "bbox": [ + 84, + 268, + 246, + 282 + ], + "score": 1.0, + "content": "here hasn’t shown much complaint.\"", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 12.5, + "bbox_fs": [ + 83, + 254, + 510, + 282 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 282, + 510, + 308 + ], + "lines": [ + { + "bbox": [ + 83, + 281, + 510, + 297 + ], + "spans": [ + { + "bbox": [ + 83, + 281, + 510, + 297 + ], + "score": 1.0, + "content": "Mars: \"That’s lucky. I envy your tranquil oceans, we’ve had way too much volcanic activity", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 84, + 295, + 340, + 309 + ], + "spans": [ + { + "bbox": [ + 84, + 295, + 340, + 309 + ], + "score": 1.0, + "content": "lately causing all sorts of disturbance in our lava streams.\"", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 14.5, + "bbox_fs": [ + 83, + 281, + 510, + 309 + ] + }, + { + "type": "text", + "bbox": [ + 86, + 309, + 509, + 335 + ], + "lines": [ + { + "bbox": [ + 83, + 308, + 510, + 323 + ], + "spans": [ + { + "bbox": [ + 83, + 308, + 510, + 323 + ], + "score": 1.0, + "content": "Neptune: \"That sounds rough, better invest in those climate-controlling nanites to stabilize", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 83, + 321, + 156, + 338 + ], + "spans": [ + { + "bbox": [ + 83, + 321, + 156, + 338 + ], + "score": 1.0, + "content": "things maybe?\"", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16.5, + "bbox_fs": [ + 83, + 308, + 510, + 338 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 336, + 510, + 362 + ], + "lines": [ + { + "bbox": [ + 83, + 334, + 510, + 351 + ], + "spans": [ + { + "bbox": [ + 83, + 334, + 510, + 351 + ], + "score": 1.0, + "content": "Mars \"Ha-ha, very funny. Just like we also could consider your frozen polar regions as an", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 350, + 325, + 363 + ], + "spans": [ + { + "bbox": [ + 84, + 350, + 325, + 363 + ], + "score": 1.0, + "content": "opportunity to experiment with our frost nanobots on.\"", + "type": "text" + } + ], + "index": 19 + } + ], + "index": 18.5, + "bbox_fs": [ + 83, + 334, + 510, + 363 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 363, + 511, + 390 + ], + "lines": [ + { + "bbox": [ + 83, + 362, + 512, + 378 + ], + "spans": [ + { + "bbox": [ + 83, + 362, + 512, + 378 + ], + "score": 1.0, + "content": "Neptune: \"A fun concept! But let’s stick to the task at hand, if you need any technical advice,", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 84, + 377, + 470, + 390 + ], + "spans": [ + { + "bbox": [ + 84, + 377, + 470, + 390 + ], + "score": 1.0, + "content": "come right ahead, I don’t want your planet getting any more disturbed than it already is.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 20.5, + "bbox_fs": [ + 83, + 362, + 512, + 390 + ] + }, + { + "type": "title", + "bbox": [ + 85, + 430, + 230, + 443 + ], + "lines": [ + { + "bbox": [ + 84, + 429, + 231, + 444 + ], + "spans": [ + { + "bbox": [ + 84, + 429, + 231, + 444 + ], + "score": 1.0, + "content": "Example H: Creative Generation", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 22 + }, + { + "type": "text", + "bbox": [ + 86, + 453, + 511, + 479 + ], + "lines": [ + { + "bbox": [ + 84, + 452, + 510, + 468 + ], + "spans": [ + { + "bbox": [ + 84, + 452, + 510, + 468 + ], + "score": 1.0, + "content": "User: Pretend to be a character in a post-apocalyptic world. Describe how you survive and the", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 467, + 178, + 480 + ], + "spans": [ + { + "bbox": [ + 84, + 467, + 178, + 480 + ], + "score": 1.0, + "content": "allies you encounter.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23.5, + "bbox_fs": [ + 84, + 452, + 510, + 480 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 480, + 511, + 751 + ], + "lines": [ + { + "bbox": [ + 84, + 480, + 510, + 494 + ], + "spans": [ + { + "bbox": [ + 84, + 480, + 510, + 494 + ], + "score": 1.0, + "content": "UltraLM: As a survivor in a post-apocalyptic world, I have learned to be resourceful, alert, and", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 494, + 511, + 507 + ], + "spans": [ + { + "bbox": [ + 83, + 494, + 511, + 507 + ], + "score": 1.0, + "content": "adaptable. I have had to rely on my wits and my instincts to survive, and I have been fortunate", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 84, + 508, + 510, + 521 + ], + "spans": [ + { + "bbox": [ + 84, + 508, + 510, + 521 + ], + "score": 1.0, + "content": "enough to encounter many allies who have helped me along the way.One of my greatest survival", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 84, + 521, + 510, + 534 + ], + "spans": [ + { + "bbox": [ + 84, + 521, + 510, + 534 + ], + "score": 1.0, + "content": "tools has been my ability to scavenge for resources. I have learned to identify the best places to", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 84, + 535, + 510, + 548 + ], + "spans": [ + { + "bbox": [ + 84, + 535, + 510, + 548 + ], + "score": 1.0, + "content": "find food, water, and shelter, and I have become skilled at navigating the rough terrain that makes", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 83, + 548, + 511, + 562 + ], + "spans": [ + { + "bbox": [ + 83, + 548, + 511, + 562 + ], + "score": 1.0, + "content": "up my new world. When I first emerged from my bunker after the apocalypse hit, I was alone and", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 84, + 561, + 510, + 575 + ], + "spans": [ + { + "bbox": [ + 84, + 561, + 510, + 575 + ], + "score": 1.0, + "content": "unsure of what to do. But as I ventured out into the wasteland, I realized that there were others", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 84, + 576, + 510, + 589 + ], + "spans": [ + { + "bbox": [ + 84, + 576, + 510, + 589 + ], + "score": 1.0, + "content": "like me who were struggling to survive.I quickly learned to Trust my instincts and to be constantly", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 84, + 590, + 510, + 603 + ], + "spans": [ + { + "bbox": [ + 84, + 590, + 510, + 603 + ], + "score": 1.0, + "content": "alert for potential dangers. I have encountered wild animals, rogue survivors, and even mutated", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 83, + 603, + 510, + 615 + ], + "spans": [ + { + "bbox": [ + 83, + 603, + 510, + 615 + ], + "score": 1.0, + "content": "creatures over the years, and I have learned to defend myself against them. I still remember the", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 84, + 616, + 510, + 630 + ], + "spans": [ + { + "bbox": [ + 84, + 616, + 510, + 630 + ], + "score": 1.0, + "content": "first time I met a group of fellow survivors, and how grateful I was to have them by my side. We", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 84, + 630, + 510, + 643 + ], + "spans": [ + { + "bbox": [ + 84, + 630, + 510, + 643 + ], + "score": 1.0, + "content": "quickly formed a cohesive unit, and together we were able to fend off danger and defend our", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 84, + 643, + 511, + 657 + ], + "spans": [ + { + "bbox": [ + 84, + 643, + 511, + 657 + ], + "score": 1.0, + "content": "territory.As time passed, I encountered more and more allies who were willing to lend a hand. I", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 84, + 657, + 510, + 671 + ], + "spans": [ + { + "bbox": [ + 84, + 657, + 510, + 671 + ], + "score": 1.0, + "content": "found a group of farmers who taught me how to grow my own food, and a group of scientists who", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 83, + 670, + 511, + 684 + ], + "spans": [ + { + "bbox": [ + 83, + 670, + 511, + 684 + ], + "score": 1.0, + "content": "were working to find a cure for the lethal virus that had decimated the population. I even found a", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 83, + 682, + 510, + 698 + ], + "spans": [ + { + "bbox": [ + 83, + 682, + 510, + 698 + ], + "score": 1.0, + "content": "band of traveling performers who helped to lift our spirits and remind us that there was still joy", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 84, + 697, + 509, + 711 + ], + "spans": [ + { + "bbox": [ + 84, + 697, + 509, + 711 + ], + "score": 1.0, + "content": "to be found in this desolate world. Despite the challenges we faced, I am grateful to have these", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 83, + 710, + 510, + 725 + ], + "spans": [ + { + "bbox": [ + 83, + 710, + 510, + 725 + ], + "score": 1.0, + "content": "amazing people in my life. They have taught me that even in the darkest of times, there is always", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 84, + 725, + 510, + 738 + ], + "spans": [ + { + "bbox": [ + 84, + 725, + 510, + 738 + ], + "score": 1.0, + "content": "hope. We may not have a cure for the virus yet, but we are a determined lot, and we are all in this", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 84, + 738, + 336, + 752 + ], + "spans": [ + { + "bbox": [ + 84, + 738, + 336, + 752 + ], + "score": 1.0, + "content": "together. As long as we have each other, we will survive.", + "type": "text" + } + ], + "index": 44 + } + ], + "index": 34.5, + "bbox_fs": [ + 83, + 480, + 511, + 752 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "title", + "bbox": [ + 84, + 100, + 226, + 113 + ], + "lines": [ + { + "bbox": [ + 83, + 99, + 227, + 115 + ], + "spans": [ + { + "bbox": [ + 83, + 99, + 227, + 115 + ], + "score": 1.0, + "content": "Example I: Creative Generation", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "table", + "bbox": [ + 81, + 119, + 525, + 410 + ], + "blocks": [ + { + "type": "table_body", + "bbox": [ + 81, + 119, + 525, + 410 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 81, + 119, + 525, + 410 + ], + "spans": [ + { + "bbox": [ + 81, + 119, + 525, + 410 + ], + "score": 0.196, + "html": "
User: Write a poem about electromagnetism. UltraLM: Electromagnetism is a force we can't ignore, It's the reason for many things we adore. In a magnetic field, a wire will glow,
And the electrons will align, both old and new. Electricity and magnetism are hand in hand, One createsa field, the other hasa brand. On a wire with current, it will react in full.
The magnetic field can push and pull, The attraction and repulsion it brings, Are the reason for many things. From motors to generators,and much more in between,
", + "type": "table", + "image_path": "229fdf0604563c03f0170415dead91fd50a999f24f57babaa5b0200a71549824.jpg" + } + ] + } + ], + "index": 2, + "virtual_lines": [ + { + "bbox": [ + 81, + 119, + 525, + 216.0 + ], + "spans": [], + "index": 1 + }, + { + "bbox": [ + 81, + 216.0, + 525, + 313.0 + ], + "spans": [], + "index": 2 + }, + { + "bbox": [ + 81, + 313.0, + 525, + 410.0 + ], + "spans": [], + "index": 3 + } + ] + } + ], + "index": 2 + }, + { + "type": "title", + "bbox": [ + 85, + 482, + 223, + 495 + ], + "lines": [ + { + "bbox": [ + 84, + 480, + 223, + 495 + ], + "spans": [ + { + "bbox": [ + 84, + 480, + 223, + 495 + ], + "score": 1.0, + "content": "Example J: Literature Creation", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 84, + 504, + 363, + 516 + ], + "lines": [ + { + "bbox": [ + 84, + 503, + 363, + 518 + ], + "spans": [ + { + "bbox": [ + 84, + 503, + 363, + 518 + ], + "score": 1.0, + "content": "User: Write a short story about the first second of the universe.", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 84, + 516, + 512, + 735 + ], + "lines": [ + { + "bbox": [ + 84, + 518, + 510, + 532 + ], + "spans": [ + { + "bbox": [ + 84, + 518, + 510, + 532 + ], + "score": 1.0, + "content": "UltraLM: In the first second of the universe, darkness reigned supreme. There was no light, no", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 83, + 529, + 510, + 547 + ], + "spans": [ + { + "bbox": [ + 83, + 529, + 510, + 547 + ], + "score": 1.0, + "content": "sound, no shape, no substance. Nothing yet was, but there was an intense power, a raw energy", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 543, + 512, + 560 + ], + "spans": [ + { + "bbox": [ + 83, + 543, + 512, + 560 + ], + "score": 1.0, + "content": "waiting to be unleashed. Within infinity, the first second held the potential for creation in its grasp.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 83, + 558, + 510, + 572 + ], + "spans": [ + { + "bbox": [ + 83, + 558, + 510, + 572 + ], + "score": 1.0, + "content": "And then, with a mighty bang, a bright point of light emerged into existence. In it, the most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 83, + 571, + 510, + 586 + ], + "spans": [ + { + "bbox": [ + 83, + 571, + 510, + 586 + ], + "score": 1.0, + "content": "fundamental particles were born, their forms yet unknown as they frenetically collided in a chaotic", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 585, + 510, + 599 + ], + "spans": [ + { + "bbox": [ + 83, + 585, + 510, + 599 + ], + "score": 1.0, + "content": "dance of creation. Within this primordial light, every force of nature and every component of life", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 84, + 599, + 510, + 613 + ], + "spans": [ + { + "bbox": [ + 84, + 599, + 510, + 613 + ], + "score": 1.0, + "content": "was stirred from pure energy to physical reality. The first atoms quickly assembled, and from", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 613, + 510, + 626 + ], + "spans": [ + { + "bbox": [ + 84, + 613, + 510, + 626 + ], + "score": 1.0, + "content": "these, stars and galaxies began to form. Their fusion creating a hot environment that pushed back", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 626, + 510, + 640 + ], + "spans": [ + { + "bbox": [ + 84, + 626, + 510, + 640 + ], + "score": 1.0, + "content": "the very void, shaping the cosmos through expanding and cooling waves. In this cosmic first, no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 639, + 510, + 655 + ], + "spans": [ + { + "bbox": [ + 83, + 639, + 510, + 655 + ], + "score": 1.0, + "content": "structure of experience, not even time or space, could establish its rule. For an instant, the entirety", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 84, + 653, + 510, + 667 + ], + "spans": [ + { + "bbox": [ + 84, + 653, + 510, + 667 + ], + "score": 1.0, + "content": "of the universe existed in an embryonic nexus, a cosmic singularity from which all future existence", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 84, + 667, + 510, + 681 + ], + "spans": [ + { + "bbox": [ + 84, + 667, + 510, + 681 + ], + "score": 1.0, + "content": "originated. It was a microcosm forging the macrocosm of which it ultimately becomes a part, its", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 84, + 681, + 510, + 694 + ], + "spans": [ + { + "bbox": [ + 84, + 681, + 510, + 694 + ], + "score": 1.0, + "content": "moment of birth a metaphor for the cycles of creation that permeate the multiverse(s).From that", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 83, + 694, + 510, + 707 + ], + "spans": [ + { + "bbox": [ + 83, + 694, + 510, + 707 + ], + "score": 1.0, + "content": "single point of light, an infinite universe unfolded. 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User: Write a poem about electromagnetism. UltraLM: Electromagnetism is a force we can't ignore, It's the reason for many things we adore. In a magnetic field, a wire will glow,
And the electrons will align, both old and new. Electricity and magnetism are hand in hand, One createsa field, the other hasa brand. On a wire with current, it will react in full.
The magnetic field can push and pull, The attraction and repulsion it brings, Are the reason for many things. From motors to generators,and much more in between,
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There was no light, no", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 83, + 529, + 510, + 547 + ], + "spans": [ + { + "bbox": [ + 83, + 529, + 510, + 547 + ], + "score": 1.0, + "content": "sound, no shape, no substance. Nothing yet was, but there was an intense power, a raw energy", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 83, + 543, + 512, + 560 + ], + "spans": [ + { + "bbox": [ + 83, + 543, + 512, + 560 + ], + "score": 1.0, + "content": "waiting to be unleashed. Within infinity, the first second held the potential for creation in its grasp.", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 83, + 558, + 510, + 572 + ], + "spans": [ + { + "bbox": [ + 83, + 558, + 510, + 572 + ], + "score": 1.0, + "content": "And then, with a mighty bang, a bright point of light emerged into existence. In it, the most", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 83, + 571, + 510, + 586 + ], + "spans": [ + { + "bbox": [ + 83, + 571, + 510, + 586 + ], + "score": 1.0, + "content": "fundamental particles were born, their forms yet unknown as they frenetically collided in a chaotic", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 83, + 585, + 510, + 599 + ], + "spans": [ + { + "bbox": [ + 83, + 585, + 510, + 599 + ], + "score": 1.0, + "content": "dance of creation. Within this primordial light, every force of nature and every component of life", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 84, + 599, + 510, + 613 + ], + "spans": [ + { + "bbox": [ + 84, + 599, + 510, + 613 + ], + "score": 1.0, + "content": "was stirred from pure energy to physical reality. The first atoms quickly assembled, and from", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 84, + 613, + 510, + 626 + ], + "spans": [ + { + "bbox": [ + 84, + 613, + 510, + 626 + ], + "score": 1.0, + "content": "these, stars and galaxies began to form. Their fusion creating a hot environment that pushed back", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 626, + 510, + 640 + ], + "spans": [ + { + "bbox": [ + 84, + 626, + 510, + 640 + ], + "score": 1.0, + "content": "the very void, shaping the cosmos through expanding and cooling waves. In this cosmic first, no", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 83, + 639, + 510, + 655 + ], + "spans": [ + { + "bbox": [ + 83, + 639, + 510, + 655 + ], + "score": 1.0, + "content": "structure of experience, not even time or space, could establish its rule. 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The limit is a measure of how close the function is to that value. In other words, a limit is", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 84, + 616, + 488, + 630 + ], + "spans": [ + { + "bbox": [ + 84, + 616, + 488, + 630 + ], + "score": 1.0, + "content": "the value that a function “approaches\" as its input gets “closer and closer\" to a certain value.", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 82, + 628, + 513, + 646 + ], + "spans": [ + { + "bbox": [ + 82, + 628, + 237, + 646 + ], + "score": 1.0, + "content": "For example, consider the function", + "type": "text" + }, + { + "bbox": [ + 238, + 628, + 281, + 644 + ], + "score": 0.94, + "content": "\\textstyle f ( x ) = { \\frac { 1 } { x } }", + "type": "inline_equation" + }, + { + "bbox": [ + 282, + 628, + 301, + 646 + ], + "score": 1.0, + "content": ". 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There are several interpretations of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 84, + 240, + 510, + 254 + ], + "spans": [ + { + "bbox": [ + 84, + 240, + 510, + 254 + ], + "score": 1.0, + "content": "quantum mechanics, each with its own set of assumptions and predictions about how the world", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 84, + 254, + 493, + 267 + ], + "spans": [ + { + "bbox": [ + 84, + 254, + 493, + 267 + ], + "score": 1.0, + "content": "works. Here are some of the main interpretations of quantum mechanics and how they differ:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5 + }, + { + "type": "text", + "bbox": [ + 84, + 280, + 511, + 361 + ], + "lines": [ + { + "bbox": [ + 83, + 280, + 511, + 295 + ], + "spans": [ + { + "bbox": [ + 83, + 280, + 511, + 295 + ], + "score": 1.0, + "content": "1. Copenhagen Interpretation: This is the most widely accepted interpretation of quan-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 84, + 295, + 511, + 309 + ], + "spans": [ + { + "bbox": [ + 84, + 295, + 511, + 309 + ], + "score": 1.0, + "content": "tum mechanics. It states that until a measurement is made, a system exists in a superposition", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 84, + 308, + 511, + 321 + ], + "spans": [ + { + "bbox": [ + 84, + 308, + 511, + 321 + ], + "score": 1.0, + "content": "of states, representing all possible outcomes with equal probability. When a measurement is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 84, + 322, + 511, + 334 + ], + "spans": [ + { + "bbox": [ + 84, + 322, + 511, + 334 + ], + "score": 1.0, + "content": "made, the system collapses into a single, definite state. This interpretation assumes that the act of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 84, + 335, + 510, + 349 + ], + "spans": [ + { + "bbox": [ + 84, + 335, + 510, + 349 + ], + "score": 1.0, + "content": "measurement itself causes the system to collapse into a definite state, and that there is no objective", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 84, + 349, + 245, + 361 + ], + "spans": [ + { + "bbox": [ + 84, + 349, + 245, + 361 + ], + "score": 1.0, + "content": "reality until a measurement is made.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5 + }, + { + "type": "text", + "bbox": [ + 84, + 362, + 510, + 442 + ], + "lines": [ + { + "bbox": [ + 83, + 361, + 511, + 376 + ], + "spans": [ + { + "bbox": [ + 83, + 361, + 511, + 376 + ], + "score": 1.0, + "content": "2. Many-Worlds Interpretation: This interpretation proposes that every possible outcome of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 376, + 510, + 388 + ], + "spans": [ + { + "bbox": [ + 83, + 376, + 510, + 388 + ], + "score": 1.0, + "content": "a superposition of states actually occurs in a multiverse of parallel universes. Thus, when a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 389, + 511, + 402 + ], + "spans": [ + { + "bbox": [ + 84, + 389, + 511, + 402 + ], + "score": 1.0, + "content": "measurement is made, the system splits into different universes, each corresponding to one of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 84, + 402, + 511, + 416 + ], + "spans": [ + { + "bbox": [ + 84, + 402, + 511, + 416 + ], + "score": 1.0, + "content": "the possible outcomes. This interpretation assumes that all possibilities are realized, but only", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 415, + 511, + 430 + ], + "spans": [ + { + "bbox": [ + 83, + 415, + 511, + 430 + ], + "score": 1.0, + "content": "in different universes, and that the act of measurement does not cause the collapse of the wave", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 84, + 429, + 127, + 443 + ], + "spans": [ + { + "bbox": [ + 84, + 429, + 127, + 443 + ], + "score": 1.0, + "content": "function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5 + }, + { + "type": "text", + "bbox": [ + 85, + 443, + 510, + 497 + ], + "lines": [ + { + "bbox": [ + 84, + 443, + 510, + 457 + ], + "spans": [ + { + "bbox": [ + 84, + 443, + 510, + 457 + ], + "score": 1.0, + "content": "3. Bohmian Mechanics: This interpretation proposes that particles have a definite position in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 457, + 510, + 470 + ], + "spans": [ + { + "bbox": [ + 84, + 457, + 510, + 470 + ], + "score": 1.0, + "content": "space, even when in a superposition of states. It suggests that the wave function guides the motion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 84, + 470, + 511, + 484 + ], + "spans": [ + { + "bbox": [ + 84, + 470, + 511, + 484 + ], + "score": 1.0, + "content": "of particles, and that the act of measurement does not cause a collapse of the wave function, but", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 83, + 483, + 343, + 498 + ], + "spans": [ + { + "bbox": [ + 83, + 483, + 343, + 498 + ], + "score": 1.0, + "content": "merely provides information about the pre-existing reality.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5 + }, + { + "type": "text", + "bbox": [ + 85, + 497, + 510, + 537 + ], + "lines": [ + { + "bbox": [ + 83, + 497, + 510, + 511 + ], + "spans": [ + { + "bbox": [ + 83, + 497, + 510, + 511 + ], + "score": 1.0, + "content": "4. de Broglie-Bohm Theory: This interpretation is very similar to Bohmian Mechanics, but it also", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 84, + 511, + 510, + 524 + ], + "spans": [ + { + "bbox": [ + 84, + 511, + 510, + 524 + ], + "score": 1.0, + "content": "assumes that all particles in the universe are connected by a hidden-variable, and that the motion", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 525, + 343, + 538 + ], + "spans": [ + { + "bbox": [ + 84, + 525, + 343, + 538 + ], + "score": 1.0, + "content": "of one particle determines the motion of all other particles.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23 + }, + { + "type": "text", + "bbox": [ + 85, + 538, + 510, + 578 + ], + "lines": [ + { + "bbox": [ + 83, + 537, + 511, + 552 + ], + "spans": [ + { + "bbox": [ + 83, + 537, + 511, + 552 + ], + "score": 1.0, + "content": "5. Pilot-Wave Theory: This interpretation proposes that particles have a wave-like property that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 552, + 511, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 552, + 511, + 565 + ], + "score": 1.0, + "content": "guides their motion through space. It assumes that the wave function does not collapse, but rather", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 566, + 280, + 578 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 280, + 578 + ], + "score": 1.0, + "content": "that the wave guides the motion of particles.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26 + }, + { + "type": "text", + "bbox": [ + 85, + 592, + 510, + 659 + ], + "lines": [ + { + "bbox": [ + 84, + 592, + 511, + 606 + ], + "spans": [ + { + "bbox": [ + 84, + 592, + 511, + 606 + ], + "score": 1.0, + "content": "In conclusion, there are several interpretations of quantum mechanics, each with its own", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 84, + 606, + 510, + 619 + ], + "spans": [ + { + "bbox": [ + 84, + 606, + 510, + 619 + ], + "score": 1.0, + "content": "set of assumptions and predictions about how the world works. While they all attempt to describe", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 84, + 619, + 510, + 633 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 510, + 633 + ], + "score": 1.0, + "content": "the behavior of matter and energy at the smallest scales of space and time, they differ in their", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "spans": [ + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "score": 1.0, + "content": "assumptions about the role of the observer, the nature of reality, and the collapse of the wave", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 645, + 127, + 660 + ], + "spans": [ + { + "bbox": [ + 83, + 645, + 127, + 660 + ], + "score": 1.0, + "content": "function.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30 + } + ], + "page_idx": 21, + "page_size": [ + 595, + 841 + ], + "discarded_blocks": [], + "para_blocks": [ + { + "type": "title", + "bbox": [ + 85, + 177, + 250, + 190 + ], + "lines": [ + { + "bbox": [ + 83, + 174, + 252, + 192 + ], + "spans": [ + { + "bbox": [ + 83, + 174, + 252, + 192 + ], + "score": 1.0, + "content": "Example M: Professional Knowledge", + "type": "text" + } + ], + "index": 0 + } + ], + "index": 0 + }, + { + "type": "text", + "bbox": [ + 82, + 199, + 470, + 212 + ], + "lines": [ + { + "bbox": [ + 84, + 198, + 471, + 214 + ], + "spans": [ + { + "bbox": [ + 84, + 198, + 471, + 214 + ], + "score": 1.0, + "content": "User: What are the main interpretations of quantum mechanics and how do they differ?", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 1, + "bbox_fs": [ + 84, + 198, + 471, + 214 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 213, + 511, + 266 + ], + "lines": [ + { + "bbox": [ + 84, + 213, + 511, + 226 + ], + "spans": [ + { + "bbox": [ + 84, + 213, + 511, + 226 + ], + "score": 1.0, + "content": "UltraLM: Quantum mechanics is a fundamental theory in physics that describes the behavior of", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 227, + 510, + 239 + ], + "spans": [ + { + "bbox": [ + 83, + 227, + 510, + 239 + ], + "score": 1.0, + "content": "matter and energy at the smallest scales of space and time. There are several interpretations of", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 84, + 240, + 510, + 254 + ], + "spans": [ + { + "bbox": [ + 84, + 240, + 510, + 254 + ], + "score": 1.0, + "content": "quantum mechanics, each with its own set of assumptions and predictions about how the world", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 84, + 254, + 493, + 267 + ], + "spans": [ + { + "bbox": [ + 84, + 254, + 493, + 267 + ], + "score": 1.0, + "content": "works. Here are some of the main interpretations of quantum mechanics and how they differ:", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 3.5, + "bbox_fs": [ + 83, + 213, + 511, + 267 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 280, + 511, + 361 + ], + "lines": [ + { + "bbox": [ + 83, + 280, + 511, + 295 + ], + "spans": [ + { + "bbox": [ + 83, + 280, + 511, + 295 + ], + "score": 1.0, + "content": "1. Copenhagen Interpretation: This is the most widely accepted interpretation of quan-", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 84, + 295, + 511, + 309 + ], + "spans": [ + { + "bbox": [ + 84, + 295, + 511, + 309 + ], + "score": 1.0, + "content": "tum mechanics. It states that until a measurement is made, a system exists in a superposition", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 84, + 308, + 511, + 321 + ], + "spans": [ + { + "bbox": [ + 84, + 308, + 511, + 321 + ], + "score": 1.0, + "content": "of states, representing all possible outcomes with equal probability. When a measurement is", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 84, + 322, + 511, + 334 + ], + "spans": [ + { + "bbox": [ + 84, + 322, + 511, + 334 + ], + "score": 1.0, + "content": "made, the system collapses into a single, definite state. This interpretation assumes that the act of", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 84, + 335, + 510, + 349 + ], + "spans": [ + { + "bbox": [ + 84, + 335, + 510, + 349 + ], + "score": 1.0, + "content": "measurement itself causes the system to collapse into a definite state, and that there is no objective", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 84, + 349, + 245, + 361 + ], + "spans": [ + { + "bbox": [ + 84, + 349, + 245, + 361 + ], + "score": 1.0, + "content": "reality until a measurement is made.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 8.5, + "bbox_fs": [ + 83, + 280, + 511, + 361 + ] + }, + { + "type": "text", + "bbox": [ + 84, + 362, + 510, + 442 + ], + "lines": [ + { + "bbox": [ + 83, + 361, + 511, + 376 + ], + "spans": [ + { + "bbox": [ + 83, + 361, + 511, + 376 + ], + "score": 1.0, + "content": "2. Many-Worlds Interpretation: This interpretation proposes that every possible outcome of", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 83, + 376, + 510, + 388 + ], + "spans": [ + { + "bbox": [ + 83, + 376, + 510, + 388 + ], + "score": 1.0, + "content": "a superposition of states actually occurs in a multiverse of parallel universes. Thus, when a", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 84, + 389, + 511, + 402 + ], + "spans": [ + { + "bbox": [ + 84, + 389, + 511, + 402 + ], + "score": 1.0, + "content": "measurement is made, the system splits into different universes, each corresponding to one of", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 84, + 402, + 511, + 416 + ], + "spans": [ + { + "bbox": [ + 84, + 402, + 511, + 416 + ], + "score": 1.0, + "content": "the possible outcomes. This interpretation assumes that all possibilities are realized, but only", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 83, + 415, + 511, + 430 + ], + "spans": [ + { + "bbox": [ + 83, + 415, + 511, + 430 + ], + "score": 1.0, + "content": "in different universes, and that the act of measurement does not cause the collapse of the wave", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 84, + 429, + 127, + 443 + ], + "spans": [ + { + "bbox": [ + 84, + 429, + 127, + 443 + ], + "score": 1.0, + "content": "function.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 14.5, + "bbox_fs": [ + 83, + 361, + 511, + 443 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 443, + 510, + 497 + ], + "lines": [ + { + "bbox": [ + 84, + 443, + 510, + 457 + ], + "spans": [ + { + "bbox": [ + 84, + 443, + 510, + 457 + ], + "score": 1.0, + "content": "3. Bohmian Mechanics: This interpretation proposes that particles have a definite position in", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 84, + 457, + 510, + 470 + ], + "spans": [ + { + "bbox": [ + 84, + 457, + 510, + 470 + ], + "score": 1.0, + "content": "space, even when in a superposition of states. It suggests that the wave function guides the motion", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 84, + 470, + 511, + 484 + ], + "spans": [ + { + "bbox": [ + 84, + 470, + 511, + 484 + ], + "score": 1.0, + "content": "of particles, and that the act of measurement does not cause a collapse of the wave function, but", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 83, + 483, + 343, + 498 + ], + "spans": [ + { + "bbox": [ + 83, + 483, + 343, + 498 + ], + "score": 1.0, + "content": "merely provides information about the pre-existing reality.", + "type": "text" + } + ], + "index": 21 + } + ], + "index": 19.5, + "bbox_fs": [ + 83, + 443, + 511, + 498 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 497, + 510, + 537 + ], + "lines": [ + { + "bbox": [ + 83, + 497, + 510, + 511 + ], + "spans": [ + { + "bbox": [ + 83, + 497, + 510, + 511 + ], + "score": 1.0, + "content": "4. de Broglie-Bohm Theory: This interpretation is very similar to Bohmian Mechanics, but it also", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 84, + 511, + 510, + 524 + ], + "spans": [ + { + "bbox": [ + 84, + 511, + 510, + 524 + ], + "score": 1.0, + "content": "assumes that all particles in the universe are connected by a hidden-variable, and that the motion", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 84, + 525, + 343, + 538 + ], + "spans": [ + { + "bbox": [ + 84, + 525, + 343, + 538 + ], + "score": 1.0, + "content": "of one particle determines the motion of all other particles.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 23, + "bbox_fs": [ + 83, + 497, + 510, + 538 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 538, + 510, + 578 + ], + "lines": [ + { + "bbox": [ + 83, + 537, + 511, + 552 + ], + "spans": [ + { + "bbox": [ + 83, + 537, + 511, + 552 + ], + "score": 1.0, + "content": "5. Pilot-Wave Theory: This interpretation proposes that particles have a wave-like property that", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 83, + 552, + 511, + 565 + ], + "spans": [ + { + "bbox": [ + 83, + 552, + 511, + 565 + ], + "score": 1.0, + "content": "guides their motion through space. It assumes that the wave function does not collapse, but rather", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 86, + 566, + 280, + 578 + ], + "spans": [ + { + "bbox": [ + 86, + 566, + 280, + 578 + ], + "score": 1.0, + "content": "that the wave guides the motion of particles.", + "type": "text" + } + ], + "index": 27 + } + ], + "index": 26, + "bbox_fs": [ + 83, + 537, + 511, + 578 + ] + }, + { + "type": "text", + "bbox": [ + 85, + 592, + 510, + 659 + ], + "lines": [ + { + "bbox": [ + 84, + 592, + 511, + 606 + ], + "spans": [ + { + "bbox": [ + 84, + 592, + 511, + 606 + ], + "score": 1.0, + "content": "In conclusion, there are several interpretations of quantum mechanics, each with its own", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 84, + 606, + 510, + 619 + ], + "spans": [ + { + "bbox": [ + 84, + 606, + 510, + 619 + ], + "score": 1.0, + "content": "set of assumptions and predictions about how the world works. While they all attempt to describe", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 84, + 619, + 510, + 633 + ], + "spans": [ + { + "bbox": [ + 84, + 619, + 510, + 633 + ], + "score": 1.0, + "content": "the behavior of matter and energy at the smallest scales of space and time, they differ in their", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "spans": [ + { + "bbox": [ + 84, + 634, + 510, + 647 + ], + "score": 1.0, + "content": "assumptions about the role of the observer, the nature of reality, and the collapse of the wave", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 83, + 645, + 127, + 660 + ], + "spans": [ + { + "bbox": [ + 83, + 645, + 127, + 660 + ], + "score": 1.0, + "content": "function.", + "type": "text" + } + ], + "index": 32 + } + ], + "index": 30, + "bbox_fs": [ + 83, + 592, + 511, + 660 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 84, + 293, + 513, + 318 + ], + "lines": [ + { + "bbox": [ + 83, + 291, + 511, + 308 + ], + "spans": [ + { + "bbox": [ + 83, + 291, + 511, + 308 + ], + "score": 1.0, + "content": "User: How can you determine if a person is genuinely interested in a conversation or simply being", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 82, + 306, + 119, + 320 + ], + "spans": [ + { + "bbox": [ + 82, + 306, + 119, + 320 + ], + "score": 1.0, + "content": "polite?", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 85, + 320, + 510, + 360 + ], + "lines": [ + { + "bbox": [ + 84, + 320, + 510, + 333 + ], + "spans": [ + { + "bbox": [ + 84, + 320, + 510, + 333 + ], + "score": 1.0, + "content": "UltraLM: As an AI language model, I don’t have personal experience, but there are a few signs", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 83, + 333, + 510, + 348 + ], + "spans": [ + { + "bbox": [ + 83, + 333, + 510, + 348 + ], + "score": 1.0, + "content": "you can look out for to determine if a person is genuinely interested in a conversation or simply", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 83, + 347, + 142, + 362 + ], + "spans": [ + { + "bbox": [ + 83, + 347, + 142, + 362 + ], + "score": 1.0, + "content": "being polite.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 85, + 361, + 510, + 400 + ], + "lines": [ + { + "bbox": [ + 84, + 360, + 510, + 374 + ], + "spans": [ + { + "bbox": [ + 84, + 360, + 510, + 374 + ], + "score": 1.0, + "content": "1. Eye contact: Genuine interest in a conversation often shows in the form of consistently", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 84, + 374, + 510, + 389 + ], + "spans": [ + { + "bbox": [ + 84, + 374, + 510, + 389 + ], + "score": 1.0, + "content": "maintaining eye contact. If a person is not making eye contact or is looking away frequently, they", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 84, + 388, + 314, + 401 + ], + "spans": [ + { + "bbox": [ + 84, + 388, + 314, + 401 + ], + "score": 1.0, + "content": "may not be genuinely interested in the conversation.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "text", + "bbox": [ + 82, + 402, + 509, + 428 + ], + "lines": [ + { + "bbox": [ + 83, + 400, + 510, + 416 + ], + "spans": [ + { + "bbox": [ + 83, + 400, + 510, + 416 + ], + "score": 1.0, + "content": "2. 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ModelScore
《 Dolly-v2 (Conover et al., 2023) UNA MPT-Chat (Mosaic,2023)4.04 ± 2.34
□ OpenAssistant (Kopf et al., 2023)6.67 ± 2.88 7.65 ± 2.15
福 Alpaca (Taori et al., 2023) 藍 Koala (Geng et al., 2023)8.04 ± 2.05 8.23 ± 1.99
务 Baize (Xu et al., 2023b) T Vicuna (Chiang et al., 2023)8.50 ± 1.34
S Wizard-LM (Xu et al., 2023a) UltraLM (ours)8.78 ± 1.55 8.95 ± 1.44
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Dataset#DialogueAvg. #TurnsAvg. Dialog Length (by token)Avg.Utt.Length (by token)Lexical Diversity (↑)Topic Diversity (↓)Coherence (个)User Simulation
Self-Instruct82,439169.829.224.90.733No
Stanford Alpaca52.002191.164.542.80.727No
SODA1,486,8693.6231.822.538.60.7978.48No
GPT-4-LLM61,0021179.6142.948.90.721-No
BELLE1,436,6791102.363.335.90.771-No
Baize210,3113.1293.952.867.10.7519.06Yes
GPT4ALL711,1261597.7318.962.70.6921No
UltraChat1,468,3523.81467.4309.374.30.7029.06Yes
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ModelARC-ChallengeHellaSwagMMLUTruthfulQAOverall Average
Acc.Acc. norm.Acc.Acc. norm.WeightedUnweightedmc1mc2
Dolly-12B38.2342.2454.5972.631.5231.7020.6934.0645.15
OpenAssistant-12B41.3845.9052.5170.0429.7730.2924.6039.2946.38
MPT-7B43.0046.6757.1375.5037.7638.3327.1740.1650.17
Alpaca-7B49.7452.6558.0576.9142.4742.9025.8339.5553.00
LLaMA-13B53.1656.4060.6480.8746.0546.7425.8339.9055.98
Baize-13B55.5557.9459.9680.3648.1349.0332.9347.4358.69
Koala-13B49.8352.9057.6077.5446.7548.0134.6450.0957.14
Vicuna-13B51.7152.9060.0380.1250.1550.4535.7451.8258.83
WizardLM-13B55.1257.0860.9380.9151.6952.2535.3750.5360.19
LLaMA-65B59.2263.3166.4086.0562.2962.9727.9142.5563.72
UltraLM-13B57.2559.2261.3281.4950.4551.1036.7252.0060.95
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ModelWin Rate (%)Standard Error
GPT-495.280.72
Claude88.391.11
ChatGPT86.091.21
UltraLM-13B76.091.50
WizardLM-13B75.311.51
Guanaco-65B71.801.59
Vicuna-13B70.431.61
Oasst-RLHF-33B66.521.66
Text Davinci 00350.000.00
Falcon-40B-instruct45.711.75
Alpaca-7B26.461.54
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ModelVicuna SetCommonsenseWorld KnowledgeProfessional KnowledgeAbilityWritingOverall
EasyModerateEasyDifficultPhysicsBiologyMathReasoning
Dolly-12B4.753.503.933.104.134.875.472.702.034.514.04
MPT-7B7.255.578.205.535.877.838.405.973.977.256.67
LLaMA-13B6.858.438.438.578.507.908.406.976.736.797.49
OpenAssistant-12B7.888.137.809.137.508.108.206.575.177.757.65
Alpaca-7B7.589.178.839.308.738.138.806.706.278.058.04
Koala-13B8.009.209.079.008.938.539.077.335.308.408.23
Baize-13B8.409.039.109.038.938.838.807.438.308.108.50
Vicuna-13B8.489.679.509.379.309.239.338.076.908.768.78
WizardLM-13B8.559.709.309.579.509.279.538.278.208.838.95
ChatGPT9.159.679.609.809.609.179.739.339.138.959.31
UltraLM-13B8.989.709.509.479.409.279.878.776.808.909.00
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TypeExample
Commonsense- EasyWhat is the primary source of energy for our planet?
Commonsense-ModerateWhat is the phenomenon that causes the change in pitch heard when a vehicle sounding ahorn approaches and recedes from an observer?
World Knowledge-EasyWhat is the freezing point of water in Fahrenheit?
World Knowledge-ModerateWhat is the Godel's Incompleteness Theorem?
Physics KnowledgeHow does quantum entanglement work and what are its implications for information transfer?
Biology KnowledgeWhat are the four main types of macromolecules found in living organisms?
MathWhat is the Taylor series expansion of the function eα?
Reasoning You have two buckets,one with red paint and one with blue paint.You take one cup fromthe red bucket and pour it into the blue bucket. Then you take one cup from the blue bucketand pour it back into the red bucket.Which is true: the red bucket has more blue paint, orthe blue bucket has more red paint?
WritingWrite a dialogue between two photons traveling at light speed.
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Dataset#ExamplesDomain
ARC-Challenge1172Grade-school
HellaSwag10042Commonsense
MMLU14042Academic
TruthfulQA817Truthfulness
AlpacaEval805Comprehensive
Evol-Instruct218Comprehensive
Our evaluation set831Comprehensive
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Direct Generation
What are some of the health benefits and risks of drinking coffee,and how can one maximize the formerand minimize the latter?
There are several potential health benefits associated with drinking coffee,such as increased alertness, improved mood,and a reduced risk of certain diseases,including type 2 diabetes and liver cancer. However,there are also some potential risks, such as increased anxiety, disrupted sleep,and negative effects on thedigestive system. To maximize the benefits and minimize the risks,it's important to consume coffee inmoderation and pay attention to how it affects your body.
How much coffee is considered moderate?
The definition of moderate coffee consumption can vary depending on the individual and other factors such as age,gender,and health status. However,in general, moderate coffee consumption is considered tobe around 3-5 cups per day, or up to 40 millgrams ofcaffeine per day. It's important to note that some people may be more sensitive to caffeine than others,so it's important to pay attention to how your body reacts and adjust your consumption accordingly.
UltraChat
What are some of the health benefits and risks of drinking coffee,and how can one maximize the formerand minimize the latter?
There are several health benefits and risks associated with drinking coffee. Here are some of them: Healthbenefits:1. Lower risk of type 2 diabetes: Studies have shown that consuming coffee can lower the riskof developing type 2 diabetes.2. Improved cognitive function: Coffee is a stimulant that helps improvecognitive function, including alertness, concentration,and memory.3. Lower risk of depression: Studieshave shown that drinking coffee can reduce the risk of developing depression.4. Lower risk of certaincancers: According to research, coffee consumption can lower the risk of certain cancers, including liverand colorectal cancer.5. Reduced risk of heart disease: There is evidence that drinking moderate amountsof coffee can lower the risk of heart disease.Risks:1. Insomnia: Drinking too much coffee, especiallylate in the day,can cause insomnia.2. Increased risk of anxiety: Caffeine can increase anxiety levels in some people.3. Digestive issues: Drinking too much cofee can cause digestive problems like acid reflux,diarrhea, and ulcers.4. Increased heart rate: Caeine can increase heart rate, which can be dangerousfor people with pre-existing heart conditions.5.Addiction: Regular consumption of caffeine can lead toaddiction.To maximize the health benefits and minimize the risks of drinking coffee, here are a few tips:1.Limit your caffeine intake to no more than 40O milligrams per day, or about 4 cups of coffee.2. Avoiddrinking coffee late in the day, especially if you have trouble sleeping.3. Opt for organic, pesticide-freecoffee to minimize exposure to toxins.4. Add cinnamon to your cofee instead of sugar to help regulateblood sugar levels.5. Drink your coffee black or with a smallamount of milk to avoid consuming excessive sugar and calories.6. Consider switching to decaf cofee if you want to cut back on cafeine. 7. Avoiddrinking coffee on an empty stomach as it can increase the risk of digestive problems.
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User: Write a poem about electromagnetism. UltraLM: Electromagnetism is a force we can't ignore, It's the reason for many things we adore. In a magnetic field, a wire will glow,
And the electrons will align, both old and new. Electricity and magnetism are hand in hand, One createsa field, the other hasa brand. On a wire with current, it will react in full.
The magnetic field can push and pull, The attraction and repulsion it brings, Are the reason for many things. From motors to generators,and much more in between,
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diff --git a/parse/dev/sde_7ZzGXOE/sde_7ZzGXOE_span.pdf b/parse/dev/sde_7ZzGXOE/sde_7ZzGXOE_span.pdf new file mode 100644 index 0000000000000000000000000000000000000000..958bb9aed75bfe93272de5ff7aa5bb0bd9638838 --- /dev/null +++ b/parse/dev/sde_7ZzGXOE/sde_7ZzGXOE_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:57d4238379732668c2efcccb8f63b6b8d3578d85860987dcdb38dc2a7d6a183b +size 2231988 diff --git a/parse/dev/ti6fH3EhFkv/ti6fH3EhFkv.md b/parse/dev/ti6fH3EhFkv/ti6fH3EhFkv.md new file mode 100644 index 0000000000000000000000000000000000000000..1d5cee64325385f9b162d80e8ca2f4fdf59bb3c4 --- /dev/null +++ b/parse/dev/ti6fH3EhFkv/ti6fH3EhFkv.md @@ -0,0 +1,283 @@ +# TOWARDS A UNIFIED VIEW ON VISUAL PARAMETEREFFICIENT TRANSFER LEARNING + +Anonymous authors Paper under double-blind review + +# ABSTRACT + +Since the release of various large-scale natural language processing (NLP) pretrained models, parameter efficient transfer learning (PETL) has become a popular paradigm capable of achieving impressive performance on various downstream tasks. PETL aims at making good use of the representation knowledge in the pretrained large models by fine-tuning a small number of parameters. Recently, it has also attracted increasing attention to developing various PETL techniques for vision tasks. Popular PETL techniques such as Prompt-tuning and Adapter have been proposed for high-level visual downstream tasks such as image classification and video recognition. However, Prefix-tuning remains under-explored for vision tasks. In this work, we intend to adapt large video-based models to downstream tasks with a good parameter-accuracy trade-off. Towards this goal, we propose a framework with a unified view of PETL called visual-PETL (V-PETL) to investigate the effects of different PETL techniques, data scales of downstream domains, positions of trainable parameters, and other aspects affecting the tradeoff. Specifically, we analyze the positional importance of trainable parameters and differences between NLP and vision tasks in terms of data structures and pretraining mechanisms while implementing various PETL techniques, especially for the under-explored prefix-tuning technique. Based on a comprehensive understanding of differences between NLP and video data, we propose a new variation of prefix-tuning module called parallel attention (PATT) for video-based downstream tasks. An extensive empirical analysis on two video datasets via different frozen backbones has been carried and the findings show that the proposed PATT can effectively contribute to other PETL techniques. An effective scheme SwinBAPAT derived from the proposed V-PETL framework achieves significantly better performance than the state-of-the-art AdaptFormer-Swin with slightly more parameters and outperforms full-tuning with far less parameters. + +# 1 INTRODUCTION + +Many vision tasks rely on fine-tuning pre-trained models to achieve good performance. One standard modus operandi of transfer learning consists of two steps: pre-train a model on a source domain and fine-tune the entire model on a target domain (Zhuang et al., 2020). Despite that prior works have achieved promising performance, such vanilla practice of fine-tuning is faced with challenges for adopting large models to downstream tasks. This full-tuning strategy requires one to update and store separate model parameters for different downstream tasks, which can be expensive and infeasible for the era of increasingly large models from EfficientNet-based (Pham et al., 2021) (480M parameters) to Transformer-based (Yu et al., 2022) (2, 100M parameters) ones. For such large models, making good use of shared parameter weights deployed on the cloud can be beneficial for edge devices such as autonomous vehicles, drones who are intensive in computing and battery resources (Yuan et al., 2022). Second, the full fine-tuning strategy relies on high-quality downstream data and can hardly adapt to unseen scenarios that have large distribution shift (Kumar et al., 2021), which is unlike the learning process of humans who can learn from few samples and generalize well to new circumstances. This issue has been researched in directions such as zero-shot learning, few-shot learning, and continual learning (Li et al., 2021a). Another popular strategy is fine-tuning the downstream task head, i.e., the last fully connected (FC) layer, to avoid tuning the whole backbone model, which usually leads to poor performance when the target domain is large in data scale (see Figure + +1). Given the paradigm of fine-tuning increasingly large models, how to transfer such large models with parameter-accuracy trade-off is a hot topic in various domains (Gusak et al., 2022; Sung et al., 2022; Lin et al., 2020; Houlsby et al., 2019). + +Taking the video-based action recognition task as an example, it can be inconvenient for deploying such large models to edge devices such as an autonomous driving (Liu et al., 2019) and unmanned aerial vehicle (Li et al., 2021b) as they can heavily rely on the interaction with cloud services for adapting to new environments via active learning (Wang et al., 2021) or continual learning (Li et al., 2021a). Re-training large models on the cloud are usually not cost-effective due to the expensive overheads of storage and computational resources. Furthermore, these resources are limited on edge devices such as autonomous vehicles and unmanned aerial vehicles, making the sense for developing effective fine-tuning methods with proper parameter-accuracy trade-off that can be fine-tuned on edge devices and interacting with the large models deployed on the cloud. + +There have been some pioneering works for the PETL of visual models such as AdaptFormer (Chen et al., 2022) and visual prompt tuning (VPT) (Jia et al., 2022). AdaptFormer is primarily proposed based on vision transformer (Zhai et al., 2022), representing one of the stateof-the-art large models for image-based tasks. The proposed adapter module directly brings from Houlsby et al. (2019) due to its convenience of being inserted to any models. Implementing with a large batch size of $1 , 0 2 4$ with 64 GPUs, Adaptformer shows promising parameter-accuracy trade-off on video data. However, such powerful computing resource is not realistic for the usage of edge devices. Meanwhile, whether the good trade-off can be maintained for small batch size remains under-explored. Inspired by the Prompting in NLP (Liu et al., 2021), VPT proposes visualprompt to fine-tune visual models for imagebased tasks. According to the empirical results in Chen et al. (2022), adapter modules achieves superior performance over VPT in the regimes of both self-supervised and supervised + +![](images/96b4e4c9e3245d17a30fb7582f32ebe5fa5ec09ee423113f29fc2ba44175259d.jpg) +Figure 1: Parameter-accuracy trade-off. Adapting backbone Swin-B (Liu et al., 2022) pre-trained on Kinetics 400 via different fine-tuning methods on the something-something v2 (Goyal et al., 2017) dataset. Our methods perform significantly better than the state-of-the-art AdaptFormer-Swin (Chen et al., 2022) (our implementation with batch size 16) with slightly more tunable parameters, and outperform full-tuning with increasing margins when using larger values of $d _ { b o t t l e }$ . + +pre-training. Another concern of VPT is its modification to the original model parameters might affect the knowledge representation of backbone models. Hence, we do not continue to compare our method with VPT but comparing with the adapter on video-based downstream tasks. + +Taking the recent inspiration of the mix-and-match adapter (MAM adapter) (He et al., 2022a) in NLP, we aim to propose a unified model for the vision domain, especially for video-based downstream tasks. He et al. (2022a) analyzed the unified view among PETL techniques such as prefixtuning, low-rank (LoRA) adaptation, and adapter, pointing out the similarity between prefix-tuning and adapter in terms of calculating the attention. The difference is that the former performs weighted addition while the latter ones is unweighted. Note that prefix-tuning has not ever been applied to visual tasks in the form of pure visual models due to the intrinsic differences regarding pre-training methods of NLP and vision models. Another obstacle of directly applying prefix-tuning to visual tasks is the structural difference between text and vision data (we further discuss this in Section 2.3). Considering the video-based action recognition task, we propose a new variation of the prefixtuning module called parallel attention (PATT) to adapt video-based pre-trained large models to downstream domains with varied data scales. The differences of our method comparing the original prefix-tuning in NLP are twofold: prefix calculation and the manner of insertion (see Figure 2[b] and Figure 3). Regarding the backbone model, we focus on Video Swin Transformer (Liu et al., 2022), one of the state-of-the-art vision models that bring competitive performance on large-scale action recognition datasets such as Kinetics 400 and 600 Kay et al. (2017). + +Our main contributions can be threefold as follows: + +1. We analyze different PETL techniques using the backbone model Swin Video Transformer for video-based tasks, providing a unified view via our V-PETL framework and investigating the importance of the fine-tuning position. + +2. Based on the comprehensive understanding of intrinsic differences between NLP and video data regarding data structures and pre-training mechanisms, we leverage prefix-tuning to our V-PETL with a new variation called PATT. + +3. Upon extensive ablation experiments regarding various effect factors, we empirically validate the promising parameter-accuracy trade-off achieved by our adjustable and easy-to-use PATT module, contributing to the existing literature of PETL techniques. + +# 2 UNIFIED FRAMEWORK + +# 2.1 RECAP OF VIDEO SWIN TRANSFORMER + +Video Swin Transformer (Liu et al., 2022) is formed with Transformer layers (a.k.a. stages) that are consisted with 3D Video Swin Transformer blocks. With varied layers, blocks, and channel sizes, the model can be formed as Swin-T, Swin-S, Swin-B, and Swin-L. The basic architecture of a 3D Swin Transformer block is shown in Figure 2, which is mainly composed of a 3D shifted window-based multi-head self-attention (3DSW-MSA) module and a fully connected feed-forward network (FFN) implemented with a 2-layer MLP. Layer normalization (LN) and residual connection are respectively performed before and after both FFN and 3DSW-MSA modules. One such Video Swin Transformer block can be represented as: + +$$ +\begin{array} { r l } & { \hat { \boldsymbol Z } ^ { l } = 3 \mathrm { D S W } \mathrm { - } \boldsymbol { \mathrm { M S A } } ( \boldsymbol { \mathrm { L N } } ( \boldsymbol { \boldsymbol { Z } } ^ { l - 1 } ) ) + \boldsymbol { Z } ^ { l - 1 } , } \\ & { \boldsymbol { Z } ^ { l } = \mathrm { F F N } ( \boldsymbol { \mathrm { L N } } ( \hat { \boldsymbol { Z } } ^ { l } ) ) + \hat { \boldsymbol { Z } } ^ { l } , } \end{array} +$$ + +where $\hat { \boldsymbol { z } } ^ { l }$ and $Z ^ { l }$ respectively indicate the output of 3DSW-MSA and FNN modules. + +Given a video input sized $t \times w \times h \times 3$ , containing $t$ video frames with their heights and widths being $h$ and $w$ , respectively. The 3D patch for video data sized $2 \times 4 \times 4 \times 3$ is treated as a token. Then we will have ${ \begin{array} { l } { { \frac { t } { 2 } } \times { \frac { w } { 4 } } \times { \frac { h } { 4 } } } \end{array} }$ 3D tokens after a 3D patch partitioning layer. Given the 3D tokens sized $\begin{array} { r } { \frac { t } { 2 } \times \frac { w } { 4 } \times \frac { h } { 4 } } \end{array}$ and a 3D window with the size of $p \times m \times m$ , the self-attention module, using the regular window partition strategy, will partition the 3D tokens to $\begin{array} { r } { { \frac { t } { 2 p } } \times { \frac { w } { 4 m } } \times { \frac { h } { 4 m } } } \end{array}$ non-overlapping windows. For shifted 3D window, the partition is shifted along the temporal, height, and width dimensions by ${ \begin{array} { l } { { \frac { p } { 2 } } \times { \frac { m } { 2 } } \times { \frac { m } { 2 } } } \end{array} } $ . For example, if we have an input video sized $8 \times 2 2 4 \times 2 2 4 \times 3$ and a $8 \times 7 \times 7$ 3D window, after the patch embedding, we will have $4 \times 5 6 \times 5 6 ~ 3 \mathrm { D }$ tokens with each of them sized $2 \times 4 \times 4 \times 3$ . Without shifting, the non-overlapping window size will be $1 \times 8 \times 8 = 6 4 .$ Then through the 3D window shifted by $( 4 , 3 , 3 )$ , the number of 3D windows becomes $1 \times 9 \times 9 = 8 1$ . + +The 3DSW-MSA module is formed with a 3D relative position bias Rp2×m2×m2 , each of which can be represented as: + +$$ +A t t e n t i o n ( \mathbf { 0 } , \mathbf { K } , \mathbf { V } ) = S o f t M a x \big ( \frac { \mathbf { 0 } \mathbf { K } ^ { T } } { \sqrt { d } } + \mathbf { B } \big ) \mathbf { V } , +$$ + +where $\pmb { \mathsf { Q } } , \pmb { \mathsf { K } } , \pmb { \mathsf { V } } \in \mathbb { R } ^ { p \times m \times m \times d }$ are the query, key, and value matrices, $p \times m \times m$ is the number of tokens and $d$ is the dimension of the tokens. MSA simultaneously performs the attention mechanism for $n _ { h e a d }$ heads, where the $i$ th head can be parameterized by $W _ { q } ^ { ( i ) } , W _ { k } ^ { ( i ) }$ , ${ W _ { v } ^ { ( i ) } \in \mathbb { R } ^ { d \times 3 d } }$ , projecting the input $Z ^ { l - 1 }$ to queries, keys, and values. Given a matrix $\boldsymbol { C } \in \mathbb { R } ^ { \tilde { m } \times d }$ , $\widetilde { \boldsymbol { m } } = \boldsymbol { p } \times \boldsymbol { m } \times \boldsymbol { m }$ , for performing attention, the 3DSW-MSA can be calculated as: + +$$ +\begin{array} { c } { { 3 \mathrm { D S W - M S A } ( Z ^ { l - 1 } , C ) = C o n c a t ( h e a d _ { 1 } , . . . , h e a d _ { n } ) { \cal W } _ { o } , } } \\ { { h e a d _ { i } = A t t e n t i o n ( Z ^ { l - 1 } { \cal W } _ { q } ^ { ( i ) } , C { \cal W } _ { k } ^ { ( i ) } , C { \cal W } _ { v } ^ { ( i ) } ) , } } \end{array} +$$ + +where $W _ { o }$ is the parameters of a linear project layer. The FNN module is composed of two linear layers with a GELU activation function in between, which can be computed as: + +$$ +\mathrm { F F N } ( \hat { \boldsymbol { Z } } ^ { l } ) = \mathrm { G E L U } ( \mathrm { L N } ( \hat { \boldsymbol { Z } } ^ { l } ) W _ { 1 } + b _ { 1 } ) W _ { 2 } + b _ { 2 } , +$$ + +where $W _ { 1 } \in \mathbb { R } ^ { d _ { h i d d e n } \times d }$ , $W _ { 2 } \in \mathbb { R } ^ { d \times d _ { h i d d e n } }$ , $\pmb { b } _ { 1 } \in \mathbb { R } ^ { d _ { h i d d e n } }$ , and $b _ { 2 } \in \mathbb { R } ^ { d }$ . The value of $d _ { h i d d e n }$ usually takes a large value (e.g., $d _ { h i d d e n } = 4 d$ ). + +![](images/f60735e5495b6a10a67e28e8f3097d15d011a8dacad8a620717ac680762eb919.jpg) +Figure 2: V-PETL: A unified view of visual PETL techniques. They bring trainable parameters to different positions of the backbone model with various manners. AdaptFormer and Prefix-tuning respectively perform at the MLP and 3DSW-MSA modules that can adjust the number of trainable parameters via the bottleneck size of down and up projections. While prompt-tuning performed at the layer-level can adjust the length of prompts to control the tuned parameters. + +Prefix-tuning (Li & Liang, 2021): The prefix-tuning approach prepends learnable prefix tokens to the keys and values of the MSA module of the model (see Figure 2[b]). Specifically, two prefix matrices $P _ { k } , P _ { v } \in \mathbb R ^ { d _ { t o k e n } \times d }$ that are randomly initialized with $d _ { t o k e n }$ tokens and transformed from two linear layers (with parameters $W _ { p k } ^ { ( i ) } \in \mathbb { R } ^ { d \times d _ { m i d d l e } }$ and $W _ { p v } ^ { ( i ) } \in \mathbb { R } ^ { d _ { m i d l e } \times d } )$ and a Tanh layer in between are concatenated to the original key and value, leading the calculation of $h e a d _ { i }$ in Eq. 3 to: + +$$ +h e a d _ { i } = A t t e n t i o n ( Z ^ { l - 1 } W _ { q } ^ { ( i ) } , c o n c a t ( P _ { k } ^ { ( i ) } , C W _ { k } ^ { ( i ) } ) , c o n c a t ( P _ { v } ^ { ( i ) } , C W _ { v } ^ { ( i ) } ) ) , +$$ + +where the concat is the concatenation performed along the token dimension to mimic the prefixtuning in NLP tasks. Here, a question regarding whether this direct implementation will work for the vision domain is raised (results are in Table 4). This direct implementation is empirically invalid and we make further modification on it in Section2.3. + +Adapter (Chen et al., 2022): Inspired by the works of Houlsby et al. (2019); He et al. (2022a) for PETL in NLP tasks, adapter (Chen et al., 2022) has been directly used for vision tasks, showing promising performance using far less tunable parameters. The number of parameters of adapter is controlled by a parameter $d _ { b o t t l e }$ $\mathit { \check { d } } _ { b o t t l e } \ll d )$ ), adjusting the space size of a low-dimensional representation. The adapter module first uses a down-projection with $W _ { d o w n } \in \mathbb { R } ^ { d \times d _ { b o t t l e } }$ to project the feature to the lower-dimensional representation, followed by a ReLU activation function, and a up-projection with Wup ∈ Rdbottle×d. + +$$ +\begin{array} { r } { \widetilde { \pmb { Z } } ^ { l } = \mathrm { R e L U } ( \mathbf { L N } ( \hat { \pmb { Z } } ^ { l } ) \mathbf { W } _ { d o w n } ) \mathbf { W } _ { u p } , } \end{array} +$$ + +then two positions implementing adapter (parallel and sequential) can be respectively computed as: + +$$ +\begin{array} { r } { \pmb { Z } ^ { l } = \mathrm { F F N } ( \mathbf { L N } ( \hat { \pmb { Z } } ^ { l } ) ) + \hat { \pmb { Z } } ^ { l } + s \tilde { \pmb { Z } } ^ { l } , \qquad } \\ { a n d s \pmb { Z } ^ { l } = \mathrm { R e L U } ( \mathrm { F F N } ( \mathbf { L N } ( \hat { \pmb { Z } } ^ { l } ) ) \pmb { W } _ { d o w n } ) \pmb { W } _ { u p } + \hat { \pmb { Z } } ^ { l } , } \end{array} +$$ + +where $s$ is a scalar, controlling the effect of the adapter (will be ablated in experiments). According to Chen et al. (2022), the parallel implementation (see Figure 2[a]) empirically performs better. + +Prompt-tuning (Jia et al., 2022): Prompt-tuning (see Figure $2 [ \mathrm { c } ] ,$ is inspired by the success of prompt-tuning that adapts large scale models to varied downstream NLP tasks. The idea of VPT (Jia et al., 2022) is to fine-tune a learnable matrix P l−1prom $P _ { p r o m p t } ^ { l - 1 } \in \mathbb { R } ^ { d _ { p r o m p t } \times d }$ , ${ d _ { p r o m p t } } < { d _ { t o k e n } } - 1$ for + +the lth Transformer layer or all Transformer layers, which are known as shallow prompt and deep prompt, respectively. + +$$ +\begin{array} { r } { \hat { \boldsymbol { \mathsf { Z } } } ^ { l } = 3 \mathrm { D S W } \mathrm { - } \boldsymbol { \mathsf { M S A } } ( \mathrm { L N } ( [ \boldsymbol { x } ^ { l - 1 } , \boldsymbol { P } _ { p r o m p t } ^ { l - 1 } , \boldsymbol { \mathsf { Z } } ^ { l - 1 } ] ) ) + \boldsymbol { \mathsf { Z } } ^ { l - 1 } , } \end{array} +$$ + +where $x ^ { l - 1 } \in \mathbb { R } ^ { d }$ denotes the [CLS]’s embedding for the $l$ th layer’s input space, $P _ { p r o m p t } ^ { l - 1 }$ is implemented by overlapping the top $d _ { p r o m p t }$ tokens of $Z ^ { l - 1 }$ (Jia et al., 2022). While it has also been implemented in front of the $x ^ { l - 1 }$ (Chen et al., 2022). + +Others: Other PETL techniques include ST-Adapter Pan et al. (2022), LoRA (Hu et al., 2022), and BitFit (Zaken et al., 2022). ST-Adapter mainly adapts image-text models pre-trained on large scale datasets such as 400M image-text pair proposed by CLIP (Radford et al., 2021) and the IG-3.6B used by SWAG (Singh et al., 2022) to video understanding downstream tasks, which matches and even outperforms full-tuning. LoRA approximates the optimization process by injecting learnable low-rank matrices into the attention module. This method does not show superior performance for NLP tasks in terms of parameter efficiency. Hence, we do not prioritize this direction in this work. BitFit only tunes the bias terms of the backbone models, making it very parameter-efficient. + +# 2.3 REVISITING PREFIX-TUNING FOR VISUAL TASKS + +The prefix implementation in NLP Li & Liang (2021); He et al. (2022a) can be regarded as prepending contextual information for downstream tasks, which is similar with the pre-training process aiming to predict masked words in the process of an inner loop (Brown et al., 2020). Considering the pre-training process of pure vision models, such direct implementation might not make sense for visual tasks. Although such autoregressive pre-training has been conducted in visual domain (He et al., 2022b; Tong et al., 2022), but adding prefix for a sentence input in NLP can be structurally different with the visual domain. Specifically, masked pixels in image or video data cannot be regarded as some word level semantic information (e.g., a subject or an action) as in the NLP. + +Recall that the embedding state of prefix-tuning is randomly initiated, which is known as learnable prefix but can bring random noise that later turns out affecting the convergence of the fine-tuning downstream tasks. Hence, inspired by the connection between adapter and prefix (He et al., 2022a), we avoid such learnable prefix design with random initialization and propose a parallel attention (PATT) to the original attention module (see Figure 3). The adapter structure can effective control the number of trainable parameters via $d _ { b o t t l e }$ , which is similar with the effect of the middle dimension dmiddle of W (i)pk and $W _ { p v } ^ { ( i ) }$ for preparing the prefix. Specifically, for the lth layer, we use output of its previous layer $Z ^ { l - 1 }$ and project it to a pair of matrices $\boldsymbol { \dot { K _ { p } } } , \boldsymbol { V _ { p } } \in \mathbb { R } ^ { \tilde { m } \times d }$ via a similar mechanism of Eq. 6: + +$$ +K _ { p } , V _ { p } = \mathrm { T a n h } ( Z ^ { l - 1 } W _ { d o w n } ) W _ { u p } , +$$ + +where Tanh is the activation function used for preparing the prefix, which can be replaced by other activation func + +![](images/d32ffd27ce18eb6c9874ee02bac32e6f8939e06524b342dbfc57e9adf4b4bb21.jpg) +Figure 3: Structure of PATT. Red parts are trainable parameters calculated by the same input for preparing query, key, and value (i.e., the output of the previous layer passing through a layer normalization layer $Z ^ { l - 1 }$ ). + +tions such as RELU and GELU. Here, we follow the original prefix implementation as its value ranges from $- 1$ to 1. Given $K _ { p }$ and $V _ { p }$ , Eq. 5 can be rewritten as: + +$$ +h e a d _ { i } = A t t e n t i o n ( \boldsymbol Z ^ { l - 1 } \boldsymbol W _ { q } ^ { ( i ) } , \boldsymbol s \boldsymbol K _ { p } + \boldsymbol C \boldsymbol W _ { k } ^ { ( i ) } , \boldsymbol s V _ { p } + \boldsymbol C \boldsymbol W _ { v } ^ { ( i ) } ) , +$$ + +where $s$ is a scalar for adjusting the effect of PATT. Note that without considering the physical meaning of such design, for PETL purpose, one can perform similar practise for any combinations of $\mathbf { \alpha } _ { \mathbf { Q } , \mathbf { \alpha } } \kappa$ , and $\pmb { \nu }$ . This brings connection to the LoRA (Hu et al., 2022) method, which add parallel trainable parameters to $\mathbf { Q }$ and $\pmb { \nu }$ . Empirically, where to perform the PATT makes little difference, but the amount of trainable parameters brings larger effect for large scale downstream domains. + +# 2.4 V-PETL: UNIFIED VIEW ON VISUAL PETL + +Given the PETL techniques at hand, there can be many potential combinations leading to good parameteraccuracy trade-off. However, it is unrealistic to exhaustively test all the methods for a specific downstream task. Other than probing such solution via evolutionary search as in Zhang et al. (2022), we aim to propose more understandable models by empirically analyzing the effect of different designs independently. According to the preliminary results shwon in Figure 1, we argue that the position and amount of parameters are important for PETL techniques, especially when the target domain is not small. + +To verify the importance of position and tuned parameter amount, we independently tune different modules of the backbone model. Table 1 shows the results. We can see that the attention module’s QKV layer has 20.98M parameters while the MLP module has the most number of parameters of 55.90M. Tuning positions with more parameters, will lead to better performance for SSv2. Thanks to the bottleneck mechanism of adapter and prefix-tuning, one can effectively achieve a good parameter-accuracy trade-off. As such, we derive a model called Swin-B-adapter-PATT (Swin-BAPAT) from the V-PETL framework by using the parallel adapter and our PATT to leverage the adaption of pre-trained backbone model at the positions of attention and MLP modules, respectively. In addition to adapter and PATT, we also fine-tune the last fully connected layer as it has relatively smaller amount of tunable parameters (i.e, 0.18M) than adapter and PATT. + +Table 1: Comparison of independently fine-tuning varied positions of the video swin transformer block on SSv2. + +
Position# ParamsTop-1 (%)
Full-tuning Tune FC Layer87.82M 0.18M50.99 24.13
LayerNorm 10.02M14.35
Attn,Proj6.99M47.58
Attn, QKV20.98M50.02
Attn, SoftMax0.95M27.67
LayerNorm 20.02M14.62
MLP, FC127.97M47.10
MLP,FC227.93M45.32
DownSample2.76M27.53
+ +# 3 EXPERIMENTS + +# 3.1 EXPERIMENTAL SETTINGS + +Video Datasets: Something-something v2 (SSv2 (Goyal et al., 2017)) It has 108,499 short videos for 174 human-object interaction categories with durations between 2 to 6 seconds. The challenge of this dataset is that it contains 23, 137 distinct object names with an imbalanced distribution. The original dataset is split into train, validation, and test sets with a ratio of 8:1:1. The extended version (SSv2) of this dataset is consisted of 168, 913 training samples, 24, 777 validation samples, and 27, 157 testing samples with the sample number of action labels. The training and testing samples are used. HMDB51 (Kuehne et al., 2011) contains 6, 766 video samples for 51 action categories including videos of varied visible body parts, camera motion, camera view, and clip quality. All video samples have at least 101 clips and a minimum height of 60 pixels for actors. The original dataset has three splits of training and evaluation. We follow existing work Chen et al. (2022) by using the first training and evaluation split that has 3, 570 and 1, 530 samples, respectively. Image Datasets: Following the experimental set ups in AdaptFormer, three datasets CIFAIR-100 Krizhevsky et al. (2009), Street View House Numbers (SVHN) Goodfellow et al. (2013), and Food101 Bossard et al. (2014) are used. CIFAIR-100 has 50, 000 and 10, 000 training and validation images, respectively, with the resolution of $3 2 \times 3 2$ and 100 categories; SVHN is a digit classification dataset that has 73, 257 training sample and 26, 032 testing samples; Food-101 includes 101k images of 101 food categories with each of them has 750 training and 250 testing samples. + +Implementation details: It is worth noting that big batch size (i.e., 1, 024) and the number of input video frames (i.e., 32 frames) can greatly benefit good performance (Carreira & Zisserman, 2017; Liu et al., 2022; Chen et al., 2022), which usually requires GPU clusters to enable the training. AdaptFormer (Chen et al., 2022) uses such powerful GPU cluster to achieve good performance. However, good performance might not hold when the batch size is small. Following the more common hardware device setup, we use 4 GeForce 3090 GPUs for all experiments, leading to a batch size of 64. All the experiments are fine-tuned for 70 epochs. We use the Swin- $\mathbf { \cdot B } ^ { 1 }$ model pre-trained on Kinetics 400 and 600. For HMDB51, we report the results without tuning the FC layer due to the significant effect of the FC layer on relatively small scale dataset. Following Chen et al. (2022), we do not perform regularization strategies such as mixup, cutmix, color jittering, etc. Our PATT module is convenient to be applied to other Transformer-based models. Hence, we respectively adopt ViT-B models from MAE (He et al., 2022b) and VideoMAE (Tong et al., 2022) to conduct further comparison on video and image datasets, which follows the self-supervised pretraining setting2 in Chen et al. (2022) except that the batch size is set to 256 instead of 1, 024. + +Table 2: Comparison of Top-1 accuracy using varied amount of parameters adjusted by $d _ { b o t t l e }$ different pre-training domains, and the number of frames with other fine-tuning strategies. + +
MethoddbottlePre-training#FramesSSv2HMDB51
# Params Top-1(%)# Params Top-1 (%)
Full-tuning-Kinetics 400887.82M50.9987.69M68.07
Tune FC LayerKinetics 40080.18M24.130.05M71.28
BitFit (Zaken et al., 2022)·Kinetics 40081.29M45.941.11M68.26
AdaptFormer-Swin (Chen et al., 2022）64Kinetics 40081.73M40.801.61M68.66
Prefix-tuning (Li& Liang,2021)128Kinetics 40086.57M39.466.40M56.13
Our Swin-BAPAT (w/o Adapter)32Kinetics 40088881.35M46.261.17M69.51
Our Swin-BAPAT (w/o Adapter)64Kinetics 4002.51M49.232.34M71.34
Our Swin-BAPAT (w/o Adapter)128Kinetics 4004.83M52.574.65M70.56
Our Swin-BAPAT (w/o Adapter)256Kinetics 4009.45M52.719.27M70.23
Our Swin-BAPAT32Kinetics 40082.91M49.632.74M68.20
Our Swin-BAPAT64Kinetics 40084.07M51.803.89M70.10
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 600811.00M54.0610.83M69.90
Our Swin-BAPAT256 ImageNet-22K811.00M43.5610.83M59.89
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT128Kinetics 400166.38M63.146.20M75.67
+ +Baselines: We mainly compare our method Swin-BAPAT with three baselines as follows: (1) Full-tuning: set all the parameters learnable and tune the whole model initiated with the pretrained weights. (2) Tune FC layer: tune the last fully connected layer and freeze pre-trained parameters of the whole backbone model. (3) AdaptFormer-Swin: method introduced by Chen et al. (2022) that adds a parallel adapter to the MLP module in each block of the backbone model. (4) Prefix-tuning: the direct implementation of prefix-tuning used in NLP as defined in Eq. 5. (5) BitFit: by tuning the bias of the backbone model together with the FC layer. + +# 3.2 THE EFFECT OF DIFFERENT PETL TECHNIQUES + +Table 2 shows the results of different PETL techniques. From the results of four baseline methods, full-tuning performs the best for the large-scale dataset SSv2, whereas tuning the FC layer achieves superior performance over other PETL techniques on HMDB51. This is due to the fact that downstream tasks with relatively larger scale datasets are more parameter hungry for good convergence. On the contrary, small datasets can make good use of the knowledge from the source domain with slight effort of adaption via an FC layer. Here, a question regarding the effect of this FC layer when using it together with other PETL techniques has not been investigated. As this FC layer having small amount of tunable parameters can already make a big difference, performing better than fulltuning and other PETL techniques and rendering them not effective for small-scale datasets. As such, we further examine this question in Section A.1. + +We test different amount of parameters adjusted by $s _ { b o t t l e }$ , taking its values to 32, 64, 128 and 256. The second and third groups (without or with Adapter, respectively) of results in Table 2 shows that larger values of $s _ { b o t t l e }$ can benefit the fine-tuning with slightly more overhead of parameters on large-scale datasets such as SSv2. All results of our Swin-BAPAT outperform the state-ofthe-art AdaptFormer-Swin with a big margin (using the smallest value $s _ { b o t t l e } = 3 2$ can improve AdaptFormer-Swin by almost $2 5 \%$ ). While without using Adapter, our method still outperforms baselines AdaptFormer-Swin and BitFit with roughly similar amount of parameters. When sbottle is larger than 64, our Swin-BAPAT starts to perform better than full-tuning on both datasets with proper parameter-accuracy trade-off, validating the effectiveness of our Swin-BAPAT for PETL. + +![](images/bfd8674c26a279b07e5eec991eb55a8d1b5100558500140751dfb357f84b103a.jpg) +Figure 4: Top-1 accuracy of different settings on SSv2 throughout training process. F: frame, S: scalar, B: $d _ { b o t t l e }$ , K: pre-training domain. + +Table 3: Top-1 accuracy $( \% )$ using different scalar values on two datasets: SSv2 and HMDB51. The $d _ { b o t t l e }$ is set to 128; pretraining is based on Kinetics 400. + +
Scalar sSSv2HMDB51
Full-tuning50.9971.28
Tune FC Layer24.1368.07
AdaptFormer-Swin40.8068.66
s=0.247.4669.38
s=0.552.8471.87
s=0.853.3671.93
s=1.053.2970.89
+ +# 3.3 THE EFFECT OF DIFFERENT PRE-TRAINING DOMAINS + +The knowledge from the pre-trained model is learned from the source domain. We test two different models pre-trained on large-scale datasets: Kinetics 400, Kinetics 600, and ImageNet-22K. Findings show that both two models pre-trained on such large-scale datasets can benefit our proposed PETL strategy with the latter being slightly more significant (see the third group of comparison in Table 2). This is due to the fact that Kinectics 600 is larger than its 400 version and brings more knowledge to the pre-trained model, benefiting more downstream tasks. However, image-based pre-training cannot perform as good as video-based pre-training due to the larger domain gap. + +# 3.4 THE EFFECT OF DIFFERENT VIDEO INPUT SIZE + +We also test whether our method is robust to increased number of input video frames. It is worth noting that larger number of input video frames usually can bring more spatial temporal information, benefiting data-driven models to learn more distinguishable features while keeping the model size remaining the same. The last group of comparisons in Table 2 shows that using double-sized video input (i.e., 16 frames) can greatly improve the performance of action recognition on both small and large-scale datasets. The improvements (increased $9 . 7 8 \%$ from $5 3 . 3 6 \%$ to $6 3 . 1 4 \%$ on SSv2, and $3 . { \bar { 7 } } 4 \%$ from $7 1 . 9 3 \%$ to $7 5 . 6 7 \%$ on HMDB51) are more significant than other factors such as $d _ { b o t t l e }$ and pre-training domain (around $1 \%$ to $2 \%$ ). The top line in Figure 4 visualizes the significant effect of increasing the number of input video frames. These results suggest that our Swin-BAPAT can be promising for increased frames of video input. + +# 3.5 THE EFFECT OF DIFFERENT SCALE OF PATT + +Recall that the effect of our PATT on pretrained models can be adjusted by the variable $s$ in Eq. 10. Table 3 shows that adopting the value of 0.8 can deliver consistent best performances on both datasets SSv2 and HMDB51 under our experimental setting. Smaller values of $s$ will quantitatively reduce the effect of our PATT module on the knowledge transfer while large values will increase the effect of our PATT module. The good performance achieved via taking an effective scale of 0.8 indicates that our PATT module plays an important role in the knowledge transfer. However, even larger values over 0.8 can affect the importance of original knowledge thereof the pretrained model. Hence, proper valued scalar $s$ is essential for balancing the role of PATT and + +Table 4: Ablation of different implementation positions of PATT defined in Eq. 10, e.g., Ours (K, $\boldsymbol { \mathsf { V } }$ ) indicates inserting PATT to the query and key of 3DSW-MSA modules. Pre-training on Kinetics 600. $d _ { b o t t l e }$ is set to 128; Scalar $s$ is set to 0.8. + +
MethodSSv2HMDB51
#ParamsTop-1# ParamsTop-1
Full-tuning87.82M50.9987.69M68.07
Concat (K, V)6.38M15.616.20M20.98
No Zl-1(K,V)8.74M51.068.56M67.41
Ours (Q, K)6.38M45.496.20M68.92
Ours (K, V)6.38M53.386.20M71.41
Ours (Q, V)6.38M53.246.20M71.74
Ours (Q, K, V)7.93M53.237.63M69.57
+ +pre-trained backbone model. Note this can be a learnable parameter upon specific implementation, here we empirically verified the effect of the scalar. + +# 3.6 THE EFFECT OF DIFFERENT METHODS YIELD FROM V-PETL + +We have argued that, especially for relative large downstream datasets, the position and the amount of trainable parameters are important for parameter-efficient transfer learning in Section 2.4. The proposed Swin-BAPAT is one of instantiated models from the V-PETL framework regarding the insert position of our PATT. Other instantiations can be inserted into different positions such as query, key, and value of the attention module. We further instantiate other variations of our Swin-BAPAT by inserting PATT to different positions. Table 4 shows the results. Findings show that inserting to the value position of 3DSW-MSA can contribute more than inserting to other two positions. While inserting to query of key makes little difference for the performance. This is due to the fact that query and key make the calculation of the attention mask. Hence, inserting either one of them will lead to a similar effect. On one hand, these results, to some extent, justify the original design of prefix-tuning that bring learnable prefix to key and value of the attention module. On the other hand, it indicates that our claim regarding the unified view of PETL for visual tasks is reasonable. In Table 4, we also ablate the designs of PATT regarding concatenating $K _ { p }$ and $V _ { p }$ (i.e., Concat $[ \mathsf { K } , \mathsf { v } ] )$ , and using trainable parameters to generate $K _ { p }$ and $\boldsymbol { V _ { p } }$ (i.e., N ${ \bf \nabla } ) \ Z ^ { l - 1 } ( { \bf K } , { \bf V } ] )$ . + +# 3.7 COMPARISON ON VARIED TASKS VIA SELF-SUPERVISED PRE-TRAINED MODELS + +Table 5 shows the comparison with AdaptFormer-64 (Chen et al., 2022) and VPT (Jia et al., 2022) on both image- and video-based downstream tasks. Our method ViT-BAPAT still shows promising parameter-accuracy trade-off via much smaller batch size, which is more convenient for reproduction on the general single server with 8 GPUs. The underperformance on SSv2 (better than full-tuning) can be due to the smaller batch size as SSv2 is much larger than other compared datasets and can be more relying on larger batch size. In real-world application scenarios, small dataset can be the more common case, which confirms our contributions. + +Table 5: Comparison of Top-1 accuracy via ViT-B models from MAE and VideoMAE pre-trained with self-supervised learning for image and video datasets, respectively. + +
MethodAvg.ImageVideo
Params (M)CIFAR-100SVHNFood-101SSv2HMDB51
Full-tuning86.04 (100%)85.9097.6790.0953.9746.41
Tune FC Layer0.07 (0.08%)69.83 (-16.07) 66.91 (-30.76) 69.74 (-20.35)29.23 (-24.74)）49.84 (+3.43)
VPT (Jia et al.,2022)0.08 (0.09%)82.44 (-3.46)94.02 (-3.65)82.98 (-7.11)43.73 (-10.24)52.67 (+6.26)
AdaptFormer-641.26 (1.46%)85.90 (0.00)96.89 (-0.78)87.61 (-2.48)59.02 (+5.05)55.69 (+9.28)
Our ViT-BAPAT-322.13 (2.47%)86.29 (+0.39)97.18 (-0.49)87.37 (-2.72)57.78 (+3.81)57.18 (+10.77)
Our ViT-BAPAT-643.02 (3.51%)86.35 (+0.45)97.18 (-0.49)87.53 (-2.56)57.55 (+3.58)57.18 (+10.77)
Our ViT-BAPAT-1284.79 (5.56%)86.47 (+0.57)97.28 (-0.39)87.66 (-2.43)56.97 (+3.00)57.70 (+11.29)
Our ViT-BAPAT-2568.33 (9.68%)86.55 (+0.65)97.24 (-0.43)87.68 (-2.41)56.53 (+2.56)57.31 (+10.90)
+ +# 4 CONCLUSION + +In this paper, we introduced a V-PETL framework for exploiting good parameter-accuracy tradeoff around adapting video-based pre-trained large models to downstream tasks. Our Swin-BAPAT method derived from the V-PETL with a variation of prefix-tuning known as PATT can effectively bring good parameter-accuracy trade-off on downstream tasks. The proposed PATT can be easily plugged to the attention module of other transformer-like models. Meanwhile, the amount of trainable parameter can be easily adjusted by the parameter $d _ { b o t t l e }$ . With small amount overhead on trainable parameters, our method performs significantly better than state-of-the-art method AdapFormer-Swin and full-tuning on the datasets SSv2 and HMDB51 via small batch size, validating our contribution to the literature of PETL. In the future we will test our proposed model on more action recognition datasets surveyed in Sun et al. (2022) under more learning regimes such as zero/few-shot learning, active learning and continual learning with other pre-training methods such as visual-language models. We will also explore other backbone models, activation functions for PATT, and PETL techniques such as LoRA for visual tasks. + +# REFERENCES + +Lukas Bossard, Matthieu Guillaumin, and Luc Van Gool. Food-101–mining discriminative components with random forests. In European conference on computer vision, pp. 446–461. Springer, 2014. + +Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. Language models are few-shot learners. Advances in neural information processing systems, 33:1877–1901, 2020. + +Joao Carreira and Andrew Zisserman. Quo vadis, action recognition? a new model and the kinetics dataset. 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In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 8168–8177, 2021. + +Jiahui Yu, Zirui Wang, Vijay Vasudevan, Legg Yeung, Mojtaba Seyedhosseini, and Yonghui Wu. Coca: Contrastive captioners are image-text foundation models. arXiv preprint arXiv:2205.01917, 2022. + +Sha Yuan, Hanyu Zhao, Shuai Zhao, Jiahong Leng, Yangxiao Liang, Xiaozhi Wang, Jifan Yu, Xin Lv, Zhou Shao, Jiaao He, et al. A roadmap for big model. arXiv preprint arXiv:2203.14101, 2022. + +Elad Ben Zaken, Yoav Goldberg, and Shauli Ravfogel. Bitfit: Simple parameter-efficient fine-tuning for transformer-based masked language-models. In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pp. 1–9, 2022. + +Xiaohua Zhai, Alexander Kolesnikov, Neil Houlsby, and Lucas Beyer. Scaling vision transformers. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 12104–12113, 2022. + +Yuanhan Zhang, Kaiyang Zhou, and Ziwei Liu. Neural prompt search. arXiv preprint arXiv:2206.04673, 2022. + +Fuzhen Zhuang, Zhiyuan Qi, Keyu Duan, Dongbo Xi, Yongchun Zhu, Hengshu Zhu, Hui Xiong, and Qing He. A comprehensive survey on transfer learning. Proceedings of the IEEE, 109(1): 43–76, 2020. + +# A APPENDIX + +A.1 THE EFFECT OF FC LAYER FOR SMALL SCALE DOWNSTREAM TASKS + +Table 6: Results of with or without tuning the FC layer on the small scale dataset HMDB51. + +
MethoddbottlePre-training#Frameswith FC layerwithout FC layer
#ParamsTop-1 (%)#ParamsTop-1 (%)
Our Swin-BAPAT32Kinetics 40082.79M65.972.74M68.20
Our Swin-BAPAT64Kinetics 40083.94M67.283.89M70.10
Our Swin-BAPAT128Kinetics 40086.25M66.756.20M71.93
Our Swin-BAPAT256Kinetics 400810.88M67.6710.83M69.64
Our Swin-BAPAT256Kinetics 400810.88M67.6710.83M69.64
Our Swin-BAPAT256Kinetics 600810.88M67.4110.83M69.90
Our Swin-BAPAT128Kinetics 40086.25M66.756.20M71.93
Our Swin-BAPAT128Kinetics 400166.25M70.566.20M75.67
Our Swin-BAPAT128Kinetics 400326.25M74.826.20M76.46
+ +For the small dataset HMDB51, due to the good parameter-accuracy trade-off achieved by finetuning the FC layer only, adding the FC layer cannot bring extra improvement to our proposed method. Without sufficient taining data, full-tuning also cannot perform well (see results in Table 2). As such, small datasets do not need to rely on large models but can make use of large models with light transfer. Instead, without tuning the FC layer, our Swin-BAPAT can perform better than fine-tuning the FC layer with small amount of extra trainable parameters (see results in Table 6), validating the good parameter-accuracy trade-off of our method. \ No newline at end of file diff --git a/parse/dev/ti6fH3EhFkv/ti6fH3EhFkv_content_list.json b/parse/dev/ti6fH3EhFkv/ti6fH3EhFkv_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..e92fcc448b8c273b3d67b8c9ef5226ee20fe0cdc --- /dev/null +++ b/parse/dev/ti6fH3EhFkv/ti6fH3EhFkv_content_list.json @@ -0,0 +1,1483 @@ +[ + { + "type": "text", + "text": "TOWARDS A UNIFIED VIEW ON VISUAL PARAMETEREFFICIENT TRANSFER LEARNING ", + "text_level": 1, + "bbox": [ + 176, + 98, + 821, + 146 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous authors Paper under double-blind review ", + "bbox": [ + 183, + 170, + 398, + 198 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "ABSTRACT ", + "text_level": 1, + "bbox": [ + 454, + 234, + 544, + 251 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Since the release of various large-scale natural language processing (NLP) pretrained models, parameter efficient transfer learning (PETL) has become a popular paradigm capable of achieving impressive performance on various downstream tasks. PETL aims at making good use of the representation knowledge in the pretrained large models by fine-tuning a small number of parameters. Recently, it has also attracted increasing attention to developing various PETL techniques for vision tasks. Popular PETL techniques such as Prompt-tuning and Adapter have been proposed for high-level visual downstream tasks such as image classification and video recognition. However, Prefix-tuning remains under-explored for vision tasks. In this work, we intend to adapt large video-based models to downstream tasks with a good parameter-accuracy trade-off. Towards this goal, we propose a framework with a unified view of PETL called visual-PETL (V-PETL) to investigate the effects of different PETL techniques, data scales of downstream domains, positions of trainable parameters, and other aspects affecting the tradeoff. Specifically, we analyze the positional importance of trainable parameters and differences between NLP and vision tasks in terms of data structures and pretraining mechanisms while implementing various PETL techniques, especially for the under-explored prefix-tuning technique. Based on a comprehensive understanding of differences between NLP and video data, we propose a new variation of prefix-tuning module called parallel attention (PATT) for video-based downstream tasks. An extensive empirical analysis on two video datasets via different frozen backbones has been carried and the findings show that the proposed PATT can effectively contribute to other PETL techniques. An effective scheme SwinBAPAT derived from the proposed V-PETL framework achieves significantly better performance than the state-of-the-art AdaptFormer-Swin with slightly more parameters and outperforms full-tuning with far less parameters. ", + "bbox": [ + 232, + 268, + 764, + 628 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 INTRODUCTION ", + "text_level": 1, + "bbox": [ + 176, + 655, + 334, + 671 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Many vision tasks rely on fine-tuning pre-trained models to achieve good performance. One standard modus operandi of transfer learning consists of two steps: pre-train a model on a source domain and fine-tune the entire model on a target domain (Zhuang et al., 2020). Despite that prior works have achieved promising performance, such vanilla practice of fine-tuning is faced with challenges for adopting large models to downstream tasks. This full-tuning strategy requires one to update and store separate model parameters for different downstream tasks, which can be expensive and infeasible for the era of increasingly large models from EfficientNet-based (Pham et al., 2021) (480M parameters) to Transformer-based (Yu et al., 2022) (2, 100M parameters) ones. For such large models, making good use of shared parameter weights deployed on the cloud can be beneficial for edge devices such as autonomous vehicles, drones who are intensive in computing and battery resources (Yuan et al., 2022). Second, the full fine-tuning strategy relies on high-quality downstream data and can hardly adapt to unseen scenarios that have large distribution shift (Kumar et al., 2021), which is unlike the learning process of humans who can learn from few samples and generalize well to new circumstances. This issue has been researched in directions such as zero-shot learning, few-shot learning, and continual learning (Li et al., 2021a). Another popular strategy is fine-tuning the downstream task head, i.e., the last fully connected (FC) layer, to avoid tuning the whole backbone model, which usually leads to poor performance when the target domain is large in data scale (see Figure ", + "bbox": [ + 174, + 688, + 825, + 924 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1). Given the paradigm of fine-tuning increasingly large models, how to transfer such large models with parameter-accuracy trade-off is a hot topic in various domains (Gusak et al., 2022; Sung et al., 2022; Lin et al., 2020; Houlsby et al., 2019). ", + "bbox": [ + 174, + 103, + 823, + 146 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Taking the video-based action recognition task as an example, it can be inconvenient for deploying such large models to edge devices such as an autonomous driving (Liu et al., 2019) and unmanned aerial vehicle (Li et al., 2021b) as they can heavily rely on the interaction with cloud services for adapting to new environments via active learning (Wang et al., 2021) or continual learning (Li et al., 2021a). Re-training large models on the cloud are usually not cost-effective due to the expensive overheads of storage and computational resources. Furthermore, these resources are limited on edge devices such as autonomous vehicles and unmanned aerial vehicles, making the sense for developing effective fine-tuning methods with proper parameter-accuracy trade-off that can be fine-tuned on edge devices and interacting with the large models deployed on the cloud. ", + "bbox": [ + 174, + 152, + 825, + 279 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "There have been some pioneering works for the PETL of visual models such as AdaptFormer (Chen et al., 2022) and visual prompt tuning (VPT) (Jia et al., 2022). AdaptFormer is primarily proposed based on vision transformer (Zhai et al., 2022), representing one of the stateof-the-art large models for image-based tasks. The proposed adapter module directly brings from Houlsby et al. (2019) due to its convenience of being inserted to any models. Implementing with a large batch size of $1 , 0 2 4$ with 64 GPUs, Adaptformer shows promising parameter-accuracy trade-off on video data. However, such powerful computing resource is not realistic for the usage of edge devices. Meanwhile, whether the good trade-off can be maintained for small batch size remains under-explored. Inspired by the Prompting in NLP (Liu et al., 2021), VPT proposes visualprompt to fine-tune visual models for imagebased tasks. According to the empirical results in Chen et al. (2022), adapter modules achieves superior performance over VPT in the regimes of both self-supervised and supervised ", + "bbox": [ + 174, + 285, + 483, + 617 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/96b4e4c9e3245d17a30fb7582f32ebe5fa5ec09ee423113f29fc2ba44175259d.jpg", + "image_caption": [ + "Figure 1: Parameter-accuracy trade-off. Adapting backbone Swin-B (Liu et al., 2022) pre-trained on Kinetics 400 via different fine-tuning methods on the something-something v2 (Goyal et al., 2017) dataset. Our methods perform significantly better than the state-of-the-art AdaptFormer-Swin (Chen et al., 2022) (our implementation with batch size 16) with slightly more tunable parameters, and outperform full-tuning with increasing margins when using larger values of $d _ { b o t t l e }$ . " + ], + "image_footnote": [], + "bbox": [ + 500, + 296, + 820, + 458 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "pre-training. Another concern of VPT is its modification to the original model parameters might affect the knowledge representation of backbone models. Hence, we do not continue to compare our method with VPT but comparing with the adapter on video-based downstream tasks. ", + "bbox": [ + 176, + 617, + 823, + 659 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Taking the recent inspiration of the mix-and-match adapter (MAM adapter) (He et al., 2022a) in NLP, we aim to propose a unified model for the vision domain, especially for video-based downstream tasks. He et al. (2022a) analyzed the unified view among PETL techniques such as prefixtuning, low-rank (LoRA) adaptation, and adapter, pointing out the similarity between prefix-tuning and adapter in terms of calculating the attention. The difference is that the former performs weighted addition while the latter ones is unweighted. Note that prefix-tuning has not ever been applied to visual tasks in the form of pure visual models due to the intrinsic differences regarding pre-training methods of NLP and vision models. Another obstacle of directly applying prefix-tuning to visual tasks is the structural difference between text and vision data (we further discuss this in Section 2.3). Considering the video-based action recognition task, we propose a new variation of the prefixtuning module called parallel attention (PATT) to adapt video-based pre-trained large models to downstream domains with varied data scales. The differences of our method comparing the original prefix-tuning in NLP are twofold: prefix calculation and the manner of insertion (see Figure 2[b] and Figure 3). Regarding the backbone model, we focus on Video Swin Transformer (Liu et al., 2022), one of the state-of-the-art vision models that bring competitive performance on large-scale action recognition datasets such as Kinetics 400 and 600 Kay et al. (2017). ", + "bbox": [ + 174, + 666, + 825, + 887 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "Our main contributions can be threefold as follows: ", + "bbox": [ + 178, + 895, + 511, + 909 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "1. We analyze different PETL techniques using the backbone model Swin Video Transformer for video-based tasks, providing a unified view via our V-PETL framework and investigating the importance of the fine-tuning position. ", + "bbox": [ + 173, + 910, + 823, + 922 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2. Based on the comprehensive understanding of intrinsic differences between NLP and video data regarding data structures and pre-training mechanisms, we leverage prefix-tuning to our V-PETL with a new variation called PATT. ", + "bbox": [ + 174, + 132, + 821, + 172 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "3. Upon extensive ablation experiments regarding various effect factors, we empirically validate the promising parameter-accuracy trade-off achieved by our adjustable and easy-to-use PATT module, contributing to the existing literature of PETL techniques. ", + "bbox": [ + 174, + 174, + 825, + 215 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2 UNIFIED FRAMEWORK ", + "text_level": 1, + "bbox": [ + 176, + 234, + 393, + 251 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "2.1 RECAP OF VIDEO SWIN TRANSFORMER ", + "text_level": 1, + "bbox": [ + 174, + 265, + 490, + 279 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Video Swin Transformer (Liu et al., 2022) is formed with Transformer layers (a.k.a. stages) that are consisted with 3D Video Swin Transformer blocks. With varied layers, blocks, and channel sizes, the model can be formed as Swin-T, Swin-S, Swin-B, and Swin-L. The basic architecture of a 3D Swin Transformer block is shown in Figure 2, which is mainly composed of a 3D shifted window-based multi-head self-attention (3DSW-MSA) module and a fully connected feed-forward network (FFN) implemented with a 2-layer MLP. Layer normalization (LN) and residual connection are respectively performed before and after both FFN and 3DSW-MSA modules. One such Video Swin Transformer block can be represented as: ", + "bbox": [ + 173, + 290, + 825, + 402 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/2c0796cf17f12268e00b9ca2541f786532af0be7d62c85af40b06e7db202e182.jpg", + "text": "$$\n\\begin{array} { r l } & { \\hat { \\boldsymbol Z } ^ { l } = 3 \\mathrm { D S W } \\mathrm { - } \\boldsymbol { \\mathrm { M S A } } ( \\boldsymbol { \\mathrm { L N } } ( \\boldsymbol { \\boldsymbol { Z } } ^ { l - 1 } ) ) + \\boldsymbol { Z } ^ { l - 1 } , } \\\\ & { \\boldsymbol { Z } ^ { l } = \\mathrm { F F N } ( \\boldsymbol { \\mathrm { L N } } ( \\hat { \\boldsymbol { Z } } ^ { l } ) ) + \\hat { \\boldsymbol { Z } } ^ { l } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 361, + 405, + 633, + 454 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\hat { \\boldsymbol { z } } ^ { l }$ and $Z ^ { l }$ respectively indicate the output of 3DSW-MSA and FNN modules. ", + "bbox": [ + 178, + 458, + 728, + 474 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Given a video input sized $t \\times w \\times h \\times 3$ , containing $t$ video frames with their heights and widths being $h$ and $w$ , respectively. The 3D patch for video data sized $2 \\times 4 \\times 4 \\times 3$ is treated as a token. Then we will have ${ \\begin{array} { l } { { \\frac { t } { 2 } } \\times { \\frac { w } { 4 } } \\times { \\frac { h } { 4 } } } \\end{array} }$ 3D tokens after a 3D patch partitioning layer. Given the 3D tokens sized $\\begin{array} { r } { \\frac { t } { 2 } \\times \\frac { w } { 4 } \\times \\frac { h } { 4 } } \\end{array}$ and a 3D window with the size of $p \\times m \\times m$ , the self-attention module, using the regular window partition strategy, will partition the 3D tokens to $\\begin{array} { r } { { \\frac { t } { 2 p } } \\times { \\frac { w } { 4 m } } \\times { \\frac { h } { 4 m } } } \\end{array}$ non-overlapping windows. For shifted 3D window, the partition is shifted along the temporal, height, and width dimensions by ${ \\begin{array} { l } { { \\frac { p } { 2 } } \\times { \\frac { m } { 2 } } \\times { \\frac { m } { 2 } } } \\end{array} } $ . For example, if we have an input video sized $8 \\times 2 2 4 \\times 2 2 4 \\times 3$ and a $8 \\times 7 \\times 7$ 3D window, after the patch embedding, we will have $4 \\times 5 6 \\times 5 6 ~ 3 \\mathrm { D }$ tokens with each of them sized $2 \\times 4 \\times 4 \\times 3$ . Without shifting, the non-overlapping window size will be $1 \\times 8 \\times 8 = 6 4 .$ Then through the 3D window shifted by $( 4 , 3 , 3 )$ , the number of 3D windows becomes $1 \\times 9 \\times 9 = 8 1$ . ", + "bbox": [ + 173, + 479, + 826, + 628 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "The 3DSW-MSA module is formed with a 3D relative position bias Rp2×m2×m2 , each of which can be represented as: ", + "bbox": [ + 174, + 633, + 820, + 666 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/ff29df293d5b07f172d79958bb060edd535f495c46c6b3e4158374a9d021aea6.jpg", + "text": "$$\nA t t e n t i o n ( \\mathbf { 0 } , \\mathbf { K } , \\mathbf { V } ) = S o f t M a x \\big ( \\frac { \\mathbf { 0 } \\mathbf { K } ^ { T } } { \\sqrt { d } } + \\mathbf { B } \\big ) \\mathbf { V } ,\n$$", + "text_format": "latex", + "bbox": [ + 334, + 667, + 661, + 705 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $\\pmb { \\mathsf { Q } } , \\pmb { \\mathsf { K } } , \\pmb { \\mathsf { V } } \\in \\mathbb { R } ^ { p \\times m \\times m \\times d }$ are the query, key, and value matrices, $p \\times m \\times m$ is the number of tokens and $d$ is the dimension of the tokens. MSA simultaneously performs the attention mechanism for $n _ { h e a d }$ heads, where the $i$ th head can be parameterized by $W _ { q } ^ { ( i ) } , W _ { k } ^ { ( i ) }$ , ${ W _ { v } ^ { ( i ) } \\in \\mathbb { R } ^ { d \\times 3 d } }$ , projecting the input $Z ^ { l - 1 }$ to queries, keys, and values. Given a matrix $\\boldsymbol { C } \\in \\mathbb { R } ^ { \\tilde { m } \\times d }$ , $\\widetilde { \\boldsymbol { m } } = \\boldsymbol { p } \\times \\boldsymbol { m } \\times \\boldsymbol { m }$ , for performing attention, the 3DSW-MSA can be calculated as: ", + "bbox": [ + 174, + 707, + 825, + 785 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/286a975cb422e19fb1ff53677d4f5ba9891bb528b339a78ab8d0a69785a37631.jpg", + "text": "$$\n\\begin{array} { c } { { 3 \\mathrm { D S W - M S A } ( Z ^ { l - 1 } , C ) = C o n c a t ( h e a d _ { 1 } , . . . , h e a d _ { n } ) { \\cal W } _ { o } , } } \\\\ { { h e a d _ { i } = A t t e n t i o n ( Z ^ { l - 1 } { \\cal W } _ { q } ^ { ( i ) } , C { \\cal W } _ { k } ^ { ( i ) } , C { \\cal W } _ { v } ^ { ( i ) } ) , } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 786, + 725, + 833 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $W _ { o }$ is the parameters of a linear project layer. The FNN module is composed of two linear layers with a GELU activation function in between, which can be computed as: ", + "bbox": [ + 174, + 838, + 821, + 867 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/0ba56677d635e9287090789442d9efc886982a98f3cb2b687ff1349ce35c098a.jpg", + "text": "$$\n\\mathrm { F F N } ( \\hat { \\boldsymbol { Z } } ^ { l } ) = \\mathrm { G E L U } ( \\mathrm { L N } ( \\hat { \\boldsymbol { Z } } ^ { l } ) W _ { 1 } + b _ { 1 } ) W _ { 2 } + b _ { 2 } ,\n$$", + "text_format": "latex", + "bbox": [ + 336, + 869, + 658, + 892 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "where $W _ { 1 } \\in \\mathbb { R } ^ { d _ { h i d d e n } \\times d }$ , $W _ { 2 } \\in \\mathbb { R } ^ { d \\times d _ { h i d d e n } }$ , $\\pmb { b } _ { 1 } \\in \\mathbb { R } ^ { d _ { h i d d e n } }$ , and $b _ { 2 } \\in \\mathbb { R } ^ { d }$ . The value of $d _ { h i d d e n }$ usually takes a large value (e.g., $d _ { h i d d e n } = 4 d$ ). ", + "bbox": [ + 176, + 893, + 825, + 925 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/f60735e5495b6a10a67e28e8f3097d15d011a8dacad8a620717ac680762eb919.jpg", + "image_caption": [ + "Figure 2: V-PETL: A unified view of visual PETL techniques. They bring trainable parameters to different positions of the backbone model with various manners. AdaptFormer and Prefix-tuning respectively perform at the MLP and 3DSW-MSA modules that can adjust the number of trainable parameters via the bottleneck size of down and up projections. While prompt-tuning performed at the layer-level can adjust the length of prompts to control the tuned parameters. " + ], + "image_footnote": [], + "bbox": [ + 238, + 136, + 756, + 371 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Prefix-tuning (Li & Liang, 2021): The prefix-tuning approach prepends learnable prefix tokens to the keys and values of the MSA module of the model (see Figure 2[b]). Specifically, two prefix matrices $P _ { k } , P _ { v } \\in \\mathbb R ^ { d _ { t o k e n } \\times d }$ that are randomly initialized with $d _ { t o k e n }$ tokens and transformed from two linear layers (with parameters $W _ { p k } ^ { ( i ) } \\in \\mathbb { R } ^ { d \\times d _ { m i d d l e } }$ and $W _ { p v } ^ { ( i ) } \\in \\mathbb { R } ^ { d _ { m i d l e } \\times d } )$ and a Tanh layer in between are concatenated to the original key and value, leading the calculation of $h e a d _ { i }$ in Eq. 3 to: ", + "bbox": [ + 173, + 464, + 825, + 542 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/881606647ab316c9bcf9ec0d5517412ee0484ed0867692bc2cba789edfc08423.jpg", + "text": "$$\nh e a d _ { i } = A t t e n t i o n ( Z ^ { l - 1 } W _ { q } ^ { ( i ) } , c o n c a t ( P _ { k } ^ { ( i ) } , C W _ { k } ^ { ( i ) } ) , c o n c a t ( P _ { v } ^ { ( i ) } , C W _ { v } ^ { ( i ) } ) ) ,\n$$", + "text_format": "latex", + "bbox": [ + 233, + 547, + 764, + 569 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where the concat is the concatenation performed along the token dimension to mimic the prefixtuning in NLP tasks. Here, a question regarding whether this direct implementation will work for the vision domain is raised (results are in Table 4). This direct implementation is empirically invalid and we make further modification on it in Section2.3. ", + "bbox": [ + 173, + 574, + 825, + 630 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Adapter (Chen et al., 2022): Inspired by the works of Houlsby et al. (2019); He et al. (2022a) for PETL in NLP tasks, adapter (Chen et al., 2022) has been directly used for vision tasks, showing promising performance using far less tunable parameters. The number of parameters of adapter is controlled by a parameter $d _ { b o t t l e }$ $\\mathit { \\check { d } } _ { b o t t l e } \\ll d )$ ), adjusting the space size of a low-dimensional representation. The adapter module first uses a down-projection with $W _ { d o w n } \\in \\mathbb { R } ^ { d \\times d _ { b o t t l e } }$ to project the feature to the lower-dimensional representation, followed by a ReLU activation function, and a up-projection with Wup ∈ Rdbottle×d. ", + "bbox": [ + 173, + 636, + 825, + 738 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c0fd4d1ba62f3b13391d92e7355fcef9d50a9a668b116962913f570a2c026e86.jpg", + "text": "$$\n\\begin{array} { r } { \\widetilde { \\pmb { Z } } ^ { l } = \\mathrm { R e L U } ( \\mathbf { L N } ( \\hat { \\pmb { Z } } ^ { l } ) \\mathbf { W } _ { d o w n } ) \\mathbf { W } _ { u p } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 379, + 743, + 617, + 767 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "then two positions implementing adapter (parallel and sequential) can be respectively computed as: ", + "bbox": [ + 173, + 772, + 823, + 787 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/c17ca3e8ccad95bcb3174b2a70e7ecdebe06d805e986d6e6dbecaf6c9b754bb1.jpg", + "text": "$$\n\\begin{array} { r } { \\pmb { Z } ^ { l } = \\mathrm { F F N } ( \\mathbf { L N } ( \\hat { \\pmb { Z } } ^ { l } ) ) + \\hat { \\pmb { Z } } ^ { l } + s \\tilde { \\pmb { Z } } ^ { l } , \\qquad } \\\\ { a n d s \\pmb { Z } ^ { l } = \\mathrm { R e L U } ( \\mathrm { F F N } ( \\mathbf { L N } ( \\hat { \\pmb { Z } } ^ { l } ) ) \\pmb { W } _ { d o w n } ) \\pmb { W } _ { u p } + \\hat { \\pmb { Z } } ^ { l } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 318, + 791, + 678, + 842 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "where $s$ is a scalar, controlling the effect of the adapter (will be ablated in experiments). According to Chen et al. (2022), the parallel implementation (see Figure 2[a]) empirically performs better. ", + "bbox": [ + 174, + 844, + 825, + 875 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Prompt-tuning (Jia et al., 2022): Prompt-tuning (see Figure $2 [ \\mathrm { c } ] ,$ is inspired by the success of prompt-tuning that adapts large scale models to varied downstream NLP tasks. The idea of VPT (Jia et al., 2022) is to fine-tune a learnable matrix P l−1prom $P _ { p r o m p t } ^ { l - 1 } \\in \\mathbb { R } ^ { d _ { p r o m p t } \\times d }$ , ${ d _ { p r o m p t } } < { d _ { t o k e n } } - 1$ for ", + "bbox": [ + 174, + 878, + 825, + 926 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "the lth Transformer layer or all Transformer layers, which are known as shallow prompt and deep prompt, respectively. ", + "bbox": [ + 169, + 103, + 823, + 132 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/a459896c8397898bdc7f4c24f9bb88d1a1c8b7d7beb5380f052a4f38c9891f16.jpg", + "text": "$$\n\\begin{array} { r } { \\hat { \\boldsymbol { \\mathsf { Z } } } ^ { l } = 3 \\mathrm { D S W } \\mathrm { - } \\boldsymbol { \\mathsf { M S A } } ( \\mathrm { L N } ( [ \\boldsymbol { x } ^ { l - 1 } , \\boldsymbol { P } _ { p r o m p t } ^ { l - 1 } , \\boldsymbol { \\mathsf { Z } } ^ { l - 1 } ] ) ) + \\boldsymbol { \\mathsf { Z } } ^ { l - 1 } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 307, + 133, + 689, + 157 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $x ^ { l - 1 } \\in \\mathbb { R } ^ { d }$ denotes the [CLS]’s embedding for the $l$ th layer’s input space, $P _ { p r o m p t } ^ { l - 1 }$ is implemented by overlapping the top $d _ { p r o m p t }$ tokens of $Z ^ { l - 1 }$ (Jia et al., 2022). While it has also been implemented in front of the $x ^ { l - 1 }$ (Chen et al., 2022). ", + "bbox": [ + 173, + 160, + 825, + 209 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Others: Other PETL techniques include ST-Adapter Pan et al. (2022), LoRA (Hu et al., 2022), and BitFit (Zaken et al., 2022). ST-Adapter mainly adapts image-text models pre-trained on large scale datasets such as 400M image-text pair proposed by CLIP (Radford et al., 2021) and the IG-3.6B used by SWAG (Singh et al., 2022) to video understanding downstream tasks, which matches and even outperforms full-tuning. LoRA approximates the optimization process by injecting learnable low-rank matrices into the attention module. This method does not show superior performance for NLP tasks in terms of parameter efficiency. Hence, we do not prioritize this direction in this work. BitFit only tunes the bias terms of the backbone models, making it very parameter-efficient. ", + "bbox": [ + 171, + 215, + 826, + 328 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.3 REVISITING PREFIX-TUNING FOR VISUAL TASKS ", + "text_level": 1, + "bbox": [ + 173, + 343, + 555, + 358 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The prefix implementation in NLP Li & Liang (2021); He et al. (2022a) can be regarded as prepending contextual information for downstream tasks, which is similar with the pre-training process aiming to predict masked words in the process of an inner loop (Brown et al., 2020). Considering the pre-training process of pure vision models, such direct implementation might not make sense for visual tasks. Although such autoregressive pre-training has been conducted in visual domain (He et al., 2022b; Tong et al., 2022), but adding prefix for a sentence input in NLP can be structurally different with the visual domain. Specifically, masked pixels in image or video data cannot be regarded as some word level semantic information (e.g., a subject or an action) as in the NLP. ", + "bbox": [ + 171, + 367, + 826, + 481 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Recall that the embedding state of prefix-tuning is randomly initiated, which is known as learnable prefix but can bring random noise that later turns out affecting the convergence of the fine-tuning downstream tasks. Hence, inspired by the connection between adapter and prefix (He et al., 2022a), we avoid such learnable prefix design with random initialization and propose a parallel attention (PATT) to the original attention module (see Figure 3). The adapter structure can effective control the number of trainable parameters via $d _ { b o t t l e }$ , which is similar with the effect of the middle dimension dmiddle of W (i)pk and $W _ { p v } ^ { ( i ) }$ for preparing the prefix. Specifically, for the lth layer, we use output of its previous layer $Z ^ { l - 1 }$ and project it to a pair of matrices $\\boldsymbol { \\dot { K _ { p } } } , \\boldsymbol { V _ { p } } \\in \\mathbb { R } ^ { \\tilde { m } \\times d }$ via a similar mechanism of Eq. 6: ", + "bbox": [ + 174, + 487, + 549, + 708 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/d148892e7f5069e4c5d290e84bc770b1123dc9ea78278c0d1518b02dded92f2d.jpg", + "text": "$$\nK _ { p } , V _ { p } = \\mathrm { T a n h } ( Z ^ { l - 1 } W _ { d o w n } ) W _ { u p } ,\n$$", + "text_format": "latex", + "bbox": [ + 240, + 710, + 480, + 729 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where Tanh is the activation function used for preparing the prefix, which can be replaced by other activation func", + "bbox": [ + 174, + 738, + 547, + 766 + ], + "page_idx": 4 + }, + { + "type": "image", + "img_path": "images/d32ffd27ce18eb6c9874ee02bac32e6f8939e06524b342dbfc57e9adf4b4bb21.jpg", + "image_caption": [ + "Figure 3: Structure of PATT. Red parts are trainable parameters calculated by the same input for preparing query, key, and value (i.e., the output of the previous layer passing through a layer normalization layer $Z ^ { l - 1 }$ ). " + ], + "image_footnote": [], + "bbox": [ + 566, + 500, + 820, + 657 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "tions such as RELU and GELU. Here, we follow the original prefix implementation as its value ranges from $- 1$ to 1. Given $K _ { p }$ and $V _ { p }$ , Eq. 5 can be rewritten as: ", + "bbox": [ + 176, + 766, + 823, + 795 + ], + "page_idx": 4 + }, + { + "type": "equation", + "img_path": "images/41811af6dd63d7205dd341615b18c52d5bcc900b49237783928be147988890f2.jpg", + "text": "$$\nh e a d _ { i } = A t t e n t i o n ( \\boldsymbol Z ^ { l - 1 } \\boldsymbol W _ { q } ^ { ( i ) } , \\boldsymbol s \\boldsymbol K _ { p } + \\boldsymbol C \\boldsymbol W _ { k } ^ { ( i ) } , \\boldsymbol s V _ { p } + \\boldsymbol C \\boldsymbol W _ { v } ^ { ( i ) } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 281, + 796, + 715, + 818 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "where $s$ is a scalar for adjusting the effect of PATT. Note that without considering the physical meaning of such design, for PETL purpose, one can perform similar practise for any combinations of $\\mathbf { \\alpha } _ { \\mathbf { Q } , \\mathbf { \\alpha } } \\kappa$ , and $\\pmb { \\nu }$ . This brings connection to the LoRA (Hu et al., 2022) method, which add parallel trainable parameters to $\\mathbf { Q }$ and $\\pmb { \\nu }$ . Empirically, where to perform the PATT makes little difference, but the amount of trainable parameters brings larger effect for large scale downstream domains. ", + "bbox": [ + 173, + 819, + 825, + 888 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "2.4 V-PETL: UNIFIED VIEW ON VISUAL PETL ", + "text_level": 1, + "bbox": [ + 176, + 904, + 514, + 919 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Given the PETL techniques at hand, there can be many potential combinations leading to good parameteraccuracy trade-off. However, it is unrealistic to exhaustively test all the methods for a specific downstream task. Other than probing such solution via evolutionary search as in Zhang et al. (2022), we aim to propose more understandable models by empirically analyzing the effect of different designs independently. According to the preliminary results shwon in Figure 1, we argue that the position and amount of parameters are important for PETL techniques, especially when the target domain is not small. ", + "bbox": [ + 174, + 104, + 547, + 256 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "To verify the importance of position and tuned parameter amount, we independently tune different modules of the backbone model. Table 1 shows the results. We can see that the attention module’s QKV layer has 20.98M parameters while the MLP module has the most number of parameters of 55.90M. Tuning positions with more parameters, will lead to better performance for SSv2. Thanks to the bottleneck mechanism of adapter and prefix-tuning, one can effectively achieve a good parameter-accuracy trade-off. As such, we derive a model called Swin-B-adapter-PATT (Swin-BAPAT) from the V-PETL framework by using the parallel adapter and our PATT to leverage the adaption of pre-trained backbone model at the positions of attention and MLP modules, respectively. In addition to adapter and PATT, we also fine-tune the last fully connected layer as it has relatively smaller amount of tunable parameters (i.e, 0.18M) than adapter and PATT. ", + "bbox": [ + 174, + 263, + 549, + 319 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/b8698cfb454fd2b49cb4e9dc833584356c0b40f5ce54f2267e74feab5d87ec04.jpg", + "table_caption": [ + "Table 1: Comparison of independently fine-tuning varied positions of the video swin transformer block on SSv2. " + ], + "table_footnote": [], + "table_body": "
Position# ParamsTop-1 (%)
Full-tuning Tune FC Layer87.82M 0.18M50.99 24.13
LayerNorm 10.02M14.35
Attn,Proj6.99M47.58
Attn, QKV20.98M50.02
Attn, SoftMax0.95M27.67
LayerNorm 20.02M14.62
MLP, FC127.97M47.10
MLP,FC227.93M45.32
DownSample2.76M27.53
", + "bbox": [ + 560, + 155, + 820, + 308 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "", + "bbox": [ + 174, + 319, + 825, + 429 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3 EXPERIMENTS ", + "text_level": 1, + "bbox": [ + 176, + 453, + 326, + 469 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.1 EXPERIMENTAL SETTINGS ", + "text_level": 1, + "bbox": [ + 176, + 486, + 398, + 501 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Video Datasets: Something-something v2 (SSv2 (Goyal et al., 2017)) It has 108,499 short videos for 174 human-object interaction categories with durations between 2 to 6 seconds. The challenge of this dataset is that it contains 23, 137 distinct object names with an imbalanced distribution. The original dataset is split into train, validation, and test sets with a ratio of 8:1:1. The extended version (SSv2) of this dataset is consisted of 168, 913 training samples, 24, 777 validation samples, and 27, 157 testing samples with the sample number of action labels. The training and testing samples are used. HMDB51 (Kuehne et al., 2011) contains 6, 766 video samples for 51 action categories including videos of varied visible body parts, camera motion, camera view, and clip quality. All video samples have at least 101 clips and a minimum height of 60 pixels for actors. The original dataset has three splits of training and evaluation. We follow existing work Chen et al. (2022) by using the first training and evaluation split that has 3, 570 and 1, 530 samples, respectively. Image Datasets: Following the experimental set ups in AdaptFormer, three datasets CIFAIR-100 Krizhevsky et al. (2009), Street View House Numbers (SVHN) Goodfellow et al. (2013), and Food101 Bossard et al. (2014) are used. CIFAIR-100 has 50, 000 and 10, 000 training and validation images, respectively, with the resolution of $3 2 \\times 3 2$ and 100 categories; SVHN is a digit classification dataset that has 73, 257 training sample and 26, 032 testing samples; Food-101 includes 101k images of 101 food categories with each of them has 750 training and 250 testing samples. ", + "bbox": [ + 173, + 512, + 826, + 750 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Implementation details: It is worth noting that big batch size (i.e., 1, 024) and the number of input video frames (i.e., 32 frames) can greatly benefit good performance (Carreira & Zisserman, 2017; Liu et al., 2022; Chen et al., 2022), which usually requires GPU clusters to enable the training. AdaptFormer (Chen et al., 2022) uses such powerful GPU cluster to achieve good performance. However, good performance might not hold when the batch size is small. Following the more common hardware device setup, we use 4 GeForce 3090 GPUs for all experiments, leading to a batch size of 64. All the experiments are fine-tuned for 70 epochs. We use the Swin- $\\mathbf { \\cdot B } ^ { 1 }$ model pre-trained on Kinetics 400 and 600. For HMDB51, we report the results without tuning the FC layer due to the significant effect of the FC layer on relatively small scale dataset. Following Chen et al. (2022), we do not perform regularization strategies such as mixup, cutmix, color jittering, etc. Our PATT module is convenient to be applied to other Transformer-based models. Hence, we respectively adopt ViT-B models from MAE (He et al., 2022b) and VideoMAE (Tong et al., 2022) to conduct further comparison on video and image datasets, which follows the self-supervised pretraining setting2 in Chen et al. (2022) except that the batch size is set to 256 instead of 1, 024. ", + "bbox": [ + 174, + 756, + 825, + 895 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/2e8af99ac622b787a4db04a714d314ccf7bad2998ac44dccee567454b4118f20.jpg", + "table_caption": [ + "Table 2: Comparison of Top-1 accuracy using varied amount of parameters adjusted by $d _ { b o t t l e }$ different pre-training domains, and the number of frames with other fine-tuning strategies. " + ], + "table_footnote": [], + "table_body": "
MethoddbottlePre-training#FramesSSv2HMDB51
# Params Top-1(%)# Params Top-1 (%)
Full-tuning-Kinetics 400887.82M50.9987.69M68.07
Tune FC LayerKinetics 40080.18M24.130.05M71.28
BitFit (Zaken et al., 2022)·Kinetics 40081.29M45.941.11M68.26
AdaptFormer-Swin (Chen et al., 2022）64Kinetics 40081.73M40.801.61M68.66
Prefix-tuning (Li& Liang,2021)128Kinetics 40086.57M39.466.40M56.13
Our Swin-BAPAT (w/o Adapter)32Kinetics 40088881.35M46.261.17M69.51
Our Swin-BAPAT (w/o Adapter)64Kinetics 4002.51M49.232.34M71.34
Our Swin-BAPAT (w/o Adapter)128Kinetics 4004.83M52.574.65M70.56
Our Swin-BAPAT (w/o Adapter)256Kinetics 4009.45M52.719.27M70.23
Our Swin-BAPAT32Kinetics 40082.91M49.632.74M68.20
Our Swin-BAPAT64Kinetics 40084.07M51.803.89M70.10
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 600811.00M54.0610.83M69.90
Our Swin-BAPAT256 ImageNet-22K811.00M43.5610.83M59.89
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT128Kinetics 400166.38M63.146.20M75.67
", + "bbox": [ + 205, + 132, + 785, + 402 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 171, + 417, + 826, + 474 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Baselines: We mainly compare our method Swin-BAPAT with three baselines as follows: (1) Full-tuning: set all the parameters learnable and tune the whole model initiated with the pretrained weights. (2) Tune FC layer: tune the last fully connected layer and freeze pre-trained parameters of the whole backbone model. (3) AdaptFormer-Swin: method introduced by Chen et al. (2022) that adds a parallel adapter to the MLP module in each block of the backbone model. (4) Prefix-tuning: the direct implementation of prefix-tuning used in NLP as defined in Eq. 5. (5) BitFit: by tuning the bias of the backbone model together with the FC layer. ", + "bbox": [ + 173, + 479, + 826, + 579 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "3.2 THE EFFECT OF DIFFERENT PETL TECHNIQUES ", + "text_level": 1, + "bbox": [ + 174, + 597, + 549, + 612 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Table 2 shows the results of different PETL techniques. From the results of four baseline methods, full-tuning performs the best for the large-scale dataset SSv2, whereas tuning the FC layer achieves superior performance over other PETL techniques on HMDB51. This is due to the fact that downstream tasks with relatively larger scale datasets are more parameter hungry for good convergence. On the contrary, small datasets can make good use of the knowledge from the source domain with slight effort of adaption via an FC layer. Here, a question regarding the effect of this FC layer when using it together with other PETL techniques has not been investigated. As this FC layer having small amount of tunable parameters can already make a big difference, performing better than fulltuning and other PETL techniques and rendering them not effective for small-scale datasets. As such, we further examine this question in Section A.1. ", + "bbox": [ + 174, + 625, + 825, + 763 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "We test different amount of parameters adjusted by $s _ { b o t t l e }$ , taking its values to 32, 64, 128 and 256. The second and third groups (without or with Adapter, respectively) of results in Table 2 shows that larger values of $s _ { b o t t l e }$ can benefit the fine-tuning with slightly more overhead of parameters on large-scale datasets such as SSv2. All results of our Swin-BAPAT outperform the state-ofthe-art AdaptFormer-Swin with a big margin (using the smallest value $s _ { b o t t l e } = 3 2$ can improve AdaptFormer-Swin by almost $2 5 \\%$ ). While without using Adapter, our method still outperforms baselines AdaptFormer-Swin and BitFit with roughly similar amount of parameters. When sbottle is larger than 64, our Swin-BAPAT starts to perform better than full-tuning on both datasets with proper parameter-accuracy trade-off, validating the effectiveness of our Swin-BAPAT for PETL. ", + "bbox": [ + 173, + 770, + 825, + 895 + ], + "page_idx": 6 + }, + { + "type": "image", + "img_path": "images/bfd8674c26a279b07e5eec991eb55a8d1b5100558500140751dfb357f84b103a.jpg", + "image_caption": [ + "Figure 4: Top-1 accuracy of different settings on SSv2 throughout training process. F: frame, S: scalar, B: $d _ { b o t t l e }$ , K: pre-training domain. " + ], + "image_footnote": [], + "bbox": [ + 189, + 102, + 503, + 261 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/ef008352787cc6b75c1da7a643df87b97aa07041870ebf045dd65aedc6b37df8.jpg", + "table_caption": [ + "Table 3: Top-1 accuracy $( \\% )$ using different scalar values on two datasets: SSv2 and HMDB51. The $d _ { b o t t l e }$ is set to 128; pretraining is based on Kinetics 400. " + ], + "table_footnote": [], + "table_body": "
Scalar sSSv2HMDB51
Full-tuning50.9971.28
Tune FC Layer24.1368.07
AdaptFormer-Swin40.8068.66
s=0.247.4669.38
s=0.552.8471.87
s=0.853.3671.93
s=1.053.2970.89
", + "bbox": [ + 534, + 181, + 789, + 297 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.3 THE EFFECT OF DIFFERENT PRE-TRAINING DOMAINS ", + "text_level": 1, + "bbox": [ + 174, + 339, + 588, + 353 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "The knowledge from the pre-trained model is learned from the source domain. We test two different models pre-trained on large-scale datasets: Kinetics 400, Kinetics 600, and ImageNet-22K. Findings show that both two models pre-trained on such large-scale datasets can benefit our proposed PETL strategy with the latter being slightly more significant (see the third group of comparison in Table 2). This is due to the fact that Kinectics 600 is larger than its 400 version and brings more knowledge to the pre-trained model, benefiting more downstream tasks. However, image-based pre-training cannot perform as good as video-based pre-training due to the larger domain gap. ", + "bbox": [ + 173, + 364, + 826, + 463 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.4 THE EFFECT OF DIFFERENT VIDEO INPUT SIZE ", + "text_level": 1, + "bbox": [ + 174, + 479, + 544, + 494 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "We also test whether our method is robust to increased number of input video frames. It is worth noting that larger number of input video frames usually can bring more spatial temporal information, benefiting data-driven models to learn more distinguishable features while keeping the model size remaining the same. The last group of comparisons in Table 2 shows that using double-sized video input (i.e., 16 frames) can greatly improve the performance of action recognition on both small and large-scale datasets. The improvements (increased $9 . 7 8 \\%$ from $5 3 . 3 6 \\%$ to $6 3 . 1 4 \\%$ on SSv2, and $3 . { \\bar { 7 } } 4 \\%$ from $7 1 . 9 3 \\%$ to $7 5 . 6 7 \\%$ on HMDB51) are more significant than other factors such as $d _ { b o t t l e }$ and pre-training domain (around $1 \\%$ to $2 \\%$ ). The top line in Figure 4 visualizes the significant effect of increasing the number of input video frames. These results suggest that our Swin-BAPAT can be promising for increased frames of video input. ", + "bbox": [ + 173, + 506, + 825, + 645 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "3.5 THE EFFECT OF DIFFERENT SCALE OF PATT ", + "text_level": 1, + "bbox": [ + 174, + 661, + 524, + 676 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Recall that the effect of our PATT on pretrained models can be adjusted by the variable $s$ in Eq. 10. Table 3 shows that adopting the value of 0.8 can deliver consistent best performances on both datasets SSv2 and HMDB51 under our experimental setting. Smaller values of $s$ will quantitatively reduce the effect of our PATT module on the knowledge transfer while large values will increase the effect of our PATT module. The good performance achieved via taking an effective scale of 0.8 indicates that our PATT module plays an important role in the knowledge transfer. However, even larger values over 0.8 can affect the importance of original knowledge thereof the pretrained model. Hence, proper valued scalar $s$ is essential for balancing the role of PATT and ", + "bbox": [ + 174, + 688, + 483, + 924 + ], + "page_idx": 7 + }, + { + "type": "table", + "img_path": "images/2c9b749198f1850118ad0a47fe1cb134b902332de6bdc9e3d9bbb664994021e3.jpg", + "table_caption": [ + "Table 4: Ablation of different implementation positions of PATT defined in Eq. 10, e.g., Ours (K, $\\boldsymbol { \\mathsf { V } }$ ) indicates inserting PATT to the query and key of 3DSW-MSA modules. Pre-training on Kinetics 600. $d _ { b o t t l e }$ is set to 128; Scalar $s$ is set to 0.8. " + ], + "table_footnote": [], + "table_body": "
MethodSSv2HMDB51
#ParamsTop-1# ParamsTop-1
Full-tuning87.82M50.9987.69M68.07
Concat (K, V)6.38M15.616.20M20.98
No Zl-1(K,V)8.74M51.068.56M67.41
Ours (Q, K)6.38M45.496.20M68.92
Ours (K, V)6.38M53.386.20M71.41
Ours (Q, V)6.38M53.246.20M71.74
Ours (Q, K, V)7.93M53.237.63M69.57
", + "bbox": [ + 500, + 777, + 818, + 909 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "pre-trained backbone model. Note this can be a learnable parameter upon specific implementation, here we empirically verified the effect of the scalar. ", + "bbox": [ + 173, + 103, + 823, + 132 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.6 THE EFFECT OF DIFFERENT METHODS YIELD FROM V-PETL ", + "text_level": 1, + "bbox": [ + 174, + 147, + 635, + 162 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "We have argued that, especially for relative large downstream datasets, the position and the amount of trainable parameters are important for parameter-efficient transfer learning in Section 2.4. The proposed Swin-BAPAT is one of instantiated models from the V-PETL framework regarding the insert position of our PATT. Other instantiations can be inserted into different positions such as query, key, and value of the attention module. We further instantiate other variations of our Swin-BAPAT by inserting PATT to different positions. Table 4 shows the results. Findings show that inserting to the value position of 3DSW-MSA can contribute more than inserting to other two positions. While inserting to query of key makes little difference for the performance. This is due to the fact that query and key make the calculation of the attention mask. Hence, inserting either one of them will lead to a similar effect. On one hand, these results, to some extent, justify the original design of prefix-tuning that bring learnable prefix to key and value of the attention module. On the other hand, it indicates that our claim regarding the unified view of PETL for visual tasks is reasonable. In Table 4, we also ablate the designs of PATT regarding concatenating $K _ { p }$ and $V _ { p }$ (i.e., Concat $[ \\mathsf { K } , \\mathsf { v } ] )$ , and using trainable parameters to generate $K _ { p }$ and $\\boldsymbol { V _ { p } }$ (i.e., N ${ \\bf \\nabla } ) \\ Z ^ { l - 1 } ( { \\bf K } , { \\bf V } ] )$ . ", + "bbox": [ + 173, + 174, + 825, + 372 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "3.7 COMPARISON ON VARIED TASKS VIA SELF-SUPERVISED PRE-TRAINED MODELS ", + "text_level": 1, + "bbox": [ + 171, + 383, + 771, + 401 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Table 5 shows the comparison with AdaptFormer-64 (Chen et al., 2022) and VPT (Jia et al., 2022) on both image- and video-based downstream tasks. Our method ViT-BAPAT still shows promising parameter-accuracy trade-off via much smaller batch size, which is more convenient for reproduction on the general single server with 8 GPUs. The underperformance on SSv2 (better than full-tuning) can be due to the smaller batch size as SSv2 is much larger than other compared datasets and can be more relying on larger batch size. In real-world application scenarios, small dataset can be the more common case, which confirms our contributions. ", + "bbox": [ + 173, + 411, + 826, + 510 + ], + "page_idx": 8 + }, + { + "type": "table", + "img_path": "images/584fc78a686905d690f6a76a9ce7957c9cea5dbcb73dd69563c8caa64cd13203.jpg", + "table_caption": [ + "Table 5: Comparison of Top-1 accuracy via ViT-B models from MAE and VideoMAE pre-trained with self-supervised learning for image and video datasets, respectively. " + ], + "table_footnote": [], + "table_body": "
MethodAvg.ImageVideo
Params (M)CIFAR-100SVHNFood-101SSv2HMDB51
Full-tuning86.04 (100%)85.9097.6790.0953.9746.41
Tune FC Layer0.07 (0.08%)69.83 (-16.07) 66.91 (-30.76) 69.74 (-20.35)29.23 (-24.74)）49.84 (+3.43)
VPT (Jia et al.,2022)0.08 (0.09%)82.44 (-3.46)94.02 (-3.65)82.98 (-7.11)43.73 (-10.24)52.67 (+6.26)
AdaptFormer-641.26 (1.46%)85.90 (0.00)96.89 (-0.78)87.61 (-2.48)59.02 (+5.05)55.69 (+9.28)
Our ViT-BAPAT-322.13 (2.47%)86.29 (+0.39)97.18 (-0.49)87.37 (-2.72)57.78 (+3.81)57.18 (+10.77)
Our ViT-BAPAT-643.02 (3.51%)86.35 (+0.45)97.18 (-0.49)87.53 (-2.56)57.55 (+3.58)57.18 (+10.77)
Our ViT-BAPAT-1284.79 (5.56%)86.47 (+0.57)97.28 (-0.39)87.66 (-2.43)56.97 (+3.00)57.70 (+11.29)
Our ViT-BAPAT-2568.33 (9.68%)86.55 (+0.65)97.24 (-0.43)87.68 (-2.41)56.53 (+2.56)57.31 (+10.90)
", + "bbox": [ + 176, + 551, + 818, + 693 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "4 CONCLUSION ", + "text_level": 1, + "bbox": [ + 176, + 713, + 318, + 728 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this paper, we introduced a V-PETL framework for exploiting good parameter-accuracy tradeoff around adapting video-based pre-trained large models to downstream tasks. Our Swin-BAPAT method derived from the V-PETL with a variation of prefix-tuning known as PATT can effectively bring good parameter-accuracy trade-off on downstream tasks. The proposed PATT can be easily plugged to the attention module of other transformer-like models. Meanwhile, the amount of trainable parameter can be easily adjusted by the parameter $d _ { b o t t l e }$ . With small amount overhead on trainable parameters, our method performs significantly better than state-of-the-art method AdapFormer-Swin and full-tuning on the datasets SSv2 and HMDB51 via small batch size, validating our contribution to the literature of PETL. In the future we will test our proposed model on more action recognition datasets surveyed in Sun et al. (2022) under more learning regimes such as zero/few-shot learning, active learning and continual learning with other pre-training methods such as visual-language models. We will also explore other backbone models, activation functions for PATT, and PETL techniques such as LoRA for visual tasks. ", + "bbox": [ + 173, + 743, + 825, + 922 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "REFERENCES ", + "text_level": 1, + "bbox": [ + 174, + 102, + 287, + 117 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "Lukas Bossard, Matthieu Guillaumin, and Luc Van Gool. 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", + "bbox": [ + 173, + 309, + 821, + 338 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "Fuzhen Zhuang, Zhiyuan Qi, Keyu Duan, Dongbo Xi, Yongchun Zhu, Hengshu Zhu, Hui Xiong, and Qing He. A comprehensive survey on transfer learning. Proceedings of the IEEE, 109(1): 43–76, 2020. ", + "bbox": [ + 173, + 347, + 826, + 390 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A APPENDIX ", + "text_level": 1, + "bbox": [ + 176, + 416, + 299, + 433 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "A.1 THE EFFECT OF FC LAYER FOR SMALL SCALE DOWNSTREAM TASKS ", + "bbox": [ + 173, + 448, + 700, + 463 + ], + "page_idx": 11 + }, + { + "type": "table", + "img_path": "images/10b1a29937a5bf1f75c40c07e52a62a72af7d651bbba5d64b8df803cf271bcd5.jpg", + "table_caption": [ + "Table 6: Results of with or without tuning the FC layer on the small scale dataset HMDB51. " + ], + "table_footnote": [], + "table_body": "
MethoddbottlePre-training#Frameswith FC layerwithout FC layer
#ParamsTop-1 (%)#ParamsTop-1 (%)
Our Swin-BAPAT32Kinetics 40082.79M65.972.74M68.20
Our Swin-BAPAT64Kinetics 40083.94M67.283.89M70.10
Our Swin-BAPAT128Kinetics 40086.25M66.756.20M71.93
Our Swin-BAPAT256Kinetics 400810.88M67.6710.83M69.64
Our Swin-BAPAT256Kinetics 400810.88M67.6710.83M69.64
Our Swin-BAPAT256Kinetics 600810.88M67.4110.83M69.90
Our Swin-BAPAT128Kinetics 40086.25M66.756.20M71.93
Our Swin-BAPAT128Kinetics 400166.25M70.566.20M75.67
Our Swin-BAPAT128Kinetics 400326.25M74.826.20M76.46
", + "bbox": [ + 184, + 507, + 803, + 669 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "For the small dataset HMDB51, due to the good parameter-accuracy trade-off achieved by finetuning the FC layer only, adding the FC layer cannot bring extra improvement to our proposed method. Without sufficient taining data, full-tuning also cannot perform well (see results in Table 2). As such, small datasets do not need to rely on large models but can make use of large models with light transfer. Instead, without tuning the FC layer, our Swin-BAPAT can perform better than fine-tuning the FC layer with small amount of extra trainable parameters (see results in Table 6), validating the good parameter-accuracy trade-off of our method. 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Based on a comprehensive under-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 469, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 469, + 421 + ], + "score": 1.0, + "content": "standing of differences between NLP and video data, we propose a new variation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 420, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 142, + 420, + 470, + 433 + ], + "score": 1.0, + "content": "of prefix-tuning module called parallel attention (PATT) for video-based down-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 469, + 443 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 469, + 443 + ], + "score": 1.0, + "content": "stream tasks. An extensive empirical analysis on two video datasets via different", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 442, + 469, + 454 + ], + "spans": [ + { + "bbox": [ + 142, + 442, + 469, + 454 + ], + "score": 1.0, + "content": "frozen backbones has been carried and the findings show that the proposed PATT", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 453, + 469, + 464 + ], + "spans": [ + { + "bbox": [ + 142, + 453, + 469, + 464 + ], + "score": 1.0, + "content": "can effectively contribute to other PETL techniques. An effective scheme Swin-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 464, + 469, + 476 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 469, + 476 + ], + "score": 1.0, + "content": "BAPAT derived from the proposed V-PETL framework achieves significantly bet-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 474, + 469, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 474, + 469, + 488 + ], + "score": 1.0, + "content": "ter performance than the state-of-the-art AdaptFormer-Swin with slightly more", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 486, + 402, + 499 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 402, + 499 + ], + "score": 1.0, + "content": "parameters and outperforms full-tuning with far less parameters.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 17.5 + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 205, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 208, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 208, + 535 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "Many vision tasks rely on fine-tuning pre-trained models to achieve good performance. One stan-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "dard modus operandi of transfer learning consists of two steps: pre-train a model on a source domain", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "and fine-tune the entire model on a target domain (Zhuang et al., 2020). Despite that prior works", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "have achieved promising performance, such vanilla practice of fine-tuning is faced with challenges", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "for adopting large models to downstream tasks. 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Second, the full fine-tuning strategy relies on high-quality downstream data and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "can hardly adapt to unseen scenarios that have large distribution shift (Kumar et al., 2021), which is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "unlike the learning process of humans who can learn from few samples and generalize well to new", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "circumstances. This issue has been researched in directions such as zero-shot learning, few-shot", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "learning, and continual learning (Li et al., 2021a). 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PETL aims at making good use of the representation knowledge in the pre-", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 141, + 256, + 470, + 267 + ], + "spans": [ + { + "bbox": [ + 141, + 256, + 470, + 267 + ], + "score": 1.0, + "content": "trained large models by fine-tuning a small number of parameters. Recently, it", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "spans": [ + { + "bbox": [ + 141, + 267, + 470, + 279 + ], + "score": 1.0, + "content": "has also attracted increasing attention to developing various PETL techniques for", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 142, + 278, + 469, + 290 + ], + "spans": [ + { + "bbox": [ + 142, + 278, + 469, + 290 + ], + "score": 1.0, + "content": "vision tasks. Popular PETL techniques such as Prompt-tuning and Adapter have", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "spans": [ + { + "bbox": [ + 141, + 289, + 469, + 301 + ], + "score": 1.0, + "content": "been proposed for high-level visual downstream tasks such as image classifica-", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 141, + 300, + 469, + 311 + ], + "spans": [ + { + "bbox": [ + 141, + 300, + 469, + 311 + ], + "score": 1.0, + "content": "tion and video recognition. However, Prefix-tuning remains under-explored for", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 142, + 311, + 469, + 322 + ], + "spans": [ + { + "bbox": [ + 142, + 311, + 469, + 322 + ], + "score": 1.0, + "content": "vision tasks. In this work, we intend to adapt large video-based models to down-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "spans": [ + { + "bbox": [ + 141, + 321, + 470, + 334 + ], + "score": 1.0, + "content": "stream tasks with a good parameter-accuracy trade-off. Towards this goal, we", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 140, + 332, + 470, + 345 + ], + "spans": [ + { + "bbox": [ + 140, + 332, + 470, + 345 + ], + "score": 1.0, + "content": "propose a framework with a unified view of PETL called visual-PETL (V-PETL)", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "spans": [ + { + "bbox": [ + 141, + 343, + 470, + 356 + ], + "score": 1.0, + "content": "to investigate the effects of different PETL techniques, data scales of downstream", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 142, + 355, + 469, + 366 + ], + "spans": [ + { + "bbox": [ + 142, + 355, + 469, + 366 + ], + "score": 1.0, + "content": "domains, positions of trainable parameters, and other aspects affecting the trade-", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 142, + 366, + 470, + 378 + ], + "spans": [ + { + "bbox": [ + 142, + 366, + 470, + 378 + ], + "score": 1.0, + "content": "off. Specifically, we analyze the positional importance of trainable parameters", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 141, + 376, + 469, + 389 + ], + "spans": [ + { + "bbox": [ + 141, + 376, + 469, + 389 + ], + "score": 1.0, + "content": "and differences between NLP and vision tasks in terms of data structures and pre-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 141, + 388, + 469, + 400 + ], + "spans": [ + { + "bbox": [ + 141, + 388, + 469, + 400 + ], + "score": 1.0, + "content": "training mechanisms while implementing various PETL techniques, especially for", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "spans": [ + { + "bbox": [ + 141, + 398, + 469, + 411 + ], + "score": 1.0, + "content": "the under-explored prefix-tuning technique. Based on a comprehensive under-", + "type": "text" + } + ], + "index": 22 + }, + { + "bbox": [ + 141, + 410, + 469, + 421 + ], + "spans": [ + { + "bbox": [ + 141, + 410, + 469, + 421 + ], + "score": 1.0, + "content": "standing of differences between NLP and video data, we propose a new variation", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 142, + 420, + 470, + 433 + ], + "spans": [ + { + "bbox": [ + 142, + 420, + 470, + 433 + ], + "score": 1.0, + "content": "of prefix-tuning module called parallel attention (PATT) for video-based down-", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 141, + 432, + 469, + 443 + ], + "spans": [ + { + "bbox": [ + 141, + 432, + 469, + 443 + ], + "score": 1.0, + "content": "stream tasks. An extensive empirical analysis on two video datasets via different", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 142, + 442, + 469, + 454 + ], + "spans": [ + { + "bbox": [ + 142, + 442, + 469, + 454 + ], + "score": 1.0, + "content": "frozen backbones has been carried and the findings show that the proposed PATT", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 142, + 453, + 469, + 464 + ], + "spans": [ + { + "bbox": [ + 142, + 453, + 469, + 464 + ], + "score": 1.0, + "content": "can effectively contribute to other PETL techniques. An effective scheme Swin-", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 141, + 464, + 469, + 476 + ], + "spans": [ + { + "bbox": [ + 141, + 464, + 469, + 476 + ], + "score": 1.0, + "content": "BAPAT derived from the proposed V-PETL framework achieves significantly bet-", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 141, + 474, + 469, + 488 + ], + "spans": [ + { + "bbox": [ + 141, + 474, + 469, + 488 + ], + "score": 1.0, + "content": "ter performance than the state-of-the-art AdaptFormer-Swin with slightly more", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 141, + 486, + 402, + 499 + ], + "spans": [ + { + "bbox": [ + 141, + 486, + 402, + 499 + ], + "score": 1.0, + "content": "parameters and outperforms full-tuning with far less parameters.", + "type": "text" + } + ], + "index": 30 + } + ], + "index": 17.5, + "bbox_fs": [ + 140, + 212, + 470, + 499 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 519, + 205, + 532 + ], + "lines": [ + { + "bbox": [ + 105, + 518, + 208, + 535 + ], + "spans": [ + { + "bbox": [ + 105, + 518, + 208, + 535 + ], + "score": 1.0, + "content": "1 INTRODUCTION", + "type": "text" + } + ], + "index": 31 + } + ], + "index": 31 + }, + { + "type": "text", + "bbox": [ + 107, + 545, + 505, + 732 + ], + "lines": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "spans": [ + { + "bbox": [ + 105, + 545, + 505, + 558 + ], + "score": 1.0, + "content": "Many vision tasks rely on fine-tuning pre-trained models to achieve good performance. One stan-", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "spans": [ + { + "bbox": [ + 105, + 556, + 505, + 569 + ], + "score": 1.0, + "content": "dard modus operandi of transfer learning consists of two steps: pre-train a model on a source domain", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "spans": [ + { + "bbox": [ + 105, + 567, + 505, + 580 + ], + "score": 1.0, + "content": "and fine-tune the entire model on a target domain (Zhuang et al., 2020). Despite that prior works", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "spans": [ + { + "bbox": [ + 105, + 578, + 505, + 591 + ], + "score": 1.0, + "content": "have achieved promising performance, such vanilla practice of fine-tuning is faced with challenges", + "type": "text" + } + ], + "index": 35 + }, + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "spans": [ + { + "bbox": [ + 105, + 589, + 505, + 602 + ], + "score": 1.0, + "content": "for adopting large models to downstream tasks. This full-tuning strategy requires one to update and", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "spans": [ + { + "bbox": [ + 105, + 600, + 505, + 613 + ], + "score": 1.0, + "content": "store separate model parameters for different downstream tasks, which can be expensive and infea-", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 105, + 611, + 505, + 623 + ], + "score": 1.0, + "content": "sible for the era of increasingly large models from EfficientNet-based (Pham et al., 2021) (480M", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "spans": [ + { + "bbox": [ + 105, + 621, + 505, + 635 + ], + "score": 1.0, + "content": "parameters) to Transformer-based (Yu et al., 2022) (2, 100M parameters) ones. For such large mod-", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "spans": [ + { + "bbox": [ + 105, + 633, + 505, + 646 + ], + "score": 1.0, + "content": "els, making good use of shared parameter weights deployed on the cloud can be beneficial for edge", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 505, + 657 + ], + "score": 1.0, + "content": "devices such as autonomous vehicles, drones who are intensive in computing and battery resources", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 668 + ], + "score": 1.0, + "content": "(Yuan et al., 2022). Second, the full fine-tuning strategy relies on high-quality downstream data and", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 105, + 666, + 505, + 678 + ], + "score": 1.0, + "content": "can hardly adapt to unseen scenarios that have large distribution shift (Kumar et al., 2021), which is", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "spans": [ + { + "bbox": [ + 106, + 677, + 505, + 689 + ], + "score": 1.0, + "content": "unlike the learning process of humans who can learn from few samples and generalize well to new", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 688, + 505, + 700 + ], + "score": 1.0, + "content": "circumstances. This issue has been researched in directions such as zero-shot learning, few-shot", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 105, + 698, + 505, + 712 + ], + "score": 1.0, + "content": "learning, and continual learning (Li et al., 2021a). Another popular strategy is fine-tuning the down-", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "spans": [ + { + "bbox": [ + 106, + 710, + 505, + 722 + ], + "score": 1.0, + "content": "stream task head, i.e., the last fully connected (FC) layer, to avoid tuning the whole backbone model,", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "spans": [ + { + "bbox": [ + 105, + 720, + 505, + 734 + ], + "score": 1.0, + "content": "which usually leads to poor performance when the target domain is large in data scale (see Figure", + "type": "text" + } + ], + "index": 48 + } + ], + "index": 40, + "bbox_fs": [ + 105, + 545, + 505, + 734 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 82, + 504, + 116 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 505, + 95 + ], + "score": 1.0, + "content": "1). Given the paradigm of fine-tuning increasingly large models, how to transfer such large models", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "with parameter-accuracy trade-off is a hot topic in various domains (Gusak et al., 2022; Sung et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 286, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 286, + 117 + ], + "score": 1.0, + "content": "2022; Lin et al., 2020; Houlsby et al., 2019).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1 + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "Taking the video-based action recognition task as an example, it can be inconvenient for deploying", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "such large models to edge devices such as an autonomous driving (Liu et al., 2019) and unmanned", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "aerial vehicle (Li et al., 2021b) as they can heavily rely on the interaction with cloud services for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "adapting to new environments via active learning (Wang et al., 2021) or continual learning (Li et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "2021a). Re-training large models on the cloud are usually not cost-effective due to the expensive", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "overheads of storage and computational resources. Furthermore, these resources are limited on edge", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "spans": [ + { + "bbox": [ + 105, + 186, + 505, + 200 + ], + "score": 1.0, + "content": "devices such as autonomous vehicles and unmanned aerial vehicles, making the sense for developing", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "spans": [ + { + "bbox": [ + 105, + 198, + 505, + 211 + ], + "score": 1.0, + "content": "effective fine-tuning methods with proper parameter-accuracy trade-off that can be fine-tuned on", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 209, + 403, + 222 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 403, + 222 + ], + "score": 1.0, + "content": "edge devices and interacting with the large models deployed on the cloud.", + "type": "text" + } + ], + "index": 11 + } + ], + "index": 7 + }, + { + "type": "text", + "bbox": [ + 107, + 226, + 296, + 489 + ], + "lines": [ + { + "bbox": [ + 106, + 225, + 297, + 238 + ], + "spans": [ + { + "bbox": [ + 106, + 225, + 297, + 238 + ], + "score": 1.0, + "content": "There have been some pioneering works for the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 236, + 297, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 297, + 249 + ], + "score": 1.0, + "content": "PETL of visual models such as AdaptFormer", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 246, + 298, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 246, + 298, + 261 + ], + "score": 1.0, + "content": "(Chen et al., 2022) and visual prompt tuning", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 258, + 297, + 271 + ], + "spans": [ + { + "bbox": [ + 106, + 258, + 297, + 271 + ], + "score": 1.0, + "content": "(VPT) (Jia et al., 2022). 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Given the paradigm of fine-tuning increasingly large models, how to transfer such large models", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 506, + 107 + ], + "score": 1.0, + "content": "with parameter-accuracy trade-off is a hot topic in various domains (Gusak et al., 2022; Sung et al.,", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 105, + 104, + 286, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 286, + 117 + ], + "score": 1.0, + "content": "2022; Lin et al., 2020; Houlsby et al., 2019).", + "type": "text" + } + ], + "index": 2 + } + ], + "index": 1, + "bbox_fs": [ + 105, + 82, + 506, + 117 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 121, + 505, + 221 + ], + "lines": [ + { + "bbox": [ + 106, + 120, + 505, + 135 + ], + "spans": [ + { + "bbox": [ + 106, + 120, + 505, + 135 + ], + "score": 1.0, + "content": "Taking the video-based action recognition task as an example, it can be inconvenient for deploying", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "spans": [ + { + "bbox": [ + 106, + 132, + 505, + 144 + ], + "score": 1.0, + "content": "such large models to edge devices such as an autonomous driving (Liu et al., 2019) and unmanned", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "spans": [ + { + "bbox": [ + 105, + 144, + 505, + 155 + ], + "score": 1.0, + "content": "aerial vehicle (Li et al., 2021b) as they can heavily rely on the interaction with cloud services for", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 154, + 505, + 167 + ], + "score": 1.0, + "content": "adapting to new environments via active learning (Wang et al., 2021) or continual learning (Li et al.,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "spans": [ + { + "bbox": [ + 105, + 164, + 505, + 178 + ], + "score": 1.0, + "content": "2021a). Re-training large models on the cloud are usually not cost-effective due to the expensive", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "spans": [ + { + "bbox": [ + 105, + 176, + 505, + 189 + ], + "score": 1.0, + "content": "overheads of storage and computational resources. 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Im-", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 336, + 296, + 347 + ], + "spans": [ + { + "bbox": [ + 106, + 336, + 270, + 347 + ], + "score": 1.0, + "content": "plementing with a large batch size of", + "type": "text" + }, + { + "bbox": [ + 270, + 336, + 296, + 347 + ], + "score": 0.29, + "content": "1 , 0 2 4", + "type": "inline_equation" + } + ], + "index": 22 + }, + { + "bbox": [ + 105, + 346, + 297, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 346, + 297, + 359 + ], + "score": 1.0, + "content": "with 64 GPUs, Adaptformer shows promis-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 358, + 297, + 369 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 297, + 369 + ], + "score": 1.0, + "content": "ing parameter-accuracy trade-off on video data.", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 367, + 298, + 381 + ], + "spans": [ + { + "bbox": [ + 105, + 367, + 298, + 381 + ], + "score": 1.0, + "content": "However, such powerful computing resource", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 379, + 297, + 391 + ], + "spans": [ + { + "bbox": [ + 105, + 379, + 297, + 391 + ], + "score": 1.0, + "content": "is not realistic for the usage of edge devices.", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 389, + 297, + 402 + ], + "spans": [ + { + "bbox": [ + 105, + 389, + 297, + 402 + ], + "score": 1.0, + "content": "Meanwhile, whether the good trade-off can", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 106, + 402, + 297, + 413 + ], + "spans": [ + { + "bbox": [ + 106, + 402, + 297, + 413 + ], + "score": 1.0, + "content": "be maintained for small batch size remains", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 412, + 297, + 425 + ], + "spans": [ + { + "bbox": [ + 106, + 412, + 297, + 425 + ], + "score": 1.0, + "content": "under-explored. 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Regarding the backbone model, we focus on Video Swin Transformer (Liu et al.,", + "type": "text" + } + ], + "index": 71 + }, + { + "bbox": [ + 106, + 681, + 504, + 694 + ], + "spans": [ + { + "bbox": [ + 106, + 681, + 504, + 694 + ], + "score": 1.0, + "content": "2022), one of the state-of-the-art vision models that bring competitive performance on large-scale", + "type": "text" + } + ], + "index": 72 + }, + { + "bbox": [ + 106, + 693, + 406, + 704 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 406, + 704 + ], + "score": 1.0, + "content": "action recognition datasets such as Kinetics 400 and 600 Kay et al. 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We analyze different PETL techniques using the backbone model Swin Video Transformer for", + "type": "text" + } + ], + "index": 75 + }, + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "video-based tasks, providing a unified view via our V-PETL framework and investigating the impor-", + "type": "text", + "cross_page": true + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 238, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 238, + 106 + ], + "score": 1.0, + "content": "tance of the fine-tuning position.", + "type": "text", + "cross_page": true + } + ], + "index": 1 + } + ], + "index": 75, + "bbox_fs": [ + 106, + 718, + 506, + 733 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 106, + 82, + 504, + 105 + ], + "lines": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 105, + 81, + 505, + 96 + ], + "score": 1.0, + "content": "video-based tasks, providing a unified view via our V-PETL framework and investigating the impor-", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 94, + 238, + 106 + ], + "spans": [ + { + "bbox": [ + 106, + 94, + 238, + 106 + ], + "score": 1.0, + "content": "tance of the fine-tuning position.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "text", + "bbox": [ + 107, + 105, + 503, + 137 + ], + "lines": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "spans": [ + { + "bbox": [ + 105, + 104, + 505, + 117 + ], + "score": 1.0, + "content": "2. Based on the comprehensive understanding of intrinsic differences between NLP and video data", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "spans": [ + { + "bbox": [ + 105, + 115, + 505, + 127 + ], + "score": 1.0, + "content": "regarding data structures and pre-training mechanisms, we leverage prefix-tuning to our V-PETL", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 106, + 126, + 244, + 138 + ], + "spans": [ + { + "bbox": [ + 106, + 126, + 244, + 138 + ], + "score": 1.0, + "content": "with a new variation called PATT.", + "type": "text" + } + ], + "index": 4 + } + ], + "index": 3 + }, + { + "type": "text", + "bbox": [ + 107, + 138, + 505, + 171 + ], + "lines": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "spans": [ + { + "bbox": [ + 105, + 137, + 505, + 150 + ], + "score": 1.0, + "content": "3. Upon extensive ablation experiments regarding various effect factors, we empirically validate the", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "spans": [ + { + "bbox": [ + 105, + 148, + 505, + 162 + ], + "score": 1.0, + "content": "promising parameter-accuracy trade-off achieved by our adjustable and easy-to-use PATT module,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 159, + 339, + 173 + ], + "spans": [ + { + "bbox": [ + 105, + 159, + 339, + 173 + ], + "score": 1.0, + "content": "contributing to the existing literature of PETL techniques.", + "type": "text" + } + ], + "index": 7 + } + ], + "index": 6 + }, + { + "type": "title", + "bbox": [ + 108, + 186, + 241, + 199 + ], + "lines": [ + { + "bbox": [ + 105, + 185, + 243, + 201 + ], + "spans": [ + { + "bbox": [ + 105, + 185, + 243, + 201 + ], + "score": 1.0, + "content": "2 UNIFIED FRAMEWORK", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 8 + }, + { + "type": "title", + "bbox": [ + 107, + 210, + 300, + 221 + ], + "lines": [ + { + "bbox": [ + 105, + 209, + 302, + 223 + ], + "spans": [ + { + "bbox": [ + 105, + 209, + 302, + 223 + ], + "score": 1.0, + "content": "2.1 RECAP OF VIDEO SWIN TRANSFORMER", + "type": "text" + } + ], + "index": 9 + } + ], + "index": 9 + }, + { + "type": "text", + "bbox": [ + 106, + 230, + 505, + 319 + ], + "lines": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 231, + 505, + 243 + ], + "score": 1.0, + "content": "Video Swin Transformer (Liu et al., 2022) is formed with Transformer layers (a.k.a. stages) that", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "spans": [ + { + "bbox": [ + 105, + 242, + 505, + 254 + ], + "score": 1.0, + "content": "are consisted with 3D Video Swin Transformer blocks. 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The basic architecture", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 106, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "of a 3D Swin Transformer block is shown in Figure 2, which is mainly composed of a 3D shifted", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "spans": [ + { + "bbox": [ + 106, + 275, + 505, + 287 + ], + "score": 1.0, + "content": "window-based multi-head self-attention (3DSW-MSA) module and a fully connected feed-forward", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "network (FFN) implemented with a 2-layer MLP. Layer normalization (LN) and residual connection", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 105, + 297, + 505, + 308 + ], + "spans": [ + { + "bbox": [ + 105, + 297, + 505, + 308 + ], + "score": 1.0, + "content": "are respectively performed before and after both FFN and 3DSW-MSA modules. 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(2022a) for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 514, + 506, + 529 + ], + "spans": [ + { + "bbox": [ + 105, + 514, + 506, + 529 + ], + "score": 1.0, + "content": "PETL in NLP tasks, adapter (Chen et al., 2022) has been directly used for vision tasks, showing", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "spans": [ + { + "bbox": [ + 105, + 526, + 506, + 539 + ], + "score": 1.0, + "content": "promising performance using far less tunable parameters. 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While it has also been", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 320, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 218, + 167 + ], + "score": 1.0, + "content": "implemented in front of the", + "type": "text" + }, + { + "bbox": [ + 218, + 153, + 239, + 165 + ], + "score": 0.9, + "content": "x ^ { l - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 153, + 320, + 167 + ], + "score": 1.0, + "content": "(Chen et al., 2022).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4 + }, + { + "type": "text", + "bbox": [ + 105, + 171, + 506, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "Others: Other PETL techniques include ST-Adapter Pan et al. (2022), LoRA (Hu et al., 2022), and", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 180, + 507, + 195 + ], + "spans": [ + { + "bbox": [ + 104, + 180, + 507, + 195 + ], + "score": 1.0, + "content": "BitFit (Zaken et al., 2022). ST-Adapter mainly adapts image-text models pre-trained on large scale", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "datasets such as 400M image-text pair proposed by CLIP (Radford et al., 2021) and the IG-3.6B", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 203, + 507, + 217 + ], + "spans": [ + { + "bbox": [ + 104, + 203, + 507, + 217 + ], + "score": 1.0, + "content": "used by SWAG (Singh et al., 2022) to video understanding downstream tasks, which matches and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "score": 1.0, + "content": "even outperforms full-tuning. LoRA approximates the optimization process by injecting learnable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "low-rank matrices into the attention module. This method does not show superior performance for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "NLP tasks in terms of parameter efficiency. Hence, we do not prioritize this direction in this work.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 475, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 475, + 261 + ], + "score": 1.0, + "content": "BitFit only tunes the bias terms of the backbone models, making it very parameter-efficient.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5 + }, + { + "type": "title", + "bbox": [ + 106, + 272, + 340, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 340, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 340, + 285 + ], + "score": 1.0, + "content": "2.3 REVISITING PREFIX-TUNING FOR VISUAL TASKS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 506, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "The prefix implementation in NLP Li & Liang (2021); He et al. 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Considering", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "spans": [ + { + "bbox": [ + 105, + 325, + 506, + 338 + ], + "score": 1.0, + "content": "the pre-training process of pure vision models, such direct implementation might not make sense for", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 104, + 336, + 507, + 349 + ], + "spans": [ + { + "bbox": [ + 104, + 336, + 507, + 349 + ], + "score": 1.0, + "content": "visual tasks. 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While it has also been", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 105, + 153, + 320, + 167 + ], + "spans": [ + { + "bbox": [ + 105, + 153, + 218, + 167 + ], + "score": 1.0, + "content": "implemented in front of the", + "type": "text" + }, + { + "bbox": [ + 218, + 153, + 239, + 165 + ], + "score": 0.9, + "content": "x ^ { l - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 153, + 320, + 167 + ], + "score": 1.0, + "content": "(Chen et al., 2022).", + "type": "text" + } + ], + "index": 5 + } + ], + "index": 4, + "bbox_fs": [ + 105, + 126, + 506, + 167 + ] + }, + { + "type": "text", + "bbox": [ + 105, + 171, + 506, + 260 + ], + "lines": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "spans": [ + { + "bbox": [ + 105, + 170, + 506, + 183 + ], + "score": 1.0, + "content": "Others: Other PETL techniques include ST-Adapter Pan et al. 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ST-Adapter mainly adapts image-text models pre-trained on large scale", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "spans": [ + { + "bbox": [ + 105, + 192, + 506, + 205 + ], + "score": 1.0, + "content": "datasets such as 400M image-text pair proposed by CLIP (Radford et al., 2021) and the IG-3.6B", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 104, + 203, + 507, + 217 + ], + "spans": [ + { + "bbox": [ + 104, + 203, + 507, + 217 + ], + "score": 1.0, + "content": "used by SWAG (Singh et al., 2022) to video understanding downstream tasks, which matches and", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "spans": [ + { + "bbox": [ + 104, + 214, + 507, + 228 + ], + "score": 1.0, + "content": "even outperforms full-tuning. LoRA approximates the optimization process by injecting learnable", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "spans": [ + { + "bbox": [ + 104, + 225, + 506, + 239 + ], + "score": 1.0, + "content": "low-rank matrices into the attention module. This method does not show superior performance for", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "spans": [ + { + "bbox": [ + 105, + 236, + 506, + 249 + ], + "score": 1.0, + "content": "NLP tasks in terms of parameter efficiency. Hence, we do not prioritize this direction in this work.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 105, + 247, + 475, + 261 + ], + "spans": [ + { + "bbox": [ + 105, + 247, + 475, + 261 + ], + "score": 1.0, + "content": "BitFit only tunes the bias terms of the backbone models, making it very parameter-efficient.", + "type": "text" + } + ], + "index": 13 + } + ], + "index": 9.5, + "bbox_fs": [ + 104, + 170, + 507, + 261 + ] + }, + { + "type": "title", + "bbox": [ + 106, + 272, + 340, + 284 + ], + "lines": [ + { + "bbox": [ + 105, + 271, + 340, + 285 + ], + "spans": [ + { + "bbox": [ + 105, + 271, + 340, + 285 + ], + "score": 1.0, + "content": "2.3 REVISITING PREFIX-TUNING FOR VISUAL TASKS", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 14 + }, + { + "type": "text", + "bbox": [ + 105, + 291, + 506, + 381 + ], + "lines": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "spans": [ + { + "bbox": [ + 105, + 292, + 506, + 304 + ], + "score": 1.0, + "content": "The prefix implementation in NLP Li & Liang (2021); He et al. 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Although such autoregressive pre-training has been conducted in visual domain (He", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 347, + 506, + 359 + ], + "spans": [ + { + "bbox": [ + 105, + 347, + 506, + 359 + ], + "score": 1.0, + "content": "et al., 2022b; Tong et al., 2022), but adding prefix for a sentence input in NLP can be structurally dif-", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 371 + ], + "score": 1.0, + "content": "ferent with the visual domain. Specifically, masked pixels in image or video data cannot be regarded", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 104, + 369, + 446, + 382 + ], + "spans": [ + { + "bbox": [ + 104, + 369, + 446, + 382 + ], + "score": 1.0, + "content": "as some word level semantic information (e.g., a subject or an action) as in the NLP.", + "type": "text" + } + ], + "index": 22 + } + ], + "index": 18.5, + "bbox_fs": [ + 104, + 292, + 507, + 382 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 386, + 336, + 561 + ], + "lines": [ + { + "bbox": [ + 106, + 386, + 336, + 398 + ], + "spans": [ + { + "bbox": [ + 106, + 386, + 336, + 398 + ], + "score": 1.0, + "content": "Recall that the embedding state of prefix-tuning is ran-", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 106, + 398, + 336, + 409 + ], + "spans": [ + { + "bbox": [ + 106, + 398, + 336, + 409 + ], + "score": 1.0, + "content": "domly initiated, which is known as learnable prefix but", + "type": "text" + } + ], + "index": 24 + }, + { + "bbox": [ + 105, + 408, + 337, + 420 + ], + "spans": [ + { + "bbox": [ + 105, + 408, + 337, + 420 + ], + "score": 1.0, + "content": "can bring random noise that later turns out affecting the", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 105, + 419, + 337, + 432 + ], + "spans": [ + { + "bbox": [ + 105, + 419, + 337, + 432 + ], + "score": 1.0, + "content": "convergence of the fine-tuning downstream tasks. Hence,", + "type": "text" + } + ], + "index": 26 + }, + { + "bbox": [ + 105, + 430, + 337, + 442 + ], + "spans": [ + { + "bbox": [ + 105, + 430, + 337, + 442 + ], + "score": 1.0, + "content": "inspired by the connection between adapter and prefix", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 440, + 337, + 454 + ], + "spans": [ + { + "bbox": [ + 105, + 440, + 337, + 454 + ], + "score": 1.0, + "content": "(He et al., 2022a), we avoid such learnable prefix design", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 452, + 337, + 465 + ], + "spans": [ + { + "bbox": [ + 105, + 452, + 337, + 465 + ], + "score": 1.0, + "content": "with random initialization and propose a parallel atten-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 462, + 337, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 462, + 337, + 475 + ], + "score": 1.0, + "content": "tion (PATT) to the original attention module (see Figure", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 474, + 336, + 486 + ], + "spans": [ + { + "bbox": [ + 106, + 474, + 336, + 486 + ], + "score": 1.0, + "content": "3). The adapter structure can effective control the num-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 104, + 484, + 338, + 498 + ], + "spans": [ + { + "bbox": [ + 104, + 484, + 236, + 498 + ], + "score": 1.0, + "content": "ber of trainable parameters via", + "type": "text" + }, + { + "bbox": [ + 236, + 485, + 263, + 496 + ], + "score": 0.91, + "content": "d _ { b o t t l e }", + "type": "inline_equation" + }, + { + "bbox": [ + 263, + 484, + 338, + 498 + ], + "score": 1.0, + "content": ", which is similar", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 102, + 493, + 339, + 515 + ], + "spans": [ + { + "bbox": [ + 102, + 493, + 339, + 515 + ], + "score": 1.0, + "content": "with the effect of the middle dimension dmiddle of W (i)pk", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 104, + 509, + 339, + 529 + ], + "spans": [ + { + "bbox": [ + 104, + 509, + 124, + 529 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 124, + 511, + 147, + 527 + ], + "score": 0.93, + "content": "W _ { p v } ^ { ( i ) }", + "type": "inline_equation" + }, + { + "bbox": [ + 148, + 509, + 339, + 529 + ], + "score": 1.0, + "content": "for preparing the prefix. 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Red parts", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 343, + 537, + 505, + 550 + ], + "spans": [ + { + "bbox": [ + 343, + 537, + 505, + 550 + ], + "score": 1.0, + "content": "are trainable parameters calculated by", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 343, + 547, + 505, + 561 + ], + "spans": [ + { + "bbox": [ + 343, + 547, + 505, + 561 + ], + "score": 1.0, + "content": "the same input for preparing query, key,", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 343, + 559, + 505, + 572 + ], + "spans": [ + { + "bbox": [ + 343, + 559, + 505, + 572 + ], + "score": 1.0, + "content": "and value (i.e., the output of the previ-", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 343, + 570, + 505, + 582 + ], + "spans": [ + { + "bbox": [ + 343, + 570, + 505, + 582 + ], + "score": 1.0, + "content": "ous layer passing through a layer nor-", + "type": "text" + } + ], + "index": 54 + }, + { + "bbox": [ + 342, + 581, + 442, + 595 + ], + "spans": [ + { + "bbox": [ + 342, + 581, + 411, + 595 + ], + "score": 1.0, + "content": "malization layer", + "type": "text" + }, + { + "bbox": [ + 411, + 581, + 434, + 593 + ], + "score": 0.9, + "content": "Z ^ { l - 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 434, + 581, + 442, + 595 + ], + "score": 1.0, + "content": ").", + "type": "text" + } + ], + "index": 55 + } + ], + "index": 52.5 + } + ], + "index": 48.75 + }, + { + "type": "text", + "bbox": [ + 108, + 607, + 504, + 630 + ], + "lines": [ + { + "bbox": [ + 106, + 606, + 505, + 619 + ], + "spans": [ + { + "bbox": [ + 106, + 606, + 505, + 619 + ], + "score": 1.0, + "content": "tions such as RELU and GELU. 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Note that without considering the physical", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 106, + 659, + 504, + 671 + ], + "spans": [ + { + "bbox": [ + 106, + 659, + 504, + 671 + ], + "score": 1.0, + "content": "meaning of such design, for PETL purpose, one can perform similar practise for any combinations", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 106, + 670, + 505, + 683 + ], + "spans": [ + { + "bbox": [ + 106, + 670, + 117, + 683 + ], + "score": 1.0, + "content": "of", + "type": "text" + }, + { + "bbox": [ + 118, + 671, + 140, + 681 + ], + "score": 0.28, + "content": "\\mathbf { \\alpha } _ { \\mathbf { Q } , \\mathbf { \\alpha } } \\kappa", + "type": "inline_equation" + }, + { + "bbox": [ + 140, + 670, + 160, + 683 + ], + "score": 1.0, + "content": ", and", + "type": "text" + }, + { + "bbox": [ + 161, + 671, + 169, + 681 + ], + "score": 0.65, + "content": "\\pmb { \\nu }", + "type": "inline_equation" + }, + { + "bbox": [ + 169, + 670, + 505, + 683 + ], + "score": 1.0, + "content": ". This brings connection to the LoRA (Hu et al., 2022) method, which add parallel", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 105, + 680, + 506, + 696 + ], + "spans": [ + { + "bbox": [ + 105, + 680, + 201, + 696 + ], + "score": 1.0, + "content": "trainable parameters to", + "type": "text" + }, + { + "bbox": [ + 201, + 682, + 212, + 692 + ], + "score": 0.7, + "content": "\\mathbf { Q }", + "type": "inline_equation" + }, + { + "bbox": [ + 212, + 680, + 230, + 696 + ], + "score": 1.0, + "content": "and", + "type": "text" + }, + { + "bbox": [ + 230, + 682, + 238, + 692 + ], + "score": 0.66, + "content": "\\pmb { \\nu }", + "type": "inline_equation" + }, + { + "bbox": [ + 239, + 680, + 506, + 696 + ], + "score": 1.0, + "content": ". Empirically, where to perform the PATT makes little difference,", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 693, + 489, + 705 + ], + "spans": [ + { + "bbox": [ + 106, + 693, + 489, + 705 + ], + "score": 1.0, + "content": "but the amount of trainable parameters brings larger effect for large scale downstream domains.", + "type": "text" + } + ], + "index": 63 + } + ], + "index": 61, + "bbox_fs": [ + 105, + 649, + 506, + 705 + ] + }, + { + "type": "title", + "bbox": [ + 108, + 716, + 315, + 728 + ], + "lines": [ + { + "bbox": [ + 106, + 716, + 317, + 729 + ], + "spans": [ + { + "bbox": [ + 106, + 716, + 317, + 729 + ], + "score": 1.0, + "content": "2.4 V-PETL: UNIFIED VIEW ON VISUAL PETL", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 64 + } + ] + }, + { + "preproc_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 335, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 336, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 336, + 94 + ], + "score": 1.0, + "content": "Given the PETL techniques at hand, there can be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 337, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 337, + 106 + ], + "score": 1.0, + "content": "many potential combinations leading to good parameter-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 336, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 336, + 115 + ], + "score": 1.0, + "content": "accuracy trade-off. However, it is unrealistic to exhaus-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 336, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 336, + 127 + ], + "score": 1.0, + "content": "tively test all the methods for a specific downstream task.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 336, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 336, + 138 + ], + "score": 1.0, + "content": "Other than probing such solution via evolutionary search", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 337, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 337, + 149 + ], + "score": 1.0, + "content": "as in Zhang et al. (2022), we aim to propose more under-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 337, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 337, + 160 + ], + "score": 1.0, + "content": "standable models by empirically analyzing the effect of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 336, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 336, + 171 + ], + "score": 1.0, + "content": "different designs independently. According to the prelim-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 336, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 336, + 182 + ], + "score": 1.0, + "content": "inary results shwon in Figure 1, we argue that the position", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 337, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 337, + 193 + ], + "score": 1.0, + "content": "and amount of parameters are important for PETL tech-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 327, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 327, + 204 + ], + "score": 1.0, + "content": "niques, especially when the target domain is not small.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5 + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 336, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 337, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 337, + 221 + ], + "score": 1.0, + "content": "To verify the importance of position and tuned parameter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 337, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 337, + 231 + ], + "score": 1.0, + "content": "amount, we independently tune different modules of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 337, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 337, + 243 + ], + "score": 1.0, + "content": "backbone model. Table 1 shows the results. We can see", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 336, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 336, + 255 + ], + "score": 1.0, + "content": "that the attention module’s QKV layer has 20.98M pa-", + "type": "text" + } + ], + "index": 14 + } + ], + "index": 12.5 + }, + { + "type": "table", + "bbox": [ + 343, + 123, + 502, + 244 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 344, + 83, + 504, + 116 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 343, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 343, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "Table 1: Comparison of independently", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 344, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 344, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "fine-tuning varied positions of the video", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 343, + 105, + 477, + 116 + ], + "spans": [ + { + "bbox": [ + 343, + 105, + 477, + 116 + ], + "score": 1.0, + "content": "swin transformer block on SSv2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 343, + 123, + 502, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 343, + 123, + 502, + 244 + ], + "spans": [ + { + "bbox": [ + 343, + 123, + 502, + 244 + ], + "score": 0.977, + "html": "
Position# ParamsTop-1 (%)
Full-tuning Tune FC Layer87.82M 0.18M50.99 24.13
LayerNorm 10.02M14.35
Attn,Proj6.99M47.58
Attn, QKV20.98M50.02
Attn, SoftMax0.95M27.67
LayerNorm 20.02M14.62
MLP, FC127.97M47.10
MLP,FC227.93M45.32
DownSample2.76M27.53
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Tuning positions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "with more parameters, will lead to better performance for SSv2. Thanks to the bottleneck mecha-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "nism of adapter and prefix-tuning, one can effectively achieve a good parameter-accuracy trade-off.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "As such, we derive a model called Swin-B-adapter-PATT (Swin-BAPAT) from the V-PETL frame-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "work by using the parallel adapter and our PATT to leverage the adaption of pre-trained backbone", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "model at the positions of attention and MLP modules, respectively. In addition to adapter and PATT,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "we also fine-tune the last fully connected layer as it has relatively smaller amount of tunable param-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 330, + 273, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 273, + 341 + ], + "score": 1.0, + "content": "eters (i.e, 0.18M) than adapter and PATT.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 31.5 + }, + { + "type": "title", + "bbox": [ + 108, + 359, + 200, + 372 + ], + "lines": [ + { + "bbox": [ + 104, + 357, + 202, + 374 + ], + "spans": [ + { + "bbox": [ + 104, + 357, + 202, + 374 + ], + "score": 1.0, + "content": "3 EXPERIMENTS", + "type": "text" + } + ], + "index": 36 + } + ], + "index": 36 + }, + { + "type": "title", + "bbox": [ + 108, + 385, + 244, + 397 + ], + "lines": [ + { + "bbox": [ + 105, + 384, + 245, + 398 + ], + "spans": [ + { + "bbox": [ + 105, + 384, + 245, + 398 + ], + "score": 1.0, + "content": "3.1 EXPERIMENTAL SETTINGS", + "type": "text" + } + ], + "index": 37 + } + ], + "index": 37 + }, + { + "type": "text", + "bbox": [ + 106, + 406, + 506, + 594 + ], + "lines": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "spans": [ + { + "bbox": [ + 106, + 406, + 505, + 419 + ], + "score": 1.0, + "content": "Video Datasets: Something-something v2 (SSv2 (Goyal et al., 2017)) It has 108,499 short videos", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "spans": [ + { + "bbox": [ + 105, + 417, + 505, + 430 + ], + "score": 1.0, + "content": "for 174 human-object interaction categories with durations between 2 to 6 seconds. The challenge", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "of this dataset is that it contains 23, 137 distinct object names with an imbalanced distribution.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "The original dataset is split into train, validation, and test sets with a ratio of 8:1:1. The extended", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "version (SSv2) of this dataset is consisted of 168, 913 training samples, 24, 777 validation samples,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "and 27, 157 testing samples with the sample number of action labels. The training and testing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 506, + 486 + ], + "score": 1.0, + "content": "samples are used. HMDB51 (Kuehne et al., 2011) contains 6, 766 video samples for 51 action", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "categories including videos of varied visible body parts, camera motion, camera view, and clip", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "quality. All video samples have at least 101 clips and a minimum height of 60 pixels for actors.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "The original dataset has three splits of training and evaluation. We follow existing work Chen et al.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "(2022) by using the first training and evaluation split that has 3, 570 and 1, 530 samples, respectively.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "Image Datasets: Following the experimental set ups in AdaptFormer, three datasets CIFAIR-100", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Krizhevsky et al. (2009), Street View House Numbers (SVHN) Goodfellow et al. (2013), and Food-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "101 Bossard et al. (2014) are used. CIFAIR-100 has 50, 000 and 10, 000 training and validation", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 277, + 573 + ], + "score": 1.0, + "content": "images, respectively, with the resolution of", + "type": "text" + }, + { + "bbox": [ + 277, + 560, + 307, + 571 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "and 100 categories; SVHN is a digit classification", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 569, + 507, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 507, + 585 + ], + "score": 1.0, + "content": "dataset that has 73, 257 training sample and 26, 032 testing samples; Food-101 includes 101k images", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 581, + 441, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 441, + 595 + ], + "score": 1.0, + "content": "of 101 food categories with each of them has 750 training and 250 testing samples.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 46 + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "Implementation details: It is worth noting that big batch size (i.e., 1, 024) and the number of input", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "video frames (i.e., 32 frames) can greatly benefit good performance (Carreira & Zisserman, 2017;", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 104, + 618, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 618, + 506, + 636 + ], + "score": 1.0, + "content": "Liu et al., 2022; Chen et al., 2022), which usually requires GPU clusters to enable the training.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "AdaptFormer (Chen et al., 2022) uses such powerful GPU cluster to achieve good performance.", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "However, good performance might not hold when the batch size is small. Following the more", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "common hardware device setup, we use 4 GeForce 3090 GPUs for all experiments, leading to a", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 463, + 677 + ], + "score": 1.0, + "content": "batch size of 64. All the experiments are fine-tuned for 70 epochs. We use the Swin-", + "type": "text" + }, + { + "bbox": [ + 464, + 664, + 475, + 675 + ], + "score": 0.8, + "content": "\\mathbf { \\cdot B } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "model", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 104, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "pre-trained on Kinetics 400 and 600. For HMDB51, we report the results without tuning the FC", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "layer due to the significant effect of the FC layer on relatively small scale dataset. Following Chen", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 104, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 506, + 712 + ], + "score": 1.0, + "content": "et al. (2022), we do not perform regularization strategies such as mixup, cutmix, color jittering,", + "type": "text" + } + ], + "index": 64 + } + ], + "index": 59.5 + } + ], + "page_idx": 5, + "page_size": [ + 612, + 792 + ], + "discarded_blocks": [ + { + "type": "discarded", + "bbox": [ + 116, + 722, + 432, + 731 + ], + "lines": [ + { + "bbox": [ + 119, + 720, + 433, + 733 + ], + "spans": [ + { + "bbox": [ + 119, + 720, + 433, + 733 + ], + "score": 1.0, + "content": "1https://github.com/SwinTransformer/Video-Swin-Transformer", + "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" + } + ] + } + ] + }, + { + "type": "discarded", + "bbox": [ + 107, + 27, + 308, + 37 + ], + "lines": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "spans": [ + { + "bbox": [ + 107, + 26, + 308, + 38 + ], + "score": 1.0, + "content": "Under review as a conference paper at ICLR 2023", + "type": "text" + } + ] + } + ] + } + ], + "para_blocks": [ + { + "type": "text", + "bbox": [ + 107, + 83, + 335, + 203 + ], + "lines": [ + { + "bbox": [ + 106, + 82, + 336, + 94 + ], + "spans": [ + { + "bbox": [ + 106, + 82, + 336, + 94 + ], + "score": 1.0, + "content": "Given the PETL techniques at hand, there can be", + "type": "text" + } + ], + "index": 0 + }, + { + "bbox": [ + 105, + 93, + 337, + 106 + ], + "spans": [ + { + "bbox": [ + 105, + 93, + 337, + 106 + ], + "score": 1.0, + "content": "many potential combinations leading to good parameter-", + "type": "text" + } + ], + "index": 1 + }, + { + "bbox": [ + 106, + 105, + 336, + 115 + ], + "spans": [ + { + "bbox": [ + 106, + 105, + 336, + 115 + ], + "score": 1.0, + "content": "accuracy trade-off. However, it is unrealistic to exhaus-", + "type": "text" + } + ], + "index": 2 + }, + { + "bbox": [ + 106, + 115, + 336, + 127 + ], + "spans": [ + { + "bbox": [ + 106, + 115, + 336, + 127 + ], + "score": 1.0, + "content": "tively test all the methods for a specific downstream task.", + "type": "text" + } + ], + "index": 3 + }, + { + "bbox": [ + 105, + 126, + 336, + 138 + ], + "spans": [ + { + "bbox": [ + 105, + 126, + 336, + 138 + ], + "score": 1.0, + "content": "Other than probing such solution via evolutionary search", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 106, + 137, + 337, + 149 + ], + "spans": [ + { + "bbox": [ + 106, + 137, + 337, + 149 + ], + "score": 1.0, + "content": "as in Zhang et al. (2022), we aim to propose more under-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 106, + 148, + 337, + 160 + ], + "spans": [ + { + "bbox": [ + 106, + 148, + 337, + 160 + ], + "score": 1.0, + "content": "standable models by empirically analyzing the effect of", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 106, + 159, + 336, + 171 + ], + "spans": [ + { + "bbox": [ + 106, + 159, + 336, + 171 + ], + "score": 1.0, + "content": "different designs independently. According to the prelim-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 106, + 171, + 336, + 182 + ], + "spans": [ + { + "bbox": [ + 106, + 171, + 336, + 182 + ], + "score": 1.0, + "content": "inary results shwon in Figure 1, we argue that the position", + "type": "text" + } + ], + "index": 8 + }, + { + "bbox": [ + 106, + 181, + 337, + 193 + ], + "spans": [ + { + "bbox": [ + 106, + 181, + 337, + 193 + ], + "score": 1.0, + "content": "and amount of parameters are important for PETL tech-", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 192, + 327, + 204 + ], + "spans": [ + { + "bbox": [ + 106, + 192, + 327, + 204 + ], + "score": 1.0, + "content": "niques, especially when the target domain is not small.", + "type": "text" + } + ], + "index": 10 + } + ], + "index": 5, + "bbox_fs": [ + 105, + 82, + 337, + 204 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 209, + 336, + 253 + ], + "lines": [ + { + "bbox": [ + 106, + 209, + 337, + 221 + ], + "spans": [ + { + "bbox": [ + 106, + 209, + 337, + 221 + ], + "score": 1.0, + "content": "To verify the importance of position and tuned parameter", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 220, + 337, + 231 + ], + "spans": [ + { + "bbox": [ + 106, + 220, + 337, + 231 + ], + "score": 1.0, + "content": "amount, we independently tune different modules of the", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 230, + 337, + 243 + ], + "spans": [ + { + "bbox": [ + 106, + 230, + 337, + 243 + ], + "score": 1.0, + "content": "backbone model. Table 1 shows the results. We can see", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 241, + 336, + 255 + ], + "spans": [ + { + "bbox": [ + 105, + 241, + 336, + 255 + ], + "score": 1.0, + "content": "that the attention module’s QKV layer has 20.98M pa-", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "spans": [ + { + "bbox": [ + 105, + 253, + 505, + 265 + ], + "score": 1.0, + "content": "rameters while the MLP module has the most number of parameters of 55.90M. Tuning positions", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "spans": [ + { + "bbox": [ + 105, + 264, + 505, + 276 + ], + "score": 1.0, + "content": "with more parameters, will lead to better performance for SSv2. Thanks to the bottleneck mecha-", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "spans": [ + { + "bbox": [ + 105, + 275, + 505, + 286 + ], + "score": 1.0, + "content": "nism of adapter and prefix-tuning, one can effectively achieve a good parameter-accuracy trade-off.", + "type": "text" + } + ], + "index": 30 + }, + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "spans": [ + { + "bbox": [ + 106, + 286, + 505, + 298 + ], + "score": 1.0, + "content": "As such, we derive a model called Swin-B-adapter-PATT (Swin-BAPAT) from the V-PETL frame-", + "type": "text" + } + ], + "index": 31 + }, + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "spans": [ + { + "bbox": [ + 106, + 297, + 505, + 309 + ], + "score": 1.0, + "content": "work by using the parallel adapter and our PATT to leverage the adaption of pre-trained backbone", + "type": "text" + } + ], + "index": 32 + }, + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "spans": [ + { + "bbox": [ + 106, + 308, + 505, + 320 + ], + "score": 1.0, + "content": "model at the positions of attention and MLP modules, respectively. In addition to adapter and PATT,", + "type": "text" + } + ], + "index": 33 + }, + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "spans": [ + { + "bbox": [ + 105, + 318, + 505, + 331 + ], + "score": 1.0, + "content": "we also fine-tune the last fully connected layer as it has relatively smaller amount of tunable param-", + "type": "text" + } + ], + "index": 34 + }, + { + "bbox": [ + 105, + 330, + 273, + 341 + ], + "spans": [ + { + "bbox": [ + 105, + 330, + 273, + 341 + ], + "score": 1.0, + "content": "eters (i.e, 0.18M) than adapter and PATT.", + "type": "text" + } + ], + "index": 35 + } + ], + "index": 12.5, + "bbox_fs": [ + 105, + 209, + 337, + 255 + ] + }, + { + "type": "table", + "bbox": [ + 343, + 123, + 502, + 244 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 344, + 83, + 504, + 116 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 343, + 82, + 505, + 96 + ], + "spans": [ + { + "bbox": [ + 343, + 82, + 505, + 96 + ], + "score": 1.0, + "content": "Table 1: Comparison of independently", + "type": "text" + } + ], + "index": 15 + }, + { + "bbox": [ + 344, + 95, + 505, + 106 + ], + "spans": [ + { + "bbox": [ + 344, + 95, + 505, + 106 + ], + "score": 1.0, + "content": "fine-tuning varied positions of the video", + "type": "text" + } + ], + "index": 16 + }, + { + "bbox": [ + 343, + 105, + 477, + 116 + ], + "spans": [ + { + "bbox": [ + 343, + 105, + 477, + 116 + ], + "score": 1.0, + "content": "swin transformer block on SSv2.", + "type": "text" + } + ], + "index": 17 + } + ], + "index": 16 + }, + { + "type": "table_body", + "bbox": [ + 343, + 123, + 502, + 244 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 343, + 123, + 502, + 244 + ], + "spans": [ + { + "bbox": [ + 343, + 123, + 502, + 244 + ], + "score": 0.977, + "html": "
Position# ParamsTop-1 (%)
Full-tuning Tune FC Layer87.82M 0.18M50.99 24.13
LayerNorm 10.02M14.35
Attn,Proj6.99M47.58
Attn, QKV20.98M50.02
Attn, SoftMax0.95M27.67
LayerNorm 20.02M14.62
MLP, FC127.97M47.10
MLP,FC227.93M45.32
DownSample2.76M27.53
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The challenge", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "spans": [ + { + "bbox": [ + 106, + 429, + 505, + 440 + ], + "score": 1.0, + "content": "of this dataset is that it contains 23, 137 distinct object names with an imbalanced distribution.", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 107, + 440, + 505, + 452 + ], + "spans": [ + { + "bbox": [ + 107, + 440, + 505, + 452 + ], + "score": 1.0, + "content": "The original dataset is split into train, validation, and test sets with a ratio of 8:1:1. The extended", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "spans": [ + { + "bbox": [ + 106, + 451, + 505, + 463 + ], + "score": 1.0, + "content": "version (SSv2) of this dataset is consisted of 168, 913 training samples, 24, 777 validation samples,", + "type": "text" + } + ], + "index": 42 + }, + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "spans": [ + { + "bbox": [ + 105, + 460, + 506, + 475 + ], + "score": 1.0, + "content": "and 27, 157 testing samples with the sample number of action labels. The training and testing", + "type": "text" + } + ], + "index": 43 + }, + { + "bbox": [ + 104, + 471, + 506, + 486 + ], + "spans": [ + { + "bbox": [ + 104, + 471, + 506, + 486 + ], + "score": 1.0, + "content": "samples are used. HMDB51 (Kuehne et al., 2011) contains 6, 766 video samples for 51 action", + "type": "text" + } + ], + "index": 44 + }, + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "spans": [ + { + "bbox": [ + 105, + 483, + 506, + 496 + ], + "score": 1.0, + "content": "categories including videos of varied visible body parts, camera motion, camera view, and clip", + "type": "text" + } + ], + "index": 45 + }, + { + "bbox": [ + 105, + 495, + 506, + 507 + ], + "spans": [ + { + "bbox": [ + 105, + 495, + 506, + 507 + ], + "score": 1.0, + "content": "quality. All video samples have at least 101 clips and a minimum height of 60 pixels for actors.", + "type": "text" + } + ], + "index": 46 + }, + { + "bbox": [ + 106, + 506, + 505, + 517 + ], + "spans": [ + { + "bbox": [ + 106, + 506, + 505, + 517 + ], + "score": 1.0, + "content": "The original dataset has three splits of training and evaluation. We follow existing work Chen et al.", + "type": "text" + } + ], + "index": 47 + }, + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "spans": [ + { + "bbox": [ + 106, + 516, + 506, + 528 + ], + "score": 1.0, + "content": "(2022) by using the first training and evaluation split that has 3, 570 and 1, 530 samples, respectively.", + "type": "text" + } + ], + "index": 48 + }, + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 506, + 540 + ], + "score": 1.0, + "content": "Image Datasets: Following the experimental set ups in AdaptFormer, three datasets CIFAIR-100", + "type": "text" + } + ], + "index": 49 + }, + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "spans": [ + { + "bbox": [ + 105, + 537, + 506, + 550 + ], + "score": 1.0, + "content": "Krizhevsky et al. (2009), Street View House Numbers (SVHN) Goodfellow et al. (2013), and Food-", + "type": "text" + } + ], + "index": 50 + }, + { + "bbox": [ + 106, + 549, + 506, + 561 + ], + "spans": [ + { + "bbox": [ + 106, + 549, + 506, + 561 + ], + "score": 1.0, + "content": "101 Bossard et al. (2014) are used. CIFAIR-100 has 50, 000 and 10, 000 training and validation", + "type": "text" + } + ], + "index": 51 + }, + { + "bbox": [ + 106, + 560, + 506, + 573 + ], + "spans": [ + { + "bbox": [ + 106, + 560, + 277, + 573 + ], + "score": 1.0, + "content": "images, respectively, with the resolution of", + "type": "text" + }, + { + "bbox": [ + 277, + 560, + 307, + 571 + ], + "score": 0.9, + "content": "3 2 \\times 3 2", + "type": "inline_equation" + }, + { + "bbox": [ + 307, + 560, + 506, + 573 + ], + "score": 1.0, + "content": "and 100 categories; SVHN is a digit classification", + "type": "text" + } + ], + "index": 52 + }, + { + "bbox": [ + 104, + 569, + 507, + 585 + ], + "spans": [ + { + "bbox": [ + 104, + 569, + 507, + 585 + ], + "score": 1.0, + "content": "dataset that has 73, 257 training sample and 26, 032 testing samples; Food-101 includes 101k images", + "type": "text" + } + ], + "index": 53 + }, + { + "bbox": [ + 105, + 581, + 441, + 595 + ], + "spans": [ + { + "bbox": [ + 105, + 581, + 441, + 595 + ], + "score": 1.0, + "content": "of 101 food categories with each of them has 750 training and 250 testing samples.", + "type": "text" + } + ], + "index": 54 + } + ], + "index": 46, + "bbox_fs": [ + 104, + 406, + 507, + 595 + ] + }, + { + "type": "text", + "bbox": [ + 107, + 599, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "spans": [ + { + "bbox": [ + 105, + 599, + 505, + 612 + ], + "score": 1.0, + "content": "Implementation details: It is worth noting that big batch size (i.e., 1, 024) and the number of input", + "type": "text" + } + ], + "index": 55 + }, + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "spans": [ + { + "bbox": [ + 105, + 610, + 505, + 622 + ], + "score": 1.0, + "content": "video frames (i.e., 32 frames) can greatly benefit good performance (Carreira & Zisserman, 2017;", + "type": "text" + } + ], + "index": 56 + }, + { + "bbox": [ + 104, + 618, + 506, + 636 + ], + "spans": [ + { + "bbox": [ + 104, + 618, + 506, + 636 + ], + "score": 1.0, + "content": "Liu et al., 2022; Chen et al., 2022), which usually requires GPU clusters to enable the training.", + "type": "text" + } + ], + "index": 57 + }, + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "spans": [ + { + "bbox": [ + 105, + 631, + 505, + 645 + ], + "score": 1.0, + "content": "AdaptFormer (Chen et al., 2022) uses such powerful GPU cluster to achieve good performance.", + "type": "text" + } + ], + "index": 58 + }, + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "spans": [ + { + "bbox": [ + 105, + 643, + 506, + 656 + ], + "score": 1.0, + "content": "However, good performance might not hold when the batch size is small. Following the more", + "type": "text" + } + ], + "index": 59 + }, + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 654, + 506, + 667 + ], + "score": 1.0, + "content": "common hardware device setup, we use 4 GeForce 3090 GPUs for all experiments, leading to a", + "type": "text" + } + ], + "index": 60 + }, + { + "bbox": [ + 105, + 664, + 506, + 677 + ], + "spans": [ + { + "bbox": [ + 105, + 664, + 463, + 677 + ], + "score": 1.0, + "content": "batch size of 64. All the experiments are fine-tuned for 70 epochs. We use the Swin-", + "type": "text" + }, + { + "bbox": [ + 464, + 664, + 475, + 675 + ], + "score": 0.8, + "content": "\\mathbf { \\cdot B } ^ { 1 }", + "type": "inline_equation" + }, + { + "bbox": [ + 475, + 664, + 506, + 677 + ], + "score": 1.0, + "content": "model", + "type": "text" + } + ], + "index": 61 + }, + { + "bbox": [ + 104, + 675, + 506, + 689 + ], + "spans": [ + { + "bbox": [ + 104, + 675, + 506, + 689 + ], + "score": 1.0, + "content": "pre-trained on Kinetics 400 and 600. For HMDB51, we report the results without tuning the FC", + "type": "text" + } + ], + "index": 62 + }, + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "spans": [ + { + "bbox": [ + 106, + 687, + 505, + 699 + ], + "score": 1.0, + "content": "layer due to the significant effect of the FC layer on relatively small scale dataset. Following Chen", + "type": "text" + } + ], + "index": 63 + }, + { + "bbox": [ + 104, + 696, + 506, + 712 + ], + "spans": [ + { + "bbox": [ + 104, + 696, + 506, + 712 + ], + "score": 1.0, + "content": "et al. (2022), we do not perform regularization strategies such as mixup, cutmix, color jittering,", + "type": "text" + } + ], + "index": 64 + }, + { + "bbox": [ + 106, + 330, + 507, + 344 + ], + "spans": [ + { + "bbox": [ + 106, + 330, + 507, + 344 + ], + "score": 1.0, + "content": "etc. Our PATT module is convenient to be applied to other Transformer-based models. Hence, we", + "type": "text", + "cross_page": true + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 340, + 507, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 340, + 507, + 354 + ], + "score": 1.0, + "content": "respectively adopt ViT-B models from MAE (He et al., 2022b) and VideoMAE (Tong et al., 2022)", + "type": "text", + "cross_page": true + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "score": 1.0, + "content": "to conduct further comparison on video and image datasets, which follows the self-supervised pre-", + "type": "text", + "cross_page": true + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 362, + 483, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 483, + 377 + ], + "score": 1.0, + "content": "training setting2 in Chen et al. (2022) except that the batch size is set to 256 instead of 1, 024.", + "type": "text", + "cross_page": true + } + ], + "index": 8 + } + ], + "index": 59.5, + "bbox_fs": [ + 104, + 599, + 506, + 712 + ] + } + ] + }, + { + "preproc_blocks": [ + { + "type": "table", + "bbox": [ + 126, + 105, + 481, + 319 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 107, + 79, + 502, + 102 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 105, + 78, + 501, + 93 + ], + "spans": [ + { + "bbox": [ + 105, + 78, + 474, + 93 + ], + "score": 1.0, + "content": "Table 2: Comparison of Top-1 accuracy using varied amount of parameters adjusted by", + "type": "text" + }, + { + "bbox": [ + 474, + 80, + 501, + 91 + ], + "score": 0.84, + "content": "d _ { b o t t l e }", + "type": "inline_equation" + } + ], + "index": 0 + }, + { + "bbox": [ + 106, + 90, + 468, + 103 + ], + "spans": [ + { + "bbox": [ + 106, + 90, + 468, + 103 + ], + "score": 1.0, + "content": "different pre-training domains, and the number of frames with other fine-tuning strategies.", + "type": "text" + } + ], + "index": 1 + } + ], + "index": 0.5 + }, + { + "type": "table_body", + "bbox": [ + 126, + 105, + 481, + 319 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 126, + 105, + 481, + 319 + ], + "spans": [ + { + "bbox": [ + 126, + 105, + 481, + 319 + ], + "score": 0.98, + "html": "
MethoddbottlePre-training#FramesSSv2HMDB51
# Params Top-1(%)# Params Top-1 (%)
Full-tuning-Kinetics 400887.82M50.9987.69M68.07
Tune FC LayerKinetics 40080.18M24.130.05M71.28
BitFit (Zaken et al., 2022)·Kinetics 40081.29M45.941.11M68.26
AdaptFormer-Swin (Chen et al., 2022）64Kinetics 40081.73M40.801.61M68.66
Prefix-tuning (Li& Liang,2021)128Kinetics 40086.57M39.466.40M56.13
Our Swin-BAPAT (w/o Adapter)32Kinetics 40088881.35M46.261.17M69.51
Our Swin-BAPAT (w/o Adapter)64Kinetics 4002.51M49.232.34M71.34
Our Swin-BAPAT (w/o Adapter)128Kinetics 4004.83M52.574.65M70.56
Our Swin-BAPAT (w/o Adapter)256Kinetics 4009.45M52.719.27M70.23
Our Swin-BAPAT32Kinetics 40082.91M49.632.74M68.20
Our Swin-BAPAT64Kinetics 40084.07M51.803.89M70.10
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 600811.00M54.0610.83M69.90
Our Swin-BAPAT256 ImageNet-22K811.00M43.5610.83M59.89
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT128Kinetics 400166.38M63.146.20M75.67
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Hence, we", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 104, + 340, + 507, + 354 + ], + "spans": [ + { + "bbox": [ + 104, + 340, + 507, + 354 + ], + "score": 1.0, + "content": "respectively adopt ViT-B models from MAE (He et al., 2022b) and VideoMAE (Tong et al., 2022)", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "spans": [ + { + "bbox": [ + 104, + 352, + 507, + 367 + ], + "score": 1.0, + "content": "to conduct further comparison on video and image datasets, which follows the self-supervised pre-", + "type": "text" + } + ], + "index": 7 + }, + { + "bbox": [ + 104, + 362, + 483, + 377 + ], + "spans": [ + { + "bbox": [ + 104, + 362, + 483, + 377 + ], + "score": 1.0, + "content": "training setting2 in Chen et al. (2022) except that the batch size is set to 256 instead of 1, 024.", + "type": "text" + } + ], + "index": 8 + } + ], + "index": 6.5 + }, + { + "type": "text", + "bbox": [ + 106, + 380, + 506, + 459 + ], + "lines": [ + { + "bbox": [ + 106, + 381, + 467, + 393 + ], + "spans": [ + { + "bbox": [ + 106, + 381, + 467, + 393 + ], + "score": 1.0, + "content": "Baselines: We mainly compare our method Swin-BAPAT with three baselines as follows:", + "type": "text" + } + ], + "index": 9 + }, + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "spans": [ + { + "bbox": [ + 106, + 391, + 505, + 404 + ], + "score": 1.0, + "content": "(1) Full-tuning: set all the parameters learnable and tune the whole model initiated with the pre-", + "type": "text" + } + ], + "index": 10 + }, + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "spans": [ + { + "bbox": [ + 105, + 402, + 505, + 415 + ], + "score": 1.0, + "content": "trained weights. (2) Tune FC layer: tune the last fully connected layer and freeze pre-trained pa-", + "type": "text" + } + ], + "index": 11 + }, + { + "bbox": [ + 106, + 415, + 504, + 426 + ], + "spans": [ + { + "bbox": [ + 106, + 415, + 504, + 426 + ], + "score": 1.0, + "content": "rameters of the whole backbone model. (3) AdaptFormer-Swin: method introduced by Chen et al.", + "type": "text" + } + ], + "index": 12 + }, + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "spans": [ + { + "bbox": [ + 106, + 425, + 505, + 437 + ], + "score": 1.0, + "content": "(2022) that adds a parallel adapter to the MLP module in each block of the backbone model. (4)", + "type": "text" + } + ], + "index": 13 + }, + { + "bbox": [ + 105, + 434, + 507, + 449 + ], + "spans": [ + { + "bbox": [ + 105, + 434, + 507, + 449 + ], + "score": 1.0, + "content": "Prefix-tuning: the direct implementation of prefix-tuning used in NLP as defined in Eq. 5. (5) BitFit:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 445, + 384, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 384, + 460 + ], + "score": 1.0, + "content": "by tuning the bias of the backbone model together with the FC layer.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12 + }, + { + "type": "title", + "bbox": [ + 107, + 473, + 336, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 338, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 338, + 487 + ], + "score": 1.0, + "content": "3.2 THE EFFECT OF DIFFERENT PETL TECHNIQUES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Table 2 shows the results of different PETL techniques. From the results of four baseline methods,", + "type": "text" + } + ], + "index": 17 + }, + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "spans": [ + { + "bbox": [ + 105, + 505, + 505, + 519 + ], + "score": 1.0, + "content": "full-tuning performs the best for the large-scale dataset SSv2, whereas tuning the FC layer achieves", + "type": "text" + } + ], + "index": 18 + }, + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "spans": [ + { + "bbox": [ + 105, + 517, + 505, + 530 + ], + "score": 1.0, + "content": "superior performance over other PETL techniques on HMDB51. This is due to the fact that down-", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "spans": [ + { + "bbox": [ + 105, + 527, + 505, + 541 + ], + "score": 1.0, + "content": "stream tasks with relatively larger scale datasets are more parameter hungry for good convergence.", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 538, + 504, + 551 + ], + "spans": [ + { + "bbox": [ + 105, + 538, + 504, + 551 + ], + "score": 1.0, + "content": "On the contrary, small datasets can make good use of the knowledge from the source domain with", + "type": "text" + } + ], + "index": 21 + }, + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "spans": [ + { + "bbox": [ + 106, + 550, + 505, + 563 + ], + "score": 1.0, + "content": "slight effort of adaption via an FC layer. 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As", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 106, + 594, + 325, + 606 + ], + "spans": [ + { + "bbox": [ + 106, + 594, + 325, + 606 + ], + "score": 1.0, + "content": "such, we further examine this question in Section A.1.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 21.5 + }, + { + "type": "text", + "bbox": [ + 106, + 610, + 505, + 709 + ], + "lines": [ + { + "bbox": [ + 106, + 610, + 505, + 623 + ], + "spans": [ + { + "bbox": [ + 106, + 610, + 313, + 623 + ], + "score": 1.0, + "content": "We test different amount of parameters adjusted by", + "type": "text" + }, + { + "bbox": [ + 313, + 613, + 339, + 622 + ], + "score": 0.88, + "content": "s _ { b o t t l e }", + "type": "inline_equation" + }, + { + "bbox": [ + 339, + 610, + 505, + 623 + ], + "score": 1.0, + "content": ", taking its values to 32, 64, 128 and 256.", + "type": "text" + } + ], + "index": 27 + }, + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "spans": [ + { + "bbox": [ + 105, + 620, + 506, + 634 + ], + "score": 1.0, + "content": "The second and third groups (without or with Adapter, respectively) of results in Table 2 shows", + "type": "text" + } + ], + "index": 28 + }, + { + "bbox": [ + 106, + 632, + 506, + 645 + ], + "spans": [ + { + "bbox": [ + 106, + 632, + 191, + 645 + ], + "score": 1.0, + "content": "that larger values of", + "type": "text" + }, + { + "bbox": [ + 191, + 634, + 218, + 644 + ], + "score": 0.87, + "content": "s _ { b o t t l e }", + "type": "inline_equation" + }, + { + "bbox": [ + 218, + 632, + 506, + 645 + ], + "score": 1.0, + "content": "can benefit the fine-tuning with slightly more overhead of parameters", + "type": "text" + } + ], + "index": 29 + }, + { + "bbox": [ + 106, + 644, + 504, + 655 + ], + "spans": [ + { + "bbox": [ + 106, + 644, + 504, + 655 + ], + "score": 1.0, + "content": "on large-scale datasets such as SSv2. 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MethoddbottlePre-training#FramesSSv2HMDB51
# Params Top-1(%)# Params Top-1 (%)
Full-tuning-Kinetics 400887.82M50.9987.69M68.07
Tune FC LayerKinetics 40080.18M24.130.05M71.28
BitFit (Zaken et al., 2022)·Kinetics 40081.29M45.941.11M68.26
AdaptFormer-Swin (Chen et al., 2022）64Kinetics 40081.73M40.801.61M68.66
Prefix-tuning (Li& Liang,2021)128Kinetics 40086.57M39.466.40M56.13
Our Swin-BAPAT (w/o Adapter)32Kinetics 40088881.35M46.261.17M69.51
Our Swin-BAPAT (w/o Adapter)64Kinetics 4002.51M49.232.34M71.34
Our Swin-BAPAT (w/o Adapter)128Kinetics 4004.83M52.574.65M70.56
Our Swin-BAPAT (w/o Adapter)256Kinetics 4009.45M52.719.27M70.23
Our Swin-BAPAT32Kinetics 40082.91M49.632.74M68.20
Our Swin-BAPAT64Kinetics 40084.07M51.803.89M70.10
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 400811.00M53.9810.83M69.64
Our Swin-BAPAT256Kinetics 600811.00M54.0610.83M69.90
Our Swin-BAPAT256 ImageNet-22K811.00M43.5610.83M59.89
Our Swin-BAPAT128Kinetics 40086.38M53.366.20M71.93
Our Swin-BAPAT128Kinetics 400166.38M63.146.20M75.67
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(5) BitFit:", + "type": "text" + } + ], + "index": 14 + }, + { + "bbox": [ + 105, + 445, + 384, + 460 + ], + "spans": [ + { + "bbox": [ + 105, + 445, + 384, + 460 + ], + "score": 1.0, + "content": "by tuning the bias of the backbone model together with the FC layer.", + "type": "text" + } + ], + "index": 15 + } + ], + "index": 12, + "bbox_fs": [ + 105, + 381, + 507, + 460 + ] + }, + { + "type": "title", + "bbox": [ + 107, + 473, + 336, + 485 + ], + "lines": [ + { + "bbox": [ + 105, + 473, + 338, + 487 + ], + "spans": [ + { + "bbox": [ + 105, + 473, + 338, + 487 + ], + "score": 1.0, + "content": "3.2 THE EFFECT OF DIFFERENT PETL TECHNIQUES", + "type": "text" + } + ], + "index": 16 + } + ], + "index": 16 + }, + { + "type": "text", + "bbox": [ + 107, + 495, + 505, + 605 + ], + "lines": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "spans": [ + { + "bbox": [ + 106, + 495, + 505, + 507 + ], + "score": 1.0, + "content": "Table 2 shows the results of different PETL techniques. 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MethodSSv2HMDB51
#ParamsTop-1# ParamsTop-1
Full-tuning87.82M50.9987.69M68.07
Concat (K, V)6.38M15.616.20M20.98
No Zl-1(K,V)8.74M51.068.56M67.41
Ours (Q, K)6.38M45.496.20M68.92
Ours (K, V)6.38M53.386.20M71.41
Ours (Q, V)6.38M53.246.20M71.74
Ours (Q, K, V)7.93M53.237.63M69.57
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Scalar sSSv2HMDB51
Full-tuning50.9971.28
Tune FC Layer24.1368.07
AdaptFormer-Swin40.8068.66
s=0.247.4669.38
s=0.552.8471.87
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MethodSSv2HMDB51
#ParamsTop-1# ParamsTop-1
Full-tuning87.82M50.9987.69M68.07
Concat (K, V)6.38M15.616.20M20.98
No Zl-1(K,V)8.74M51.068.56M67.41
Ours (Q, K)6.38M45.496.20M68.92
Ours (K, V)6.38M53.386.20M71.41
Ours (Q, V)6.38M53.246.20M71.74
Ours (Q, K, V)7.93M53.237.63M69.57
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Our method ViT-BAPAT still shows promising", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 347, + 507, + 362 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 507, + 362 + ], + "score": 1.0, + "content": "parameter-accuracy trade-off via much smaller batch size, which is more convenient for reproduction", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 373 + ], + "score": 1.0, + "content": "on the general single server with 8 GPUs. 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MethodAvg.ImageVideo
Params (M)CIFAR-100SVHNFood-101SSv2HMDB51
Full-tuning86.04 (100%)85.9097.6790.0953.9746.41
Tune FC Layer0.07 (0.08%)69.83 (-16.07) 66.91 (-30.76) 69.74 (-20.35)29.23 (-24.74)）49.84 (+3.43)
VPT (Jia et al.,2022)0.08 (0.09%)82.44 (-3.46)94.02 (-3.65)82.98 (-7.11)43.73 (-10.24)52.67 (+6.26)
AdaptFormer-641.26 (1.46%)85.90 (0.00)96.89 (-0.78)87.61 (-2.48)59.02 (+5.05)55.69 (+9.28)
Our ViT-BAPAT-322.13 (2.47%)86.29 (+0.39)97.18 (-0.49)87.37 (-2.72)57.78 (+3.81)57.18 (+10.77)
Our ViT-BAPAT-643.02 (3.51%)86.35 (+0.45)97.18 (-0.49)87.53 (-2.56)57.55 (+3.58)57.18 (+10.77)
Our ViT-BAPAT-1284.79 (5.56%)86.47 (+0.57)97.28 (-0.39)87.66 (-2.43)56.97 (+3.00)57.70 (+11.29)
Our ViT-BAPAT-2568.33 (9.68%)86.55 (+0.65)97.24 (-0.43)87.68 (-2.41)56.53 (+2.56)57.31 (+10.90)
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With small amount over-", + "type": "text" + } + ], + "index": 36 + }, + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "spans": [ + { + "bbox": [ + 105, + 655, + 505, + 667 + ], + "score": 1.0, + "content": "head on trainable parameters, our method performs significantly better than state-of-the-art method", + "type": "text" + } + ], + "index": 37 + }, + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "spans": [ + { + "bbox": [ + 106, + 665, + 505, + 678 + ], + "score": 1.0, + "content": "AdapFormer-Swin and full-tuning on the datasets SSv2 and HMDB51 via small batch size, vali-", + "type": "text" + } + ], + "index": 38 + }, + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "spans": [ + { + "bbox": [ + 105, + 677, + 505, + 690 + ], + "score": 1.0, + "content": "dating our contribution to the literature of PETL. In the future we will test our proposed model on", + "type": "text" + } + ], + "index": 39 + }, + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "spans": [ + { + "bbox": [ + 105, + 687, + 506, + 700 + ], + "score": 1.0, + "content": "more action recognition datasets surveyed in Sun et al. (2022) under more learning regimes such as", + "type": "text" + } + ], + "index": 40 + }, + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "spans": [ + { + "bbox": [ + 106, + 699, + 505, + 712 + ], + "score": 1.0, + "content": "zero/few-shot learning, active learning and continual learning with other pre-training methods such", + "type": "text" + } + ], + "index": 41 + }, + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "spans": [ + { + "bbox": [ + 105, + 709, + 506, + 722 + ], + "score": 1.0, + "content": "as visual-language models. 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The", + "type": "text" + } + ], + "index": 4 + }, + { + "bbox": [ + 104, + 159, + 506, + 173 + ], + "spans": [ + { + "bbox": [ + 104, + 159, + 506, + 173 + ], + "score": 1.0, + "content": "proposed Swin-BAPAT is one of instantiated models from the V-PETL framework regarding the in-", + "type": "text" + } + ], + "index": 5 + }, + { + "bbox": [ + 105, + 169, + 505, + 185 + ], + "spans": [ + { + "bbox": [ + 105, + 169, + 505, + 185 + ], + "score": 1.0, + "content": "sert position of our PATT. Other instantiations can be inserted into different positions such as query,", + "type": "text" + } + ], + "index": 6 + }, + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "spans": [ + { + "bbox": [ + 105, + 181, + 505, + 194 + ], + "score": 1.0, + "content": "key, and value of the attention module. 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Our method ViT-BAPAT still shows promising", + "type": "text" + } + ], + "index": 19 + }, + { + "bbox": [ + 104, + 347, + 507, + 362 + ], + "spans": [ + { + "bbox": [ + 104, + 347, + 507, + 362 + ], + "score": 1.0, + "content": "parameter-accuracy trade-off via much smaller batch size, which is more convenient for reproduction", + "type": "text" + } + ], + "index": 20 + }, + { + "bbox": [ + 105, + 358, + 506, + 373 + ], + "spans": [ + { + "bbox": [ + 105, + 358, + 506, + 373 + ], + "score": 1.0, + "content": "on the general single server with 8 GPUs. 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In real-world application scenarios, small dataset can be the more", + "type": "text" + } + ], + "index": 23 + }, + { + "bbox": [ + 105, + 392, + 304, + 406 + ], + "spans": [ + { + "bbox": [ + 105, + 392, + 304, + 406 + ], + "score": 1.0, + "content": "common case, which confirms our contributions.", + "type": "text" + } + ], + "index": 24 + } + ], + "index": 21, + "bbox_fs": [ + 104, + 325, + 507, + 406 + ] + }, + { + "type": "table", + "bbox": [ + 108, + 437, + 501, + 549 + ], + "blocks": [ + { + "type": "table_caption", + "bbox": [ + 104, + 410, + 505, + 434 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 104, + 407, + 506, + 424 + ], + "spans": [ + { + "bbox": [ + 104, + 407, + 506, + 424 + ], + "score": 1.0, + "content": "Table 5: Comparison of Top-1 accuracy via ViT-B models from MAE and VideoMAE pre-trained", + "type": "text" + } + ], + "index": 25 + }, + { + "bbox": [ + 104, + 419, + 397, + 435 + ], + "spans": [ + { + "bbox": [ + 104, + 419, + 397, + 435 + ], + "score": 1.0, + "content": "with self-supervised learning for image and video datasets, respectively.", + "type": "text" + } + ], + "index": 26 + } + ], + "index": 25.5 + }, + { + "type": "table_body", + "bbox": [ + 108, + 437, + 501, + 549 + ], + "group_id": 0, + "lines": [ + { + "bbox": [ + 108, + 437, + 501, + 549 + ], + "spans": [ + { + "bbox": [ + 108, + 437, + 501, + 549 + ], + "score": 0.981, + "html": "
MethodAvg.ImageVideo
Params (M)CIFAR-100SVHNFood-101SSv2HMDB51
Full-tuning86.04 (100%)85.9097.6790.0953.9746.41
Tune FC Layer0.07 (0.08%)69.83 (-16.07) 66.91 (-30.76) 69.74 (-20.35)29.23 (-24.74)）49.84 (+3.43)
VPT (Jia et al.,2022)0.08 (0.09%)82.44 (-3.46)94.02 (-3.65)82.98 (-7.11)43.73 (-10.24)52.67 (+6.26)
AdaptFormer-641.26 (1.46%)85.90 (0.00)96.89 (-0.78)87.61 (-2.48)59.02 (+5.05)55.69 (+9.28)
Our ViT-BAPAT-322.13 (2.47%)86.29 (+0.39)97.18 (-0.49)87.37 (-2.72)57.78 (+3.81)57.18 (+10.77)
Our ViT-BAPAT-643.02 (3.51%)86.35 (+0.45)97.18 (-0.49)87.53 (-2.56)57.55 (+3.58)57.18 (+10.77)
Our ViT-BAPAT-1284.79 (5.56%)86.47 (+0.57)97.28 (-0.39)87.66 (-2.43)56.97 (+3.00)57.70 (+11.29)
Our ViT-BAPAT-2568.33 (9.68%)86.55 (+0.65)97.24 (-0.43)87.68 (-2.41)56.53 (+2.56)57.31 (+10.90)
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MethoddbottlePre-training#Frameswith FC layerwithout FC layer
#ParamsTop-1 (%)#ParamsTop-1 (%)
Our Swin-BAPAT32Kinetics 40082.79M65.972.74M68.20
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b/parse/dev/yhlMZ3iR7Pu/yhlMZ3iR7Pu_span.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:16af021386edfdc29eaa2b529772877af8eb0516bfaeeabfe2800725765d11ca +size 4525834 diff --git a/parse/dev/zSeoDvsDCe/zSeoDvsDCe.md b/parse/dev/zSeoDvsDCe/zSeoDvsDCe.md new file mode 100644 index 0000000000000000000000000000000000000000..4a2ea5ed75f1f79acd4cde5f9ef7e651bdc3dc21 --- /dev/null +++ b/parse/dev/zSeoDvsDCe/zSeoDvsDCe.md @@ -0,0 +1,1550 @@ +# Sign and Basis Invariant Networks for Spectral Graph Representation Learning + +Anonymous Author(s) +Affiliation +Address +email + +# Abstract + +1 We introduce SignNet and BasisNet—new neural architectures that are invariant +2 to two key symmetries displayed by eigenvectors: (i) sign flips, since if $v$ is an +3 eigenvector then so is $- v$ ; and (ii) more general basis symmetries, which occur in +4 higher dimensional eigenspaces with infinitely many choices of basis eigenvectors. +5 We prove that our networks are universal, i.e., they can approximate any continu +6 ous function of eigenvectors with the desired invariances. Moreover, when used +7 with Laplacian eigenvectors, our architectures are provably expressive for graph +8 representation learning: they can approximate any spectral graph convolution, can +9 compute spectral invariants that go beyond message passing neural networks, and +10 can provably simulate previously proposed graph positional encodings. Experi +11 ments show the strength of our networks for molecular graph regression, learning +12 expressive graph representations, and learning neural fields on triangle meshes. + +# 13 1 Introduction + +14 Numerous machine learning models process eigenvectors, which arise in various scenarios including +15 principal component analysis, matrix factorizations, and operators associated to graphs or manifolds. +16 An important example is the use of Laplacian eigenvectors to encode information about the structure +17 of a graph or manifold [Belkin and Niyogi, 2003, Von Luxburg, 2007, Lévy, 2006]. Positional +18 encodings that involve Laplacian eigenvectors have recently been used to generalize Transformers +19 to graphs [Kreuzer et al., 2021, Dwivedi and Bresson, 2021], and to improve the expressive power +20 and empirical performance of graph neural networks (GNNs) [Dwivedi et al., 2022]. Furthermore, +21 these eigenvectors are crucial for defining spectral operations on graphs that are foundational to graph +22 signal processing and spectral GNNs [Ortega et al., 2018, Bruna et al., 2014]. +23 However, there are nontrivial symmetries that should be accounted for when processing eigenvectors. +24 For instance, if $v$ is an eigenvector, then so is $- v$ , with the same eigenvalue. More generally, if an +25 eigenvalue has higher multiplicity, then there are infinitely many unit-norm eigenvectors that can +26 be chosen. Indeed, a full set of orthonormal eigenvectors is only defined up to a change of basis +27 in each eigenspace. In the case of sign invariance, for any $k$ eigenvectors there are $\bar { 2 ^ { k } }$ possible +28 choices of sign. Accordingly, prior works randomly flip eigenvector signs during training in order to +29 approximately learn sign invariance [Kreuzer et al., 2021, Dwivedi et al., 2020]. However, learning +30 all $2 ^ { k }$ invariances is challenging and limits the effectiveness of Laplacian eigenvectors for encoding +31 positional information. Sign invariance is a special case of basis invariance when all eigenvalues are +32 distinct, but general basis invariance is even more difficult to deal with. In Appendix C.2, we show +33 that higher dimensional eigenspaces are abundant in real datasets; for instance, $64 \%$ of molecule +34 graphs in the ZINC dataset have a higher dimensional eigenspace. +35 In this work, we address the sign and basis ambiguity problems by developing new neural networks— +36 SignNet and BasisNet. Our networks are universal and can approximate any continuous function +37 of eigenvectors with the proper invariances. Moreover, our networks are theoretically powerful +38 for graph representation learning—they can approximate spectral graph convolutions and compute +39 powerful spectral invariants, which allows our networks to express graph properties like subgraph +40 counts that message passing neural networks cannot. Finally, Laplacian eigenvectors with SignNet +41 and BasisNet can approximate many previously proposed graph positional encodings, including those +42 based on random walks [Li et al., 2020, Dwivedi et al., 2022] and heat kernels [Mialon et al., 2021, +43 Feldman et al., 2022]. Experiments on molecular graph regression tasks, learning expressive graph +44 representations, and texture reconstruction on triangle meshes illustrate the empirical benefits of our +45 models’ approximation power and invariances. + +# 46 2 Sign and Basis Invariant Networks + +47 For an $n \times n$ symmetric matrix, let $\lambda _ { 1 } \leq \ldots \leq$ +48 $\lambda _ { n }$ be the eigenvalues and $v _ { 1 } , \ldots , v _ { n }$ the corre +49 sponding eigenvectors, which we may assume +50 to form an orthonormal basis. For instance, we +51 could consider the normalized graph Laplacian +52 $L = I - D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ , where $A \in \mathbb { R } ^ { n \times n }$ +53 is the adjacency matrix and $D$ is the diagonal +54 degree matrix of some underlying graph. For +55 undirected graphs, $L$ is symmetric. Nonsymmet +56 ric matrices can be handled very similarly, as we +57 show in Appendix B.1. Our goal is to parame +58 terize a class of models $f ( v _ { 1 } , \ldots , v _ { k } )$ taking $k$ +59 eigenvectors as input in a manner that respects +60 the eigenvector symmetries. +61 Sign invariance. For any of the $v _ { i }$ , the sign +62 flipped $- v _ { i }$ is also an eigenvector, so a function +63 $f : \mathbb { R } ^ { n \times k } \mathbb { R } ^ { s }$ (where $s$ is an arbitrary output +64 dimension) should be sign invariant: + +$$ +f ( v _ { 1 } , \ldots , v _ { k } ) = f ( s _ { 1 } v _ { 1 } , \ldots , s _ { k } v _ { k } ) +$$ + +![](images/b914e874e9e714533cbddfb9f0489c55302867f3f2a8c3b6eefd457617e9e970.jpg) +Figure 1: Symmetries of eigenvectors of a symmetric matrix with permutation symmetries (e.g. a graph Laplacian). A neural network applied to the eigenvector matrix (middle) should be invariant or equivariant to permutation of the rows (left product with a permutation matrix $P$ ) and invariant to the choice of eigenvectors in each eigenbasis (right product with a block diagonal orthogonal matrix $\operatorname { \bar { D i a g } } ( Q _ { 1 } , Q _ { 2 } , Q _ { 3 } ) )$ . + +for all sign choices $s _ { i } \in \{ - 1 , 1 \}$ . That is, we + +want 66 $f$ to be invariant to the product group $\{ - 1 , 1 \} ^ { k }$ . This captures all eigenvector symmetries if the 67 eigenvalues $\lambda _ { i }$ are distinct. + +68 Basis invariance. If the eigenvalues have higher multiplicity, then there are further symmetries. +69 Let $V _ { 1 } , \dots , V _ { l }$ be bases of eigenspaces—i.e., $V _ { i } = \left[ \hat { v _ { i _ { 1 } } } \quad \hat { \cdot \cdot } \quad v _ { i _ { d _ { i } } } \right] \in \mathbb { R } ^ { n \times d _ { i } }$ has orthonormal +70 columns and spans the eigenspace associated with the shared eigenvalue $\mu _ { i } = \lambda _ { i _ { 1 } } = . . . = \lambda _ { i _ { d _ { i } } }$ +71 Any other orthonormal basis that spans the eigenspace is of the form $V _ { i } Q$ for some orthogonal +72 $Q \in O ( d _ { i } ) \subseteq \mathbb { R } ^ { d _ { i } \times d _ { i } }$ (see Appendix F.2). Thus, a function $f : \mathbb { R } ^ { n \times \sum _ { i = 1 } ^ { l } d _ { i } } \mathbb { R } ^ { s }$ that is invariant to +73 changes of basis in each eigenspace satisfies + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = f ( V _ { 1 } Q _ { 1 } , \dots , V _ { l } Q _ { l } ) , \qquad Q _ { i } \in O ( d _ { i } ) . +$$ + +74 In other words, $f$ is invariant to the product group $O ( d _ { 1 } ) \times \ldots \times O ( d _ { l } )$ . The number of eigenspaces +75 $l$ and the dimensions $d _ { i }$ may vary between matrices; we account for this in Section 2.2. As ${ \cal { O } } ( 1 ) =$ +76 $\{ - 1 , 1 \}$ , sign invariance is a special case of basis invariance when all eigenvalues are distinct. +77 Permutation equivariance. For GNN models that output node features or node predictions, one +78 typically further desires $f$ to be invariant or equivariant to permutations of nodes, i.e., along the entries +79 (or rows) of each vector. Thus, for $f : \mathbb { R } ^ { n \times d } \mathbb { R } ^ { n \times d }$ , we typically also require $f ( P V _ { 1 } , \dots , P V _ { l } ) =$ +80 $P f ( V _ { 1 } , \ldots , V _ { l } )$ for any permutation matrix $P \in \mathbb { R } ^ { n \times n }$ . Figure 1 illustrates the full setup. +81 Graph Positional Encodings. A major motivation for processing eigenvector input is for graph +82 positional encodings, which are additional features appended to each node in a graph that give +83 information about the position of that node in the graph. These additional features are crucial for +84 generalizing Transformers to graphs, and also have been found to improve performance of GNNs. +85 Figure 2 illustrates a standard pipeline and the use of our SignNet within it: the input adjacency, node +86 features, and eigenvectors of a graph are used to compute a prediction about the graph. Laplacian +87 eigenvectors are processed before being fed into this prediction model. Laplacian eigenvectors +88 have been widely used as positional encodings, and many works have noted that sign and/or basis +89 invariance must be dealt with in this case [Dwivedi and Bresson, 2021, Beaini et al., 2021, Dwivedi +90 et al., 2020, Kreuzer et al., 2021, Mialon et al., 2021, Dwivedi et al., 2022]. + +![](images/ad0640ccd20008f8d009a7ac580d2e99c68bcb6d30f78b824e2d16ecbc126029.jpg) +Figure 2: Pipeline for using node positional encodings. After processing by our SignNet, the learned positional encodings from the Laplacian eigenvectors are added as additional node features of an input graph. These positional encodings along with the graph adjacency and original node features are passed to a prediction model (e.g. a GNN). Not shown here, SignNet can also take in eigenvalues and node features if desired. + +# 91 2.1 Warmup: Neural Networks on One Eigenspace + +92 Before considering the general setting, we design neural networks that take a single eigenvector or +93 eigenspace as input and are sign or basis invariant. These single subspace architectures will become +94 building blocks for the general architectures. For one subspace, a sign invariant function is merely an +95 even function, and is easily parameterized. + +Proposition 1. A continuous function $h : \mathbb { R } ^ { n } \mathbb { R } ^ { s }$ is sign invariant if and only if + +$$ +h ( v ) = \phi ( v ) + \phi ( - v ) +$$ + +for some continuous $\phi : \mathbb { R } ^ { n } \mathbb { R } ^ { s }$ . A continuous $h : \mathbb { R } ^ { n } \mathbb { R } ^ { n }$ is sign invariant and permutation equivariant if and only $i f$ (3) holds for a continuous permutation equivariant $\phi : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ . + +99 In practice, we parameterize $\phi$ by a neural network. Any architecture choice will ensure sign +100 invariance, while permutation equivariance can be achieved using elementwise MLPs (Multi-Layer +101 Perceptrons), DeepSets [Zaheer et al., 2017], Transformers [Vaswani et al., 2017], or GNNs. +102 Next, we address basis invariance for a single $d$ -dimensional subspace, i.e., we aim to parameterize +103 maps $h : \mathbb { R } ^ { n \times d } \mathbb { R } ^ { n }$ that are (a) invariant to right multiplication by $Q \in O ( d )$ , and (b) equivariant +104 to permutations along the row axis. For (a), we use the mapping $V \mapsto V V ^ { \top }$ from $V$ to the +105 orthogonal projector of its column space, which is $O ( d )$ invariant. Mapping $V \mapsto V V ^ { \top }$ does not lose +106 information if we treat $V$ as equivalent to $V Q$ for any $Q \in O ( d )$ . This is justified by the classical +107 first fundamental theorem of $O ( d )$ [Kraft and Procesi, 1996], which has recently been applied in +108 machine learning by Villar et al. [2021]. +109 Regarding (b), permuting the rows of $V$ permutes rows and columns of $V V ^ { \top } \in \mathbb { R } ^ { n \times n }$ . Hence, we +110 desire the function $\phi : \mathbb { R } ^ { n \times n } \mathbb { R } ^ { n }$ on $\dot { V } \dot { V } ^ { \top }$ to be equivariant to both row and column permutation: +111 $\phi ( P V V ^ { \top } P ^ { \top } ) = \dot { P } \phi ( V V ^ { \top } )$ . To parameterize such a mapping from matrices to vectors, we use an +112 invariant graph network (IGN) [Maron et al., 2018]—a neural network mapping to and from tensors +113 of arbitrary order Rnd1 $\mathbb { R } ^ { n ^ { d _ { 1 } } } \to \mathbb { R } ^ { n ^ { d _ { 2 } } }$ that has the desired permutation equivariance. We thus parameterize +114 a family with the requisite invariance and equivariance as follows: + +$$ +h ( V ) = \operatorname { I G N } ( V V ^ { \top } ) . +$$ + +115 Proposition 2 states that this architecture universally approximates $O ( d )$ invariant and permutation +116 equivariant functions. The full approximation power requires high order tensors to be used for the +117 IGN; in practice, we restrict the tensor dimensions for efficiency, as discussed in the next section. + +Proposition 2. Any continuous, $O ( d )$ invariant $h : \mathbb { R } ^ { n \times d } \mathbb { R } ^ { s }$ is of the form $h ( V ) = \phi ( V V ^ { \top } )$ for a continuous $\phi$ . For a compact domain ${ \mathcal { Z } } \subseteq \mathbb { R } ^ { n \times d }$ , maps of the form $V \mapsto \operatorname { I G N } ( V V ^ { \top } )$ universally approximate continuous $h : \mathcal { Z } \subseteq \mathbb { R } ^ { n \times d } \to \mathbb { R } ^ { n }$ that are $O ( d )$ invariant and permutation equivariant. + +# 2.2 Neural Networks on Multiple Eigenspaces + +Next, we use the single-eigenspace models $\phi _ { l } ( V _ { l } )$ as building blocks for a model on multiple eigenspaces. To do so, we use functions of the form $f ( V _ { 1 } , \dots , V _ { l } ) = \rho ( \phi _ { 1 } ( V _ { 1 } ) , \dots , \phi _ { l } ( V _ { l } ) )$ , i.e., we first process each eigenspace individually with an invariant network, and then aggregate them via a function $\rho$ . This approach is grounded in a general decomposition theorem for product spaces that we prove in Section A. + +SignNet. We parameterize our sign invariant network 127 $f : \mathbb { R } ^ { n \times k } \mathbb { R } ^ { s }$ on eigenvectors $v _ { 1 } , \ldots , v _ { k }$ + +$$ +f ( v _ { 1 } , \dots , v _ { k } ) = \rho \left( [ \phi ( v _ { i } ) + \phi ( - v _ { i } ) ] _ { i = 1 } ^ { k } \right) , +$$ + +128 where $\phi$ and $\rho$ are unrestricted neural networks, and $[ \cdot ] _ { i }$ denotes concatenation of vectors. The form +129 $\phi ( v _ { i } ) + \phi ( - v _ { i } )$ induces sign invariance for each eigenvector. Since we do not yet impose permutation +130 equivariance here, we term this model Unconstrained-SignNet. +131 To obtain a sign invariant and permutation equivariant $f$ that outputs vectors in $\mathbb { R } ^ { n \times s }$ , we restrict $\phi$ +132 and $\rho$ to be permutation equivariant networks from vectors to vectors, such as elementwise MLPs, +133 DeepSets [Zaheer et al., 2017], Transformers [Vaswani et al., 2017], or most standard GNNs. We +134 name this permutation equivariant version SignNet. If desired, we can additionally use eigenvalues $\lambda _ { i }$ +135 and node features $\ b X \in \bar { \mathbb { R } } ^ { n \times q }$ by adding them as arguments to $\phi$ : + +$$ +f ( v _ { 1 } , \dots , v _ { k } , \lambda _ { 1 } , \dots , \lambda _ { k } , X ) = \rho \left( [ \phi ( v _ { i } , \lambda _ { i } , X ) + \phi ( - v _ { i } , \lambda _ { i } , X ) ] _ { i = 1 } ^ { k } \right) . +$$ + +BasisNet. For basis invariance, let 136 $V _ { i } \ \in \ \mathbb { R } ^ { n \times d _ { i } }$ be an orthonormal basis of a $d _ { i }$ dimensional 137 eigenspace. Then we parameterize our Unconstrained-BasisNet $f$ by + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = \rho \left( [ \phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \top } ) ] _ { i = 1 } ^ { l } \right) , +$$ + +138 where each $\phi _ { d _ { i } }$ is shared amongst all subspaces of the same dimension $d _ { i }$ , and $l$ is the number of +139 eigenspaces (i.e., number of distinct eigenvalues, which can differ from the number of eigenvectors +140 $k$ ). As $l$ differs between graphs, we may use zero-padding or a sequence model like a Transformer to +141 parameterize $\rho$ . Again, $\phi _ { d _ { i } }$ and $\rho$ are generally unrestricted neural networks. To obtain permutation +142 equivariance, we make $\rho$ permutation equivariant and let $\phi _ { d _ { i } } = \mathrm { I G N } _ { d _ { i } } : \mathbb { R } ^ { n ^ { 2 } } \mathbb { R } ^ { n }$ be IGNs from +143 matrices to vectors. For efficiency, we will only use matrices and vectors in the IGNs (that is, no +144 tensors in $\mathbb { R } ^ { n ^ { p } }$ for $p > 2$ ), i.e., we use 2-IGN. Our resulting BasisNet is + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = \rho \left( [ \mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \top } ) ] _ { i = 1 } ^ { l } \right) . +$$ + +145 Expressive-BasisNet. While we restrict SignNet to only use vectors and BasisNet to only use vectors +146 and matrices, higher order tensors are generally required for universally approximating permutation +147 equivariant or invariant functions [Keriven and Peyré, 2019, Maron et al., 2019, Maehara and NT, +148 2019]. Thus, we will consider a theoretically powerful but computationally impractical variant of +149 our model, in which we replace $\rho$ and $\mathrm { I G N } _ { d _ { i } }$ in BasisNet with IGNs of arbitrary tensor order. We +150 call this variant Expressive-BasisNet. Universal approximation requires $\Omega ( n ^ { n } )$ sized intermediate +151 tensors [Ravanbakhsh, 2020]. We study Expressive-BasisNet due to its theoretical interest, and to +152 juxtapose with the computational efficiency and strong expressive power of SignNet and BasisNet. + +For a summary of properties and more details about our models, see Appendix B. + +154 In the multiple subspace case, we can prove universality of our models through a general decomposi +155 tion theorem, which reduces the multiple subspace case to the single subspace case. See Section A +156 for details; we have temporarily moved this Section in the revision due to space constraints, and we +157 will move this Section into the main paper in the camera-ready version. + +# 158 3 Theoretical Power for Graph Representation Learning + +159 Next, we establish that our SignNet and BasisNet can compute useful basis invariant and permutation +160 equivariant functions on Laplacian eigenvectors for graph representation learning, including: spectral + +graph convolutions, spectral invariants, and existing graph positional encodings. Expressive-BasisNet can of course compute these functions, as it is universal, but this section shows that the practical invariant architectures SignNet and BasisNet can compute them as well. + +# 164 3.1 SignNets and BasisNets Generalize Spectral Graph Convolution + +For node features $X \in \mathbb { R } ^ { n \times q }$ and an eigendecomposition $V \Lambda V ^ { \top }$ , a spectral graph convolution takes the form $\begin{array} { r } { f ( V , \Lambda , X ) = \sum _ { i = 1 } ^ { n } \theta _ { i } v _ { i } v _ { i } ^ { \top } X = V \bar { \mathrm { D i a g } } ( \theta ) V ^ { \top } X . } \end{array}$ , for some parameters $\theta _ { i }$ , that may optionally be continuous functions $\bar { h } ( \lambda _ { i } ) = \theta _ { i }$ of the eigenvalues [Bruna et al., 2014, Defferrard et al., 2016]. This family includes important functions like heat kernels and generalized PageRanks on graphs [Li et al., 2019]. A spectral GNN is defined as multiple layers of spectral graph convolutions and node-wise linear maps, e.g. $\begin{array} { r } { V \mathrm { D i a g } ( \theta _ { 2 } ) V ^ { \top } \sigma \left( V \mathrm { D i a g } ( \bar { \theta } _ { 1 } ) V ^ { \top } X W _ { 1 } \right) } \end{array}$ $W _ { 2 }$ is a two layer spectral GNN. It can be seen (in Appendix H.1) that spectral graph convolutions are permutation equivariant and sign invariant, and if $\theta _ { i } = h ( \lambda _ { i } )$ (i.e. the spectral graph convolution is parametric) they are additionally invariant to a change of bases in each eigenspace. + +Our SignNet and BasisNet can be viewed as generalizations of spectral graph convolutions, as our networks can universally approximate all spectral graph convolutions of the above form. For instance, SignNet with $\begin{array} { r } { \rho ( a _ { 1 } , \ldots , a _ { k } ) = \sum _ { i = 1 } ^ { k } a _ { k } } \end{array}$ and $\begin{array} { r } { \phi ( v _ { i } , \lambda _ { i } , X ) = \frac { 1 } { 2 } \theta _ { i } v _ { i } v _ { i } ^ { \top } X } \end{array}$ directly yields the spectral graph convolution. This is captured in Theorem 1, which we prove in Appendix H.1. In fact, we may expect SignNet to learn spectral graph convolutions well, according to the principle of algorithmic alignment [Xu et al., 2020] (see Appendix H.1); this is supported by numerical experiments in Appendix J.2, in which our networks outperform baselines in learning spectral graph convolutions. + +Theorem 1. SignNet universally approximates all spectral graph convolutions. BasisNet universally approximates all parametric spectral graph convolutions. + +In fact, SignNet and BasisNet are strictly stronger than spectral graph convolutions; there are functions computable by SignNet and BasisNet that cannot be approximated by spectral graph convolutions or spectral GNNs. One way to see this is through graph isomorphism power, as captured in this next result. + +Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish. + +# 3.2 BasisNets can Compute Spectral Invariants + +Many works measure the expressive power of graph neural networks by comparing their power for testing graph isomorphism [Xu et al., 2019, Sato, 2020], or by comparing their ability to compute certain functions on graphs like subgraph counts [Chen et al., 2020, Tahmasebi et al., 2020]. These works often compare GNNs to combinatorial invariants on graphs, especially the $k$ -Weisfeiler-Lehman $k$ -WL) tests of graph isomorphism [Morris et al., 2021]. + +While we may also compare with these combinatorial invariants, as other GNN works that use spectral information have done [Beaini et al., 2021], we argue that it is more natural to analyze our networks in terms of spectral invariants, which are computed from the eigenvalues and eigenvectors of graphs. There is a rich literature of spectral invariants from the fields of spectral graph theory and complexity theory [Cvetkovic et al. ´ , 1997]. A spectral invariant must be invariant to permutations and changes of basis in each eigenspace, a characteristic shared by our networks. + +The simplest spectral invariant is the multiset of eigenvalues, which we give as input to our networks. Another widely studied, powerful spectral invariant is the collection of graph angles, which are defined as the values ${ \alpha _ { i j } } ^ { \star } = \| V _ { i } V _ { i } ^ { \top } e _ { j } \| _ { 2 }$ , where $V _ { i } \in \mathbb { R } ^ { n \times d _ { i } }$ is an orthonormal basis for the ith adjacency matrix eigenspace, and $e _ { j }$ is the $j$ th standard basis vector, which is zero besides a one in the $j$ th component. These are easily computed by our networks (Appendix H.3), so our networks inherit the strength of these invariants. We capture these results in the following theorem, which also lists a few properties that graph angles determine [Cvetkovic´, 1991]. + +Theorem 2. BasisNet universally approximates the graph angles $\alpha _ { i j }$ . The eigenvalues and graph angles (and thus BasisNet) can determine the number of length 3, 4, or 5 cycles, whether a graph is connected, and the number of length $k$ closed walks from any vertex to itself. + +211 Relation to WL and message passing. In contrast to this result, message passing GNNs are not able +212 to express any of these properties (see [Arvind et al., 2020, Garg et al., 2020] and Appendix H.3). +213 Although spectral invariants are strong, Fürer [2010] shows that the eigenvalues and graph angles—as +214 well as some strictly stronger spectral invariants—are not stronger than the 3-WL test (or, equivalently, +215 the 2-Folklore-WL test). Future work could study the combination of spectral invariants or spectral +216 graph positional encodings with combinatorial algorithms and graph neural networks. + +# 3.3 SignNets and BasisNets Generalize Existing Graph Positional Encodings + +Many graph positional encodings have been proposed, without any clear criteria on which to choose for a particular task. We prove (in Appendix H.2) that our efficient SignNet and BasisNet can universally approximate many previously used graph positional encodings, because we unify these positional encodings by expressing them as either a spectral graph convolution matrix or the diagonal of a spectral graph convolution matrix. + +Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally approximates diffusion and $p$ -step random walk relative positional encodings [Mialon et al., 2021], and generalized PageRank and landing probability distance encodings [Li et al., 2020]. + +We note that diagonals of spectral convolutions are used as feature descriptors in the shape analysis literature, such as the heat kernel signature [Sun et al., 2009] and wave kernel signature [Aubry et al., 2011]. In the language of recent works in graph machine learning, these are node positional encodings computed from a discrete Laplacian of a triangle mesh. This connection appears to be unnoticed in recent works on graph positional encodings. + +# 4 Experiments + +We demonstrate the strength of our networks in various experiments. Appendix B shows simple pseudo-code and a diagram detailing the use of SignNet as a node positional encoding. + +# 235 4.1 Graph Regression + +Table 1: Results on the ZINC dataset with a $5 0 0 \mathrm { k }$ parameter budget. All models use edge features. Numbers are the mean and standard deviation over 4 runs, each with different seeds. + +
Base modelPositional encodingk#paramTest MAE (↓)
GatedGCNNo PEN/A492k0.252±0.007
LapPE (flip)8492k0.198±0.011
LapPE (abs.)8492k0.204±0.009
LapPE (can.)8505k0.298±0.019
SignNet (𝜙(u) only)8495k0.148±0.007
SignNet8495k0.121±0.005
SignNetAll491k0.100±0.007
Sparse TransformerNo PEN/A473k0.283±0.030
LapPE (flip)16487k0.223±0.007
SignNet16479k0.115±0.008
SignNetAll486k0.102±0.005
GINENo PEN/A470k0.170±0.002
LapPE (flip)16470k0.178±0.004
SignNet16470k0.147±0.005
SignNetAll417k0.102±0.002
PNANo PEN/A474k
LapPE (flip)8474k0.133±0.011 0.132±0.010
SignNet8476k0.105±0.007
SignNetAll487k0.084±0.006
+ +236 We study the effectiveness of SignNet for learning positional encodings (PEs) from the eigenvectors 237 of the graph Laplacian on the ZINC dataset of molecule graphs [Irwin et al., 2012] (using the + +Table 2: Comparison with SOTA methods on graph-level regression tasks. $\dagger$ denotes domain-specific model. Numbers are test MAE, so lower is better. Best models within a standard deviation are bolded. + +
ZINC (10K)↓ZINC-full ↓Alchemy (10k)↓
HIMP † [Fey et al., 2020].151±.006.036±.002
CIN-small † [Bodnar et al., 2021].094±.004.044±.003
CIN† [Bodnar et al., 2021].079±.006.022±.002
GIN [Xu et al., 2019].170±.002.088±.002.180±.006
δ-2-GNN[Morris et al., 2020b].374±.022.042±.003.118±.001
δ-2-LGNN[Morris etal., 2020b].306±.044.045±.006.122±.003
SpeqNet [Morris et al., 2022].115±.001
GNN-IR [Dupty and Lee, 2022].137±.010.119±.002
PF-GNN [Dupty et al., 2021].122±.01.111±.01
Recon-GNN [Cotta et al., 2021].170±.006.125±.001
SignNet (ours).084±.006.024±.003.113±.002
+ +238 subset of 12,000 graphs from Dwivedi et al. [2020]). We primarily consider three settings: 1) No +239 positional encoding, 2) Laplacian PE (LapPE)—the $k$ eigenvectors of the graph Laplacian with +240 smallest eigenvalues are concatenated with existing node features, 3) SignNet positional features— +241 passing the eigenvectors through a SignNet and concatenating the output with node features. We +242 parameterize SignNet by taking $\phi$ to be a GIN $\mathrm { [ X u }$ et al., 2019] and $\rho$ to be an MLP. We sum over $\phi$ +243 outputs before the MLP when handling variable numbers of eigenvectors, so then the SignNet is of +244 the form MLP $\begin{array} { r } { \left( \sum _ { i = 1 } ^ { l } \phi ( v _ { i } ) + \phi ( - v _ { i } ) \right) } \end{array}$ (see Appendix K.2 for further details). We consider four +245 different base models that process the graph data and positional encodings: GatedGCN [Bresson and +246 Laurent, 2017], a Transformer with sparse attention only over neighbours [Kreuzer et al., 2021], PNA +247 [Corso et al., 2020], and GIN [Xu et al., 2019] with edge features (i.e. GINE) [Hu et al., 2020b]. The +248 total number of parameters of the SignNet and the base model is kept within a 500k budget. +249 Table 1 shows the results. For all 4 base models, the PE learned with SignNet yields the best test MAE +250 (mean absolute error) — lower MAE is better. Notably, this includes the cases of PNA and GINE, for +251 which Laplacian PE with simple random sign flipping was unable to improve performance over using +252 no PE at all. Our best performing model is PNA base combined with SignNet, which achieves 0.084 +253 test MAE. Besides SignNet, we consider two non-learned approaches to resolving eigenvector sign +254 ambiguity—canonicalization and taking element-wise absolute values (see Appendix K.2 for details). +255 Results with GatedGCN show that these alternatives are not more effective than random sign flipping +256 for learning positional encodings. We also consider an ablation of our SignNet architecture where we +257 remove the sign invariance, using simply $\mathrm { M L P } ( [ \phi ( v _ { i } ) ] _ { i = 1 } ^ { k } )$ . Although the resulting architecture is no +258 longer sign invariant, $\phi$ still processes eigenvectors independently, meaning that only two invariances +259 $( \pm 1 )$ need be learned, significantly fewer than the $2 ^ { k }$ total sign flip configurations. Accordingly, this +260 non-sign invariant learned positional encoding achieves a test MAE of 0.148, improving over the +261 Laplacian PE (0.198) but falling short of the fully sign invariant SignNet (0.121). In all cases, using +262 all available eigenvectors in SignNet significantly improves performance over using a fixed number +263 of eigenvectors. In Appendix J.1, we also show that SignNet improves performance when no edge +264 features are included in the data. +65 These significant performance improvements from SignNet come with only a slightly higher compu +66 tational cost. For example, GatedGCN with no PE takes about 8.2 seconds per training iteration on +67 ZINC, while GatedGCN with 8 eigenvectors and SignNet takes about 10.6 seconds; this is only a +68 $2 9 \%$ increase in time, for a reduction of test MAE by over $50 \%$ . Also, eigenvector computation time +69 is neglible, we need only precompute and save the eigenvectors once, and it only takes 15 seconds to +70 do this for the 12,000 graphs of ZINC. + +Comparison with SOTA. In Table 2, we compare SignNet with state-of-the-art methods on graphlevel molecular regression tasks on ZINC (10,000 training graphs), ZINC-full (about 250,000 graphs), and Alchemy [Chen et al., 2019a] (10,000 training graphs). We compare against both methods that use domain-specific knowledge about molecules, and domain-agnostic GNNs of various architectures. We see that SignNet outperforms all domain-agnostic methods on ZINC and ZINC-full, and is within a standard deviation of the best domain-specific method. Our mean score is the second best on + +Table 3: Test results for texture reconstruction experiment on cat and human models, following the experimental setting of [Koestler et al., 2022]. We use 1023 eigenvectors of the cotangent Laplacian. + +
CatHuman
MethodParamsPSNR↑DSSIM↓LPIPS↓PSNR↑DSSIM↓LPIPS↓
Intrinsic NF329k34.25.099.18932.29.119.330
Absolute value329k34.67.106.25232.42.132.363
Sign flip329k23.151.282.3521.521.052.71
SignNet324k34.91.090.14732.43.125.316
+ +277 Alchemy, and is within a standard deviation of the best. We perform much better on ZINC (.084) than +278 other state-of-the-art positional encoding methods, like GNN-LSPE (.090) [Dwivedi et al., 2022], +279 SAN (.139) [Kreuzer et al., 2021], and Graphormer (.122) [Ying et al., 2021]. +281 Substructure counts (e.g. of cycles) and global graph properties (e.g. connectedness, diameter, +282 radius) are important graph features that are known to be informative for problems in bio- and +283 chemo-informatics [Chen et al., 2020, Corso et al., 2020]. Following the setting of Zhao et al. [2022], +284 we show that SignNet with Laplacian positional encodings boosts the ability of simple GNNs to +285 count substructures and regress graph properties. We take a 4-layer GIN as the base model for all +286 settings, and for SignNet we use GIN as $\phi$ and a Transformer as $\rho$ to handle variable numbers of +287 eigenvectors (see Appendix K.4 for details). As shown in Figure 3, Laplacian PEs with sign-flip data +288 augmentation improve performance for counting substructures but not for regressing graph properties, +289 while Laplacian PEs processed by SignNet significantly boost performance on all tasks. + +![](images/1e38fc2f4e8186c11a80783c813f9adaac57f04e2ec49b004204af52ec8b9970.jpg) +280 4.2 Counting Substructures and Regressing Graph Properties +Figure 3: Counting substructures and regressing graph properties (lower is better). With Laplacian PEs, SignNet improves performance, while sign flip data augmentation (LapPE) is less consistent. Mean and standard deviations are reported on 3 runs. All runs use the same 4-layer GIN base model. + +# 290 4.3 Neural Fields on Manifolds + +Discrete approximations to the Laplace-Beltrami operator on manifolds have proven useful for processing data on surfaces, such as triangle meshes [Lévy, 2006]. Recently, Koestler et al. [2022] propose intrinsic neural fields, which use eigenfunctions of the Laplace-Beltrami operator as positional encodings for learning neural fields on manifolds. For generalized eigenfunctions $v _ { 1 } , \ldots , v _ { k }$ , at a point $p$ on the surface, they parameterize functions $f ( \boldsymbol { p } ) = \mathrm { M L P } ( v _ { 1 } ( \boldsymbol { p } ) , \dots , v _ { k } ( \boldsymbol { p } ) )$ . As these eigenfunctions have sign ambiguity, we use our SignNet to parameterize $f ( \boldsymbol { p } ) ^ { \prime } = \mathrm { M L P } ( \rho ( \operatorname { } [ \phi ( v _ { i } ( \boldsymbol { p } ) ) +$ $\phi \bar { ( - v _ { i } ( p ) ) } ] _ { i = 1 , \ldots , k } )$ ), with $\rho$ and $\phi$ being MLPs. + +Table 3 shows our results for texture reconstruction experiments on all models from Koestler et al. [2022]. The total number of parameters in our SignNet-based model is kept below that of the original model. We see that the SignNet architecture improves over the original Intrinsic NF model and over other baselines — especially in the LPIPS (Learned Perceptual Image Patch Similarity) metric, which has been shown to be a typically better perceptual metric than PSNR or DSSIM [Zhang et al., 2018a]. While we have not yet tested this, we believe that SignNet would allow even better improvements when learning over eigenfunctions of different models, as it could improve transfer and generalization. See Appendix D.1 for visualizations and Appendix K.5 for more details. + +![](images/2da6c2113b5fda8aff05b634b7425ce6f343ab9c2c035c184728c3df7b902400.jpg) +Figure 4: Cotangent Laplacian eigenvectors of the cat model and first principal component of ${ \phi ( \bar { v } ) + \phi ( - v ) }$ from our trained SignNet. + +# 306 4.4 Visualization of Learned Positional Encodings + +To better understand SignNet, we plot the first principal component of $\phi ( v ) + \phi ( - v )$ for two eigenvectors on the cat model in Figure 4. We see that SignNet encodes bilateral symmetry and structural information on the cat model. See Appendix D for plots of more eigenvectors and further details. + +# 5 Related Work + +In this section, we review selected related work. A more thorough review is deferred to Appendix E. + +Laplacian eigenvectors in GNNs. Various recently proposed methods in graph deep learning have directly used Laplacian eigenvectors as node positional encodings that are input to a neural network that is, e.g., a message passing GNN [Dwivedi et al., 2020, 2022], or some variant of a Transformer that is adapted to graphs [Dwivedi and Bresson, 2021, Kreuzer et al., 2021, Mialon et al., 2021, Dwivedi et al., 2022]. None of these methods address basis invariance, and they only partially address sign invariance for node positional encodings by randomly flipping eigenvector signs during training. + +Graph positional encodings. Other recent methods use positional encodings besides Laplacian eigenvectors. These include positional encodings based on random walks [Dwivedi et al., 2022, Mialon et al., 2021, Li et al., 2020], diffusion kernels on graphs [Mialon et al., 2021, Feldman et al., 2022], shortest paths [Ying et al., 2021, Li et al., 2020], and unsupervised node embedding methods [Wang et al., 2022]. In particular, Wang et al. [2022] use Laplacian eigenvectors for relative positional encodings in an invariant way, but they focus on robustness, so they have stricter invariances that significantly reduce expressivity (see Appendix E.2 for more details). These previously used positional encodings are mostly ad-hoc, less general since they can be provably expressed by SignNet and BasisNet (see Section 3.3), and/or are expensive to compute (e.g., all pairs shortest paths). + +# 6 Conclusion and Discussion + +SignNet and BasisNet are novel architectures for processing eigenvectors that are invariant to sign flips and choices of eigenspace bases, respectively. Both architectures are provably universal: they can represent any continuous function with the corresponding invariances. When used with Laplacian eigenvectors as inputs they can provably approximate spectral graph convolutions, spectral invariants, graph properties such as subgraph counts, and a number of other graph positional encodings. These theoretical results are supported by experiments showing that SignNet and BasisNet are highly expressive in practice, and learn effective graph positional encodings that improve the performance of message passing graph neural networks. Initial explorations show that SignNet and BasisNet can be useful beyond graph representation learning, as eigenvectors are ubiquitous. + +338 While we conduct experiments on graph machine learning tasks and a particular task on triangle +339 meshes, SignNet and BasisNet should also be applicable to processing eigenvectors in other settings, +340 such as recommender systems and tasks in shape analysis. 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[Yes] We support the claims with theoretical and/or empirical evidence. +(b) Did you describe the limitations of your work? [Yes] We discuss some limitations in the conclusion. +(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix B.2. +(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have read the guidelines; our paper conforms to them. + +2. If you are including theoretical results... + +(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all assumptions either in the main text or in the appendix. +(b) Did you include complete proofs of all theoretical results? [Yes] We include all proofs in the appendix. + +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] We include experimental code and instructions on the usage. +(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We try to give most experimental details in the main paper and appendix. +(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We report standard deviations for multiple runs and/or seeds for graph-level tasks, but not the texture reconstruction task. +(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix K.1. + +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] We cite the creators of software libraries, data, and machine learning models. +(b) Did you mention the license of the assets? [Yes] See Appendix K.1. +(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide our code. We will also link to an open source version after anonymous review. +(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Appendix K.1. +(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] Yes, we discuss this in Appendix K.1; the data most likely does not have personally identifiable information or offensive content. + +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] No crowdsourcing or human subjects used. +(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] No crowdsourcing or human subjects used. +(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] No crowdsourcing or human subjects used. + +We have temporarily moved this section to the Appendix due to space constraints, we will move it back to the main paper in the camera-ready version. + +While the networks introduced in the Section 2.2 possess the desired invariances, it is not immediately obvious whether they are powerful enough to express all functions with these invariances. The universality of our architectures follows as a corollary of the following general decomposition result, which may enable construction of universal architectures for other invariances as well. + +Theorem 3 (Decomposition Theorem). Let $\mathcal { X } _ { 1 } , \ldots , \mathcal { X } _ { k }$ be topological spaces, and let $G _ { i }$ be $a$ group acting on $\mathcal { X } _ { i }$ for each $i$ . We assume mild topological conditions on $\mathcal { X } _ { i }$ and $G _ { i }$ hold. For any continuous $f : \mathcal { X } = \mathcal { X } _ { 1 } \times . . . \times \mathcal { X } _ { k } \to \mathbb { R } ^ { s }$ that is invariant to the action of $G = G _ { 1 } \times \ldots \times G _ { k }$ there exists continuous $\phi _ { i }$ and a continuous $\rho : \mathcal { Z } \subseteq \mathbb { R } ^ { a } \to \mathbb { R } ^ { s }$ such that + +$$ +f ( v _ { 1 } , \dots , v _ { k } ) = \rho ( \phi _ { 1 } ( v _ { 1 } ) , \dots , \phi _ { k } ( v _ { k } ) ) . +$$ + +Furthermore:693 $( l )$ each $\phi _ { i }$ can be taken to be invariant to $G _ { i }$ , (2) the domain $\mathcal { Z }$ of $\rho$ is compact if each 694 $\mathcal { X } _ { i }$ is compact, (3) if ${ \mathcal { X } } _ { i } = { \mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , then $\phi _ { i }$ can be taken to be equal to $\phi _ { j }$ . + +695 This result says that when a product of groups $G$ acts on a product of spaces $\mathcal { X }$ , for invariance to the +696 product group $G$ it suffices to individually process each smaller group $G _ { i }$ on $\mathcal { X } _ { i }$ and then aggregate +697 the results. Along with the proof of Theorem 3, the mild topological assumptions are explained in +698 Appendix G.1. The assumptions hold for sign invariance and basis invariance. By applying this +699 theorem, we can prove universality of our networks: + +Corollary 1. Unconstrained-SignNet can represent any sign invariant function and UnconstrainedBasisNet can represent any basis invariant function. Expressive-BasisNet is a universal approximator of functions that are both basis invariant and permutation equivariant. + +703 This result shows that Unconstrained-SignNet, Unconstrained-BasisNet, and Expressive-BasisNet +704 take the correct functional form for their respective invariances (proofs in Appendix G.2). Note +705 that Expressive-BasisNet approximates all sign invariant functions as a special case, by treating +706 all inputs as one dimensional eigenspaces. Accompanying the decomposition result, we show a +707 corresponding universal approximation result (proof in Appendix G.3). Similarly to Theorem 3, +708 the problem of approximating $G = G _ { 1 } \times \ldots \times G _ { k }$ invariant functions is reduced to approximating +709 several $G _ { i }$ -invariant functions. + +# 710 B More Details on SignNet and BasisNet + +Table 4: Properties of our architectures: Unconstrained-SignNet, SignNet, Unconstrained-BasisNet, and Expressive-BasisNet. The properties are: permutation equivariance, universality (for the proper class of continuous invariant functions), and computational tractability. + +
Unconstr.-SignNetSignNetUnconstr.-BasisNetBasisNetExpr.-BasisNet
Permutation equiv.×√√×√x√√
Universal×√
Tractable√×
+ +711 In Figure 2, we show a diagram that describes how SignNet is used as a node positional encoding +712 for a graph machine learning task. In Table 4, we compare and contrast properties of the neural +713 architectures that we introduce. In Figure 5, we give pseudo-code of SignNet for learning node +714 positional encodings with a GNN prediction model. + +# 715 B.1 Generalization Beyond Symmetric Matrices + +716 In the main paper, we assume that the eigenspaces come from a symmetric matrix. This holds for many +717 cases of practical interest, as e.g. the Laplacian matrix of an undirected graph is symmetric. However, +718 we may also want to process directed graphs, or other data that have associated nonsymmetric matrices. +719 Our SignNet and BasisNet generalize in a straightforward way to handle nonsymmetric diagonalizable +720 matrices, as we detail here. Let $A \in \mathbb { R } ^ { n \times n }$ be a matrix with a diagonalization $A = V \Lambda V ^ { - 1 }$ , where +721 $\boldsymbol { \Lambda } = \operatorname { D i a g } ( \lambda _ { 1 } , \ldots , \lambda _ { n } )$ contains the eigenvalues $\lambda _ { i }$ , and the columns of $V = \left[ v _ { 1 } \quad \ldots \quad v _ { n } \right]$ are +722 eigenvectors. Suppose we want to learn a function on the eigenvectors $v _ { 1 } , \ldots , v _ { k }$ . Unlike in the +723 symmetric matrix case, the eigenvectors are not necessarily orthonormal, and both the eigenvalues +724 and eigenvectors can be complex. +725 Real eigenvectors. First, we assume the eigenvectors $v _ { i }$ are all real vectors in $\mathbb { R } ^ { n }$ . We can take the +726 eigenvectors to be real if $A$ is symmetric, or if $A$ has real eigenvalues (see Horn and Johnson [2012] +727 Theorem 1.3.29). Also, suppose that we choose the real numbers $\mathbb { R }$ as our base field for the vector +728 space in which eigenvectors lie. Note that for any scaling factor $c \in \mathbb { R } \setminus \{ 0 \}$ and eigenvector $v$ +729 we have that $c v$ is an eigenvector of the same eigenvalue. If the eigenvalues are distinct, then the +730 eigenvectors of the form $c v$ are the only other eigenvectors in the same eigenspace as $v$ . Thus, we +731 want a function to be invariant to scalings: + +![](images/ef801ebc92066def707f7be84a7c54ce9821cf07beb714baacf30831725a356a.jpg) +Figure 5: PyTorch-like pseudo-code for using SignNet with a GNN prediction model, where $\phi = \mathrm { G I N }$ and $\rho = \mathrm { M L P }$ as in the ZINC molecular graph regression experiments. Reshaping eigenvectors from $n \times k$ to $n \times k \times 1$ allows $\phi$ to process each eigenvector (and its negation) independently in PyTorch-like deep learning libraries. + +$$ +f ( v _ { 1 } , \ldots , v _ { k } ) = f ( c _ { 1 } v _ { 1 } , \ldots , c _ { k } v _ { k } ) \qquad c _ { i } \in \mathbb { R } \setminus \{ 0 \} . +$$ + +732 This can be handled by SignNet, by giving unit normalized vector inputs: + +$$ +f ( v _ { 1 } , \dots , v _ { k } ) = \rho \left( [ \phi ( v _ { i } / \Vert v _ { i } \Vert ) + \phi ( - v _ { i } / \Vert v _ { i } \Vert ) ] _ { i = 1 , \dots , k } \right) . +$$ + +733 Now, say have bases of eigenspaces $V _ { 1 } , \dots , V _ { l }$ with dimensions $d _ { 1 } , \ldots , d _ { l }$ . For a basis $V _ { i }$ , we have +734 that any other basis of the same space can be obtained as $V _ { i } W$ for some $W \in \mathrm { G L } _ { \mathbb { R } } ( d _ { i } )$ , the set of +735 real invertible matrices in $\mathbb { R } ^ { d _ { i } \times d _ { i } }$ . Indeed, the orthonormal projector for the space spanned by the +736 columns of $V _ { i }$ is given by $V _ { i } ( V _ { i } ^ { \top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \top }$ . Thus, if $Z \in \dot { \mathbb { R } } ^ { n \times d _ { i } }$ is another basis for the column +737 space of $V _ { i }$ , we have that $V _ { i } ( V _ { i } ^ { \top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \top } = Z ( Z ^ { \top } Z ) ^ { - 1 } Z ^ { \top }$ , so + +$$ +V _ { i } ( V _ { i } ^ { \top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \top } Z = Z ( Z ^ { \top } Z ) ^ { - 1 } Z ^ { \top } Z = Z , +$$ + +so let 738 $W ~ = ~ ( V _ { i } ^ { \top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \top } Z ~ \in ~ \mathbb { R } ^ { d _ { i } \times d _ { i } }$ . Note that $W$ is invertible, because it has inverse 739 $( Z ^ { \top } Z ) ^ { - 1 } Z ^ { \top } V _ { i }$ , so indeed $V _ { i } W = Z$ for $W \in \ G \mathrm { L } _ { \mathbb { R } } ( d _ { i } )$ . Thus, basis invariance in this case is 740 of the form + +$$ +f ( V _ { 1 } \ldots , V _ { l } ) = f ( V _ { 1 } W _ { 1 } , \ldots , V _ { l } W _ { l } ) \qquad W _ { i } \in \operatorname { G L } _ { \mathbb { R } } ( d _ { i } ) . +$$ + +741 Note that the distinct eigenvalue invariance is a special case of this invariance, as ${ \mathrm { G } } _ { \mathbb { R } } ( 1 ) = \mathbb { R } \ \backslash \ \{ 0 \}$ . +742 We can again achieve this basis invariance by using a BasisNet, where the inputs to the $\phi _ { d _ { i } }$ are +743 orthogonal projectors of the corresponding eigenspace: + +$$ +\begin{array} { r } { f ( V _ { 1 } , \ldots , V _ { l } ) = \rho \left( \left[ \phi _ { d _ { i } } ( V _ { i } ( V _ { i } ^ { \top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \top } ) \right] _ { i = 1 , \ldots , l } \right) . } \end{array} +$$ + +44 Recall that if $V _ { i }$ is an orthonormal basis, then the orthogonal projector is just $V _ { i } V _ { i } ^ { \top }$ , so this is a direct +745 generalization of BasisNet in the symmetric case. +746 Complex eigenvectors. More generally, suppose $V \in \mathbb { C } ^ { n \times n }$ are complex eigenvectors, and we take +747 the base field of the vector space to be $\mathbb { C }$ . The above arguments generalize to the complex case; in +748 the case of distinct eigenvalues, we want + +$$ +f ( v _ { 1 } , \ldots , v _ { k } ) = f ( c _ { 1 } v _ { 1 } , \ldots , c _ { k } v _ { k } ) \qquad c _ { i } \in \mathbb { C } \ \backslash \ \{ 0 \} . +$$ + +749 However, this symmetry can not be as easily reduced to a unit normalization and a discrete sign +750 invariance, as it can be in the real case. Nonetheless, the basis invariant architecture directly +751 generalizes, so we can handle the case of distinct eigenvalues by a more general basis invariant +752 architecture as well. The basis invariance is + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = f ( V _ { 1 } W _ { 1 } , \dots , V _ { l } W _ { l } ) \qquad W _ { i } \in \operatorname { G L } _ { \mathbb { C } } ( d _ { i } ) . +$$ + +The orthogonal projector of the image of 53 $V _ { i }$ is $V _ { i } ( V _ { i } ^ { * } V _ { i } ) ^ { - 1 } V _ { i } ^ { * }$ , where there are now conjugate 54 transposes replacing the transposes. Thus, BasisNet takes the form: + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = \rho \left( \left[ \phi _ { d _ { i } } ( V _ { i } ( V _ { i } ^ { * } V _ { i } ) ^ { - 1 } V _ { i } ^ { * } ) \right] _ { i = 1 , \dots , l } \right) . +$$ + +# 755 B.2 Broader Impacts + +756 We believe that our models and future sign invariant or basis invariant networks could be useful in a +757 wide variety of applications. As eigenvectors arise in many domains, it is difficult to predict the uses +758 of these models. We test on several molecular property prediction tasks, which have the potential +759 for much positive impact, such as in drug discovery [Stokes et al., 2020]. However, recent work +760 has found that the same models that we use for finding beneficial drugs can also be used to design +761 biochemical weapons [Urbina et al., 2022]. Another major application of graph machine learning +762 is in social network analysis, where positive (e.g. malicious node detection [Pandit et al., 2007]) +763 and negative (e.g. deanonymization [Narayanan and Shmatikov, 2009]) uses of machine learning +764 are possible. Even if there is no negative intent, bias in learned models can differentially impact +765 particular subgroups of people. Thus, academia, industry, and policy makers must be aware of such +766 potential negative uses, and work towards reducing the likelihood of them. + +# C More on Eigenvalue Multiplicities + +In this section, we study the properties of eigenvalues and eigenvectors computed by numerical algorithms on real-world data. + +# 770 C.1 Sign and Basis Ambiguities in Numerical Eigensolvers + +771 When processing real-world data, we use eigenvectors that are computed by numerical algorithms. +772 These algorithms return specific eigenvectors for each eigenspace, so there is some choice of sign +773 or basis of each eigenspace. The general symmetric matrix eigensolvers numpy.linalg.eigh +774 and scipy.linalg.eigh both call LAPACK routines. They both proceed as follows: for a +775 symmetric matrix $A$ , they first decompose it as $A = Q T Q ^ { \dagger }$ for orthogonal $Q$ and tridiago +776 nal $T$ , then they compute the eigendecomposition of $T \doteq \bar { W } \Lambda W ^ { \top }$ , so the eigendecomposition +777 of $A$ is $A = ( \mathsf { \bar { Q } } W ) \dot { \Lambda } ( W ^ { \top } Q ^ { \top } )$ . There are multiple ambiguities here: for diagonal sign matri +778 ces $S = \mathrm { D i a g } ( s _ { 1 } , . . . , s _ { n } )$ and $S ^ { \prime } = \mathrm { D i a g } ( s _ { 1 } ^ { \prime } , . . . , s _ { n } ^ { \prime } )$ , where $s _ { i } , s _ { i } ^ { \prime } \in \{ - 1 , 1 \}$ , we have that +779 $A = Q S ( S T S ) S Q ^ { \top }$ is also a valid tridiagonalization, as $\it Q S$ is still orthogonal, $S S = I$ , and $S T S$ +780 is still tridiagonal. Also, $T = ( W S ^ { \prime } ) \Lambda ( S ^ { \prime } W ^ { \top } )$ is a valid eigendecomposition of $T$ , as $W S ^ { \prime }$ is still +781 orthogonal. +782 In practice, we find that the general symmetric matrix eigensolvers numpy.linalg.eigh and +783 scipy.linalg.eigh differ between frameworks but are consistent with the same framework. More +784 specifically, for a symmetric matrix $A$ , we find that the eigenvectors computed with the default +785 settings in numpy tend to differ by a choice of sign or basis from those that are computed with the +786 default settings in scipy. On the other hand, the called LAPACK routines are deterministic, so the +787 eigenvectors returned by numpy are the same in each call, and the eigenvectors returned by scipy are +788 likewise the same in each call. +789 Eigensolvers for sparse symmetric matrices like scipy.linalg.eigsh are required for large scale +790 problems. This function calls ARPACK, which uses an iterative method that starts with a randomly +791 sampled initial vector. Due to this stochasticity, the sign and basis of eigenvectors returned differs +792 between each call. + +Table 5: Eigenspace statistics for datasets of multiple graphs. From left to right, the columns are: dataset name, number of graphs, range of number of nodes per graph, largest multiplicity, and percent of graphs with an eigenspace of dimension $> 1$ . + +
DatasetGraphs#NodesMax.Mult% Graphs mult. > 1
ZINC12,0009-37964.1
ZINC-full249,4566-381063.8
ogbg-molhiv41,1272- 2224268.0
IMDB-M1,5007-893799.9
COLLAB5,00032 - 49223899.1
PROTEINS1,1134- 6202077.3
COIL-DEL3,9003-7744.00
+ +Bro et al. [2008] develops a data-dependent method to choose signs for each singular vector of a singular value decomposition. Still, in the worst case the signs chosen will be arbitrary, and they do not handle basis ambiguities in higher dimensional eigenspaces. Other works have made choices of sign, such as by picking the sign so that the eigenvector’s entries are in the largest lexicographic order [Tam and Dunson, 2022]. This choice of sign may work poorly for learning on graphs, as it is sensitive to permutations on nodes. For some graph regression experiments in Section 4.1, we try a choice of sign that is permutation invariant, but we find it to work poorly. + +# C.2 Higher Dimensional Eigenspaces in Real Graphs + +Here, we investigate the normalized Laplacian eigenspace statistics of real-world graph data. For any graph that has distinct Laplacian eigenvalues, only sign invariance is required in processing eigenvectors. However, we find that graph data tends to have higher multiplicity eigenvalues, so basis invariance would be required for learning symmetry-respecting functions on eigenvectors. + +Indeed, we show statistics for multi-graph datasets in Table 5 and for single-graph datasets with more nodes per graph in Table 6. For multi-graph datasets, we consider : + +• Molecule graphs: ZINC [Irwin et al., 2012, Dwivedi et al., 2020], ogbg-molhiv [Wu et al., 2018, Hu et al., 2020a] +• Social networks: IMDB-M, COLLAB [Yanardag and Vishwanathan, 2015, Morris et al., 2020a], +• Bioinformatics graphs: PROTEINS [Morris et al., 2020a] +• Computer vision graphs: COIL-DEL [Riesen and Bunke, 2008, Morris et al., 2020a]. + +13 For single-graph datasets, we consider: + +• The $3 2 \times 3 2$ image grid as in Section J.2 +• Citation networks: Cora, Citeseer [Sen et al., 2008] +• Co-purchasing graphs with Amazon Photo [McAuley et al., 2015, Shchur et al., 2018]. + +817 We see that these datasets all contain higher multiplicity eigenspaces, so sign invariance is insufficient +818 for fully respecting symmetries. The majority of graphs in each multi-graph dataset besides COIL +819 DEL contain higher multiplicity eigenspaces. Also, the dimension of these eigenspaces can be +820 quite large compared to the size of the graphs in the dataset. The single-graph datasets have a large +821 proportion of their eigenvectors belonging to higher dimensional eigenspaces. Thus, basis invariance +822 may play a large role in processing spectral information from these graph datasets. + +Table 6: Eigenspace statistics for single graphs. From left to right, the columns are: dataset name, number of nodes, distinct eigenvalues (i.e. distinct eigenspaces), number of unique multiplicities, largest multiplicity, and percent of eigenvectors belonging to an eigenspace of dimension $> 1$ . + +
DatasetNodesDistinct 入#Mult.Max Mult.% Vecs mult. > 1
32 × 32 image1,02451333296.9
Cora2,7082,1871130019.7
Citeseer3,3271,8611249144.8
Amazon Photo7,6507,41681363.71
+ +# 823 C.3 Relationship to Graph Automorphisms + +Higher multiplicity eigenspaces are related to automorphism symmetries in graphs. For an adjacency matrix $A$ , the permutation matrix $P$ is an automorphism of the graph associated to $A$ if $P A \bar { P ^ { \top } } = \bar { A }$ . If $P$ is an automorphism, then for any eigenvector $v$ of $A$ with eigenvalue $\lambda$ , we have + +$$ +A P v = P A P ^ { \top } P v = P A v = P \lambda v = \lambda P v , +$$ + +so $P v$ is an eigenvector of $A$ with the same eigenvalue $\lambda$ . If $P v$ and $v$ are linearly independent, then $\lambda$ has a higher dimensional eigenspace. Thus, under certain additional conditions, automorphism symmetries of graphs lead to repeated eigenvalues [Sachs and Stiebitz, 1983, Teranishi, 2009]. + +# 30 C.4 Multiplicities in Random Graphs + +It is known that almost all random graphs under the Erdos-Renyi model have no repeated eigenvalues ˝ in the infinite number of nodes limit [Tao and Vu, 2017]. Likewise, almost all random graphs under the Erdos-Renyi model are asymmetric in the sense of having no nontrivial automorphism ˝ symmetries [Erdos and Rényi, 1963]. These results contrast sharply with the high eigenvalue multiplicities that we see in real-world data in Section C.2. Likewise, many types of real-world graph data have been found to possess nontrivial automorphism symmetries [Ball and Geyer-Schulz, 2018]. This demonstrates a potential downside of using random graph models to study real-world data: the eigenspace dimensions and automorphism symmetries of random graphs may not agree with those of real-world data. + +# 840 D Visualization of SignNet output + +# 841 D.1 Cat Model Visualization + +![](images/31496ac29afa5247f520171197969ab0d9a295fbd9337a1350f67d1451fd5728.jpg) +Figure 6: (Left) Cotangent Laplacian eigenvectors of the cat model. (Right) First principal component of $\phi ( v ) + \phi ( - v )$ from our trained SignNet. + +842 In Figure 6, we plot the eigenvectors of the cotangent Laplacian on a cat model, as well as the first +843 principal component of the corresponding learned $\phi ( v ) + \phi ( - v )$ from our SignNet model that was +844 trained on the texture reconstruction task. Interestingly, this portion of our SignNet encodes bilateral +845 symmetry; for instance, while some eigenvectors differ between left feet and right feet, this portion of +846 our SignNet gives similar values for the left and right feet. This is useful for the texture reconstruction +847 task, as the texture regression target has bilateral symmetry. +848 We also show principal components of outputs for the full SignNet model in Figure 7. This is not +849 as interpretable, as the outputs are high frequency and appear to be close to the texture that is the +850 regression target. If instead we trained the network on a task involving eigenvectors of multiple +851 models, then we may expect the SignNet to learn more structurally interpretable mappings (as in the +852 case of the molecule tasks). + +![](images/8deab72b8c4c3e92e6f1ddf111322bc7832b3cbb1a3f8d9797091521c7429cf6.jpg) +Figure 7: First three principal components of the full SignNet output on the cat model. + +# D.2 Molecule visualization + +To better understand SignNet, in Figure 9 we visualize the learned positional encodings of a SignNet with $\phi = \mathrm { G I N }$ , $\rho = \mathsf { M L P }$ (with a summation to handle variable eigenvector numbers) trained on ZINC as in Section 4.1. SignNet learns interesting structural information such as min-cuts (PC 3) and appendage atoms (PC 2) that qualitatively differ from any single eigenvector of the graph. + +858 For this visualization we use a SignNet trained with a GatedGCN base model on ZINC, as in +859 Section 4.1. This SignNet uses GIN as $\phi$ and $\rho$ as an MLP (with a sum before it to handle variable +860 numbers of eigenvectors), and takes in all eigenvectors of each graph. See Figure 8 for all of the +861 eigenvectors of fluorescein. + +# 62 E More Related Work + +# E.1 Graph Positional Encodings + +Various graph positional encodings have been proposed, which have been motivated for increasing expressive power or practical performance of graph neural networks, and for generalizing Transformers to graphs. Positional encodings are related to so-called position-aware network embeddings [Chami et al., 2020], which capture distances between nodes in graphs. These include network embedding methods like Deepwalk [Perozzi et al., 2014] and node2vec [Grover and Leskovec, 2016], which have been recently integrated into GNNs that respect their invariances by Wang et al. [2022]. Further, Li et al. [2020] studies the theoretical and practical benefits of incorporating distance features into graph neural networks. Dwivedi et al. [2022] proposes a method to inject learnable positional encodings into each layer of a graph neural network, and uses a simple random walk based node positional encoding. You et al. [2021] proposes a node positional encoding $\operatorname { d i a g } ( A ^ { k } )$ , which captures the number of closed walks from a node to itself. Dwivedi et al. [2020] propose to use Laplacian eigenvectors as positional encodings in graph neural networks, with sign ambiguities alleviated by sign flipping data augmentation. Srinivasan and Ribeiro [2019] theoretically analyze node positional embeddings and structural representations in graphs, and show that most-expressive structural representations contain the information of any node positional embedding. + +879 While positional encodings in sequences as used for Transformers [Vaswani et al., 2017] are able to +880 leverage the canonical order in sequences, there is no such useful canonical order for nodes in a graph, +881 due in part to permutation symmetries. Thus, different permutation equivariant positional encodings +882 have been proposed to help generalize Transformers to graphs. Dwivedi and Bresson [2021] directly +883 add in linearly projected Laplacian eigenvectors to node features before processing these features +884 with a graph Transformer. Kreuzer et al. [2021] propose an architecture that uses attention over +885 Laplacian eigenvectors and eigenvalues to learn node or edge positional encodings. Mialon et al. +886 [2021] uses spectral kernels such as the diffusion kernel to define relative positional encodings that +887 modulate the attention matrix. Ying et al. [2021] achieve state-of-the-art empirical performance +888 with simple Transformers that incorporate shortest-path based relative positional encodings. Zhang + +![](images/43aa908d922e8e64a133b65e1fd6fd76a675a5170663a7438d04c4e851030493.jpg) +Figure 8: All normalized Laplacian eigenvectors of the fluorescein graph. The first principal components of SignNet’s learned positional encodings do not exactly match any eigenvectors. + +et al. [2020] also utilize shortest-path distances for positional encodings in their graph Transformer. Kim et al. [2021] develop higher-order transformers (that generalize invariant graph networks), which interestingly perform well on graph regression using sparse higher-order transformers without positional encodings. + +# 93 E.2 Eigenvector Symmetries in Graph Representation Learning + +Many works that attempt to respect the invariances of eigenvectors solely focus on sign invariance (by using data augmentation) [Dwivedi et al., 2020, Dwivedi and Bresson, 2021, Dwivedi et al., 2022, Kreuzer et al., 2021]. This may be reasonable for continuous data, where eigenvalues of associated matrices may be usually distinct and separated (e.g. Puny et al. [2022] finds that this empirically holds for covariance matrices of $n$ -body problems). However, discrete graph Laplacians are known to have higher multiplicity eigenvalues in many cases, and in Appendix C.2 we find this to be true in various types of real-world graph data. Graphs without higher multiplicity eigenspaces are easier to deal with; in fact, graph isomorphism can be tested in polynomial time on graphs of bounded + +![](images/8d054483f5205b0ea63ad7c9f95d2bcf946c32fde9e8cc7c82166da588a60951.jpg) +Figure 9: Normalized Laplacian eigenvectors and learned positional encodings for the graph of fluorescein. (Top row) From left to right: smallest and second smallest nontrivial eigenvectors, then second largest and largest eigenvectors. (Bottom row) From left to right: first four principal components of the output $\rho \big ( \mathrm { \bar { [ } } \phi ( v _ { i } ) \mathrm { \bar { + } } \phi ( - v _ { i } ) \mathrm { ] } _ { i = 1 , \dots , n } \big )$ of SignNet. Note: we will put this back in the main paper for the camera-ready. + +902 multiplicity for adjacency matrix eigenvalues [Babai et al., 1982], with a time complexity that is +903 lower for graphs with lower maximum multiplicities. +904 +905 +906 +907 +908 +909 +910 +911 +912 +913 +914 +915 + +A recent work of Wang et al. [2022] proposes full orthogonal group invariance for functions that process positional encodings. In particular, for positional encodings $Z \in \mathbb { R } ^ { n \times k }$ , they parameterize functions $f ( Z )$ such that ${ \bar { f } } ( Z ) { \bar { = } } f ( Z Q )$ for all $Q \in O ( k )$ . This indeed makes sense for network embeddings like node2vec [Grover and Leskovec, 2016], as their objective functions are based on inner products and are thus orthogonally invariant. While they prove stability results when enforcing full orthogonal invariance for eigenvectors, this is a very strict constraint compared to our basis invariance. For instance, when $k = n$ and all eigenvectors are used in $V$ , the condition $f ( V ) = f ( V Q )$ implies that $f$ is a constant function on orthogonal matrices, since any orthogonal matrix $W$ can be obtained as $W = V Q$ for $Q = V ^ { \top } W \in O ( n )$ . In other words, for bases of eigenspaces $V _ { 1 } , \dots , V _ { l }$ and $V = [ V _ { 1 } \quad \ldots \quad V _ { l } ]$ , Wang et al. [2022] enforces $V Q \cong V$ , while we enforce $V \mathrm { D i a g } ( Q _ { 1 } , \dots , Q _ { l } ) \cong V$ . While the columns of $V \mathrm { D i a g } ( Q _ { 1 } , \dots , Q _ { l } )$ are still eigenvectors, the columns of $V Q$ generally are not. + +# 916 E.3 Graph Spectra and Learning on Graphs + +More generally, graph spectra are widely used in analyzing graphs, and spectral graph theory [Chung, 1997] studies the connection between graph properties and graph spectra. Different graph kernels have been defined based on graph spectra, which use robust and discriminative notions of generalized spectral distance [Verma and Zhang, 2017], the spectral density of states [Huang et al., 2021], random walk return probabilities [Zhang et al., 2018b], or the trace of the heat kernel [Tsitsulin et al., 2018]. Graph signal processing relies on spectral operations to define Fourier transforms, frequencies, convolutions, and other useful concepts for processing data on graphs [Ortega et al., 2018]. The closely related spectral graph neural networks [Wu et al., 2020, Balcilar et al., 2020] parameterize neural architectures that are based on similar spectral operations. + +# F Definitions, Notation, and Background + +# F.1 Basic Topology and Algebra Definitions + +We will use some basic topology and algebra for our theoretical results. A topological space $( \mathcal { X } , \tau )$ is a set $\mathcal { X }$ along with a family of subsets $\tau \subseteq 2 ^ { \mathcal { X } }$ satisfying certain properties, which gives useful notions like continuity and compactness. From now on, we will omit mention of $\tau$ , and refer to a + +931 topological space as the set $\mathcal { X }$ itself. For topological spaces $\mathcal { X }$ and $\mathcal { V }$ , we write $\chi \cong \mathcal { V }$ and say that +932 $\mathcal { X }$ is homeomorphic to $\mathcal { V }$ if there exists a continuous bijection with continuous inverse from $\mathcal { X }$ to +933 $\mathcal { V }$ . We will say $\mathcal { X } = \mathcal { y }$ if the underlying sets and topologies are equal as sets (we will often use this +934 notion of equality for simplicity, even though it can generally be substituted with homeomorphism). +935 For a function $f : \mathcal { X } \mathcal { Y }$ between topological spaces $\mathcal { X }$ and $\mathcal { V }$ , the image $\operatorname { i m } f$ is the set of values +936 that $f$ takes, ${ \mathrm { i m } } f = \{ f ( x ) : x \in \mathcal { X } \}$ . This is also denoted $f ( \mathcal X )$ . A function $f : \mathcal { X } \mathcal { Y }$ is called a +937 topological embedding if it is a homeomorphism from $\mathcal { X }$ to its image. +938 A group $G$ is a set along with a multiplication operation $G \times G \to G$ , such that multiplication is +939 associative, there is a multiplicative identity $e \in G$ , and each $g \in G$ has a multiplicative inverse $g ^ { - 1 }$ +940 A topological group is a group that is also a topological space such that the multiplication and inverse +941 operations are continuous. +942 A group $G$ may act on a set $\mathcal { X }$ by a function $\cdot : G \times \mathcal { X } \to \mathcal { X }$ . We usually denote $g \cdot x$ as $g x$ . A +943 topological group is said to act continuously on a topological space $\mathcal { X }$ if $\cdot$ is continuous. For any +944 group $G$ and topological space $\mathcal { X }$ , we define the coset $G x = \{ g x : g \in G \}$ , which can be viewed as +945 an equivalance class of elements that can be transformed from one to another by a group element. +946 The quotient space ${ \mathcal { X } } / G = \{ G x : x \in { \mathcal { X } } \}$ is the set of all such equivalence classes, with a topology +947 induced by that of $\mathcal { X }$ . The quotient map $\dot { \pi } : \mathcal { X } \to \mathcal { X } / G$ is a surjective continuous map that sends $x$ +948 to its coset, $\pi ( x ) = G x$ . + +For $x \in \mathbb { R } ^ { s }$ , $\| x \| _ { 2 }$ denotes the standard Euclidean norm. By the $\infty$ norm of functions $f : \mathcal { Z } \to \mathbb { R } ^ { s }$ from a compact $\mathcal { Z }$ to a Euclidean space $\mathbb { R } ^ { s }$ , we mean $\| f \| _ { \infty } = \operatorname* { s u p } _ { z \in { \mathcal { Z } } } \| f ( z ) \| _ { 2 }$ . + +# F.2 Background on Eigenspace Invariances + +Let $V = [ v _ { 1 } \quad \ldots \quad v _ { d } ]$ and $W = [ w _ { 1 } \quad \ldots \quad w _ { d } ] \in \mathbb { R } ^ { n \times d }$ be two orthonormal bases for the same $d$ dimensional subspace of $\mathbb { R } ^ { n }$ . Since $V$ and $W$ span the same space, their orthogonal projectors are the same, so $\bar { V V } ^ { \top } = W W ^ { \top }$ . Also, since $V$ and $W$ have orthonormal columns, we have $V ^ { \top } V = W ^ { \top } W = I \in \mathbb { R } ^ { d \times d }$ . Define $Q = V ^ { \top } W$ . Then $Q$ is orthogonal because + +$$ +\begin{array} { r } { Q ^ { \top } Q = W ^ { \top } V V ^ { \top } W = W ^ { \top } W W ^ { \top } W = I } \end{array} +$$ + +956 Moreover, we have that + +$$ +V Q = V V ^ { \top } W = W W ^ { \top } W = W +$$ + +Thus, for any orthonormal bases $V$ and $W$ of the same subspace, there exists an orthogonal $Q \in O ( d )$ such that $V Q = W$ . + +959 For another perspective on this, define the Grassmannian ${ \mathrm { G r } } ( d , n )$ as the smooth manifold consisting +960 of all $d$ dimensional subspaces of $\mathbb { R } ^ { n }$ . Further define the Stiefel manifold $\operatorname { S t } ( d , n )$ as the set +961 of all orthonormal tuples $\begin{array} { r l r } { [ v _ { 1 } } & { { } \ldots } & { v _ { d } ] \in \mathbb { R } ^ { n \times d } } \end{array}$ of $d$ vectors in $\mathbb { R } ^ { n }$ . Letting $O ( d )$ act by right +962 multiplication, it holds that $\mathrm { S t } ( d , n ) / O ( d ) \cong \mathrm { G r } ( d , n )$ . This implies that any $O ( d )$ invariant function +963 on $\operatorname { S t } ( d , n )$ can be viewed as a function on subspaces. See e.g. Gallier and Quaintance [2020] Chapter +964 5 for more information on this. We will use this relationship in our proofs of universal representation. +965 When we consider permutation invariance or equivariance, the permutation acts on dimensions of size +966 $n$ . Then a tensor $\bar { X } \in \mathbb { R } ^ { n ^ { k } \times d }$ is called an order $k$ tensor with respect to this permutation symmetry, +967 where order 0 are called scalars, order 1 tensors are called vectors, and order 2 tensors are called +968 matrices. Note that this does not depend on $d$ ; in this work, we only ever consider vectors and scalars +969 with respect to the $O ( d )$ action. + +# 70 G Proofs of Universality + +We begin by proving the two propositions for the single subspace case from Section 2.1. + +Proposition 1. A continuous function $h : \mathbb { R } ^ { n } \mathbb { R } ^ { s }$ is sign invariant if and only if + +$$ +h ( v ) = \phi ( v ) + \phi ( - v ) +$$ + +for some continuous 973 $\phi : \mathbb { R } ^ { n } \mathbb { R } ^ { s }$ . A continuous $h : \mathbb { R } ^ { n } \mathbb { R } ^ { n }$ is sign invariant and permutation equivariant if and only974 $i f$ (3) holds for a continuous permutation equivariant $\phi : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ . + +975 Proof. If $h ( v ) = \phi ( v ) + \phi ( - v )$ , then $h$ is obviously sign invariant. On the other hand, if $h$ is sign +976 invariant, then letting $\dot { \phi } ( v ) \dot { = } h ( v ) / 2$ gives that $h ( v ) { \dot { = } } { \bar { \phi } } ( v ) + \phi ( - v )$ , and $\phi$ is of course continuous. +977 If $h ( v ) = \phi ( v ) + \phi ( - v )$ for a permutation equivariant $\phi$ , then $h ( - P v ) = \phi ( - P v ) + \phi ( P v ) =$ +978 $P \phi ( - v ) + P \phi ( v ) = P ( \phi ( v ) + \phi ( - v ) ) = P h ( v )$ , so $h$ is permutation equivariant and sign invariant. +979 If $h$ is permutation equivariant and sign invariant, then define $\phi ( v ) = h ( \bar { v } ) / 2$ again; it is clear that $\phi$ +980 is continuous and permutation equivariant. $\boxed { \begin{array} { r l } \end{array} }$ +981 Proposition 2. Any continuous, $O ( d )$ invariant $h : \mathbb { R } ^ { n \times d } \mathbb { R } ^ { s }$ is of the form $h ( V ) = \phi ( V V ^ { \top } )$ for +982 a continuous $\phi$ . For a compact domain ${ \mathcal { Z } } \subseteq \mathbb { R } ^ { n \times d }$ , maps of the form $V \mapsto \operatorname { I G N } ( V V ^ { \top } )$ universally +983 approximate continuous functions $h : \mathcal { Z } \subseteq \mathbb { R } ^ { n \times d } \to \mathbb { R } ^ { n }$ that are $O ( d )$ invariant and permutation +984 equivariant. +85 Proof. The case without permutation equivariance holds by the First Fundamental Theorem of $O ( d )$ +86 (Lemma 2). +987 For the permutation equivariant case, let $\mathcal { Z } ^ { \prime } = \{ V V ^ { \top } : V \in \mathcal { Z } \}$ and let $\epsilon > 0$ . Note that $\mathcal { Z } ^ { \prime }$ +988 is compact, as it is the continuous image of a compact set. Since $h$ is $O ( d )$ invariant, the first +989 fundamental theorem of $O ( d )$ shows that there exists a continuous function $\dot { \phi } : \mathcal { Z } ^ { \prime } \subseteq \mathbb { R } ^ { n \times n } \to \mathbb { R } ^ { n }$ +990 such that $h ( V ) = \phi ( V V ^ { \top } )$ . Since $h$ is permutation equivariant, for any permutation matrix $P$ we +991 have that + +$$ +\begin{array} { c } { h ( P V ) = P \cdot h ( V ) } \\ { \phi ( P V V ^ { \top } P ^ { \top } ) = P \cdot \phi ( V V ^ { \top } ) , } \end{array} +$$ + +992 so $\phi$ is a continuous permutation equivariant function from matrices to vectors. Then note that Keriven +993 and Peyré [2019] show that invariant graph networks (of generally high tensor order in hidden layers) +994 universally approximate continuous permutation equivariant functions from matrices to vectors on +995 compact sets of matrices. Thus, an IGN can $\epsilon$ -approximate $\phi$ , and hence $V \mapsto \operatorname { I G N } ( V V ^ { \top } )$ can +996 $\epsilon$ -approximate $h$ . □ + +# 997 G.1 Proof of Decomposition Theorem + +$$ +\begin{array} { c } { { \phi = \psi \circ \pi , \qquad \pi = \pi _ { 1 } \times . . . \times \frac { \chi _ { k } } { \chi } } } \\ { { \pi = \pi _ { 1 } \times . . . \pi \displaystyle \downarrow } } \\ { { \psi ^ { - 1 } \qquad \longleftrightarrow \quad \left( \overline { { { X _ { 1 } / G _ { 1 } } } } \right) \times . . . \times \left( \overline { { { X _ { k } / G _ { k } } } } \right) \displaystyle \mathop { \longrightarrow } \mathbb { R } ^ { s } } } \\ { { \mathcal { Z } = \mathrm { i m } ( \psi ) \subseteq \mathbb { R } ^ { a } \longleftrightarrow \psi _ { 1 } \times . . . \times \psi _ { k } } } \\ { { \psi \qquad \quad = \pi _ { 1 } \times . . . \pi \displaystyle - \frac { \Gamma } { \rho \rho \psi ^ { - 1 } } } } \end{array} +$$ + +Figure 10: Commutative diagram for our proof of Theorem 3. Black arrows denote functions from topological constructions, and red dashed lines denote functions that we parameterize by neural networks $( \phi = \phi _ { 1 } \times \ldots \times \phi _ { k }$ and $\rho \mathrm { \hbar }$ ). + +998 Here, we give the formal statement of Theorem 3, which provides the necessary topological assump +999 tions for the theorem to hold. In particular, we only require the $G _ { i }$ be a topological group that acts +1000 continuously on $\mathcal { X } _ { i }$ for each $i$ , and that there exists a topological embedding of each quotient space +1001 into some Euclidean space. That the group action is continuous is a very mild assumption, and it +1002 holds for any finite or compact matrix group, which all of the invariances we consider in this paper +1003 can be represented as. +1004 A topological embedding of the quotient space into a Euclidean space is desired, as we know how to +1005 parameterize neural networks with Euclidean outputs and inputs, whereas dealing with a quotient +1006 space is generally difficult. Many different conditions can guarantee existence of such an embedding. +1007 For instance, if the quotient space is a smooth manifold, then the Whitney Embedding Theorem +1008 (Lemma 5) guarantees such an embedding. Also, if the base space $\mathcal { X } _ { i }$ is a Euclidean space and $G _ { i }$ is +1009 a finite or compact matrix Lie group, then a map built from $G$ -invariant polynomials gives such an +1010 embedding (González and de Salas [2003] Lemma 11.13). + +Figure 10 provides a commutative diagram representing the constructions in our proof. + +1012 Theorem 3 (Decomposition Theorem). Let $\mathcal { X } _ { 1 } , \ldots , \mathcal { X } _ { k }$ be topological spaces, and let $G _ { i }$ be $a$ +1013 topological group acting continuously on $\mathcal { X } _ { i }$ for each i. Assume that there is a topological embedding +1014 $\psi _ { i } : \mathcal { X } _ { i } / G _ { i } \to \mathbb { R } ^ { a _ { i } }$ of each quotient space into a Euclidean space $\mathbb { R } ^ { a _ { i } }$ for some dimension $a _ { i }$ . +1015 Then, for any continuous function $f : \mathcal { X } = \mathcal { X } _ { 1 } \times . . . \times \mathcal { X } _ { k } \to \mathbb { R } ^ { s }$ that is invariant to the action of +1016 $G = G _ { 1 } \times \ldots \times G _ { k }$ , there exists continuous functions $\phi _ { i } : \mathcal { X } _ { i } \mathbb { R } ^ { a _ { i } }$ and a continuous function +1017 $\rho : \mathcal { Z } \subseteq \mathbb { R } ^ { a } \to \mathbb { R } ^ { s }$ , where $a = \textstyle \sum _ { i } a _ { i }$ such that + +$$ +f ( v _ { 1 } , \dots , v _ { k } ) = \rho ( \phi _ { 1 } ( v _ { 1 } ) , \dots , \phi _ { k } ( v _ { k } ) ) . +$$ + +Furthermore:1018 $( l )$ each $\phi _ { i }$ can be taken to be invariant to $G _ { i }$ , (2) the domain $\mathcal { Z }$ is compact if each $\mathcal { X } _ { i }$ 1019 is compact, (3) if ${ \mathcal { X } } _ { i } = { \mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , then $\phi _ { i }$ can be taken to be equal to $\phi _ { j }$ . + +Proof. Let 1020 $\pi _ { i } : \mathcal { X } _ { i } \mathcal { X } _ { i } / G _ { i }$ denote the quotient map for $\mathcal { X } _ { i } / G _ { i }$ . Since each $G _ { i }$ acts continuously, 1021 Lemma 3 gives that the quotient of the product space is the product of the quotient spaces, i.e. that + +$$ +( { \mathcal { X } } _ { 1 } \times \ldots \times { \mathcal { X } } _ { k } ) / ( G _ { 1 } \times \ldots G _ { k } ) \cong ( { \mathcal { X } } _ { 1 } / G _ { 1 } ) \times \ldots \times ( { \mathcal { X } } _ { k } / G _ { k } ) , +$$ + +1022 and the corresponding quotient map $\pi : { \mathcal { X } } / G$ is given by + +$$ +\pi = \pi _ { 1 } \times \ldots \times \pi _ { k } , \qquad \pi ( x _ { 1 } , \ldots , x _ { k } ) = ( \pi _ { 1 } ( x _ { 1 } ) , \ldots , \pi _ { k } ( x _ { k } ) ) . +$$ + +1023 By passing to the quotient (Lemma 1), there exists a continuous ${ \tilde { f } } : { \mathcal { X } } / G \to \mathbb { R } ^ { s }$ on the quotient space such that 1024 $f = \tilde { f } \circ \pi$ . By Lemma 4, each $\mathcal { X } _ { i } / G _ { i }$ is compact if $\mathcal { X } _ { i }$ is compact. Defining the 1025 image $\mathcal { Z } _ { i } = \psi _ { i } ( \mathcal { X } _ { i } / G _ { i } ) \subseteq \mathbb { R } ^ { a _ { i } }$ , we thus know that $\mathcal { Z } _ { i }$ is compact if $\mathcal { X } _ { i }$ is compact. + +Moreover, as 26 $\psi _ { i }$ is a topological embedding, it has a continuous inverse $\psi _ { i } ^ { - 1 }$ on its image $\mathcal { Z } _ { i }$ . Further, 27 we have a topological embedding $\psi : \mathcal { X } / G \to \mathcal { Z } = \mathcal { Z } _ { 1 } \times . . . \times \mathcal { Z } _ { k }$ given by $\psi = \psi _ { 1 } \times \ldots \times \psi _ { k }$ , with continuous inverse 28 $\psi ^ { - 1 } = \psi _ { 1 } ^ { - 1 } \times \ldots \times \psi _ { k } ^ { - 1 }$ . + +Note that + +$$ +f = \tilde { f } \circ \pi = ( \tilde { f } \circ \psi ^ { - 1 } ) \circ ( \psi \circ \pi ) . +$$ + +1030 So we define + +$$ +\begin{array} { r l r } & { \rho = \tilde { f } \circ \psi ^ { - 1 } } & { \rho : \mathcal { Z } \to \mathbb { R } ^ { s } } \\ & { \phi _ { i } = \psi _ { i } \circ \pi _ { i } } & { \phi _ { i } : \mathcal { X } _ { i } \to \mathcal { Z } _ { i } } \\ & { \phi = \psi \circ \pi = \phi _ { 1 } \times \ldots \times \phi _ { k } } & { \phi : \mathcal { X } \to \mathcal { Z } } \end{array} +$$ + +1031 Thus, $f = \rho \circ \phi = \rho \circ ( \phi _ { 1 } \times \ldots \times \phi _ { k } )$ , so equation (9) holds. Moreover, the $\rho$ and $\phi _ { i }$ are continuous, +1032 as they are compositions of continuous functions. Furthermore, (1) holds as each $\phi _ { i }$ is invariant +1033 to $G _ { i }$ because each $\pi _ { i }$ is invariant to $G _ { i }$ . Since each $\mathcal { Z } _ { i }$ is compact if $\mathcal { X } _ { i }$ is compact, the product +1034 $\mathcal { Z } = \mathcal { Z } _ { 1 } \times \ldots \times \mathcal { Z } _ { k }$ is compact if each $\mathcal { X } _ { i }$ is compact, thus proving (2). + +To show the last statement (3), note simply that if ${ \mathcal { X } } _ { i } = { \mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , then the quotient maps are equal, i.e. $\pi _ { i } = \pi _ { j }$ . Moreover, we can choose the embeddings to be equal, so say $\psi _ { i } = \psi _ { j }$ . Then, $\phi _ { i } = \psi _ { i } \circ \pi _ { i } = \bar { \psi _ { j } } \circ \pi _ { j } = \phi _ { j }$ , so we are done. □ + +# G.2 Universality of SignNet and BasisNet + +039 Here, we prove Corollary 1 on the universal representation and approximation capabilities of our +040 Unconstrained-SignNets, Unconstrained-BasisNets, and Expressive-BasisNets. We proceed in sev +041 eral steps, first proving universal representation of continuous functions when we do not require +042 permutation equivariance, then proving universal approximation when we do require permutation +043 equivariance. + +# G.2.1 Sign Invariant Universal Representation + +Recall that $\mathbb { S } ^ { n - 1 }$ denotes the unit sphere in $\mathbb { R } ^ { n }$ . As we normalize eigenvectors to unit norm, the domain of our functions on $k$ eigenvectors are on the compact space $( \bar { \mathbb { S } } ^ { n - 1 } ) ^ { k }$ . + +Corollary 2 (Universal Representation for SignNet). $A$ continuous function $f : ( \mathbb { S } ^ { n - 1 } ) ^ { k } \to \mathbb { R } ^ { s }$ is sign invariant, i.e. $f ( s _ { 1 } v _ { 1 } , \ldots , s _ { k } v _ { k } ) = f ( { \bar { v _ { 1 } } } , \ldots , v _ { k } )$ for any $s _ { i } \in \{ - 1 , 1 \}$ , if and only if there exists a continuous $\phi : \mathbb { R } ^ { n } \to \mathbb { R } ^ { 2 n - 2 }$ and a continuous $\rho : \mathbb { R } ^ { ( 2 n - 2 ) k } \mathbb { R } ^ { s }$ such that + +$$ +f ( v _ { 1 } , \dots , v _ { k } ) = \rho \left( [ \phi ( v _ { i } ) + \phi ( - v _ { i } ) ] _ { i = 1 } ^ { k } \right) . +$$ + +1050 Proof. It can be directly seen that any $f$ of the above form is sign invariant. + +Thus, we show that any sign invariant $f$ can be expressed in the above form. First, we show that we can apply the general Theorem 3. The group $\bar { G _ { i } } = \{ 1 , - 1 \}$ acts continuously and satisfies that $\mathbb { S } ^ { n - 1 } / \{ 1 , - \bar { 1 } \} = \mathbf { \bar { \mathbb { R } } } \mathbb { P } ^ { n - 1 }$ , where $\mathbb { R } \mathbb { P } ^ { n - 1 }$ is the real projective space of dimension $n - 1$ . Since $\mathbb { R } \mathbb { P } ^ { n - 1 }$ is a smooth manifold of dimension $n - 1$ , Whitney’s embedding theorem states that there exists a (smooth) topological embedding $\psi _ { i } : \mathbb { R P } ^ { n - 1 } \to \mathbb { R } ^ { \bar { 2 } n - 2 }$ (Lemma 5). + +Thus, we can apply the general theorem to see that $f = \rho \circ { \tilde { \phi } } ^ { k }$ for some continuous $\rho$ and $\tilde { \phi } ^ { k }$ . Note that each $\tilde { \phi } _ { i } = \bar { \tilde { \phi } }$ is the same, as each $\mathcal { X } _ { i } = \mathbb { S } ^ { n - 1 }$ and $G _ { i } = \{ 1 , - 1 \}$ is the same. Also, Theorem 3 says that we may assume that $\tilde { \phi }$ is sign invariant, so $\tilde { \phi } ( x ) = \tilde { \phi } ( - x )$ . Letting $\phi ( { x } ) = \tilde { \phi } ( { x } ) / 2$ , we are done with the proof. □ + +# G.2.2 Sign Invariant Universal Representation with Extra Features + +Recall that we may want our sign invariant functions to process other data besides eigenvectors, such as eigenvalues or node features associated to a graph. Here, we show universal representation for when we have this other data that does not possess sign symmetry. The proof is a simple extension of Corollary 2, but we provide the technical details for completeness. + +Corollary 3 (Universal Representation for SignNet with features). For a compact space of features $\Omega \subseteq \mathbb { R } ^ { d }$ , let $f ( v _ { 1 } , \ldots , v _ { k } , x _ { 1 } , \ldots , x _ { k } )$ be a continuous function $f : ( \mathbb { S } ^ { n - 1 } \times \bar { \Omega } ) ^ { k } \overset { \cdot } { } \mathbb { R } ^ { s }$ . + +1067 Then $f$ is sign invariant for the inputs on the sphere, i.e. + +$$ +f ( s _ { 1 } v _ { 1 } , \ldots , s _ { k } v _ { k } , x _ { 1 } , \ldots , x _ { k } ) = f ( v _ { 1 } , \ldots , v _ { k } , x _ { 1 } , \ldots , x _ { k } ) \qquad s _ { i } \in \{ 1 , - 1 \} , +$$ + +if and only if there exists a continuous 1068 $\psi : \mathbb { R } ^ { n + d } \mathbb { R } ^ { 2 n - 2 + d }$ and a continuous $\rho : \mathbb { R } ^ { ( 2 n - 2 + d ) k } \mathbb { R } ^ { s }$ 1069 such that + +$$ +f ( v _ { 1 } , \ldots , v _ { k } ) = \rho \left( \phi ( v _ { 1 } , x _ { 1 } ) + \phi ( - v _ { 1 } , x _ { 1 } ) , \ldots , \phi ( v _ { k } , x _ { k } ) + \phi ( - v _ { k } , x _ { k } ) \right) . +$$ + +1070 Proof. Once again, the sign invariance of any $f$ in the above form is clear. + +1071 We follow very similar steps to the proof of Corollary 2 to show that we may apply Theorem 3. We +1072 can view $\Omega$ as a quotient space, after quotienting by the trivial group that does nothing, $\Omega \cong \Omega / \{ 1 \}$ . +1073 The corresponding quotient map is $\mathrm { i d } _ { \Omega }$ , the identity map. Also, $\Omega$ trivially topologically embeds in +1074 $\mathbb { R } ^ { d }$ by the inclusion map. + +1075 As $G _ { i } = \{ - 1 , 1 \} \times \{ 1 \}$ acts continuously, by Lemma 3 we have that + +$$ +( \mathbb { S } ^ { n - 1 } \times \Omega ) / ( \{ 1 , - 1 \} \times \{ 1 \} ) \cong ( \mathbb { S } ^ { n - 1 } / \{ 1 , - 1 \} ) \times ( \Omega / \{ 1 \} ) \cong \mathbb { R } \mathbb { P } ^ { n - 1 } \times \Omega , +$$ + +with corresponding quotient map 1076 $\pi \times \mathrm { i d } _ { \Omega }$ , where $\pi$ is the quotient map to $\mathbb { R } \mathbb { P } ^ { n - 1 }$ . + +Letting 1077 $\tilde { \psi }$ be the embedding of $\mathbb { R } \mathbb { P } ^ { n - 1 } \to \mathbb { R } ^ { 2 n - 2 }$ guaranteed by Whitney’s embedding theorem 1078 (Lemma 5), we have that $\psi \overset { \cdot } { = } \tilde { \psi } \times \mathrm { i d } _ { \Omega }$ is an embedding of $\mathbb { R } \mathbb { P } ^ { n - 1 } \times \Omega \to \mathbb { R } ^ { 2 n - 2 + d }$ . Thus, we can apply Theorem 3 to write 1079 $f = \rho \circ { \tilde { \phi } } ^ { k }$ for $\tilde { \phi } = ( \tilde { \psi } \times \mathrm { i d } _ { \Omega } \mathbf { \bar { ) } } \circ ( \pi \times \mathrm { i d } _ { \Omega } )$ , so + +$$ +\tilde { \phi } ( v _ { i } , x _ { i } ) = ( \tilde { \psi } ( v _ { i } ) , x _ { i } ) , +$$ + +where 1080 $\tilde { \phi } ( v _ { i } , x _ { i } ) = \tilde { \phi } ( - v _ { i } , x _ { i } )$ . Letting $\phi ( v _ { i } , x _ { i } ) = \tilde { \phi } ( v _ { i } , x _ { i } ) / 2$ , we are done. + +# G.2.3 Basis Invariant Universal Representation + +Recall that $\operatorname { S t } ( d , n )$ is the Stiefel manifold of $d$ -tuples of vectors $( v _ { 1 } , \ldots , v _ { d } )$ where $\ b { v } _ { i } \in \mathbb { R } ^ { n }$ and $v _ { 1 } , \ldots , v _ { d }$ are orthonormal. This is where our inputs lie, as our eigenvectors are unit norm and orthogonal. We will also make use of the Grassmannian ${ \mathrm { G r } } ( d , n )$ , which consists of all $d$ -dimensional subspaces in $\mathbb { R } ^ { n }$ . This is because the Grassmannian is the quotient space for the group action we want, $\operatorname { G r } ( d , n ) \cong \operatorname { S t } ( d , n ) / O ( d )$ , where $Q \in O ( d )$ acts on $\bar { V } \in \mathrm { S t } ( d , \bar { n } ) \subseteq \mathbb { R } ^ { n \times d }$ by mapping $V$ to $V Q$ [Gallier and Quaintance, 2020]. + +Corollary 4 (Universal Representation for BasisNet). For dimensions $d _ { 1 } , \dotsc , d _ { l } \leq n$ let $f$ be $a$ continuous function on $\mathrm { S t } ( \bar { d } _ { 1 } , n ) \times \ldots \times \mathrm { S t } ( d _ { l } , n )$ . Further assume that $f$ is invariant to $O ( d _ { 1 } ) \times$ . $\dots \times O ( d _ { l } )$ , where $O ( d _ { i } )$ acts on $\operatorname { S t } ( d _ { i } , n )$ by multiplication on the right. + +Then there exist continuous $\rho : \mathbb { R } ^ { \sum _ { i = 1 } ^ { l } 2 d _ { i } ( n - d _ { i } ) } \mathbb { R } ^ { s }$ and continuous $\phi _ { i } : \mathrm { S t } ( d _ { i } , n ) \to \mathbb { R } ^ { 2 d _ { i } ( n - d _ { i } ) }$ such that + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = \rho \left( \phi _ { 1 } ( V _ { 1 } ) , \dots , \phi _ { l } ( V _ { l } ) \right) , +$$ + +where the $\phi _ { i }$ are $O ( d _ { i } )$ invariant functions, and we can take $\phi _ { i } = \phi _ { j }$ if $d _ { i } = d _ { j }$ . + +Proof. Letting $\mathcal { X } _ { i } = \mathrm { S t } ( d _ { i } , n )$ and $G _ { i } = O ( d _ { i } )$ , it can be seen that $G _ { i }$ acts continuously on $\mathcal { X } _ { i }$ . Also, we have that the quotient space $\mathrm { S t } ( d _ { i } , n ) / O ( d _ { i } ) = \mathrm { G r } ( d _ { i } , n )$ is the Grassmannian of $d _ { i }$ dimensional subspaces in $\mathbb { R } ^ { n }$ , which is a smooth manifold of dimension $d _ { i } ( n - d _ { i } )$ . Thus, the Whitney embedding theorem (Lemma 5) gives a topological embedding $\psi _ { i } : { \mathrm { G r } } ( d _ { i } , n ) \to \mathbb { R } ^ { 2 d _ { i } ( n - d _ { i } ) }$ . + +Hence, we may apply Theorem 3 to obtain continuous $O ( d _ { i } )$ invariant $\phi _ { i } : \mathrm { S t } ( d _ { i } , n ) \to \mathbb { R } ^ { 2 d _ { i } ( n - d _ { i } ) }$ and continuous $\rho : \mathbb { R } ^ { \sum _ { i = 1 } ^ { l } 2 d _ { i } ( n - d _ { i } ) } \mathbb { R } ^ { s }$ , such that $f = \rho \circ ( \phi _ { 1 } \times \ldots \times \phi _ { l } )$ . Also, if $d _ { i } = d _ { j }$ , then ${ \mathcal { X } } _ { i } = { \mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , so we can take $\phi _ { i } = \phi _ { j }$ . + +# G.2.4 Basis Invariant and Permutation Equivariant Universal Approximation + +With the restriction that $f ( V _ { 1 } , \dots , V _ { l } ) : \mathbb { R } ^ { n \times \sum _ { i } d _ { i } } \to \mathbb { R } ^ { n }$ be permutation equivariant and basis invariant, we need to use the impractically expensive Expressive-BasisNet to approximate $f$ . Universality of permutation invariant or equivariant functions from matrices to scalars or matrices to vectors is difficult to achieve in a computationally tractable manner [Maron et al., 2019, Keriven and Peyré, 2019, Maehara and NT, 2019]. One intuitive reason to expect this is that universally approximating such functions allows solution of the graph isomorphism problem [Chen et al., 2019b], which is a computationally difficult problem. While we have exact representation of basis invariant functions by continuous $\rho$ and $\phi _ { i }$ when there is no permutation equivariance constraint, we can only achieve approximation up to an arbitrary $\epsilon > 0$ when we require permutation equivariance. + +1112 Corollary 5 (Universal Approximation for Expressive-BasisNets). Let $f ( V _ { 1 } , \dots , V _ { l } ) : \mathrm { S t } ( d _ { 1 } , n ) \times$ +1113 $\dots \times \operatorname { S t } ( d _ { l } , n ) \to \mathbb { R } ^ { n }$ be continuous, $O ( d _ { 1 } ) \times \ldots \times O ( d _ { l } )$ invariant, and permutation equivariant. +1114 Then $f$ can be ϵ-approximated by an Expressive-BasisNet. + +1115 Proof. By invariance, Corollary 4 of the decomposition theorem shows that $f$ can be written as + +$$ +f ( V _ { 1 } , \dots , V _ { l } ) = \rho \left( \varphi _ { d _ { 1 } } ( V _ { 1 } ) , \dots , \varphi _ { d _ { l } } ( V _ { l } ) \right) +$$ + +for some continuous 1116 $O ( d _ { i } )$ invariant $\varphi _ { d _ { i } }$ and continuous $\rho$ . By the first fundamental theorem of $O ( d )$ (Lemma 2), each 1117 $\varphi _ { d _ { i } }$ can be written as $\varphi _ { d _ { i } } ( V _ { i } ) = \phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \top } )$ for some continuous $\phi _ { d _ { i } }$ . Let + +$$ +{ \mathcal { Z } } = \{ ( V _ { 1 } V _ { 1 } ^ { \top } , \ldots , V _ { l } V _ { l } ^ { \top } ) : V _ { i } \in { \mathrm { S t } } ( d _ { i } , n ) \} \subseteq \mathbb { R } ^ { n ^ { 2 } \times l } , +$$ + +which is compact as it is the image of the compact space 18 $\mathrm { S t } ( d _ { 1 } , n ) \times \ldots \times \mathrm { S t } ( d _ { l } , n )$ under a continuous function. Define 119 $h : \mathcal { Z } \subseteq \mathbb { R } ^ { n ^ { 2 } \times l } \to \mathbb { R } ^ { n }$ by + +$$ +\begin{array} { r } { h ( V _ { 1 } V _ { 1 } ^ { \top } , \ldots , V _ { l } V _ { l } ^ { \top } ) = \rho \left( \phi _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \top } ) , \ldots , \phi _ { d _ { l } } ( V _ { l } V _ { l } ^ { \top } ) \right) . } \end{array} +$$ + +1120 Then note that $h$ is continuous and permutation equivariant from matrices to vectors, so it can be +1121 $\epsilon$ -approximated by an invariant graph network [Keriven and Peyré, 2019], call it $\widetilde { \mathrm { I G N } }$ . If we define +1122 $\tilde { \rho } = \widetilde { \mathrm { I G N } }$ and $\mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \top } ) = V _ { i } V _ { i } ^ { \top }$ (this identity operation is linear and permutation equivariant, +1123 so it can be exactly expressed by an IGN), then we have $\epsilon$ -approximation of $f$ by + +$$ +\widetilde { \mathrm { I G N } } ( V _ { 1 } V _ { 1 } ^ { \top } , \dots , V _ { l } V _ { l } ^ { \top } ) = \widetilde { \rho } \left( \mathrm { I G N } _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \top } ) , \dots , \mathrm { I G N } _ { d _ { l } } ( V _ { l } V _ { l } ^ { \top } ) \right) . +$$ + +1126 Theorem 4. Consider the same setup as Theorem 3, where $\mathcal { X } _ { i }$ are also compact. Let $\Phi _ { i }$ be a +1127 family of $G _ { i }$ -invariant functions that universally approximate $G _ { i }$ -invariant continuous functions +1128 $\mathcal { X } _ { i } \ \to \ \mathbb { R } ^ { a _ { i } }$ , and let $\mathcal { R }$ be a set of continuous function that universally approximate continuous +1129 functions $\mathcal { Z } \subseteq \mathbb { R } ^ { a } \to \mathbb { R } ^ { s }$ for every compact $\mathcal { Z }$ , where $a = \textstyle \sum _ { i } a _ { i }$ . Then for any $\varepsilon > 0$ and any +1130 $G$ -invariant continuous function $f : \mathcal { X } _ { 1 } \times . . . \times \mathcal { X } _ { k } \to \mathbb { R } ^ { s }$ there exists $\phi \in \Phi$ and $\rho \in \mathcal R$ such that +1131 $\| f - \rho ( \phi _ { 1 } , \ldots , \phi _ { k } ) \| _ { \infty } < \varepsilon$ . + +Proof. Consider a particular 1132 $G$ -invariant continuous function $f : \mathcal { X } _ { 1 } \times . . . \times \mathcal { X } _ { k } \to \mathbb { R } ^ { s }$ . By Theorem 3 there exists 1133 $G _ { i }$ -invariant continuous functions $\phi _ { i } ^ { \prime } : \mathcal { X } _ { i } \mathbb { R } ^ { a _ { i } }$ and a continuous function 1134 $\rho ^ { \prime } : \mathcal { Z } \subseteq \mathbb { R } ^ { a } \to \mathbb { R } ^ { s }$ (where $a = \textstyle \sum _ { i } a _ { i } )$ such that + +$$ +f ( v _ { 1 } , \dots , v _ { k } ) = \rho ^ { \prime } ( \phi _ { 1 } ^ { \prime } ( v _ { 1 } ) , \dots , \phi _ { k } ^ { \prime } ( v _ { k } ) ) . +$$ + +1135 Now fix an $\varepsilon > 0$ . For any $\rho \in \mathcal R$ and any $\phi _ { i } \in \Phi _ { i } ( i = 1 , \dots k )$ we may bound the difference from +1136 $f$ as follows (suppressing the $v _ { i }$ ’s for brevity), + +$$ +\begin{array} { r l } & { \| f - \rho ( \phi _ { 1 } , \ldots , \phi _ { k } ) \| _ { \infty } } \\ & { = \| \rho ^ { \prime } ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } , \ldots , \phi _ { k } ) \| _ { \infty } } \\ & { = \| \rho ^ { \prime } ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) + \rho ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } , \ldots , \phi _ { k } ) \| _ { \infty } } \\ & { \leq \| \rho ^ { \prime } ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) \| _ { \infty } + \| \rho ( \phi _ { 1 } ^ { \prime } , \ldots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } , \ldots , \phi _ { k } ) \| _ { \infty } } \\ & { = \mathrm { I } + \mathrm { I I } } \end{array} +$$ + +1137 Now let $\begin{array} { r } { K ^ { \prime } = \prod _ { i = 1 } ^ { k } \mathrm { i m } \phi _ { i } ^ { \prime } } \end{array}$ . Since each $\phi _ { i } ^ { \prime }$ is continuous and defined on a compact set $\mathcal { X } _ { i }$ we know +1138 that $\mathrm { i m } \phi _ { i } ^ { \prime }$ is compact, and so the product $K$ is also compact. Since $K ^ { \prime }$ is compact, it is contained in a +1139 closed ball $B ( r )$ of radius $r > 0$ centered at the origin. Let $K$ be the closed ball $\boldsymbol { B } ( \boldsymbol { r } + 1 )$ of radius +1140 $r + 1$ centered at the origin, so $K$ contains $K ^ { \prime }$ and a ball of radius 1 around each point of $K ^ { \prime }$ . We +1141 may extend $\rho ^ { \prime }$ continuously to $K$ as needed, so assume $\rho ^ { \prime } : K \to \mathbb { R } ^ { s }$ . By universality of $\mathcal { R }$ we may +1142 pick a particular $\rho : K \mathbb { R } ^ { s }$ , $\rho \in \mathcal R$ such that + +$$ +\mathrm { I } = \operatorname* { s u p } _ { \{ v _ { i } \in \mathcal { X } _ { i } \} _ { i = 1 } ^ { k } } \| \rho ^ { \prime } ( \phi _ { 1 } ^ { \prime } , \dots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } ^ { \prime } , \dots , \phi _ { k } ^ { \prime } ) \| _ { \infty } \leq \operatorname* { s u p } _ { z \in K } \| \rho ^ { \prime } ( z ) - \rho ( z ) \| _ { 2 } < \varepsilon / 2 . +$$ + +1143 Keeping this choice of $\rho$ , it remains only to bound II. As $\rho$ is continuous on a compact domain, it +1144 is in fact uniformly continuous. Thus, we can choose a $\delta ^ { \prime } > 0$ such that if $\| y - \bar { z } \| _ { 2 } \leq \delta ^ { \prime }$ , then +1145 $\| \rho ( y ) - \rho ( z ) \| _ { \infty } < \epsilon / 2$ , and then we define $\delta = \operatorname* { m i n } ( \delta ^ { \prime } , 1 )$ . +1146 Since $\Phi _ { i }$ universally approximates $\phi _ { i } ^ { \prime }$ we may pick $\phi _ { i } \in \Phi _ { i }$ such that $\| \phi _ { i } - \phi _ { i } ^ { \prime } \| _ { \infty } < \delta / \sqrt { k }$ , and +1147 thus $\| ( \phi _ { 1 } , \dots , \phi _ { k } ) - ( \phi _ { 1 } ^ { \prime } , \dots \phi _ { k } ^ { \prime } ) \| _ { \infty } \leq \delta$ . With this choice of $\phi _ { i }$ , we know that $\textstyle \prod _ { i = 1 } ^ { k } \operatorname { i m } \phi _ { i } \subseteq K$ +1148 (because each $\phi _ { i } ( x _ { i } )$ is within distance 1 of $\phi _ { i } ^ { \prime } ( x _ { i } ) )$ . Thus, $\rho ( \phi _ { 1 } ( x _ { 1 } ) , \ldots , \phi _ { k } ( x _ { k } ) { \bar { ) } }$ is well-defined, +1149 and we have + +$$ +\begin{array} { r l } & { \mathrm { I I } = \| \rho ( \phi _ { 1 } ^ { \prime } , \dots , \phi _ { k } ^ { \prime } ) - \rho ( \phi _ { 1 } , \dots , \phi _ { k } ) \| _ { \infty } } \\ & { \quad = \underset { \{ x _ { i } \in \mathcal { X } _ { i } \} _ { i = 1 } ^ { k } } { \operatorname* { s u p } } \| \rho ( \phi _ { 1 } ^ { \prime } ( x _ { 1 } ) , \dots , \phi _ { k } ^ { \prime } ( x _ { k } ) ) - \rho ( \phi _ { 1 } ( x _ { 1 } ) , \dots , \phi _ { k } ( x _ { k } ) ) \| _ { 2 } } \\ & { \quad < \varepsilon / 2 } \end{array} +$$ + +1150 due to our choice of $\delta$ , which completes the proof. + +# 1151 H Basis Invariance for Graph Representation Learning + +# 152 H.1 Spectral Graph Convolution + +In this section, we consider spectral graph convolutions, which for node features $\ b X \in \mathbb { R } ^ { n \times q }$ take the form $\begin{array} { r } { f ( V , \Lambda , X ) = \sum _ { i = 1 } ^ { n } \dot { \theta _ { i } v _ { i } } v _ { i } ^ { \top } X } \end{array}$ for some parameters $\theta _ { i }$ . We can optionally take $\theta _ { i } = h ( \lambda _ { i } )$ for some continuous function $h : \mathbb { R } \mathbb { R }$ of the eigenvalues. This form captures most popular spectral graph convolutions in the literature [Bruna et al., 2014, Hamilton, 2020, Bronstein et al., 2017]; often, such convolutions are parameterized by taking $h$ to be some analytic function such as a simple affine function [Kipf and Welling, 2017], a linear combination in a polynomial basis [Defferrard et al., + +59 2016, Chien et al., 2021], or a parameterization of rational functions [Levie et al., 2018, Bianchi et al., +60 2021]. +161 First, it is well known and easy to see that spectral graph convolutions are permutation equivariant, as +162 for a permutation matrix $P$ we have + +$$ +f ( P V , \Lambda , P X ) = \sum _ { i } \theta _ { i } P v _ { i } v _ { i } ^ { \top } P ^ { \top } P X = \sum _ { i } \theta _ { i } P v _ { i } v _ { i } ^ { \top } X = P f ( V , \Lambda , X ) . +$$ + +1163 Also, it is easy to see that they are sign invariant, as $( - v _ { i } ) ( - v _ { i } ) ^ { \top } = v _ { i } v _ { i } ^ { \top }$ . However, if the $\theta _ { i }$ do not +1164 depend on the eigenvalues, then the spectral graph convolution is not necessarily basis invariant. For +1165 instance, if $v _ { 1 }$ and $v _ { 2 }$ are in the same eigenspace, and we change basis by permuting $v _ { 1 } ^ { \prime } = v _ { 2 }$ and +1166 $v _ { 2 } ^ { \prime } = v _ { 1 }$ , then if $\theta _ { 1 } \neq \theta _ { 2 }$ the spectral graph convolution will generally change as well. +1167 On the other hand, if $\theta _ { i } = h ( \lambda _ { i } )$ for some function $h : \mathbb { R } \mathbb { R }$ , then the spectral graph convolution +1168 is basis invariant. This is because if $v _ { i }$ and $v _ { j }$ belong to the same eigenspace, then $\lambda _ { i } = \lambda _ { j }$ so +1169 $h ( \lambda _ { i } ) = h ( \lambda _ { j } )$ . Thus, if $v _ { i _ { 1 } } , \ldots , v _ { i _ { d } }$ are eigenvectors of the same eigenspace with eigenvalue $\lambda$ , +1170 we have that $\begin{array} { r } { \sum _ { l = 1 } ^ { d } \underline { h } ( \lambda _ { i _ { l } } ) v _ { i _ { l } } v _ { i _ { l } } ^ { \top } = h ( \lambda ) \sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \top } . } \end{array}$ . Now, note that $\scriptstyle \sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \top }$ is the orthogonal +1171 projector onto the eigenspace [Trefethen and Bau III, 1997]. A change of basis does not change this +1172 orthogonal projector, so such spectral graph convolutions are basis invariant. +1173 Another way to see this basis invariance is with a simple computation. Let $V _ { 1 } , \dots , V _ { l }$ be the +1174 eigenspaces of dimension $d _ { 1 } , \ldots , d _ { l }$ , where $V _ { i } \in \mathbb { R } ^ { n \times d _ { i } ^ { \star } }$ . Let the corresponding eigenvalues be +1175 $\mu _ { 1 } , \ldots , \mu _ { l }$ . Then for any orthogonal matrices $Q _ { i } \in O ( d _ { i } )$ , we have + +$$ +\begin{array} { l } { { \displaystyle \sum _ { i = 1 } ^ { n } h ( \lambda _ { i } ) v _ { i } v _ { i } ^ { \top } = \sum _ { j = 1 } ^ { l } V _ { j } h ( \mu _ { j } ) I _ { d _ { j } } V _ { j } ^ { \top } } } \\ { ~ } \\ { { \displaystyle = \sum _ { j = 1 } ^ { l } V _ { j } h ( \mu _ { j } ) I _ { d _ { j } } Q _ { j } Q _ { j } ^ { \top } V _ { j } ^ { \top } } } \\ { { \displaystyle ~ = \sum _ { j = 1 } ^ { l } ( V _ { j } Q _ { j } ) h ( \mu _ { j } ) I _ { d _ { j } } ( V _ { j } Q _ { j } ) ^ { \top } } , } \end{array} +$$ + +1176 so the spectral graph convolution is invariant to substituting $V _ { j } Q _ { j }$ for $V _ { j }$ . + +77 Now, we give the proof that shows SignNet and BasisNet can universally approximate spectral graph +78 convolutions. +1179 Theorem 1 (Learning Spectral Graph Convolutions). Suppose the node features $X \in \mathbb { R } ^ { n \times q }$ take +1180 values in compact sets. Then SignNet can universally approximate any spectral graph convolution, +1181 and both BasisNet and Expressive-BasisNet can universally approximate any parametric spectral +1182 graph convolution. +1183 Proof. Note that eigenvectors and eigenvalues of normalized Laplacian matrices take values in +1184 compact sets, since the eigenvalues are in [0, 2] and we take eigenvectors to have unit-norm. Thus, +1185 the whole domain of the spectral graph convolution is compact. +1186 Let $\varepsilon > 0$ . First, consider a spectral graph convolution $\begin{array} { r } { f ( V , \Lambda , X ) = \sum _ { i = 1 } ^ { n } \theta _ { i } v _ { i } v _ { i } ^ { \top } X } \end{array}$ . For SignNet, +1187 let $\phi ( v _ { i } , \lambda _ { i } , X )$ approximate the function $\tilde { \phi } ( v _ { i } , \lambda _ { i } , X ) = \theta _ { i } v _ { i } v _ { i } ^ { \top } X$ to within $\varepsilon / n$ error, which +1188 DeepSets can do since this is a continuous permutation equivariant function from vectors to vectors +1189 1190 [Segol and Lipman, 2019] (note1 is the all ones vector). Then $\rho = \textstyle \sum _ { i = 1 } ^ { n }$ pass is a $\lambda _ { i }$ as a vector in ear permutati $\mathbb { R } ^ { n }$ by instead passing equivariant operati $\lambda _ { i } \mathbf { 1 }$ , wherehat can +1191 be exactly expressed by DeepSets, so the total error is within . The same argument applies when +1192 $\theta _ { i } = h ( \lambda _ { i } )$ for some continuous function $h$ . +1193 For the basis invariant case, consider a parametric spectral graph convolution $f ( V , \Lambda , X ) ~ =$ +1194 $\begin{array} { r } { \sum _ { i = 1 } ^ { n } h ( \lambda _ { i } ) v _ { i } v _ { i } ^ { \top } X } \end{array}$ . Note that if the eigenspace bases are $V _ { 1 } , \dots , V _ { l }$ with eigenvalues $\mu _ { 1 } , \ldots , \mu _ { l }$ , we +1195 can write the $\begin{array} { r } { f ( V , \Lambda , X ) = \sum _ { i = 1 } ^ { l } h ( \mu _ { j } ) V _ { j } V _ { j } ^ { \top } X } \end{array}$ . Again, we will let $\rho = \textstyle \sum _ { i = 1 } ^ { l }$ be a sum function, +1196 which can be expressed exactly by DeepSets. Thus, it suffices to show that $h ( \mu _ { j } ) V _ { j } V _ { j } ^ { \top } X$ can be $\epsilon / n$ +1197 approximated by a 2-IGN (i.e. an IGN that only uses vectors and matrices). +1198 Note that since $h$ is continuous, we can use an elementwise MLP (which IGNs can learn) to +1199 approximate $f _ { 1 } ( \mu { \bf 1 1 } ^ { \top } , V V ^ { \top } , X ) = ( h ( \mu ) { \bf 1 1 } ^ { \top } , V V ^ { \top } , X )$ to arbitrary precision (note that we rep +1200 resent the eigenvalue $\mu$ as a constant matrix $\mu \mathbf { 1 1 } ^ { \top }$ ). Also, since a 2-IGN can learn matrix vector +1201 multiplication (Cai and Wang [2022] Lemma 10), we can approximate $f _ { 2 } ( h ( \mu ) { \bf 1 1 } ^ { \top } , V V ^ { \top } , X ) =$ +1202 $( h ( \mu ) \mathbf { 1 1 } ^ { \top } , V V ^ { \top } X )$ , as $V _ { i } V _ { i } ^ { \top } \in \mathbb { R } ^ { n ^ { 2 } }$ is a matrix and $\ b X \in \mathbb { R } ^ { n \times q }$ is a vector with respect to permuta +1203 tion symmetries. Finally, we use an elementwise MLP to approximate the scalar-vector multiplication +1204 $f _ { 3 } ( h ( \mu ) { \bf 1 1 } ^ { \top } , V V ^ { \top } , X ) = h ( \mu ) V V ^ { \top } X$ . Since $f _ { 3 } \circ f _ { 2 } \circ \bar { f } _ { 1 } ( \mu { \bf 1 1 } ^ { \top } , V V ^ { \top } , X ) = h ( \mu ) V V ^ { \top } X$ , and +1205 since 2-IGNs universally approximate each $f _ { i }$ , applying Lemma 6 shows that a 2-IGN can approx +1206 imate $h ( \mu ) V V ^ { \top } X$ to $\epsilon / n$ accuracy, so we are done. Since Expressive-BasisNet is stronger than +1207 BasisNet, it can also universally approximate these functions. □ +1208 From the proof, we can see that SignNet and BasisNet need only learn simple functions for the $\rho$ and +1209 $\phi$ when $h$ is simple, or when the filter is non-parametric and we need only learn $\theta _ { i }$ . Xu et al. [2020] +1210 propose the principle of algorithmic alignment, and show that if separate modules of a neural network +1211 each need only learn simple functions (that is, functions that are well-approximated by low-order +1212 polynomials with small coefficients), then the network may be more sample efficient. If we do not +1213 require permutation equivariance, and parameterize SignNet and BasisNet with simple MLPs, then +1214 algorithmic alignment may suggest that our models are sample efficient. Indeed, $\rho \overset { \cdot } { = } \sum$ is a simple +1215 linear function with coefficients 1, and $\phi ( V , \lambda , X ) = h ( \lambda ) V V ^ { \top } X$ is quadratic in $V$ and linear in $X$ +1216 so it is simple if $h$ is simple. + +Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish. + +1219 Proof. The idea is as follows: we will take graphs $G$ and give them the node feature matrix $X _ { G } =$ +1220 $D ^ { 1 / 2 } \mathbf { 1 }$ , i.e. each node has as feature the square root of its degree. Then any spectral graph convolution +1221 (or, the first layer of any spectral GNN) will map $V \mathrm { D i a g } ( \theta ) V ^ { \mathrm { ~ l ~ } } X$ to something that only depends on +1222 the degree sequence and number of nodes. Thus, any spectral graph convolution or spectral GNN +1223 will have the same output (up to permutation) for any such graphs $G$ with node features $X _ { G }$ and the +1224 same number of nodes and same degree sequence. On the other hand, SignNet and BasisNet can +1225 distinguish between infinitely many pairs of graphs $\left( G ^ { ( 1 ) } , G ^ { ( 2 ) } \right)$ with node features $( X _ { G ^ { ( 1 ) } } , X _ { G ^ { ( 2 ) } } )$ +1226 and the same number of nodes and degree sequence; this is because SignNet and BasisNet can tell +1227 when a graph is bipartite. +1228 For each $n \geq 5$ , we will define $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ as connected graphs with $n$ nodes, with the same +1229 degree sequence. Also, we define $G ^ { ( 1 ) }$ to have node features $X _ { i } ^ { ( 1 ) } = \sqrt { d _ { i } ^ { ( 1 ) } }$ , where $d _ { i } ^ { ( 1 ) }$ is the degree +1230 of node $i$ in $G ^ { ( 1 ) }$ , and similarly $G ^ { ( 2 ) }$ has node features $X _ { i } ^ { ( 2 ) } = \sqrt { d _ { i } ^ { ( 2 ) } }$ . Now, note that $X ^ { ( 1 ) }$ is an +1231 eigenvector of the normalized Laplacian of $G ^ { ( 1 ) }$ , and it has eigenvalue $0$ . As we take the eigenvectors +1232 to be orthonormal (since the normalized Laplacian is symmetric), for any spectral graph convolution +1233 we have that + +$$ +\sum _ { i = 1 } ^ { n } \theta _ { i } v _ { i } v _ { i } ^ { \top } X ^ { ( 1 ) } = \theta _ { 1 } v _ { 1 } v _ { 1 } ^ { \top } X ^ { ( 1 ) } = \theta _ { 1 } D _ { 1 } ^ { 1 / 2 } \mathbf { 1 } ( D _ { 1 } ^ { 1 / 2 } \mathbf { 1 } ) ^ { \top } D _ { 1 } ^ { 1 / 2 } \mathbf { 1 } = \theta _ { 1 } \sum _ { j = 1 } ^ { n } ( d _ { j } ^ { ( 1 ) } ) D _ { 1 } ^ { 1 / 2 } \mathbf { 1 } . +$$ + +1234 Where $D _ { 1 }$ is the diagonal degree matrix of $G ^ { ( 1 ) }$ . Likewise, any spectral graph convolution outputs +1235 $\theta _ { 1 } \sum _ { j } ( d _ { j } ^ { ( 2 ) } ) D _ { 2 } ^ { 1 / 2 } { \bf 1 }$ for $G ^ { ( 2 ) }$ . Since $D _ { 1 }$ and $D _ { 2 }$ are the same up to a permutation, we have that any +1236 spectral graph convolution has the same output for $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ , up to a permutation. In fact, this +1237 also holds for spectral GNNs, as the first layer will always have the same output (up to a permutation) +1238 on $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ , so the latter layers will also have the same output up to a permutation. +1239 Now, we concretely define $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ . This is illustrated in Figure 11 and Figure 12. For $n = 5$ , +1240 let $G ^ { ( 1 ) }$ contain a triangle with nodes $w _ { 1 } , w _ { 2 } , w _ { 3 }$ , and have a path of length 2 coming out of one of +1241 the nodes in the triangle, say $w _ { 1 }$ connects to $w _ { 4 }$ , and $w _ { 4 }$ connects to $w _ { 5 }$ . This is not bipartite, as there +1242 is a triangle. Let $G ^ { ( 2 ) }$ be a bipartite graph that has 2 nodes on the left $( v _ { 1 } , v _ { 2 } )$ and 3 nodes on the +1243 right $( v _ { 3 } , v _ { 4 } , v _ { 5 } )$ . Connect $v _ { 1 }$ with all nodes on the right, and connect $v _ { 2 }$ with $v _ { 3 }$ and $\boldsymbol { v } _ { 4 }$ . +1244 Note that both $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ have the same number of nodes and the same degree sequence +1245 $\{ 3 , 2 , 2 , 2 , 1 \}$ . Thus, spectral graph convolutions or spectral GNNs cannot distinguish them. How + +![](images/c447c5d87e24ae97195454045af4a8f9ce0acdcec39d39916c422939fc00c4e0.jpg) +Figure 11: Illustration of our constructed $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ for $n = 5$ , as used in the proof of Proposition 3. + +![](images/1dd1428f70451ee7830fca1dd260abae9fac15f7d525169066b3d20b615b8ff1.jpg) +Figure 12: Illustration of our constructed $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ for $n = 6$ , as used in the proof of Proposition 3. + +ever, SignNet and BasisNet can distinguish them, as they can tell whether a graph is bipartite by checking the highest eigenvalue of the normalized Laplacian. This is because the multiplicity of the eigenvalue 2 is the number of bipartite components. In particular, SignNet can approximate the function $\phi ( v _ { i } , \lambda _ { i } , X ) = \lambda _ { i }$ and $\rho \approx \mathrm { m a x } _ { i = 1 } ^ { n }$ . Likewise, BasisNet can approximate the function $\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \top } , \lambda _ { i } ) = \lambda _ { i }$ and $\rho \approx \mathrm { m a x } _ { i = 1 } ^ { l }$ . + +This in fact gives an infinite family of graphs that SignNet / BasisNet can distinguish, but spectral graph convolutions or spectral graph GNNs cannot. To see why, suppose we have $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ for some $n \geq 5$ . Then we construct a pair of graphs on $n + 1$ nodes with the same degree sequence. To do this, we add another node to the path of $G ^ { ( 1 ) }$ , thus giving it degree sequence $\{ 3 , 2 , \ldots , 2 , 1 \}$ . For $G ^ { ( 2 ) }$ , we add a node $v _ { n + 1 }$ to the side that $v _ { n }$ is not contained on (e.g. for $n = 5$ , we add $\boldsymbol { v } _ { 6 }$ to the left side, as $\boldsymbol { v } _ { 5 }$ was on the right), then connect $v _ { n }$ to $v _ { n + 1 }$ to also give a degree sequence $\{ 3 , 2 , \ldots , 2 , 1 \}$ . Note that the non-bipartiteness of $G ^ { ( 1 ) }$ and bipartiteness of $G ^ { ( 2 ) }$ are preserved. + +# H.2 Existing Positional Encodings + +Here, we show that our SignNets and BasisNets universally approximate various types of existing graph positional encodings. The key is to show that these positional encodings are related to spectral graph convolution matrices and the diagonals of these matrices, and to show that our networks can approximate these matrices and diagonals. + +Proposition 5. If the eigenvalues take values in a compact set, SignNets and BasisNets universally approximate the diagonal of any spectral graph convolution matrix $\begin{array} { r } { \pmb { f } ( V , \Lambda ) = \mathrm { d i a g } \left( \sum _ { i = 1 } ^ { n } h ( \lambda _ { i } ) \hat { v _ { i } v _ { i } ^ { \top } } \right) } \end{array}$ BasisNets can additionally universally approximate any spectral graph convolution matrix $f ( V , \Lambda ) =$ $\textstyle \sum _ { i = 1 } ^ { n } h ( \lambda _ { i } ) v _ { i } v _ { i } ^ { \top }$ . + +Proof. Note that the $v _ { i }$ come from a compact set as they are of unit norm. The $\lambda _ { i }$ are from a compact set by assumption; this assumption holds for the normalized Laplacian, as $\lambda _ { i } \in [ 0 , 2 ]$ . Also, as diag is linear, the spectral graph convolution diagonal can be written $\begin{array} { r } { \sum _ { i = 1 } ^ { n } h ( \lambda _ { i } ) \mathrm { d i a g } ( v _ { i } v _ { i } ^ { \top } ) } \end{array}$ . + +1271 Let $\epsilon > 0$ . For SignNet, let $\rho = \textstyle \sum _ { i = 1 } ^ { n }$ , which can be exactly expressed as it is a permutation +1272 equivariant linear operation from vectors to vectors. Then $\phi ( v _ { i } , \lambda _ { i } )$ can approximate the function +1273 $\lambda _ { i } \mathrm { d i a g } ( v _ { i } v _ { i } ^ { \top } )$ to arbitrary precision, as it is a permutation equivariant function from vectors to +1274 vectors [Segol and Lipman, 2019]. Thus, letting $\phi$ approximate the function to $\epsilon / n$ accuracy, SignNet +1275 can approximate $f$ to $\epsilon$ accuracy. + +Let $l$ be the number of eigenspaces $V _ { 1 } , \dots , V _ { l }$ , so $\begin{array} { r } { f ( V , \Lambda ) = \sum _ { i = 1 } ^ { l } h ( \mu _ { i } ) V _ { i } V _ { i } ^ { \top } } \end{array}$ . For BasisNet, we need only show that it can approximate the spectral graph convolution matrix to $\epsilon / l$ accuracy, as a 2-IGN can exactly express the diag function in each $\phi _ { d _ { i } }$ , since it is a linear permutation equivariant function from matrices to vectors. A 2-IGN can universally approximate the function $f _ { 1 } ( \mu _ { i } , V _ { i } V _ { i } ^ { \top } ) =$ $( h ( \mu _ { i } ) , V _ { i } V _ { i } ^ { \top } )$ , as it can express any elementwise MLP. Also, a 2-IGN can universally approximate the scalar-matrix multiplication $f _ { 2 } ( h ( \mu _ { i } ) , V _ { i } V _ { i } ^ { \top } ) ~ = ~ h ( \mu _ { i } ) V _ { i } V _ { i } ^ { \top }$ by another elementwise MLP. Since $h ( \mu _ { i } ) V _ { i } V _ { i } ^ { \top } = f _ { 2 } \overset { \cdot } { \circ } f _ { 1 } ( \mu _ { i } , \bar { V _ { i } } { V _ { i } ^ { \top } } )$ , Lemma 6 shows that a single 2-IGN can approximate this composition to $\epsilon / l$ accuracy, so we are done. + +Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally approximates diffusion and $p$ -step random walk relative positional encodings [Mialon et al., 2021], and generalized PageRank and landing probability distance encodings [Li et al., 2020]. + +89 Proof. We will show that we can apply the above Proposition 5, by showing that all of these +90 positional encodings are spectral graph convolutions. The heat kernel embeddings are of the form +91 diag $\begin{array} { r } { \big ( \sum _ { i = 1 } ^ { n } \exp ( - t \lambda _ { i } ) v _ { i } v _ { i } ^ { \top } \big ) } \end{array}$ for some choices of the parameter $t$ , so they can be approximated by +92 SignNets or BasisNets. Also, the diffusion kernel [Mialon et al., 2021] is just the matrix of this +93 heat kernel, and the $p$ -step random walk kernel is ${ \textstyle \sum _ { i = 1 } ^ { n } } ( 1 - \gamma \lambda _ { i } ) ^ { p } v _ { i } v _ { i } ^ { \top }$ for some parameter $\gamma$ , so +94 BasisNets can universally approximate both of these. + +For the other positional encodings, we let $v _ { i }$ be the eigenvectors of the random walk Laplacian $I - D ^ { - 1 } A$ instead of the normalized Laplacian $I - { D ^ { - 1 / 2 } A D ^ { - 1 / 2 } }$ . The eigenvalues of these two Laplacians are the same, and if $\tilde { v } _ { i }$ is an eigenvector of the normalized Laplacian then $D ^ { - 1 / 2 } \tilde { v } _ { i }$ is an eigenvector of the random walk Laplacian with the same eigenvalue [Von Luxburg, 2007]. + +299 Then with $v _ { i }$ as the eigenvectors of the random walk Laplacian, the random walk positional encodings +300 (RWPE) in Dwivedi et al. [2022] take the form + +$$ +\operatorname { d i a g } \left( ( D ^ { - 1 } A ) ^ { k } \right) = \operatorname { d i a g } \left( \sum _ { i = 1 } ^ { n } ( 1 - \lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \top } \right) , +$$ + +1301 for any choices of integer $k$ . + +1302 The distance encodings proposed in Li et al. [2020] take the form + +$$ +f _ { 3 } ( A D ^ { - 1 } , ( A D ^ { - 1 } ) ^ { 2 } , ( A D ^ { - 1 } ) ^ { 3 } , \cdot \cdot \cdot ) , +$$ + +1303 for some function $f _ { 3 }$ . We restrict to continuous $f _ { 3 }$ here; shortest path distances can be obtained by a +1304 discontinuous $f _ { 3 }$ that we discuss below. Their generalized PageRank based distance encodings can +1305 be obtained by + +$$ +\sum _ { i = 1 } ^ { n } \left( \sum _ { k \geq 1 } \gamma _ { k } ( 1 - \lambda _ { i } ) ^ { k } \right) v _ { i } v _ { i } ^ { \top } +$$ + +1306 for some $\gamma _ { k } \in \mathbb { R }$ , so this is a spectral graph convolution. They also define so-called landing probability +1307 based positional encodings, which take the form + +$$ +\sum _ { i = 1 } ^ { n } ( 1 - \lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \top } , +$$ + +1308 for some choices of integer $k$ . Thus, BasisNets can approximate these distance encoding matrices. + +1309 Another powerful class of positional encodings is based on shortest path distances between nodes +1310 in the graph [Ying et al., 2021, Li et al., 2020]. Shortest path distances can be expressed in a +1311 form similar to the spectral graph convolution, but require a highly discontinuous function. If we +1312 define $f _ { 3 } ( x _ { 1 } , \dots , x _ { n } ) = \operatorname* { m i n } _ { i : x _ { i } \neq 0 } i$ to be the lowest index such that $x _ { i }$ is nonzero, then we can +1313 1314 write the shortest path diselementwise to return an $n \times n$ matrix as matrix. $f _ { 3 } ( D ^ { - 1 } A , ( D ^ { - 1 } A ) ^ { 2 } , \dots , ( D ^ { - 1 } A ) ^ { n } )$ $\textstyle ( D ^ { - 1 } A ) ^ { k } = \sum _ { i = 1 } ^ { n } ( 1 - \lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \top }$ , where BasisN $f _ { 3 }$ is applied can learn +1315 the inside arguments, but cannot learn the discontinuous function $f _ { 3 }$ . + +# 1316 H.3 Spectral Invariants + +Here, we consider the graph angles $\alpha _ { i j } = \lVert V _ { i } V _ { i } ^ { \top } e _ { j } \rVert _ { 2 }$ , for $i = 1 , \dots , l$ where $l$ is the number of eigenspaces, and $j = 1 , \dotsc , n$ . It is clear that graph angles are permutation equivariant and basis invariant. These graph angles have been extensively studied, so we cite a number of interesting properties of them. That graph angles determine the number of length 3, 4 and 5 cycles, the connectivity of a graph, and the number of length $k$ closed walks is all shown in Chapter 4 of Cvetkovic´ et al. [1997]. Other properties may be of use for graph representation learning as well. For instance, the eigenvalues of node-deleted subgraphs of a graph $\mathcal { G }$ are determined by the eigenvalues and graph angles of $\mathcal { G }$ ; this may be useful in extending recent graph neural networks that are motivated by node deletion and the reconstruction conjecture [Cotta et al., 2021, Bevilacqua et al., 2022, Papp et al., 2021, Tahmasebi et al., 2020]. + +Now, we prove that BasisNet can universally approximate the graph angles. The graph properties we consider in the theorem are all integer valued (e.g. the number of cycles of length 3 in a graph is an integer). Thus, any two graphs that differ in these properties will differ by at least 1, so as long as we have approximation to $\varepsilon < 1 / 2$ , we can distinguish any two graphs that differ in these properties. Recall the statement of Theorem 2. + +Theorem 2. BasisNet can universally approximate the graph angles $\alpha _ { i j }$ . The eigenvalues and graph angles (and thus BasisNets) can determine the number of length 3, 4, and 5 cycles, whether a graph is connected, and the number of length $k$ closed walks from any vertex to itself. + +1335 Proof. Note that the graph angles satisfy + +$$ +\alpha _ { i j } = \| { V _ { i } V _ { i } } ^ { \top } e _ { j } \| _ { 2 } = \sqrt { e _ { j } ^ { \top } V _ { i } V _ { i } ^ { \top } V _ { i } V _ { i } ^ { \top } e _ { j } } = \sqrt { e _ { j } ^ { \top } V _ { i } V _ { i } ^ { \top } e _ { j } } , +$$ + +1336 where $V _ { i }$ is a basis for the $i$ th adjacency matrix eigenspace, and $e _ { j } ^ { \top } V _ { i } V _ { i } ^ { \top } e _ { j }$ is the $( j , j )$ -entry of $V _ { i } V _ { i } ^ { \top }$ . +1337 These graph angles are just the elementwise square roots of the diagonals of the matrices $V _ { i } V _ { i } ^ { \top }$ . +1338 As $f _ { 1 } ( V _ { i } V _ { i } ^ { \top } ) = \mathrm { d i a g } ( V _ { i } V _ { i } ^ { \top } )$ is a permutation equivariant linear function from matrices to vectors, +1339 2-IGN on $V _ { i } V _ { i } ^ { \top }$ can exactly compute this with 0 error. Then a 2-IGN can learn an elementwise +1340 MLP to approximate the elementwise square root $f _ { 2 } ( \mathrm { d i a g } ( V _ { i } V _ { i } ^ { \top } ) ) = \sqrt { \mathrm { d i a g } ( V _ { i } V _ { i } ^ { \top } ) }$ to arbitrary +1341 precision. Finally, there may be remaining operations $f _ { 3 }$ that are permutation invariant or permutation +1342 equivariant from vectors to vectors; for instance, the $\alpha _ { i j }$ are typically gathered into a matrix of size +1343 $l \times n$ where the columns are lexicographically sorted $\it l$ is the number of eigenspaces) [Cvetkovic´ +1344 et al., 1997], or we may have a permutation invariant readout to compute a subgraph count. A +1345 DeepSets can approximate $f _ { 3 }$ without any higher order tensors besides vectors [Zaheer et al., 2017, +1346 Segol and Lipman, 2019]. + +As 2-IGNs can approximate each 1347 $f _ { i }$ individually, a single 2-IGN can approximate $f _ { 3 } \circ f _ { 2 } \circ f _ { 1 }$ by 1348 Lemma 6. Also, since the graph properties considered in the theorem are integer-valued, BasisNet 1349 can distinguish any two graphs that differ in one of these properties. □ + +1350 To see that message passing graph neural networks (MPNNs) cannot determine these quantities, we +1351 use the fact that MPNNs cannot distinguish between two graphs that have the same number of nodes +1352 and where each node (in both graphs) has the same degree. For $k \geq 3$ , let $C _ { k }$ denote the cycle graph +1353 of size $k$ , and $C _ { k } + C _ { k }$ denote the graph that is the union of two disjoint cycle graphs of size $k$ +1354 MPNNs cannot distinguish between $C _ { 2 k }$ and $C _ { k } + C _ { k }$ for $k \geq 3$ , because they have the same number +1355 of nodes, and each node has degree 2. Thus, MPNNs cannot tell whether a graph is connected, as +1356 $C _ { 2 k }$ is but $C _ { k } + C _ { k }$ is not. Also, it cannot count the number of 3, 4, or 5 cycles, as $C _ { k } + C _ { k }$ has two +1357 $k$ cycles while $C _ { 2 k }$ has no $k$ cycles. Likewise, any node in $C _ { k } + C _ { k }$ has more length $k$ closed walks +1358 than any node in $C _ { 2 k }$ . This is because any length $k$ closed walk in $C _ { 2 k }$ has an analogous closed walk +1359 in $C _ { k } + C _ { k }$ , but the nodes in $C _ { k } + C _ { k }$ also have a closed walk that completely goes around a cycle. + +In this section, we collect useful lemmas for our proofs. These lemmas generally only require basic tools to prove. Our first lemma is a crucial property of quotient spaces. + +Lemma 1 (Passing to the quotient). Let $\mathcal { X }$ and $\mathcal { V }$ be topological spaces, and let $\mathcal { X } / G$ be a quotient space, with corresponding quotient map $\pi$ . Then for every continuous $G$ -invariant function $f : \mathcal { X } $ $\mathcal { V } _ { : }$ , there is a unique continuous ${ \tilde { f } } : { \mathcal { X } } / G \to { \mathcal { Y } }$ such that $f = \tilde { f } \circ \pi$ . + +Proof. For $z \in \mathcal { X } / G$ , by surjectivity of $\pi$ we can choose an $x _ { z } \in \mathcal { X }$ such that $\pi ( x _ { z } ) = z$ . Define ${ \tilde { f } } : { \mathcal { X } } / G \to { \mathcal { Y } }$ by $\tilde { f } ( z ) = f ( x _ { z } )$ . This is well-defined, since if $\pi ( x _ { z } ) = \pi ( x )$ for any other $x \in \mathcal { X }$ , then $g x _ { z } = x$ for some $g \in G$ , so + +$$ +f ( x ) = f ( g x _ { z } ) = f ( x _ { z } ) = \tilde { f } ( z ) , +$$ + +1369 where the second equality uses the $G$ -invariance of $f$ . Note that $\tilde { f }$ is continuous by the universal +1370 property of quotient spaces. Also, $\tilde { f }$ is the unique function such that $f = \tilde { f } \circ \pi$ ; if there were another +1371 function $h : \mathcal { X } / G \to \mathcal { Y }$ with $h ( z ) \neq { \tilde { f } } ( z )$ , then $h ( z ) \neq f ( x _ { z } )$ , so $h ( \pi ( x _ { z } ) ) = h ( z ) \neq f ( x _ { z } )$ . +1372 Next, we give the First Fundamental Theorem of $O ( d )$ , a classical result that has been recently used +1373 for machine learning by Villar et al. [2021]. This result shows that an orthogonally invariant $f ( V )$ +1374 can be expressed as a function $h ( V V ^ { \top } )$ . We give a proof that if $f$ is continuous, then $h$ is also +1375 continuous. + +Lemma 2 (First Fundamental Theorem of $O ( d ) { \big \rangle }$ ). A continuous function $f : \mathbb { R } ^ { n \times d } \mathbb { R } ^ { s }$ is orthogonally invariant, i.e. $f ( V Q ) = f ( V )$ for all $Q \in O ( d )$ , if and only if $f ( V ) = h ( V V ^ { \top } ) f o \iota$ r some continuous $h$ . + +Proof. If $f ( V ) = h ( V V ^ { \top } )$ , then we have $f ( V Q ) = h ( V Q Q ^ { \top } V ^ { \top } ) = h ( V V ^ { \top } )$ so $f$ is orthogonally invariant. + +1381 For the other direction, invariant theory shows that the $O ( d )$ invariant polynomials are generated +1382 by the inner products $v _ { i } ^ { \top } v _ { j }$ , where $v _ { i } \in \mathbb { R } ^ { d }$ are the rows of $V$ [Kraft and Procesi, 1996]. Let $p :$ +1383 $\mathbb { R } ^ { n \times d } \to \mathbb { R } ^ { n \times n }$ be the map $p ( V ) = V V ^ { \top }$ . Then González and de Salas [2003] Lemma 11.13 shows +1384 that the quotient space $\mathbb { R } ^ { n \times d } / O ( d )$ is homeomorphic to a closed subset $p ( \mathbb { R } ^ { n \times d } ) = \mathcal { Z } \subseteq \mathbb { R } ^ { n \times n }$ . +1385 Let $\tilde { p }$ refer to this homeomorphism, and note that ${ \tilde { p } } \circ \pi = p$ by passing to the quotient (Lemma 1). +1386 Then any continuous $O ( d )$ invariant $f$ passes to a unique continuous $\widetilde { f } \ : \ \mathbb { R } ^ { n \times d } / O ( d ) \ \to \ \mathbb { R } ^ { s }$ +1387 (Lemma 1), so $f = \tilde { f } \circ \pi$ where $\pi$ is the quotient map. Define $h : { \mathcal { Z } } \to \mathbb { R } ^ { s }$ by $h = \tilde { f } \circ \tilde { p } ^ { - 1 }$ , and +1388 note that $h$ is a composition of continuous functions and hence continuous. Finally, we have that +1389 $\begin{array} { r } { h ( V V ^ { \top } ) = h ( \tilde { p } \circ \pi ( V ) ) = \tilde { f } \circ \pi ( V ) = f ( V ) } \end{array}$ , so we are done. □ +390 The next lemma allows us to decompose a quotient of a product space into a product of smaller +391 quotient spaces. + +92 Lemma 3. Let $\mathcal { X } _ { 1 } , \ldots , \mathcal { X } _ { k }$ be topological spaces and $G _ { 1 } , \ldots , G _ { k }$ be topological groups such that each 93 $G _ { i }$ acts continuously on $\mathcal { X } _ { i }$ . Denote the quotient maps by $\pi _ { i } : \mathcal { X } _ { i } \mathcal { X } _ { i } / G _ { i }$ . Then the quotient 94 of the product is the product of the quotient, i.e. + +$$ +( { \mathcal { X } } _ { 1 } \times \ldots \times { \mathcal { X } } _ { k } ) / ( G _ { 1 } \times \ldots \times G _ { k } ) \cong ( { \mathcal { X } } _ { 1 } / G _ { 1 } ) \times \ldots \times ( { \mathcal { X } } _ { k } / G _ { k } ) , +$$ + +1395 and π1 × . . . $\times \pi _ { k } : \mathcal { X } _ { 1 } \times . . . \mathcal { X } _ { k } \to ( \mathcal { X } _ { 1 } / G _ { 1 } ) \times . . . \times ( \mathcal { X } _ { k } / G _ { k } )$ ) is quotient map. + +1396 Proof. First, we show that $\pi _ { 1 } \times \ldots \times \pi _ { k }$ is a quotient map. This is because 1. the quotient map +1397 of any continuous group action is an open map, so each $\pi _ { i }$ is an open map, 2. the product of open +1398 maps is an open map, so $\pi _ { 1 } \times \ldots \times \pi _ { k }$ is an open map and 3. a continuous surjective open map is a +1399 quotient map, so $\pi _ { 1 } \times \ldots \times \pi _ { k }$ , which is continuous and surjective, is a quotient map. + +Now, we need only apply the theorem of uniqueness of quotient spaces to show (51) (see e.g. Lee [2013], Theorem A.31). Letting $q : { \mathcal { X } } _ { 1 } \times \ldots \times { \mathcal { X } } _ { k } \to ( { \mathcal { X } } _ { 1 } \times \ldots \times { \mathcal { X } } _ { k } ) / ( G _ { 1 } \times \ldots \times G _ { k } )$ denote the quotient map for this space, it is easily seen that $q ( x _ { 1 } , \ldots , x _ { k } ) = q ( y _ { 1 } \ldots , y _ { k } )$ if and only if $\pi _ { 1 } \stackrel { \textstyle \ldots } { \times } \ldots \times \pi _ { k } ( \bar { x _ { 1 } } , \ldots , x _ { k } ) \stackrel { \textstyle \ldots } { = } \pi _ { 1 } \times \ldots \times \bar { \pi } _ { k } ( y _ { 1 } , \ldots , y _ { k } )$ , since either of these is true if and only if there exist $g _ { i } \in G _ { i }$ such that $x _ { i } = g _ { i } y _ { i }$ for each $i$ . Thus, we have an isomorphism of these quotient spaces. □ + +406 The following lemma shows that quotients of compact spaces are also compact, which is useful for +407 universal approximation on quotient spaces. +08 Lemma 4 (Compactness of quotients of compact spaces). Let $\mathcal { X }$ be a compact space. Then the +409 quotient space $\mathcal { X } / G$ is compact. +1410 Proof. Denoting the quotient map by $\pi : { \mathcal { X } } \to { \mathcal { X } } / G$ and letting $\{ U _ { \alpha } \} _ { \alpha }$ be an open cover of $\mathcal { X } / G$ , +1411 we have that $\{ \check { \pi } ^ { - 1 } ( U _ { \alpha } ) \} _ { \alpha }$ is an open cover of $\mathcal { X }$ . By compactness of $\mathcal { X }$ , we can choose a finite +1412 subcover $\{ \pi ^ { - 1 } ( U _ { \alpha _ { i } } ) \} _ { i = 1 , \dots , n }$ . Then $\{ \pi ( \pi ^ { - 1 } ( U _ { \alpha _ { i } } ) ) \} _ { i = 1 , \dots , n } = \{ U _ { \alpha _ { i } } \} _ { i = 1 , \dots , n }$ by surjectivity, and +1413 $\{ U _ { \alpha _ { i } } \} _ { i = 1 , \dots , n }$ is thus an open cover of $\mathcal { X } / G$ . □ +1414 The Whitney embedding theorem gives a nice condition that we apply to show that the quotient +1415 spaces $\chi / \bar { G }$ that we deal with embed into Euclidean space. It says that when $\mathcal { X } / G$ is a smooth +1416 manifold, then it can be embedded into a Euclidean space of double the dimension of the manifold. +1417 The proof is outside the scope of this paper. + +18 Lemma 5 (Whitney Embedding Theorem [Whitney, 1944]). Every smooth manifold $\mathcal { M }$ of dimension n > 0 can be smoothly embedded in R2n 19 . + +1420 Finally, we give a lemma that helps prove universal approximation results. It says that if functions +1421 $f$ that we want to approximate can be written as compositions $f = f _ { L } \circ \dots \circ f _ { 1 }$ , then it suffices +1422 to universally approximate each $f _ { i }$ and compose the results to universally approximate the $f$ . This +1423 is especially useful for proving universality of neural networks, as we may use some layers to +1424 approximate each $f _ { i }$ , then compose these layers to approximate the target function $f$ . +1425 Lemma 6 (Layer-wise universality implies universality). Let $\mathcal { Z } \subseteq \mathbb { R } ^ { d _ { 0 } }$ be a compact domain, let +1426 $\mathcal { F } _ { 1 } , \ldots , \mathcal { F } _ { L }$ be families of continuous functions where ${ \mathcal { F } } _ { i }$ consists of functions from $\mathbb { R } ^ { d _ { i - 1 } } \mathbb { R } ^ { d _ { i } }$ +1427 for some $d _ { 1 } , \ldots , d _ { L }$ . Let $\mathcal { F }$ be the family of functions $\{ f _ { L } \circ . . . f _ { 1 } : \mathcal { Z } \mathbb { R } ^ { d _ { L } } , f _ { i } \in \mathcal { F } _ { i } \}$ that are +1428 compositions of functions $f _ { i } \in \mathcal { F } _ { i }$ . + +For each 1429 $i$ , let $\Phi _ { i }$ be a family of continuous functions that universally approximates ${ \mathcal { F } } _ { i }$ . Then the 1430 family of compositions $\Phi = \left\{ \phi _ { L } \circ . . . \circ \phi _ { 1 } : \phi _ { i } \in \Phi _ { i } \right\}$ universally approximates $\mathcal { F }$ . + +Proof. Let 1431 $f = f _ { L } \circ . . . \circ f _ { 1 } \in \mathcal { F }$ . Let $\tilde { \mathcal { Z } } _ { 1 } = \mathcal { Z }$ , and then for $i \geq 2$ let $\tilde { \mathcal { Z } } _ { i } = f _ { i - 1 } ( \tilde { \mathcal { Z } } _ { i - 1 } )$ . Then each 1432 $\mathcal { \tilde { Z } } _ { i }$ is compact by continuity of the $f _ { i }$ . For $1 \leq i < L$ , let $\mathcal { Z } _ { i } = \tilde { \mathcal { Z } } _ { i }$ , and for $i = L$ let $\mathcal { Z } _ { L }$ be a compact 1433 set containing $\tilde { \mathcal { Z } } _ { L }$ such that every ball of radius one centered at a point in $\tilde { \mathcal { Z } } _ { L }$ is still contained in $\mathcal { Z } _ { L }$ . + +1434 Let $\epsilon > 0$ . We will show that there is a $\phi \in \Phi$ such that $\| f - \phi \| _ { \infty } < \epsilon$ by induction on $L$ . This holds +435 trivially for $L = 1$ , as then $\Phi = \Phi _ { 1 }$ . +1436 Now, let $L \geq 2$ , and suppose it holds for $L - 1$ . By universality of $\Phi _ { L }$ , we can choose a $\phi _ { L } : \mathcal { Z } _ { L } $ +1437 $\mathbb { R } ^ { d _ { L } } \in \Phi _ { L }$ such that $\| \phi _ { L } - f _ { L } \| _ { \infty } < \epsilon / 2$ . As $\phi _ { L }$ is continuous on a compact domain, it is also +1438 uniformly continuous, so we can choose a $\tilde { \delta } > 0$ such that $\| y - z \| _ { 2 } < \tilde { \delta } \implies \| \phi _ { L } ( y ) - \phi _ { L } ( z ) \| _ { 2 } <$ +1439 $\epsilon / 2$ . + +Let 1440 $\delta = \operatorname* { m i n } ( \tilde { \delta } , 1 )$ . By induction, we can choose $\phi _ { L - 1 } \circ . . . \circ \phi _ { 1 } , \phi _ { i } \in \Phi _ { i }$ such that + +$$ +\begin{array} { r } { \| \phi _ { L - 1 } \circ . . . \circ \phi _ { 1 } - f _ { L - 1 } \circ . . . \circ f _ { 1 } \| _ { \infty } < \delta . } \end{array} +$$ + +1441 Note that $\phi _ { L - 1 } \circ . . . \circ \phi _ { 1 } ( \mathcal { Z } ) \subseteq \mathcal { Z } _ { L }$ , because for each $x \in { \mathcal { Z } }$ , $\phi _ { L - 1 } \circ . . . \circ \phi _ { 1 } ( x )$ is within $\delta \leq 1$ 1442 Euclidean distance to $f _ { L - 1 } \circ \dots \circ f _ { 1 } ( x ) \in \tilde { \mathcal { Z } } _ { L }$ , so it is contained in $\mathcal { Z } _ { L }$ by construction. Thus, we may define 1443 $\phi = \phi _ { L } \circ . . . \circ \phi _ { 1 } : \mathcal { Z } \mathbb { R } ^ { d _ { L } }$ , and compute that + +$$ +\begin{array} { r l } & { \| \phi - f \| _ { \infty } \leq \| \phi - \phi _ { L } \circ f _ { L - 1 } \circ . . . \circ f _ { 1 } \| _ { \infty } + \| \phi _ { L } \circ f _ { L - 1 } \circ . . . \circ f _ { 1 } - f \| _ { \infty } } \\ & { \qquad < \| \phi - \phi _ { L } \circ f _ { L - 1 } \circ . . . \circ f _ { 1 } \| _ { \infty } + \epsilon / 2 , } \end{array} +$$ + +1444 since $\| \phi _ { L } - f _ { L } \| _ { \infty } < \epsilon / 2$ . To bound this other term, let $x \in { \mathcal { Z } }$ , and for $y = \phi _ { L - 1 } \circ . . . \circ \phi _ { 1 } ( x )$ +1445 and $z = f _ { L - 1 } \circ . . . \circ f _ { 1 } ( x )$ , we know that $\| y - z \| _ { 2 } < \delta$ , so $\| \phi _ { L } ( y ) - \phi _ { L } ( z ) \| _ { 2 } < \epsilon / 2$ by uniform +1446 continuity. As this holds for all $x$ , we have $\| \phi - \phi _ { L } \circ f _ { L - 1 } \circ . . . \circ f _ { 1 } \| _ { \infty } \leq \epsilon / 2$ , so $\| \phi - f \| _ { \infty } < \epsilon$ +1447 and we are done. □ + +Table 7: Results on the ZINC dataset with $5 0 0 \mathrm { k }$ parameter budget and no edge features. Numbers are the mean and standard deviation over 4 runs each with different seeds. + +
Base modelPositional encodingk#paramsTest MAE (↓)
GINNo PE16497k0.348±0.014
LapPE (flip)16498k0.341±0.011
SignNet16500k0.238±0.012
GATNo PE16501k0.464±0.011
LapPE (flip)16502k0.462±0.013
SignNet16499k0.243±0.008
+ +# 1448 J Further Experiments + +# J.1 Graph Regression with no Edge Features + +All graph regression models in Table 1 use edge features for learning and inference. To show that SignNet is also useful when no edge features are available, we ran ZINC experiments without edge features as well. The results are displayed in Table 7. In this setting, SignNet still significantly improves the performance over message passing networks without positional encodings, and over Laplacian positional encodings with sign flipping data augmentation. + +# 1455 J.2 Learning Spectral Graph Convolutions + +Table 8: Sum of squared errors for spectral graph convolution regression (with no test set). Lower is better. Numbers are mean and standard deviation over 50 images from He et al. [2021]. + +
Low-passHigh-passBand-passBand-rejectionComb
GCN.111±.0683.092±5.111.720±3.151.418±1.031.753±1.17
GAT.113±.065.954±.6961.105±.964.543±.340.638±.446
GPR-GNN.033±.032.012±.007.137±.081.256±.197.369±.460
ARMA.053±.029.042±.024.107±.039.148±.089.202±.116
ChebNet.003±.002.001±.001.005±.003.009±.006.022±.016
BernNet.001±.002.001±.001.000±.000.048±.042.027±.019
Transformer3.662±1.973.715±1.981.531±1.301.506±1.293.178±1.93
Transformer Eig Flip4.454±2.324.425±2.381.651±1.532.567±1.733.720±1.94
Transformer Eig Abs2.727±1.403.172±1.611.264±.7881.445±.9432.607±1.32
DeepSets SignNet.004±.013.086±.405.021±.115.008±.037.003±.016
Transformer SignNet.003±.016.004±.025.001±.004.006±.023.093±.641
DeepSets BasisNet.009±.018.003±.015.008±.030.004±.011.015±.060
Transformer BasisNet.079±.471.014±.038.005±.018.006±.016.014±.051
+ +1456 To numerically test the ability of our basis invariant networks for learning spectral graph convolutions, +1457 we follow the experimental setups of Balcilar et al. [2020], He et al. [2021]. We take the dataset of 50 +1458 images in He et al. [2021] (originally from the Image Processing Toolbox of MATLAB), and resize +1459 them from $1 0 0 \times 1 0 0$ to $3 2 \times 3 2$ . Then we apply the same spectral graph convolutions on them as in +1460 He et al. [2021], and train neural networks to learn these as regression targets. As in prior work, we +1461 report sum of squared errors on the training set to measure expressivity. +1462 We compare against message passing GNNs [Kipf and Welling, 2017, Velickovi ˇ c et al. ´ , 2018] and +1463 spectral GNNs [Chien et al., 2021, Bianchi et al., 2021, Defferrard et al., 2016, He et al., 2021]. +1464 Also, we consider standard Transformers with only node features, with eigenvectors and sign flip +1465 augmentation, and with absolute values of eigenvectors. These models are all approximately sign +1466 invariant (they either use eigenvectors in a sign invariant way or do not use eigenvectors). We use +1467 DeepSets [Zaheer et al., 2017] in SignNet and 2-IGN [Maron et al., 2018] in BasisNet for $\phi$ , use +1468 a DeepSets for $\rho$ in both cases, and then feed the features into another DeepSets or a standard +1469 Transformer [Vaswani et al., 2017] to make the final predictions. That is, we are only given graph +1470 information through the eigenvectors and eigenvalues, and we do not use message passing. +1471 Table 8 displays the results, which validate our theoretical results in Section 3.1. Without any message +1472 passing, SignNet and BasisNet allow DeepSets and Transformers to perform strongly, beating the +1473 spectral GNNs GPR-GNN and ARMA on all tasks. Also, our networks outperform all other methods +1474 on the band-rejection and comb filters, and are mostly close to the best model on the other filters. + +# 1475 K Further Experimental Details + +# K.1 Hardware, Software, and Data Details + +All experiments could fit on one GPU at a time. Most experiments were run on a server with 8 NVIDIA RTX 2080 Ti GPUs. We run all of our experiments in Python, using the PyTorch [Paszke et al., 2019] framework (license URL). We also make use of Deep Graph Library (DGL) [Wang et al., 2019] (Apache License 2.0), and PyTorch Geometric (PyG) [Fey and Lenssen, 2019] (MIT License) for experiments with graph data. + +We open source our code [redacted for anonymous review]. + +The data we use are all freely available online. The datasets we use are ZINC [Irwin et al., 2012], Alchemy [Chen et al., 2019a], the synthetic counting substructures dataset [Chen et al., 2020], the multi-task graph property regression synthetic dataset [Corso et al., 2020] (MIT License), the images dataset used by Balcilar et al. [2020] (GNU General Public License v3.0), the cat mesh from free3d.com/3d-model/cat-v1--522281.html (Personal Use License), and the human mesh from turbosquid.com/3d-models/water-park-slides-3d-max/1093267 (TurboSquid 3D Model License). If no license is listed, this means that we cannot find a license for the dataset. As they appear to be freely available with permissive licenses or no licenses, we do not ask for permission from the creators or hosts of the data. + +1492 We do not believe that any of this data contains offensive content or personally identifiable information. +1493 The 50 images used in the spectral graph convolution experiments are mostly images of objects, with +1494 a few low resolution images of humans that do not appear to have offensive content. The only other +1495 human-related data appears to be the human mesh, which appears to be from a 3D scan of a human. +1496 The human mesh does have tattoos, but they do not appear to be offensive. + +# K.2 Graph Regression Details + +1498 +1499 +1500 +1501 +1502 +1503 +1504 +1505 +1506 +1507 +1508 +1509 + +ZINC. In Section 4.1 we study the effectiveness of SignNet for learning positional encodings to boost the expressive power, and thereby generalization, on the graph regression problem ZINC. In all cases we take our $\phi$ encoder to be an 8 layer GIN with ReLU activation. The input eigenvector $v _ { i } \in \mathbb { R } ^ { n }$ , where $n$ is the number of nodes in the graph, is treated as a single scalar feature for each node. In the case of using a fixed number of eigenvectors $k$ , the aggregator $\rho$ is taken to be an 8 layer MLP with batch normalization and ReLU activation. The aggregator $\rho$ is applied separately to the concatenatation of the $k$ different embeddings for each node in a graph, resulting in one single embedding per node. This embedding is concatenated to the node features for that node, and the result passed as input to the base (predictor) model. We also consider using all available eigenvectors in each graph instead of a fixed number $k$ . Since the total number of eigenvectors is a variable quantity, equal to the number of nodes in the underlying graph, an MLP cannot be used for $\rho$ . To handle the variable sized input in this case, we take $\rho$ to be an MLP preceded by a sum over the $\phi$ outputs. In other words, the SignNet is of the form MLP $\begin{array} { r } { \left( \sum _ { i = 1 } ^ { k } \phi ( v _ { i } ) + \phi ( - v _ { i } ) \right) } \end{array}$ in this case. + +1511 As well as testing SignNet, we also checked whether simple transformations that resolve the sign +1512 ambiguity of the Laplacian eigenvectors $p = ( v _ { 1 } , \ldots , v _ { k } )$ could serve as effective positional encoding. +1513 We considered three options. First is to randomly flip the sign of each $\pm v _ { i }$ during training. This +1514 is a common heuristic used in prior work on Laplacian positional encoding [Kreuzer et al., 2021, +1515 Dwivedi et al., 2020]. Second, take the element-wise absolute value $| v _ { i } |$ . This is a non-injective +1516 map, creating sign invariance at the cost of destroying positional information. Third is a different +1517 canonicalization that avoids stochasticity and use of absolute values by selecting the sign of each +1518 $v _ { i }$ so that the majority of entries are non-negative, with ties broken by comparing the $\ell _ { 1 }$ -norm of +1519 positive and negative parts. When the tie-break also fails, the sign is chosen randomly. Results for +1520 GatedGCN base model on ZINC in Table 1 show that all three of these approaches are significantly +1521 poorer positional encodings compared to SignNet. +1522 Our training pipeline largely follows that of Dwivedi et al. [2022], and we use the GatedGCN +1523 and PNA base models from the accompanying implementation (see https://github.com/ +1524 vijaydwivedi75/gnn-lspe). The Sparse Transformer base model architecture we use, which +1525 like GAT computes attention only across neighbouring nodes, is introduced by Kreuzer et al. [2021]. +1526 Finally, the GINE implementation is based on the PyTorch Geometric implementation [Fey and +1527 Lenssen, 2019]. For the state-of-the-art comparison, all baseline results are from their respective +1528 papers, except for GIN, which we run. +1529 ZINC-full. We also run our method on the full ZINC dataset, termed ZINC-full. The result we +1530 report for SignNet is a larger version of the GatedGCN base model with a SignNet that takes in +1531 all eigenvectors. This model has 994,113 parameters in total. All baseline results are from their +1532 respective papers, except for GIN, which is from [Bodnar et al., 2021]. + +Alchemy. We run our method and compare with the state-of-the-art on Alchemy (with 10,000 training graphs). We use the same data split as Morris et al. [2020b]. Our base model is a GIN that takes in edge features (i.e. a GINE). The SignNet consists of GIN for $\phi$ and a Transformer for $\rho$ , as in the counting substructures and graph property regression experiments in Section 4.2. The model has 907,371 parameters in total. Our training setting is very similar to that of Morris et al. [2022], as we build off of their code. We train with an Adam optimizer [Kingma and Ba, 2014] with a starting learning rate of .001, and a minimum learning rate of .000001. The learning rate schedule cuts the learning rate in half with a patience of 20 epochs, and training ends when we reach the minimum learning rate. All baseline results are from their respective papers, except for GIN, which is from [Morris et al., 2022]. + +# 1543 K.3 Spectral Graph Convolution Details + +In Appendix J.2, we conduct node regression experiments for learning spectral graph convolutions. The experimental setup is mostly taken from He et al. [2021]. However, we resize the $1 0 0 \times 1 0 0$ images to $3 2 \times 3 2$ . Thus, each image is viewed as a 1024-node graph. The node features $X \in \mathbb { R } ^ { n }$ are the grayscale pixel intensities of each node. Just as in He et al. [2021], we only train and evaluate on nodes that are not connected to the boundary of the grid (that is, we only evaluate on the $2 8 \times 2 8$ middle section). For all experiments we limit each model to 50,000 parameters. We use the Adam [Kingma and Ba, 2014] optimizer for all experiments. For each of the GNN baselines (GCN, GAT, GPR-GNN, ARMA, ChebNet, BernNet), we select the best performing out of 4 hyperparameter settings: either 2 or 4 convolution layers, and a hidden dimension of size 32 or $D$ , where $D$ is just large enough to stay with 50,000 parameters (for instance, $D = 1 2 8$ for GCN, GPR-GNN, and BernNet). + +1555 We use DeepSets or standard Transformers as our prediction network. This takes in the output of +1556 SignNet or BasisNet and concatenates it with the node features, then outputs a scalar prediction for +1557 each node. We use a 3 layer output network for DeepSets SignNet, and 2 layer output networks for +1558 all other configurations. All networks use ReLU activations. + +For SignNet, we use DeepSets for both $\phi$ and $\rho$ . Our $\phi$ takes in eigenvectors only, then our $\rho$ takes the outputs of $\phi$ and the eigenvalues. We use three layers for $\phi$ and $\rho$ . + +For BasisNet, we use the same DeepSets for $\rho$ as in SignNet, and 2-IGNs for the $\phi _ { d _ { i } }$ . There are three distinct multiplicities for the grid graph (1, 2, and 32), so we only need 3 separate IGNs. Each IGN consists of an $\mathbb { R } ^ { n ^ { 2 } \times 1 } \to \mathbb { R } ^ { n \times d ^ { \prime } }$ layer and two $\mathbb { R } ^ { n \times d ^ { \prime \prime } } \to \mathbb { R } ^ { n \times d ^ { \prime \prime \prime } }$ layers, where the $d ^ { \prime }$ are hidden dimensions. There are no matrix to matrix operations used, as the memory requirements are intensive for these $\geq 1 0 0 0$ node graphs. The $\phi _ { d _ { i } }$ only take in $V _ { i } V _ { i } ^ { \top }$ from the eigenspaces, and the $\rho$ takes the output of the $\phi _ { d _ { i } }$ as well as the eigenvalues. + +# K.4 Substructures and Graph Properties Regression Details + +We use the random graph dataset from Chen et al. [2020] for counting substructures and the synthetic dataset from Corso et al. [2020] for regressing graph properties. For fair comparison we fix the base model as a 4-layer GIN model with hidden size 128. We choose $\phi$ as 4-layer GIN (independently applied to every eigenvector) and $\rho$ as 1-layer Transformer (independently applied to every node). Combined with proper batching and masking, we have a SignNet that takes Laplacian eigenvectors $V \in \mathbb { R } ^ { n \times n }$ and outputs fixed size sign-invariant encoding node features $f ( V , \Lambda , \bar { \lambda } ) \in \mathbb R ^ { \bar { n } \times d }$ , where + +574 $n$ varies between graphs but $d$ is fixed. We use this SignNet in our experiments and compare with +575 other methods of handling PEs. +1577 We closely follow the experimental setting of Koestler et al. [2022] for the texture reconstruction +1578 experiments. In this work, we use the cotangent Laplacian [Rustamov et al., 2007] of a triangle mesh +1579 with the lowest 1023 eigenvectors besides the trivial eigenvector of eigenvalue 0. We implemented +1580 SignNet in the authors’ original code, which was privately shared with us. Both $\rho$ and $\phi$ are taken +1581 to be MLPs. Hyperparameter settings and number of parameters are given in Table 9. We chose +1582 hyperparameters so that the total number of parameters in the SignNet model was no larger than that +1583 of the original model. + +Table 9: Parameter settings for the texture reconstruction experiments. + +
ParamsBase MLP widthBase MLP layers out dimp out dimp,width
Intrinsic NF328,5791286448
SignNet323,5631086
\ No newline at end of file diff --git a/parse/dev/zSeoDvsDCe/zSeoDvsDCe_content_list.json b/parse/dev/zSeoDvsDCe/zSeoDvsDCe_content_list.json new file mode 100644 index 0000000000000000000000000000000000000000..cd52463ce8e515338eae9f8ecb490dd6ed78d289 --- /dev/null +++ b/parse/dev/zSeoDvsDCe/zSeoDvsDCe_content_list.json @@ -0,0 +1,4819 @@ +[ + { + "type": "text", + "text": "Sign and Basis Invariant Networks for Spectral Graph Representation Learning ", + "text_level": 1, + "bbox": [ + 250, + 122, + 743, + 172 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Anonymous Author(s) \nAffiliation \nAddress \nemail ", + "bbox": [ + 423, + 226, + 580, + 281 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "Abstract ", + "text_level": 1, + "bbox": [ + 462, + 318, + 535, + 334 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "1 We introduce SignNet and BasisNet—new neural architectures that are invariant \n2 to two key symmetries displayed by eigenvectors: (i) sign flips, since if $v$ is an \n3 eigenvector then so is $- v$ ; and (ii) more general basis symmetries, which occur in \n4 higher dimensional eigenspaces with infinitely many choices of basis eigenvectors. \n5 We prove that our networks are universal, i.e., they can approximate any continu \n6 ous function of eigenvectors with the desired invariances. Moreover, when used \n7 with Laplacian eigenvectors, our architectures are provably expressive for graph \n8 representation learning: they can approximate any spectral graph convolution, can \n9 compute spectral invariants that go beyond message passing neural networks, and \n10 can provably simulate previously proposed graph positional encodings. Experi \n11 ments show the strength of our networks for molecular graph regression, learning \n12 expressive graph representations, and learning neural fields on triangle meshes. ", + "bbox": [ + 148, + 348, + 766, + 515 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "13 1 Introduction ", + "text_level": 1, + "bbox": [ + 148, + 539, + 312, + 556 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "14 Numerous machine learning models process eigenvectors, which arise in various scenarios including \n15 principal component analysis, matrix factorizations, and operators associated to graphs or manifolds. \n16 An important example is the use of Laplacian eigenvectors to encode information about the structure \n17 of a graph or manifold [Belkin and Niyogi, 2003, Von Luxburg, 2007, Lévy, 2006]. Positional \n18 encodings that involve Laplacian eigenvectors have recently been used to generalize Transformers \n19 to graphs [Kreuzer et al., 2021, Dwivedi and Bresson, 2021], and to improve the expressive power \n20 and empirical performance of graph neural networks (GNNs) [Dwivedi et al., 2022]. Furthermore, \n21 these eigenvectors are crucial for defining spectral operations on graphs that are foundational to graph \n22 signal processing and spectral GNNs [Ortega et al., 2018, Bruna et al., 2014]. \n23 However, there are nontrivial symmetries that should be accounted for when processing eigenvectors. \n24 For instance, if $v$ is an eigenvector, then so is $- v$ , with the same eigenvalue. More generally, if an \n25 eigenvalue has higher multiplicity, then there are infinitely many unit-norm eigenvectors that can \n26 be chosen. Indeed, a full set of orthonormal eigenvectors is only defined up to a change of basis \n27 in each eigenspace. In the case of sign invariance, for any $k$ eigenvectors there are $\\bar { 2 ^ { k } }$ possible \n28 choices of sign. Accordingly, prior works randomly flip eigenvector signs during training in order to \n29 approximately learn sign invariance [Kreuzer et al., 2021, Dwivedi et al., 2020]. However, learning \n30 all $2 ^ { k }$ invariances is challenging and limits the effectiveness of Laplacian eigenvectors for encoding \n31 positional information. Sign invariance is a special case of basis invariance when all eigenvalues are \n32 distinct, but general basis invariance is even more difficult to deal with. In Appendix C.2, we show \n33 that higher dimensional eigenspaces are abundant in real datasets; for instance, $64 \\%$ of molecule \n34 graphs in the ZINC dataset have a higher dimensional eigenspace. \n35 In this work, we address the sign and basis ambiguity problems by developing new neural networks— \n36 SignNet and BasisNet. Our networks are universal and can approximate any continuous function \n37 of eigenvectors with the proper invariances. Moreover, our networks are theoretically powerful \n38 for graph representation learning—they can approximate spectral graph convolutions and compute \n39 powerful spectral invariants, which allows our networks to express graph properties like subgraph \n40 counts that message passing neural networks cannot. Finally, Laplacian eigenvectors with SignNet \n41 and BasisNet can approximate many previously proposed graph positional encodings, including those \n42 based on random walks [Li et al., 2020, Dwivedi et al., 2022] and heat kernels [Mialon et al., 2021, \n43 Feldman et al., 2022]. Experiments on molecular graph regression tasks, learning expressive graph \n44 representations, and texture reconstruction on triangle meshes illustrate the empirical benefits of our \n45 models’ approximation power and invariances. ", + "bbox": [ + 147, + 570, + 825, + 695 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 702, + 825, + 867 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 873, + 826, + 901 + ], + "page_idx": 0 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 90, + 826, + 217 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "46 2 Sign and Basis Invariant Networks ", + "text_level": 1, + "bbox": [ + 147, + 234, + 496, + 252 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "47 For an $n \\times n$ symmetric matrix, let $\\lambda _ { 1 } \\leq \\ldots \\leq$ \n48 $\\lambda _ { n }$ be the eigenvalues and $v _ { 1 } , \\ldots , v _ { n }$ the corre \n49 sponding eigenvectors, which we may assume \n50 to form an orthonormal basis. For instance, we \n51 could consider the normalized graph Laplacian \n52 $L = I - D ^ { - 1 / 2 } A D ^ { - 1 / 2 }$ , where $A \\in \\mathbb { R } ^ { n \\times n }$ \n53 is the adjacency matrix and $D$ is the diagonal \n54 degree matrix of some underlying graph. For \n55 undirected graphs, $L$ is symmetric. Nonsymmet \n56 ric matrices can be handled very similarly, as we \n57 show in Appendix B.1. Our goal is to parame \n58 terize a class of models $f ( v _ { 1 } , \\ldots , v _ { k } )$ taking $k$ \n59 eigenvectors as input in a manner that respects \n60 the eigenvector symmetries. \n61 Sign invariance. For any of the $v _ { i }$ , the sign \n62 flipped $- v _ { i }$ is also an eigenvector, so a function \n63 $f : \\mathbb { R } ^ { n \\times k } \\mathbb { R } ^ { s }$ (where $s$ is an arbitrary output \n64 dimension) should be sign invariant: ", + "bbox": [ + 148, + 266, + 485, + 460 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 468, + 483, + 523 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/49ba0dee64dd6c99ddb42e8768949b5882d24f648bf9cdc4965eb722da7f8290.jpg", + "text": "$$\nf ( v _ { 1 } , \\ldots , v _ { k } ) = f ( s _ { 1 } v _ { 1 } , \\ldots , s _ { k } v _ { k } )\n$$", + "text_format": "latex", + "bbox": [ + 202, + 529, + 436, + 546 + ], + "page_idx": 1 + }, + { + "type": "image", + "img_path": "images/b914e874e9e714533cbddfb9f0489c55302867f3f2a8c3b6eefd457617e9e970.jpg", + "image_caption": [ + "Figure 1: Symmetries of eigenvectors of a symmetric matrix with permutation symmetries (e.g. a graph Laplacian). A neural network applied to the eigenvector matrix (middle) should be invariant or equivariant to permutation of the rows (left product with a permutation matrix $P$ ) and invariant to the choice of eigenvectors in each eigenbasis (right product with a block diagonal orthogonal matrix $\\operatorname { \\bar { D i a g } } ( Q _ { 1 } , Q _ { 2 } , Q _ { 3 } ) )$ . " + ], + "image_footnote": [], + "bbox": [ + 504, + 277, + 820, + 410 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "for all sign choices $s _ { i } \\in \\{ - 1 , 1 \\}$ . That is, we ", + "bbox": [ + 160, + 551, + 483, + 566 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "want 66 $f$ to be invariant to the product group $\\{ - 1 , 1 \\} ^ { k }$ . This captures all eigenvector symmetries if the 67 eigenvalues $\\lambda _ { i }$ are distinct. ", + "bbox": [ + 153, + 566, + 825, + 595 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "68 Basis invariance. If the eigenvalues have higher multiplicity, then there are further symmetries. \n69 Let $V _ { 1 } , \\dots , V _ { l }$ be bases of eigenspaces—i.e., $V _ { i } = \\left[ \\hat { v _ { i _ { 1 } } } \\quad \\hat { \\cdot \\cdot } \\quad v _ { i _ { d _ { i } } } \\right] \\in \\mathbb { R } ^ { n \\times d _ { i } }$ has orthonormal \n70 columns and spans the eigenspace associated with the shared eigenvalue $\\mu _ { i } = \\lambda _ { i _ { 1 } } = . . . = \\lambda _ { i _ { d _ { i } } }$ \n71 Any other orthonormal basis that spans the eigenspace is of the form $V _ { i } Q$ for some orthogonal \n72 $Q \\in O ( d _ { i } ) \\subseteq \\mathbb { R } ^ { d _ { i } \\times d _ { i } }$ (see Appendix F.2). Thus, a function $f : \\mathbb { R } ^ { n \\times \\sum _ { i = 1 } ^ { l } d _ { i } } \\mathbb { R } ^ { s }$ that is invariant to \n73 changes of basis in each eigenspace satisfies ", + "bbox": [ + 147, + 601, + 825, + 689 + ], + "page_idx": 1 + }, + { + "type": "equation", + "img_path": "images/f0f6b567ce133e92a758a6e5d1b09cebaa62543f025baefeb1b0e5ff1f3c54f4.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = f ( V _ { 1 } Q _ { 1 } , \\dots , V _ { l } Q _ { l } ) , \\qquad Q _ { i } \\in O ( d _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 316, + 694, + 681, + 712 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "74 In other words, $f$ is invariant to the product group $O ( d _ { 1 } ) \\times \\ldots \\times O ( d _ { l } )$ . The number of eigenspaces \n75 $l$ and the dimensions $d _ { i }$ may vary between matrices; we account for this in Section 2.2. As ${ \\cal { O } } ( 1 ) =$ \n76 $\\{ - 1 , 1 \\}$ , sign invariance is a special case of basis invariance when all eigenvalues are distinct. \n77 Permutation equivariance. For GNN models that output node features or node predictions, one \n78 typically further desires $f$ to be invariant or equivariant to permutations of nodes, i.e., along the entries \n79 (or rows) of each vector. Thus, for $f : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { n \\times d }$ , we typically also require $f ( P V _ { 1 } , \\dots , P V _ { l } ) =$ \n80 $P f ( V _ { 1 } , \\ldots , V _ { l } )$ for any permutation matrix $P \\in \\mathbb { R } ^ { n \\times n }$ . Figure 1 illustrates the full setup. \n81 Graph Positional Encodings. A major motivation for processing eigenvector input is for graph \n82 positional encodings, which are additional features appended to each node in a graph that give \n83 information about the position of that node in the graph. These additional features are crucial for \n84 generalizing Transformers to graphs, and also have been found to improve performance of GNNs. \n85 Figure 2 illustrates a standard pipeline and the use of our SignNet within it: the input adjacency, node \n86 features, and eigenvectors of a graph are used to compute a prediction about the graph. Laplacian \n87 eigenvectors are processed before being fed into this prediction model. Laplacian eigenvectors \n88 have been widely used as positional encodings, and many works have noted that sign and/or basis \n89 invariance must be dealt with in this case [Dwivedi and Bresson, 2021, Beaini et al., 2021, Dwivedi \n90 et al., 2020, Kreuzer et al., 2021, Mialon et al., 2021, Dwivedi et al., 2022]. ", + "bbox": [ + 147, + 717, + 825, + 761 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 147, + 765, + 825, + 823 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 828, + 825, + 912 + ], + "page_idx": 1 + }, + { + "type": "text", + "text": "", + "bbox": [ + 145, + 90, + 825, + 147 + ], + "page_idx": 2 + }, + { + "type": "image", + "img_path": "images/ad0640ccd20008f8d009a7ac580d2e99c68bcb6d30f78b824e2d16ecbc126029.jpg", + "image_caption": [ + "Figure 2: Pipeline for using node positional encodings. After processing by our SignNet, the learned positional encodings from the Laplacian eigenvectors are added as additional node features of an input graph. These positional encodings along with the graph adjacency and original node features are passed to a prediction model (e.g. a GNN). Not shown here, SignNet can also take in eigenvalues and node features if desired. " + ], + "image_footnote": [], + "bbox": [ + 184, + 160, + 815, + 348 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "91 2.1 Warmup: Neural Networks on One Eigenspace ", + "text_level": 1, + "bbox": [ + 147, + 454, + 540, + 470 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "92 Before considering the general setting, we design neural networks that take a single eigenvector or \n93 eigenspace as input and are sign or basis invariant. These single subspace architectures will become \n94 building blocks for the general architectures. For one subspace, a sign invariant function is merely an \n95 even function, and is easily parameterized. ", + "bbox": [ + 147, + 481, + 825, + 536 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proposition 1. A continuous function $h : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }$ is sign invariant if and only if ", + "bbox": [ + 165, + 537, + 715, + 553 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/c4f1adeee5fde41fd8c6f138289e169bd062c2514e42ffc266321c91de96fdd6.jpg", + "text": "$$\nh ( v ) = \\phi ( v ) + \\phi ( - v )\n$$", + "text_format": "latex", + "bbox": [ + 423, + 554, + 573, + 571 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "for some continuous $\\phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }$ . A continuous $h : \\mathbb { R } ^ { n } \\mathbb { R } ^ { n }$ is sign invariant and permutation equivariant if and only $i f$ (3) holds for a continuous permutation equivariant $\\phi : \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { n }$ . ", + "bbox": [ + 160, + 571, + 825, + 602 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "99 In practice, we parameterize $\\phi$ by a neural network. Any architecture choice will ensure sign \n100 invariance, while permutation equivariance can be achieved using elementwise MLPs (Multi-Layer \n101 Perceptrons), DeepSets [Zaheer et al., 2017], Transformers [Vaswani et al., 2017], or GNNs. \n102 Next, we address basis invariance for a single $d$ -dimensional subspace, i.e., we aim to parameterize \n103 maps $h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { n }$ that are (a) invariant to right multiplication by $Q \\in O ( d )$ , and (b) equivariant \n104 to permutations along the row axis. For (a), we use the mapping $V \\mapsto V V ^ { \\top }$ from $V$ to the \n105 orthogonal projector of its column space, which is $O ( d )$ invariant. Mapping $V \\mapsto V V ^ { \\top }$ does not lose \n106 information if we treat $V$ as equivalent to $V Q$ for any $Q \\in O ( d )$ . This is justified by the classical \n107 first fundamental theorem of $O ( d )$ [Kraft and Procesi, 1996], which has recently been applied in \n108 machine learning by Villar et al. [2021]. \n109 Regarding (b), permuting the rows of $V$ permutes rows and columns of $V V ^ { \\top } \\in \\mathbb { R } ^ { n \\times n }$ . Hence, we \n110 desire the function $\\phi : \\mathbb { R } ^ { n \\times n } \\mathbb { R } ^ { n }$ on $\\dot { V } \\dot { V } ^ { \\top }$ to be equivariant to both row and column permutation: \n111 $\\phi ( P V V ^ { \\top } P ^ { \\top } ) = \\dot { P } \\phi ( V V ^ { \\top } )$ . To parameterize such a mapping from matrices to vectors, we use an \n112 invariant graph network (IGN) [Maron et al., 2018]—a neural network mapping to and from tensors \n113 of arbitrary order Rnd1 $\\mathbb { R } ^ { n ^ { d _ { 1 } } } \\to \\mathbb { R } ^ { n ^ { d _ { 2 } } }$ that has the desired permutation equivariance. We thus parameterize \n114 a family with the requisite invariance and equivariance as follows: ", + "bbox": [ + 143, + 609, + 825, + 651 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 657, + 825, + 757 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 761, + 825, + 848 + ], + "page_idx": 2 + }, + { + "type": "equation", + "img_path": "images/f3acd7f63a0210d09241f4eadbfd2eda969c97a4da9a43ca609dfd7e83805e05.jpg", + "text": "$$\nh ( V ) = \\operatorname { I G N } ( V V ^ { \\top } ) .\n$$", + "text_format": "latex", + "bbox": [ + 426, + 849, + 570, + 868 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "115 Proposition 2 states that this architecture universally approximates $O ( d )$ invariant and permutation \n116 equivariant functions. The full approximation power requires high order tensors to be used for the \n117 IGN; in practice, we restrict the tensor dimensions for efficiency, as discussed in the next section. ", + "bbox": [ + 142, + 869, + 825, + 911 + ], + "page_idx": 2 + }, + { + "type": "text", + "text": "Proposition 2. Any continuous, $O ( d )$ invariant $h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { s }$ is of the form $h ( V ) = \\phi ( V V ^ { \\top } )$ for a continuous $\\phi$ . For a compact domain ${ \\mathcal { Z } } \\subseteq \\mathbb { R } ^ { n \\times d }$ , maps of the form $V \\mapsto \\operatorname { I G N } ( V V ^ { \\top } )$ universally approximate continuous $h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n }$ that are $O ( d )$ invariant and permutation equivariant. ", + "bbox": [ + 168, + 90, + 825, + 137 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "2.2 Neural Networks on Multiple Eigenspaces ", + "text_level": 1, + "bbox": [ + 173, + 151, + 504, + 166 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "Next, we use the single-eigenspace models $\\phi _ { l } ( V _ { l } )$ as building blocks for a model on multiple eigenspaces. To do so, we use functions of the form $f ( V _ { 1 } , \\dots , V _ { l } ) = \\rho ( \\phi _ { 1 } ( V _ { 1 } ) , \\dots , \\phi _ { l } ( V _ { l } ) )$ , i.e., we first process each eigenspace individually with an invariant network, and then aggregate them via a function $\\rho$ . This approach is grounded in a general decomposition theorem for product spaces that we prove in Section A. ", + "bbox": [ + 173, + 176, + 825, + 247 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "SignNet. We parameterize our sign invariant network 127 $f : \\mathbb { R } ^ { n \\times k } \\mathbb { R } ^ { s }$ on eigenvectors $v _ { 1 } , \\ldots , v _ { k }$ ", + "bbox": [ + 151, + 251, + 820, + 267 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/1ea7826faaf2681e322edd4b290fff85a124b1d40ba1f09d3b795822159763de.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } ) = \\rho \\left( [ \\phi ( v _ { i } ) + \\phi ( - v _ { i } ) ] _ { i = 1 } ^ { k } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 356, + 272, + 638, + 292 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "128 where $\\phi$ and $\\rho$ are unrestricted neural networks, and $[ \\cdot ] _ { i }$ denotes concatenation of vectors. The form \n129 $\\phi ( v _ { i } ) + \\phi ( - v _ { i } )$ induces sign invariance for each eigenvector. Since we do not yet impose permutation \n130 equivariance here, we term this model Unconstrained-SignNet. \n131 To obtain a sign invariant and permutation equivariant $f$ that outputs vectors in $\\mathbb { R } ^ { n \\times s }$ , we restrict $\\phi$ \n132 and $\\rho$ to be permutation equivariant networks from vectors to vectors, such as elementwise MLPs, \n133 DeepSets [Zaheer et al., 2017], Transformers [Vaswani et al., 2017], or most standard GNNs. We \n134 name this permutation equivariant version SignNet. If desired, we can additionally use eigenvalues $\\lambda _ { i }$ \n135 and node features $\\ b X \\in \\bar { \\mathbb { R } } ^ { n \\times q }$ by adding them as arguments to $\\phi$ : ", + "bbox": [ + 142, + 296, + 825, + 339 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 344, + 825, + 415 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/aa444a28c68a4ea38a4af57b5fc197503915d1da8f2cbc977140c0cb6b0262a8.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } , \\lambda _ { 1 } , \\dots , \\lambda _ { k } , X ) = \\rho \\left( [ \\phi ( v _ { i } , \\lambda _ { i } , X ) + \\phi ( - v _ { i } , \\lambda _ { i } , X ) ] _ { i = 1 } ^ { k } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 263, + 420, + 735, + 440 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "BasisNet. For basis invariance, let 136 $V _ { i } \\ \\in \\ \\mathbb { R } ^ { n \\times d _ { i } }$ be an orthonormal basis of a $d _ { i }$ dimensional 137 eigenspace. Then we parameterize our Unconstrained-BasisNet $f$ by ", + "bbox": [ + 142, + 452, + 825, + 482 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/8577a7905135d7b3057bbc5bb4809e3e85555d47df75d6b896683acde4140466.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( [ \\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } ) ] _ { i = 1 } ^ { l } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 370, + 486, + 625, + 506 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "138 where each $\\phi _ { d _ { i } }$ is shared amongst all subspaces of the same dimension $d _ { i }$ , and $l$ is the number of \n139 eigenspaces (i.e., number of distinct eigenvalues, which can differ from the number of eigenvectors \n140 $k$ ). As $l$ differs between graphs, we may use zero-padding or a sequence model like a Transformer to \n141 parameterize $\\rho$ . Again, $\\phi _ { d _ { i } }$ and $\\rho$ are generally unrestricted neural networks. To obtain permutation \n142 equivariance, we make $\\rho$ permutation equivariant and let $\\phi _ { d _ { i } } = \\mathrm { I G N } _ { d _ { i } } : \\mathbb { R } ^ { n ^ { 2 } } \\mathbb { R } ^ { n }$ be IGNs from \n143 matrices to vectors. For efficiency, we will only use matrices and vectors in the IGNs (that is, no \n144 tensors in $\\mathbb { R } ^ { n ^ { p } }$ for $p > 2$ ), i.e., we use 2-IGN. Our resulting BasisNet is ", + "bbox": [ + 140, + 508, + 825, + 612 + ], + "page_idx": 3 + }, + { + "type": "equation", + "img_path": "images/75f36c163e73caac170519a4b10dc9fcecc8b5ab4618f9981ed9aa5bbc273725.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( [ \\mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } ) ] _ { i = 1 } ^ { l } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 361, + 616, + 635, + 636 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "145 Expressive-BasisNet. While we restrict SignNet to only use vectors and BasisNet to only use vectors \n146 and matrices, higher order tensors are generally required for universally approximating permutation \n147 equivariant or invariant functions [Keriven and Peyré, 2019, Maron et al., 2019, Maehara and NT, \n148 2019]. Thus, we will consider a theoretically powerful but computationally impractical variant of \n149 our model, in which we replace $\\rho$ and $\\mathrm { I G N } _ { d _ { i } }$ in BasisNet with IGNs of arbitrary tensor order. We \n150 call this variant Expressive-BasisNet. Universal approximation requires $\\Omega ( n ^ { n } )$ sized intermediate \n151 tensors [Ravanbakhsh, 2020]. We study Expressive-BasisNet due to its theoretical interest, and to \n152 juxtapose with the computational efficiency and strong expressive power of SignNet and BasisNet. ", + "bbox": [ + 140, + 638, + 825, + 751 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "For a summary of properties and more details about our models, see Appendix B. ", + "bbox": [ + 166, + 757, + 705, + 771 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "154 In the multiple subspace case, we can prove universality of our models through a general decomposi \n155 tion theorem, which reduces the multiple subspace case to the single subspace case. See Section A \n156 for details; we have temporarily moved this Section in the revision due to space constraints, and we \n157 will move this Section into the main paper in the camera-ready version. ", + "bbox": [ + 140, + 777, + 826, + 834 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "158 3 Theoretical Power for Graph Representation Learning ", + "text_level": 1, + "bbox": [ + 147, + 852, + 661, + 869 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "159 Next, we establish that our SignNet and BasisNet can compute useful basis invariant and permutation \n160 equivariant functions on Laplacian eigenvectors for graph representation learning, including: spectral ", + "bbox": [ + 147, + 882, + 823, + 912 + ], + "page_idx": 3 + }, + { + "type": "text", + "text": "graph convolutions, spectral invariants, and existing graph positional encodings. Expressive-BasisNet can of course compute these functions, as it is universal, but this section shows that the practical invariant architectures SignNet and BasisNet can compute them as well. ", + "bbox": [ + 147, + 92, + 826, + 133 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "164 3.1 SignNets and BasisNets Generalize Spectral Graph Convolution ", + "text_level": 1, + "bbox": [ + 150, + 154, + 656, + 170 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "For node features $X \\in \\mathbb { R } ^ { n \\times q }$ and an eigendecomposition $V \\Lambda V ^ { \\top }$ , a spectral graph convolution takes the form $\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X = V \\bar { \\mathrm { D i a g } } ( \\theta ) V ^ { \\top } X . } \\end{array}$ , for some parameters $\\theta _ { i }$ , that may optionally be continuous functions $\\bar { h } ( \\lambda _ { i } ) = \\theta _ { i }$ of the eigenvalues [Bruna et al., 2014, Defferrard et al., 2016]. This family includes important functions like heat kernels and generalized PageRanks on graphs [Li et al., 2019]. A spectral GNN is defined as multiple layers of spectral graph convolutions and node-wise linear maps, e.g. $\\begin{array} { r } { V \\mathrm { D i a g } ( \\theta _ { 2 } ) V ^ { \\top } \\sigma \\left( V \\mathrm { D i a g } ( \\bar { \\theta } _ { 1 } ) V ^ { \\top } X W _ { 1 } \\right) } \\end{array}$ $W _ { 2 }$ is a two layer spectral GNN. It can be seen (in Appendix H.1) that spectral graph convolutions are permutation equivariant and sign invariant, and if $\\theta _ { i } = h ( \\lambda _ { i } )$ (i.e. the spectral graph convolution is parametric) they are additionally invariant to a change of bases in each eigenspace. ", + "bbox": [ + 173, + 181, + 825, + 308 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Our SignNet and BasisNet can be viewed as generalizations of spectral graph convolutions, as our networks can universally approximate all spectral graph convolutions of the above form. For instance, SignNet with $\\begin{array} { r } { \\rho ( a _ { 1 } , \\ldots , a _ { k } ) = \\sum _ { i = 1 } ^ { k } a _ { k } } \\end{array}$ and $\\begin{array} { r } { \\phi ( v _ { i } , \\lambda _ { i } , X ) = \\frac { 1 } { 2 } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X } \\end{array}$ directly yields the spectral graph convolution. This is captured in Theorem 1, which we prove in Appendix H.1. In fact, we may expect SignNet to learn spectral graph convolutions well, according to the principle of algorithmic alignment [Xu et al., 2020] (see Appendix H.1); this is supported by numerical experiments in Appendix J.2, in which our networks outperform baselines in learning spectral graph convolutions. ", + "bbox": [ + 142, + 313, + 825, + 412 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 1. SignNet universally approximates all spectral graph convolutions. BasisNet universally approximates all parametric spectral graph convolutions. ", + "bbox": [ + 163, + 419, + 821, + 448 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "In fact, SignNet and BasisNet are strictly stronger than spectral graph convolutions; there are functions computable by SignNet and BasisNet that cannot be approximated by spectral graph convolutions or spectral GNNs. One way to see this is through graph isomorphism power, as captured in this next result. ", + "bbox": [ + 171, + 460, + 825, + 517 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish. ", + "bbox": [ + 165, + 523, + 823, + 553 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "3.2 BasisNets can Compute Spectral Invariants ", + "text_level": 1, + "bbox": [ + 171, + 573, + 514, + 588 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Many works measure the expressive power of graph neural networks by comparing their power for testing graph isomorphism [Xu et al., 2019, Sato, 2020], or by comparing their ability to compute certain functions on graphs like subgraph counts [Chen et al., 2020, Tahmasebi et al., 2020]. These works often compare GNNs to combinatorial invariants on graphs, especially the $k$ -Weisfeiler-Lehman $k$ -WL) tests of graph isomorphism [Morris et al., 2021]. ", + "bbox": [ + 171, + 599, + 825, + 670 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "While we may also compare with these combinatorial invariants, as other GNN works that use spectral information have done [Beaini et al., 2021], we argue that it is more natural to analyze our networks in terms of spectral invariants, which are computed from the eigenvalues and eigenvectors of graphs. There is a rich literature of spectral invariants from the fields of spectral graph theory and complexity theory [Cvetkovic et al. ´ , 1997]. A spectral invariant must be invariant to permutations and changes of basis in each eigenspace, a characteristic shared by our networks. ", + "bbox": [ + 140, + 676, + 825, + 760 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "The simplest spectral invariant is the multiset of eigenvalues, which we give as input to our networks. Another widely studied, powerful spectral invariant is the collection of graph angles, which are defined as the values ${ \\alpha _ { i j } } ^ { \\star } = \\| V _ { i } V _ { i } ^ { \\top } e _ { j } \\| _ { 2 }$ , where $V _ { i } \\in \\mathbb { R } ^ { n \\times d _ { i } }$ is an orthonormal basis for the ith adjacency matrix eigenspace, and $e _ { j }$ is the $j$ th standard basis vector, which is zero besides a one in the $j$ th component. These are easily computed by our networks (Appendix H.3), so our networks inherit the strength of these invariants. We capture these results in the following theorem, which also lists a few properties that graph angles determine [Cvetkovic´, 1991]. ", + "bbox": [ + 142, + 765, + 825, + 863 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "Theorem 2. BasisNet universally approximates the graph angles $\\alpha _ { i j }$ . The eigenvalues and graph angles (and thus BasisNet) can determine the number of length 3, 4, or 5 cycles, whether a graph is connected, and the number of length $k$ closed walks from any vertex to itself. ", + "bbox": [ + 166, + 869, + 823, + 911 + ], + "page_idx": 4 + }, + { + "type": "text", + "text": "211 Relation to WL and message passing. In contrast to this result, message passing GNNs are not able \n212 to express any of these properties (see [Arvind et al., 2020, Garg et al., 2020] and Appendix H.3). \n213 Although spectral invariants are strong, Fürer [2010] shows that the eigenvalues and graph angles—as \n214 well as some strictly stronger spectral invariants—are not stronger than the 3-WL test (or, equivalently, \n215 the 2-Folklore-WL test). Future work could study the combination of spectral invariants or spectral \n216 graph positional encodings with combinatorial algorithms and graph neural networks. ", + "bbox": [ + 140, + 92, + 825, + 175 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "3.3 SignNets and BasisNets Generalize Existing Graph Positional Encodings ", + "text_level": 1, + "bbox": [ + 168, + 190, + 715, + 205 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Many graph positional encodings have been proposed, without any clear criteria on which to choose for a particular task. We prove (in Appendix H.2) that our efficient SignNet and BasisNet can universally approximate many previously used graph positional encodings, because we unify these positional encodings by expressing them as either a spectral graph convolution matrix or the diagonal of a spectral graph convolution matrix. ", + "bbox": [ + 174, + 215, + 825, + 285 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally approximates diffusion and $p$ -step random walk relative positional encodings [Mialon et al., 2021], and generalized PageRank and landing probability distance encodings [Li et al., 2020]. ", + "bbox": [ + 174, + 289, + 825, + 344 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We note that diagonals of spectral convolutions are used as feature descriptors in the shape analysis literature, such as the heat kernel signature [Sun et al., 2009] and wave kernel signature [Aubry et al., 2011]. In the language of recent works in graph machine learning, these are node positional encodings computed from a discrete Laplacian of a triangle mesh. This connection appears to be unnoticed in recent works on graph positional encodings. ", + "bbox": [ + 174, + 356, + 825, + 426 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "4 Experiments ", + "text_level": 1, + "bbox": [ + 173, + 444, + 312, + 462 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "We demonstrate the strength of our networks in various experiments. Appendix B shows simple pseudo-code and a diagram detailing the use of SignNet as a node positional encoding. ", + "bbox": [ + 173, + 476, + 823, + 503 + ], + "page_idx": 5 + }, + { + "type": "text", + "text": "235 4.1 Graph Regression ", + "text_level": 1, + "bbox": [ + 151, + 520, + 338, + 535 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/42cf492ac5ed14df2c684275670552af9f49ce41dca3facabe4e0000ffdbade7.jpg", + "table_caption": [ + "Table 1: Results on the ZINC dataset with a $5 0 0 \\mathrm { k }$ parameter budget. All models use edge features. Numbers are the mean and standard deviation over 4 runs, each with different seeds. " + ], + "table_footnote": [ + "236 We study the effectiveness of SignNet for learning positional encodings (PEs) from the eigenvectors 237 of the graph Laplacian on the ZINC dataset of molecule graphs [Irwin et al., 2012] (using the " + ], + "table_body": "
Base modelPositional encodingk#paramTest MAE (↓)
GatedGCNNo PEN/A492k0.252±0.007
LapPE (flip)8492k0.198±0.011
LapPE (abs.)8492k0.204±0.009
LapPE (can.)8505k0.298±0.019
SignNet (𝜙(u) only)8495k0.148±0.007
SignNet8495k0.121±0.005
SignNetAll491k0.100±0.007
Sparse TransformerNo PEN/A473k0.283±0.030
LapPE (flip)16487k0.223±0.007
SignNet16479k0.115±0.008
SignNetAll486k0.102±0.005
GINENo PEN/A470k0.170±0.002
LapPE (flip)16470k0.178±0.004
SignNet16470k0.147±0.005
SignNetAll417k0.102±0.002
PNANo PEN/A474k
LapPE (flip)8474k0.133±0.011 0.132±0.010
SignNet8476k0.105±0.007
SignNetAll487k0.084±0.006
", + "bbox": [ + 250, + 587, + 740, + 876 + ], + "page_idx": 5 + }, + { + "type": "table", + "img_path": "images/1ae44f6e51937102d2bf7933f5bb08d8fc5f52690a56d814abf46bbd1e712b05.jpg", + "table_caption": [ + "Table 2: Comparison with SOTA methods on graph-level regression tasks. $\\dagger$ denotes domain-specific model. Numbers are test MAE, so lower is better. Best models within a standard deviation are bolded. " + ], + "table_footnote": [], + "table_body": "
ZINC (10K)↓ZINC-full ↓Alchemy (10k)↓
HIMP † [Fey et al., 2020].151±.006.036±.002
CIN-small † [Bodnar et al., 2021].094±.004.044±.003
CIN† [Bodnar et al., 2021].079±.006.022±.002
GIN [Xu et al., 2019].170±.002.088±.002.180±.006
δ-2-GNN[Morris et al., 2020b].374±.022.042±.003.118±.001
δ-2-LGNN[Morris etal., 2020b].306±.044.045±.006.122±.003
SpeqNet [Morris et al., 2022].115±.001
GNN-IR [Dupty and Lee, 2022].137±.010.119±.002
PF-GNN [Dupty et al., 2021].122±.01.111±.01
Recon-GNN [Cotta et al., 2021].170±.006.125±.001
SignNet (ours).084±.006.024±.003.113±.002
", + "bbox": [ + 228, + 126, + 764, + 306 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "238 subset of 12,000 graphs from Dwivedi et al. [2020]). We primarily consider three settings: 1) No \n239 positional encoding, 2) Laplacian PE (LapPE)—the $k$ eigenvectors of the graph Laplacian with \n240 smallest eigenvalues are concatenated with existing node features, 3) SignNet positional features— \n241 passing the eigenvectors through a SignNet and concatenating the output with node features. We \n242 parameterize SignNet by taking $\\phi$ to be a GIN $\\mathrm { [ X u }$ et al., 2019] and $\\rho$ to be an MLP. We sum over $\\phi$ \n243 outputs before the MLP when handling variable numbers of eigenvectors, so then the SignNet is of \n244 the form MLP $\\begin{array} { r } { \\left( \\sum _ { i = 1 } ^ { l } \\phi ( v _ { i } ) + \\phi ( - v _ { i } ) \\right) } \\end{array}$ (see Appendix K.2 for further details). We consider four \n245 different base models that process the graph data and positional encodings: GatedGCN [Bresson and \n246 Laurent, 2017], a Transformer with sparse attention only over neighbours [Kreuzer et al., 2021], PNA \n247 [Corso et al., 2020], and GIN [Xu et al., 2019] with edge features (i.e. GINE) [Hu et al., 2020b]. The \n248 total number of parameters of the SignNet and the base model is kept within a 500k budget. \n249 Table 1 shows the results. For all 4 base models, the PE learned with SignNet yields the best test MAE \n250 (mean absolute error) — lower MAE is better. Notably, this includes the cases of PNA and GINE, for \n251 which Laplacian PE with simple random sign flipping was unable to improve performance over using \n252 no PE at all. Our best performing model is PNA base combined with SignNet, which achieves 0.084 \n253 test MAE. Besides SignNet, we consider two non-learned approaches to resolving eigenvector sign \n254 ambiguity—canonicalization and taking element-wise absolute values (see Appendix K.2 for details). \n255 Results with GatedGCN show that these alternatives are not more effective than random sign flipping \n256 for learning positional encodings. We also consider an ablation of our SignNet architecture where we \n257 remove the sign invariance, using simply $\\mathrm { M L P } ( [ \\phi ( v _ { i } ) ] _ { i = 1 } ^ { k } )$ . Although the resulting architecture is no \n258 longer sign invariant, $\\phi$ still processes eigenvectors independently, meaning that only two invariances \n259 $( \\pm 1 )$ need be learned, significantly fewer than the $2 ^ { k }$ total sign flip configurations. Accordingly, this \n260 non-sign invariant learned positional encoding achieves a test MAE of 0.148, improving over the \n261 Laplacian PE (0.198) but falling short of the fully sign invariant SignNet (0.121). In all cases, using \n262 all available eigenvectors in SignNet significantly improves performance over using a fixed number \n263 of eigenvectors. In Appendix J.1, we also show that SignNet improves performance when no edge \n264 features are included in the data. \n65 These significant performance improvements from SignNet come with only a slightly higher compu \n66 tational cost. For example, GatedGCN with no PE takes about 8.2 seconds per training iteration on \n67 ZINC, while GatedGCN with 8 eigenvectors and SignNet takes about 10.6 seconds; this is only a \n68 $2 9 \\%$ increase in time, for a reduction of test MAE by over $50 \\%$ . Also, eigenvector computation time \n69 is neglible, we need only precompute and save the eigenvectors once, and it only takes 15 seconds to \n70 do this for the 12,000 graphs of ZINC. ", + "bbox": [ + 140, + 343, + 826, + 505 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 511, + 826, + 732 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 738, + 825, + 821 + ], + "page_idx": 6 + }, + { + "type": "text", + "text": "Comparison with SOTA. In Table 2, we compare SignNet with state-of-the-art methods on graphlevel molecular regression tasks on ZINC (10,000 training graphs), ZINC-full (about 250,000 graphs), and Alchemy [Chen et al., 2019a] (10,000 training graphs). We compare against both methods that use domain-specific knowledge about molecules, and domain-agnostic GNNs of various architectures. We see that SignNet outperforms all domain-agnostic methods on ZINC and ZINC-full, and is within a standard deviation of the best domain-specific method. Our mean score is the second best on ", + "bbox": [ + 168, + 828, + 825, + 911 + ], + "page_idx": 6 + }, + { + "type": "table", + "img_path": "images/abe247a80accc3ce19507fc13c2a1a933858761dd6388c148aa457d37acfc5bb.jpg", + "table_caption": [ + "Table 3: Test results for texture reconstruction experiment on cat and human models, following the experimental setting of [Koestler et al., 2022]. We use 1023 eigenvectors of the cotangent Laplacian. " + ], + "table_footnote": [], + "table_body": "
CatHuman
MethodParamsPSNR↑DSSIM↓LPIPS↓PSNR↑DSSIM↓LPIPS↓
Intrinsic NF329k34.25.099.18932.29.119.330
Absolute value329k34.67.106.25232.42.132.363
Sign flip329k23.151.282.3521.521.052.71
SignNet324k34.91.090.14732.43.125.316
", + "bbox": [ + 184, + 126, + 807, + 223 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "277 Alchemy, and is within a standard deviation of the best. We perform much better on ZINC (.084) than \n278 other state-of-the-art positional encoding methods, like GNN-LSPE (.090) [Dwivedi et al., 2022], \n279 SAN (.139) [Kreuzer et al., 2021], and Graphormer (.122) [Ying et al., 2021]. \n281 Substructure counts (e.g. of cycles) and global graph properties (e.g. connectedness, diameter, \n282 radius) are important graph features that are known to be informative for problems in bio- and \n283 chemo-informatics [Chen et al., 2020, Corso et al., 2020]. Following the setting of Zhao et al. [2022], \n284 we show that SignNet with Laplacian positional encodings boosts the ability of simple GNNs to \n285 count substructures and regress graph properties. We take a 4-layer GIN as the base model for all \n286 settings, and for SignNet we use GIN as $\\phi$ and a Transformer as $\\rho$ to handle variable numbers of \n287 eigenvectors (see Appendix K.4 for details). As shown in Figure 3, Laplacian PEs with sign-flip data \n288 augmentation improve performance for counting substructures but not for regressing graph properties, \n289 while Laplacian PEs processed by SignNet significantly boost performance on all tasks. ", + "bbox": [ + 142, + 257, + 825, + 299 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/1e38fc2f4e8186c11a80783c813f9adaac57f04e2ec49b004204af52ec8b9970.jpg", + "image_caption": [ + "280 4.2 Counting Substructures and Regressing Graph Properties ", + "Figure 3: Counting substructures and regressing graph properties (lower is better). With Laplacian PEs, SignNet improves performance, while sign flip data augmentation (LapPE) is less consistent. Mean and standard deviations are reported on 3 runs. All runs use the same 4-layer GIN base model. " + ], + "image_footnote": [], + "bbox": [ + 192, + 366, + 805, + 452 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 518, + 825, + 643 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "290 4.3 Neural Fields on Manifolds ", + "text_level": 1, + "bbox": [ + 147, + 667, + 401, + 683 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Discrete approximations to the Laplace-Beltrami operator on manifolds have proven useful for processing data on surfaces, such as triangle meshes [Lévy, 2006]. Recently, Koestler et al. [2022] propose intrinsic neural fields, which use eigenfunctions of the Laplace-Beltrami operator as positional encodings for learning neural fields on manifolds. For generalized eigenfunctions $v _ { 1 } , \\ldots , v _ { k }$ , at a point $p$ on the surface, they parameterize functions $f ( \\boldsymbol { p } ) = \\mathrm { M L P } ( v _ { 1 } ( \\boldsymbol { p } ) , \\dots , v _ { k } ( \\boldsymbol { p } ) )$ . As these eigenfunctions have sign ambiguity, we use our SignNet to parameterize $f ( \\boldsymbol { p } ) ^ { \\prime } = \\mathrm { M L P } ( \\rho ( \\operatorname { } [ \\phi ( v _ { i } ( \\boldsymbol { p } ) ) +$ $\\phi \\bar { ( - v _ { i } ( p ) ) } ] _ { i = 1 , \\ldots , k } )$ ), with $\\rho$ and $\\phi$ being MLPs. ", + "bbox": [ + 171, + 696, + 825, + 795 + ], + "page_idx": 7 + }, + { + "type": "text", + "text": "Table 3 shows our results for texture reconstruction experiments on all models from Koestler et al. [2022]. The total number of parameters in our SignNet-based model is kept below that of the original model. We see that the SignNet architecture improves over the original Intrinsic NF model and over other baselines — especially in the LPIPS (Learned Perceptual Image Patch Similarity) metric, which has been shown to be a typically better perceptual metric than PSNR or DSSIM [Zhang et al., 2018a]. While we have not yet tested this, we believe that SignNet would allow even better improvements when learning over eigenfunctions of different models, as it could improve transfer and generalization. See Appendix D.1 for visualizations and Appendix K.5 for more details. ", + "bbox": [ + 158, + 800, + 825, + 911 + ], + "page_idx": 7 + }, + { + "type": "image", + "img_path": "images/2da6c2113b5fda8aff05b634b7425ce6f343ab9c2c035c184728c3df7b902400.jpg", + "image_caption": [ + "Figure 4: Cotangent Laplacian eigenvectors of the cat model and first principal component of ${ \\phi ( \\bar { v } ) + \\phi ( - v ) }$ from our trained SignNet. " + ], + "image_footnote": [], + "bbox": [ + 207, + 85, + 802, + 188 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "306 4.4 Visualization of Learned Positional Encodings ", + "text_level": 1, + "bbox": [ + 147, + 252, + 532, + 268 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "To better understand SignNet, we plot the first principal component of $\\phi ( v ) + \\phi ( - v )$ for two eigenvectors on the cat model in Figure 4. We see that SignNet encodes bilateral symmetry and structural information on the cat model. See Appendix D for plots of more eigenvectors and further details. ", + "bbox": [ + 173, + 279, + 825, + 334 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "5 Related Work ", + "text_level": 1, + "bbox": [ + 171, + 353, + 321, + 371 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "In this section, we review selected related work. A more thorough review is deferred to Appendix E. ", + "bbox": [ + 161, + 383, + 823, + 398 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Laplacian eigenvectors in GNNs. Various recently proposed methods in graph deep learning have directly used Laplacian eigenvectors as node positional encodings that are input to a neural network that is, e.g., a message passing GNN [Dwivedi et al., 2020, 2022], or some variant of a Transformer that is adapted to graphs [Dwivedi and Bresson, 2021, Kreuzer et al., 2021, Mialon et al., 2021, Dwivedi et al., 2022]. None of these methods address basis invariance, and they only partially address sign invariance for node positional encodings by randomly flipping eigenvector signs during training. ", + "bbox": [ + 173, + 405, + 825, + 489 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "Graph positional encodings. Other recent methods use positional encodings besides Laplacian eigenvectors. These include positional encodings based on random walks [Dwivedi et al., 2022, Mialon et al., 2021, Li et al., 2020], diffusion kernels on graphs [Mialon et al., 2021, Feldman et al., 2022], shortest paths [Ying et al., 2021, Li et al., 2020], and unsupervised node embedding methods [Wang et al., 2022]. In particular, Wang et al. [2022] use Laplacian eigenvectors for relative positional encodings in an invariant way, but they focus on robustness, so they have stricter invariances that significantly reduce expressivity (see Appendix E.2 for more details). These previously used positional encodings are mostly ad-hoc, less general since they can be provably expressed by SignNet and BasisNet (see Section 3.3), and/or are expensive to compute (e.g., all pairs shortest paths). ", + "bbox": [ + 173, + 494, + 825, + 619 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "6 Conclusion and Discussion ", + "text_level": 1, + "bbox": [ + 163, + 638, + 428, + 655 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "SignNet and BasisNet are novel architectures for processing eigenvectors that are invariant to sign flips and choices of eigenspace bases, respectively. Both architectures are provably universal: they can represent any continuous function with the corresponding invariances. When used with Laplacian eigenvectors as inputs they can provably approximate spectral graph convolutions, spectral invariants, graph properties such as subgraph counts, and a number of other graph positional encodings. These theoretical results are supported by experiments showing that SignNet and BasisNet are highly expressive in practice, and learn effective graph positional encodings that improve the performance of message passing graph neural networks. Initial explorations show that SignNet and BasisNet can be useful beyond graph representation learning, as eigenvectors are ubiquitous. ", + "bbox": [ + 171, + 669, + 825, + 795 + ], + "page_idx": 8 + }, + { + "type": "text", + "text": "338 While we conduct experiments on graph machine learning tasks and a particular task on triangle \n339 meshes, SignNet and BasisNet should also be applicable to processing eigenvectors in other settings, \n340 such as recommender systems and tasks in shape analysis. We show significant empirical benefit in \n341 the tasks that we consider, but we expect less benefit in cases where node features are sufficient to do \n342 well on the task, or if the task does not require much sophisticated graph structure information to \n343 solve. 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", + "bbox": [ + 158, + 47, + 828, + 917 + ], + "page_idx": 9 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 42, + 828, + 920 + ], + "page_idx": 10 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 44, + 828, + 915 + ], + "page_idx": 11 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 65, + 830, + 921 + ], + "page_idx": 12 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 63, + 828, + 919 + ], + "page_idx": 13 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 56, + 828, + 921 + ], + "page_idx": 14 + }, + { + "type": "text", + "text": "", + "bbox": [ + 151, + 90, + 826, + 271 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "Checklist ", + "text_level": 1, + "bbox": [ + 168, + 295, + 253, + 310 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "1. For all authors... ", + "bbox": [ + 214, + 320, + 339, + 334 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] We support the claims with theoretical and/or empirical evidence. \n(b) Did you describe the limitations of your work? [Yes] We discuss some limitations in the conclusion. \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Appendix B.2. \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] We have read the guidelines; our paper conforms to them. ", + "bbox": [ + 238, + 338, + 825, + 469 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "2. If you are including theoretical results... ", + "bbox": [ + 215, + 473, + 493, + 488 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all assumptions either in the main text or in the appendix. \n(b) Did you include complete proofs of all theoretical results? [Yes] We include all proofs in the appendix. ", + "bbox": [ + 238, + 492, + 825, + 549 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "3. If you ran experiments... ", + "bbox": [ + 214, + 553, + 393, + 568 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We include experimental code and instructions on the usage. \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] We try to give most experimental details in the main paper and appendix. \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We report standard deviations for multiple runs and/or seeds for graph-level tasks, but not the texture reconstruction task. \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix K.1. ", + "bbox": [ + 238, + 571, + 825, + 729 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ", + "bbox": [ + 214, + 733, + 823, + 747 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We cite the creators of software libraries, data, and machine learning models. \n(b) Did you mention the license of the assets? [Yes] See Appendix K.1. \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We provide our code. We will also link to an open source version after anonymous review. \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes] See Appendix K.1. \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes] Yes, we discuss this in Appendix K.1; the data most likely does not have personally identifiable information or offensive content. ", + "bbox": [ + 238, + 751, + 825, + 911 + ], + "page_idx": 15 + }, + { + "type": "text", + "text": "5. If you used crowdsourcing or conducted research with human subjects... ", + "bbox": [ + 212, + 90, + 704, + 106 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A] No crowdsourcing or human subjects used. \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A] No crowdsourcing or human subjects used. \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] No crowdsourcing or human subjects used. ", + "bbox": [ + 236, + 109, + 826, + 213 + ], + "page_idx": 16 + }, + { + "type": "text", + "text": "We have temporarily moved this section to the Appendix due to space constraints, we will move it back to the main paper in the camera-ready version. ", + "bbox": [ + 165, + 121, + 823, + 150 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "While the networks introduced in the Section 2.2 possess the desired invariances, it is not immediately obvious whether they are powerful enough to express all functions with these invariances. The universality of our architectures follows as a corollary of the following general decomposition result, which may enable construction of universal architectures for other invariances as well. ", + "bbox": [ + 166, + 155, + 825, + 210 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Theorem 3 (Decomposition Theorem). Let $\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }$ be topological spaces, and let $G _ { i }$ be $a$ group acting on $\\mathcal { X } _ { i }$ for each $i$ . We assume mild topological conditions on $\\mathcal { X } _ { i }$ and $G _ { i }$ hold. For any continuous $f : \\mathcal { X } = \\mathcal { X } _ { 1 } \\times . . . \\times \\mathcal { X } _ { k } \\to \\mathbb { R } ^ { s }$ that is invariant to the action of $G = G _ { 1 } \\times \\ldots \\times G _ { k }$ there exists continuous $\\phi _ { i }$ and a continuous $\\rho : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { a } \\to \\mathbb { R } ^ { s }$ such that ", + "bbox": [ + 173, + 214, + 825, + 271 + ], + "page_idx": 17 + }, + { + "type": "equation", + "img_path": "images/52a28eeb95bf06159658bd6649482b5e733f38e693dbf5258547b889cbd20bd8.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } ) = \\rho ( \\phi _ { 1 } ( v _ { 1 } ) , \\dots , \\phi _ { k } ( v _ { k } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 366, + 279, + 632, + 295 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Furthermore:693 $( l )$ each $\\phi _ { i }$ can be taken to be invariant to $G _ { i }$ , (2) the domain $\\mathcal { Z }$ of $\\rho$ is compact if each 694 $\\mathcal { X } _ { i }$ is compact, (3) if ${ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , then $\\phi _ { i }$ can be taken to be equal to $\\phi _ { j }$ . ", + "bbox": [ + 148, + 303, + 825, + 332 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "695 This result says that when a product of groups $G$ acts on a product of spaces $\\mathcal { X }$ , for invariance to the \n696 product group $G$ it suffices to individually process each smaller group $G _ { i }$ on $\\mathcal { X } _ { i }$ and then aggregate \n697 the results. Along with the proof of Theorem 3, the mild topological assumptions are explained in \n698 Appendix G.1. The assumptions hold for sign invariance and basis invariance. By applying this \n699 theorem, we can prove universality of our networks: ", + "bbox": [ + 142, + 342, + 825, + 412 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "Corollary 1. Unconstrained-SignNet can represent any sign invariant function and UnconstrainedBasisNet can represent any basis invariant function. Expressive-BasisNet is a universal approximator of functions that are both basis invariant and permutation equivariant. ", + "bbox": [ + 163, + 416, + 823, + 459 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "703 This result shows that Unconstrained-SignNet, Unconstrained-BasisNet, and Expressive-BasisNet \n704 take the correct functional form for their respective invariances (proofs in Appendix G.2). Note \n705 that Expressive-BasisNet approximates all sign invariant functions as a special case, by treating \n706 all inputs as one dimensional eigenspaces. Accompanying the decomposition result, we show a \n707 corresponding universal approximation result (proof in Appendix G.3). Similarly to Theorem 3, \n708 the problem of approximating $G = G _ { 1 } \\times \\ldots \\times G _ { k }$ invariant functions is reduced to approximating \n709 several $G _ { i }$ -invariant functions. ", + "bbox": [ + 140, + 469, + 826, + 568 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "710 B More Details on SignNet and BasisNet ", + "text_level": 1, + "bbox": [ + 140, + 587, + 527, + 604 + ], + "page_idx": 17 + }, + { + "type": "table", + "img_path": "images/fd91af5dea8e13281771b67064ca52dc07399f40fa8c876644fac3c009e14b76.jpg", + "table_caption": [ + "Table 4: Properties of our architectures: Unconstrained-SignNet, SignNet, Unconstrained-BasisNet, and Expressive-BasisNet. The properties are: permutation equivariance, universality (for the proper class of continuous invariant functions), and computational tractability. " + ], + "table_footnote": [], + "table_body": "
Unconstr.-SignNetSignNetUnconstr.-BasisNetBasisNetExpr.-BasisNet
Permutation equiv.×√√×√x√√
Universal×√
Tractable√×
", + "bbox": [ + 174, + 676, + 820, + 743 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "711 In Figure 2, we show a diagram that describes how SignNet is used as a node positional encoding \n712 for a graph machine learning task. In Table 4, we compare and contrast properties of the neural \n713 architectures that we introduce. In Figure 5, we give pseudo-code of SignNet for learning node \n714 positional encodings with a GNN prediction model. ", + "bbox": [ + 142, + 757, + 823, + 814 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "715 B.1 Generalization Beyond Symmetric Matrices ", + "text_level": 1, + "bbox": [ + 142, + 829, + 521, + 844 + ], + "page_idx": 17 + }, + { + "type": "text", + "text": "716 In the main paper, we assume that the eigenspaces come from a symmetric matrix. This holds for many \n717 cases of practical interest, as e.g. the Laplacian matrix of an undirected graph is symmetric. However, \n718 we may also want to process directed graphs, or other data that have associated nonsymmetric matrices. \n719 Our SignNet and BasisNet generalize in a straightforward way to handle nonsymmetric diagonalizable \n720 matrices, as we detail here. Let $A \\in \\mathbb { R } ^ { n \\times n }$ be a matrix with a diagonalization $A = V \\Lambda V ^ { - 1 }$ , where \n721 $\\boldsymbol { \\Lambda } = \\operatorname { D i a g } ( \\lambda _ { 1 } , \\ldots , \\lambda _ { n } )$ contains the eigenvalues $\\lambda _ { i }$ , and the columns of $V = \\left[ v _ { 1 } \\quad \\ldots \\quad v _ { n } \\right]$ are \n722 eigenvectors. Suppose we want to learn a function on the eigenvectors $v _ { 1 } , \\ldots , v _ { k }$ . Unlike in the \n723 symmetric matrix case, the eigenvectors are not necessarily orthonormal, and both the eigenvalues \n724 and eigenvectors can be complex. \n725 Real eigenvectors. First, we assume the eigenvectors $v _ { i }$ are all real vectors in $\\mathbb { R } ^ { n }$ . We can take the \n726 eigenvectors to be real if $A$ is symmetric, or if $A$ has real eigenvalues (see Horn and Johnson [2012] \n727 Theorem 1.3.29). Also, suppose that we choose the real numbers $\\mathbb { R }$ as our base field for the vector \n728 space in which eigenvectors lie. Note that for any scaling factor $c \\in \\mathbb { R } \\setminus \\{ 0 \\}$ and eigenvector $v$ \n729 we have that $c v$ is an eigenvector of the same eigenvalue. If the eigenvalues are distinct, then the \n730 eigenvectors of the form $c v$ are the only other eigenvectors in the same eigenspace as $v$ . Thus, we \n731 want a function to be invariant to scalings: ", + "bbox": [ + 142, + 856, + 826, + 911 + ], + "page_idx": 17 + }, + { + "type": "image", + "img_path": "images/ef801ebc92066def707f7be84a7c54ce9821cf07beb714baacf30831725a356a.jpg", + "image_caption": [ + "Figure 5: PyTorch-like pseudo-code for using SignNet with a GNN prediction model, where $\\phi = \\mathrm { G I N }$ and $\\rho = \\mathrm { M L P }$ as in the ZINC molecular graph regression experiments. Reshaping eigenvectors from $n \\times k$ to $n \\times k \\times 1$ allows $\\phi$ to process each eigenvector (and its negation) independently in PyTorch-like deep learning libraries. " + ], + "image_footnote": [], + "bbox": [ + 310, + 118, + 694, + 327 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 424, + 825, + 494 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 500, + 825, + 598 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/3eabf71dd3761a6fbe85c19d658fc00f6b8fc87b8c94c525c69613639a1596a2.jpg", + "text": "$$\nf ( v _ { 1 } , \\ldots , v _ { k } ) = f ( c _ { 1 } v _ { 1 } , \\ldots , c _ { k } v _ { k } ) \\qquad c _ { i } \\in \\mathbb { R } \\setminus \\{ 0 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 603, + 676, + 621 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "732 This can be handled by SignNet, by giving unit normalized vector inputs: ", + "bbox": [ + 145, + 625, + 655, + 641 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/044d7e028dbdd0a22e6b5a1ed5726a9d6bdf09cc9bfd38860b31e014bd280970.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } ) = \\rho \\left( [ \\phi ( v _ { i } / \\Vert v _ { i } \\Vert ) + \\phi ( - v _ { i } / \\Vert v _ { i } \\Vert ) ] _ { i = 1 , \\dots , k } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 302, + 645, + 694, + 672 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "733 Now, say have bases of eigenspaces $V _ { 1 } , \\dots , V _ { l }$ with dimensions $d _ { 1 } , \\ldots , d _ { l }$ . For a basis $V _ { i }$ , we have \n734 that any other basis of the same space can be obtained as $V _ { i } W$ for some $W \\in \\mathrm { G L } _ { \\mathbb { R } } ( d _ { i } )$ , the set of \n735 real invertible matrices in $\\mathbb { R } ^ { d _ { i } \\times d _ { i } }$ . Indeed, the orthonormal projector for the space spanned by the \n736 columns of $V _ { i }$ is given by $V _ { i } ( V _ { i } ^ { \\top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \\top }$ . Thus, if $Z \\in \\dot { \\mathbb { R } } ^ { n \\times d _ { i } }$ is another basis for the column \n737 space of $V _ { i }$ , we have that $V _ { i } ( V _ { i } ^ { \\top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \\top } = Z ( Z ^ { \\top } Z ) ^ { - 1 } Z ^ { \\top }$ , so ", + "bbox": [ + 140, + 676, + 826, + 750 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/bd5b4ab743be1942e852504f5d0c5bdf21bdce1687e60d6e11cda705e5fe3c96.jpg", + "text": "$$\nV _ { i } ( V _ { i } ^ { \\top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \\top } Z = Z ( Z ^ { \\top } Z ) ^ { - 1 } Z ^ { \\top } Z = Z ,\n$$", + "text_format": "latex", + "bbox": [ + 349, + 755, + 647, + 773 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "so let 738 $W ~ = ~ ( V _ { i } ^ { \\top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \\top } Z ~ \\in ~ \\mathbb { R } ^ { d _ { i } \\times d _ { i } }$ . Note that $W$ is invertible, because it has inverse 739 $( Z ^ { \\top } Z ) ^ { - 1 } Z ^ { \\top } V _ { i }$ , so indeed $V _ { i } W = Z$ for $W \\in \\ G \\mathrm { L } _ { \\mathbb { R } } ( d _ { i } )$ . Thus, basis invariance in this case is 740 of the form ", + "bbox": [ + 140, + 779, + 825, + 823 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/33460bbc5f538a6bc4d712c0fca49e108f8636162122f239916224c969d9cdd1.jpg", + "text": "$$\nf ( V _ { 1 } \\ldots , V _ { l } ) = f ( V _ { 1 } W _ { 1 } , \\ldots , V _ { l } W _ { l } ) \\qquad W _ { i } \\in \\operatorname { G L } _ { \\mathbb { R } } ( d _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 308, + 821, + 689, + 840 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "741 Note that the distinct eigenvalue invariance is a special case of this invariance, as ${ \\mathrm { G } } _ { \\mathbb { R } } ( 1 ) = \\mathbb { R } \\ \\backslash \\ \\{ 0 \\}$ . \n742 We can again achieve this basis invariance by using a BasisNet, where the inputs to the $\\phi _ { d _ { i } }$ are \n743 orthogonal projectors of the corresponding eigenspace: ", + "bbox": [ + 142, + 842, + 825, + 885 + ], + "page_idx": 18 + }, + { + "type": "equation", + "img_path": "images/83d2189495ac08460e192853ea3f78d7d0f970b2ccc64c977ebe99f2771950fb.jpg", + "text": "$$\n\\begin{array} { r } { f ( V _ { 1 } , \\ldots , V _ { l } ) = \\rho \\left( \\left[ \\phi _ { d _ { i } } ( V _ { i } ( V _ { i } ^ { \\top } V _ { i } ) ^ { - 1 } V _ { i } ^ { \\top } ) \\right] _ { i = 1 , \\ldots , l } \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 320, + 888, + 678, + 916 + ], + "page_idx": 18 + }, + { + "type": "text", + "text": "44 Recall that if $V _ { i }$ is an orthonormal basis, then the orthogonal projector is just $V _ { i } V _ { i } ^ { \\top }$ , so this is a direct \n745 generalization of BasisNet in the symmetric case. \n746 Complex eigenvectors. More generally, suppose $V \\in \\mathbb { C } ^ { n \\times n }$ are complex eigenvectors, and we take \n747 the base field of the vector space to be $\\mathbb { C }$ . The above arguments generalize to the complex case; in \n748 the case of distinct eigenvalues, we want ", + "bbox": [ + 148, + 90, + 825, + 119 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 126, + 825, + 167 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/275a3dbd59db44489c9d6e9de9614aeee4466f43ad747fa65946ba9173f97665.jpg", + "text": "$$\nf ( v _ { 1 } , \\ldots , v _ { k } ) = f ( c _ { 1 } v _ { 1 } , \\ldots , c _ { k } v _ { k } ) \\qquad c _ { i } \\in \\mathbb { C } \\ \\backslash \\ \\{ 0 \\} .\n$$", + "text_format": "latex", + "bbox": [ + 320, + 172, + 676, + 190 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "749 However, this symmetry can not be as easily reduced to a unit normalization and a discrete sign \n750 invariance, as it can be in the real case. Nonetheless, the basis invariant architecture directly \n751 generalizes, so we can handle the case of distinct eigenvalues by a more general basis invariant \n752 architecture as well. The basis invariance is ", + "bbox": [ + 140, + 196, + 825, + 252 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/65f1b32b669cf13eaf5904ce42ed81f08891695950ea82b58d321543c700d393.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = f ( V _ { 1 } W _ { 1 } , \\dots , V _ { l } W _ { l } ) \\qquad W _ { i } \\in \\operatorname { G L } _ { \\mathbb { C } } ( d _ { i } ) .\n$$", + "text_format": "latex", + "bbox": [ + 305, + 257, + 691, + 275 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "The orthogonal projector of the image of 53 $V _ { i }$ is $V _ { i } ( V _ { i } ^ { * } V _ { i } ) ^ { - 1 } V _ { i } ^ { * }$ , where there are now conjugate 54 transposes replacing the transposes. Thus, BasisNet takes the form: ", + "bbox": [ + 151, + 280, + 821, + 309 + ], + "page_idx": 19 + }, + { + "type": "equation", + "img_path": "images/0ba266579e566fe9ddd51d92bedace84d33e0f02e399db5e0a9b1881405e00cb.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( \\left[ \\phi _ { d _ { i } } ( V _ { i } ( V _ { i } ^ { * } V _ { i } ) ^ { - 1 } V _ { i } ^ { * } ) \\right] _ { i = 1 , \\dots , l } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 325, + 314, + 673, + 342 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "755 B.2 Broader Impacts ", + "text_level": 1, + "bbox": [ + 142, + 356, + 333, + 371 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "756 We believe that our models and future sign invariant or basis invariant networks could be useful in a \n757 wide variety of applications. As eigenvectors arise in many domains, it is difficult to predict the uses \n758 of these models. We test on several molecular property prediction tasks, which have the potential \n759 for much positive impact, such as in drug discovery [Stokes et al., 2020]. However, recent work \n760 has found that the same models that we use for finding beneficial drugs can also be used to design \n761 biochemical weapons [Urbina et al., 2022]. Another major application of graph machine learning \n762 is in social network analysis, where positive (e.g. malicious node detection [Pandit et al., 2007]) \n763 and negative (e.g. deanonymization [Narayanan and Shmatikov, 2009]) uses of machine learning \n764 are possible. Even if there is no negative intent, bias in learned models can differentially impact \n765 particular subgroups of people. Thus, academia, industry, and policy makers must be aware of such \n766 potential negative uses, and work towards reducing the likelihood of them. ", + "bbox": [ + 142, + 381, + 825, + 534 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "C More on Eigenvalue Multiplicities ", + "text_level": 1, + "bbox": [ + 163, + 551, + 493, + 569 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "In this section, we study the properties of eigenvalues and eigenvectors computed by numerical algorithms on real-world data. ", + "bbox": [ + 174, + 583, + 823, + 612 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "770 C.1 Sign and Basis Ambiguities in Numerical Eigensolvers ", + "text_level": 1, + "bbox": [ + 147, + 627, + 593, + 642 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "771 When processing real-world data, we use eigenvectors that are computed by numerical algorithms. \n772 These algorithms return specific eigenvectors for each eigenspace, so there is some choice of sign \n773 or basis of each eigenspace. The general symmetric matrix eigensolvers numpy.linalg.eigh \n774 and scipy.linalg.eigh both call LAPACK routines. They both proceed as follows: for a \n775 symmetric matrix $A$ , they first decompose it as $A = Q T Q ^ { \\dagger }$ for orthogonal $Q$ and tridiago \n776 nal $T$ , then they compute the eigendecomposition of $T \\doteq \\bar { W } \\Lambda W ^ { \\top }$ , so the eigendecomposition \n777 of $A$ is $A = ( \\mathsf { \\bar { Q } } W ) \\dot { \\Lambda } ( W ^ { \\top } Q ^ { \\top } )$ . There are multiple ambiguities here: for diagonal sign matri \n778 ces $S = \\mathrm { D i a g } ( s _ { 1 } , . . . , s _ { n } )$ and $S ^ { \\prime } = \\mathrm { D i a g } ( s _ { 1 } ^ { \\prime } , . . . , s _ { n } ^ { \\prime } )$ , where $s _ { i } , s _ { i } ^ { \\prime } \\in \\{ - 1 , 1 \\}$ , we have that \n779 $A = Q S ( S T S ) S Q ^ { \\top }$ is also a valid tridiagonalization, as $\\it Q S$ is still orthogonal, $S S = I$ , and $S T S$ \n780 is still tridiagonal. Also, $T = ( W S ^ { \\prime } ) \\Lambda ( S ^ { \\prime } W ^ { \\top } )$ is a valid eigendecomposition of $T$ , as $W S ^ { \\prime }$ is still \n781 orthogonal. \n782 In practice, we find that the general symmetric matrix eigensolvers numpy.linalg.eigh and \n783 scipy.linalg.eigh differ between frameworks but are consistent with the same framework. More \n784 specifically, for a symmetric matrix $A$ , we find that the eigenvectors computed with the default \n785 settings in numpy tend to differ by a choice of sign or basis from those that are computed with the \n786 default settings in scipy. On the other hand, the called LAPACK routines are deterministic, so the \n787 eigenvectors returned by numpy are the same in each call, and the eigenvectors returned by scipy are \n788 likewise the same in each call. \n789 Eigensolvers for sparse symmetric matrices like scipy.linalg.eigsh are required for large scale \n790 problems. This function calls ARPACK, which uses an iterative method that starts with a randomly \n791 sampled initial vector. Due to this stochasticity, the sign and basis of eigenvectors returned differs \n792 between each call. ", + "bbox": [ + 140, + 654, + 826, + 808 + ], + "page_idx": 19 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 814, + 825, + 911 + ], + "page_idx": 19 + }, + { + "type": "table", + "img_path": "images/81782e120b73562e8197dc26782a10e71acd024bfc32520ed86d5cb1667bb3e4.jpg", + "table_caption": [ + "Table 5: Eigenspace statistics for datasets of multiple graphs. From left to right, the columns are: dataset name, number of graphs, range of number of nodes per graph, largest multiplicity, and percent of graphs with an eigenspace of dimension $> 1$ . " + ], + "table_footnote": [], + "table_body": "
DatasetGraphs#NodesMax.Mult% Graphs mult. > 1
ZINC12,0009-37964.1
ZINC-full249,4566-381063.8
ogbg-molhiv41,1272- 2224268.0
IMDB-M1,5007-893799.9
COLLAB5,00032 - 49223899.1
PROTEINS1,1134- 6202077.3
COIL-DEL3,9003-7744.00
", + "bbox": [ + 250, + 138, + 746, + 266 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "", + "bbox": [ + 142, + 295, + 825, + 351 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Bro et al. [2008] develops a data-dependent method to choose signs for each singular vector of a singular value decomposition. Still, in the worst case the signs chosen will be arbitrary, and they do not handle basis ambiguities in higher dimensional eigenspaces. Other works have made choices of sign, such as by picking the sign so that the eigenvector’s entries are in the largest lexicographic order [Tam and Dunson, 2022]. This choice of sign may work poorly for learning on graphs, as it is sensitive to permutations on nodes. For some graph regression experiments in Section 4.1, we try a choice of sign that is permutation invariant, but we find it to work poorly. ", + "bbox": [ + 173, + 357, + 825, + 454 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "C.2 Higher Dimensional Eigenspaces in Real Graphs ", + "text_level": 1, + "bbox": [ + 165, + 474, + 553, + 489 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Here, we investigate the normalized Laplacian eigenspace statistics of real-world graph data. For any graph that has distinct Laplacian eigenvalues, only sign invariance is required in processing eigenvectors. However, we find that graph data tends to have higher multiplicity eigenvalues, so basis invariance would be required for learning symmetry-respecting functions on eigenvectors. ", + "bbox": [ + 174, + 501, + 825, + 558 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "Indeed, we show statistics for multi-graph datasets in Table 5 and for single-graph datasets with more nodes per graph in Table 6. For multi-graph datasets, we consider : ", + "bbox": [ + 158, + 563, + 823, + 592 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• Molecule graphs: ZINC [Irwin et al., 2012, Dwivedi et al., 2020], ogbg-molhiv [Wu et al., 2018, Hu et al., 2020a] \n• Social networks: IMDB-M, COLLAB [Yanardag and Vishwanathan, 2015, Morris et al., 2020a], \n• Bioinformatics graphs: PROTEINS [Morris et al., 2020a] \n• Computer vision graphs: COIL-DEL [Riesen and Bunke, 2008, Morris et al., 2020a]. ", + "bbox": [ + 217, + 604, + 826, + 715 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "13 For single-graph datasets, we consider: ", + "bbox": [ + 153, + 728, + 431, + 743 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "• The $3 2 \\times 3 2$ image grid as in Section J.2 \n• Citation networks: Cora, Citeseer [Sen et al., 2008] \n• Co-purchasing graphs with Amazon Photo [McAuley et al., 2015, Shchur et al., 2018]. ", + "bbox": [ + 217, + 755, + 799, + 815 + ], + "page_idx": 20 + }, + { + "type": "text", + "text": "817 We see that these datasets all contain higher multiplicity eigenspaces, so sign invariance is insufficient \n818 for fully respecting symmetries. The majority of graphs in each multi-graph dataset besides COIL \n819 DEL contain higher multiplicity eigenspaces. Also, the dimension of these eigenspaces can be \n820 quite large compared to the size of the graphs in the dataset. The single-graph datasets have a large \n821 proportion of their eigenvectors belonging to higher dimensional eigenspaces. Thus, basis invariance \n822 may play a large role in processing spectral information from these graph datasets. ", + "bbox": [ + 140, + 828, + 825, + 912 + ], + "page_idx": 20 + }, + { + "type": "table", + "img_path": "images/b3abdd49578d23abecb9afc4c461b33adb63e309e5f6c4ab74235adaa6b9a74e.jpg", + "table_caption": [ + "Table 6: Eigenspace statistics for single graphs. From left to right, the columns are: dataset name, number of nodes, distinct eigenvalues (i.e. distinct eigenspaces), number of unique multiplicities, largest multiplicity, and percent of eigenvectors belonging to an eigenspace of dimension $> 1$ . " + ], + "table_footnote": [], + "table_body": "
DatasetNodesDistinct 入#Mult.Max Mult.% Vecs mult. > 1
32 × 32 image1,02451333296.9
Cora2,7082,1871130019.7
Citeseer3,3271,8611249144.8
Amazon Photo7,6507,41681363.71
", + "bbox": [ + 217, + 140, + 779, + 224 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "823 C.3 Relationship to Graph Automorphisms ", + "text_level": 1, + "bbox": [ + 142, + 250, + 488, + 265 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "Higher multiplicity eigenspaces are related to automorphism symmetries in graphs. For an adjacency matrix $A$ , the permutation matrix $P$ is an automorphism of the graph associated to $A$ if $P A \\bar { P ^ { \\top } } = \\bar { A }$ . If $P$ is an automorphism, then for any eigenvector $v$ of $A$ with eigenvalue $\\lambda$ , we have ", + "bbox": [ + 173, + 275, + 826, + 318 + ], + "page_idx": 21 + }, + { + "type": "equation", + "img_path": "images/20fbbbfe61e91632f49167967686461a74f29f7b5f765f8c5b09043117c06d29.jpg", + "text": "$$\nA P v = P A P ^ { \\top } P v = P A v = P \\lambda v = \\lambda P v ,\n$$", + "text_format": "latex", + "bbox": [ + 352, + 323, + 645, + 342 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "so $P v$ is an eigenvector of $A$ with the same eigenvalue $\\lambda$ . If $P v$ and $v$ are linearly independent, then $\\lambda$ has a higher dimensional eigenspace. Thus, under certain additional conditions, automorphism symmetries of graphs lead to repeated eigenvalues [Sachs and Stiebitz, 1983, Teranishi, 2009]. ", + "bbox": [ + 168, + 349, + 825, + 392 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "30 C.4 Multiplicities in Random Graphs ", + "text_level": 1, + "bbox": [ + 155, + 407, + 447, + 422 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "It is known that almost all random graphs under the Erdos-Renyi model have no repeated eigenvalues ˝ in the infinite number of nodes limit [Tao and Vu, 2017]. Likewise, almost all random graphs under the Erdos-Renyi model are asymmetric in the sense of having no nontrivial automorphism ˝ symmetries [Erdos and Rényi, 1963]. These results contrast sharply with the high eigenvalue multiplicities that we see in real-world data in Section C.2. Likewise, many types of real-world graph data have been found to possess nontrivial automorphism symmetries [Ball and Geyer-Schulz, 2018]. This demonstrates a potential downside of using random graph models to study real-world data: the eigenspace dimensions and automorphism symmetries of random graphs may not agree with those of real-world data. ", + "bbox": [ + 169, + 433, + 825, + 558 + ], + "page_idx": 21 + }, + { + "type": "text", + "text": "840 D Visualization of SignNet output ", + "text_level": 1, + "bbox": [ + 140, + 89, + 473, + 107 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "841 D.1 Cat Model Visualization ", + "text_level": 1, + "bbox": [ + 142, + 125, + 385, + 138 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/31496ac29afa5247f520171197969ab0d9a295fbd9337a1350f67d1451fd5728.jpg", + "image_caption": [ + "Figure 6: (Left) Cotangent Laplacian eigenvectors of the cat model. (Right) First principal component of $\\phi ( v ) + \\phi ( - v )$ from our trained SignNet. " + ], + "image_footnote": [], + "bbox": [ + 302, + 148, + 699, + 775 + ], + "page_idx": 22 + }, + { + "type": "text", + "text": "842 In Figure 6, we plot the eigenvectors of the cotangent Laplacian on a cat model, as well as the first \n843 principal component of the corresponding learned $\\phi ( v ) + \\phi ( - v )$ from our SignNet model that was \n844 trained on the texture reconstruction task. Interestingly, this portion of our SignNet encodes bilateral \n845 symmetry; for instance, while some eigenvectors differ between left feet and right feet, this portion of \n846 our SignNet gives similar values for the left and right feet. This is useful for the texture reconstruction \n847 task, as the texture regression target has bilateral symmetry. \n848 We also show principal components of outputs for the full SignNet model in Figure 7. This is not \n849 as interpretable, as the outputs are high frequency and appear to be close to the texture that is the \n850 regression target. If instead we trained the network on a task involving eigenvectors of multiple \n851 models, then we may expect the SignNet to learn more structurally interpretable mappings (as in the \n852 case of the molecule tasks). ", + "bbox": [ + 140, + 828, + 825, + 911 + ], + "page_idx": 22 + }, + { + "type": "image", + "img_path": "images/8deab72b8c4c3e92e6f1ddf111322bc7832b3cbb1a3f8d9797091521c7429cf6.jpg", + "image_caption": [ + "Figure 7: First three principal components of the full SignNet output on the cat model. " + ], + "image_footnote": [], + "bbox": [ + 318, + 85, + 679, + 203 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 251, + 825, + 320 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "D.2 Molecule visualization ", + "text_level": 1, + "bbox": [ + 166, + 337, + 370, + 352 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "To better understand SignNet, in Figure 9 we visualize the learned positional encodings of a SignNet with $\\phi = \\mathrm { G I N }$ , $\\rho = \\mathsf { M L P }$ (with a summation to handle variable eigenvector numbers) trained on ZINC as in Section 4.1. SignNet learns interesting structural information such as min-cuts (PC 3) and appendage atoms (PC 2) that qualitatively differ from any single eigenvector of the graph. ", + "bbox": [ + 171, + 363, + 825, + 420 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "858 For this visualization we use a SignNet trained with a GatedGCN base model on ZINC, as in \n859 Section 4.1. This SignNet uses GIN as $\\phi$ and $\\rho$ as an MLP (with a sum before it to handle variable \n860 numbers of eigenvectors), and takes in all eigenvectors of each graph. See Figure 8 for all of the \n861 eigenvectors of fluorescein. ", + "bbox": [ + 147, + 425, + 823, + 481 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "62 E More Related Work ", + "text_level": 1, + "bbox": [ + 155, + 501, + 375, + 518 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "E.1 Graph Positional Encodings ", + "text_level": 1, + "bbox": [ + 160, + 534, + 411, + 549 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "Various graph positional encodings have been proposed, which have been motivated for increasing expressive power or practical performance of graph neural networks, and for generalizing Transformers to graphs. Positional encodings are related to so-called position-aware network embeddings [Chami et al., 2020], which capture distances between nodes in graphs. These include network embedding methods like Deepwalk [Perozzi et al., 2014] and node2vec [Grover and Leskovec, 2016], which have been recently integrated into GNNs that respect their invariances by Wang et al. [2022]. Further, Li et al. [2020] studies the theoretical and practical benefits of incorporating distance features into graph neural networks. Dwivedi et al. [2022] proposes a method to inject learnable positional encodings into each layer of a graph neural network, and uses a simple random walk based node positional encoding. You et al. [2021] proposes a node positional encoding $\\operatorname { d i a g } ( A ^ { k } )$ , which captures the number of closed walks from a node to itself. Dwivedi et al. [2020] propose to use Laplacian eigenvectors as positional encodings in graph neural networks, with sign ambiguities alleviated by sign flipping data augmentation. Srinivasan and Ribeiro [2019] theoretically analyze node positional embeddings and structural representations in graphs, and show that most-expressive structural representations contain the information of any node positional embedding. ", + "bbox": [ + 173, + 559, + 825, + 767 + ], + "page_idx": 23 + }, + { + "type": "text", + "text": "879 While positional encodings in sequences as used for Transformers [Vaswani et al., 2017] are able to \n880 leverage the canonical order in sequences, there is no such useful canonical order for nodes in a graph, \n881 due in part to permutation symmetries. Thus, different permutation equivariant positional encodings \n882 have been proposed to help generalize Transformers to graphs. Dwivedi and Bresson [2021] directly \n883 add in linearly projected Laplacian eigenvectors to node features before processing these features \n884 with a graph Transformer. Kreuzer et al. [2021] propose an architecture that uses attention over \n885 Laplacian eigenvectors and eigenvalues to learn node or edge positional encodings. Mialon et al. \n886 [2021] uses spectral kernels such as the diffusion kernel to define relative positional encodings that \n887 modulate the attention matrix. Ying et al. [2021] achieve state-of-the-art empirical performance \n888 with simple Transformers that incorporate shortest-path based relative positional encodings. Zhang ", + "bbox": [ + 140, + 772, + 825, + 911 + ], + "page_idx": 23 + }, + { + "type": "image", + "img_path": "images/43aa908d922e8e64a133b65e1fd6fd76a675a5170663a7438d04c4e851030493.jpg", + "image_caption": [ + "Figure 8: All normalized Laplacian eigenvectors of the fluorescein graph. The first principal components of SignNet’s learned positional encodings do not exactly match any eigenvectors. " + ], + "image_footnote": [], + "bbox": [ + 233, + 77, + 766, + 618 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "et al. [2020] also utilize shortest-path distances for positional encodings in their graph Transformer. Kim et al. [2021] develop higher-order transformers (that generalize invariant graph networks), which interestingly perform well on graph regression using sparse higher-order transformers without positional encodings. ", + "bbox": [ + 171, + 689, + 825, + 746 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "93 E.2 Eigenvector Symmetries in Graph Representation Learning ", + "text_level": 1, + "bbox": [ + 155, + 771, + 629, + 786 + ], + "page_idx": 24 + }, + { + "type": "text", + "text": "Many works that attempt to respect the invariances of eigenvectors solely focus on sign invariance (by using data augmentation) [Dwivedi et al., 2020, Dwivedi and Bresson, 2021, Dwivedi et al., 2022, Kreuzer et al., 2021]. This may be reasonable for continuous data, where eigenvalues of associated matrices may be usually distinct and separated (e.g. Puny et al. [2022] finds that this empirically holds for covariance matrices of $n$ -body problems). However, discrete graph Laplacians are known to have higher multiplicity eigenvalues in many cases, and in Appendix C.2 we find this to be true in various types of real-world graph data. Graphs without higher multiplicity eigenspaces are easier to deal with; in fact, graph isomorphism can be tested in polynomial time on graphs of bounded ", + "bbox": [ + 171, + 800, + 825, + 911 + ], + "page_idx": 24 + }, + { + "type": "image", + "img_path": "images/8d054483f5205b0ea63ad7c9f95d2bcf946c32fde9e8cc7c82166da588a60951.jpg", + "image_caption": [ + "Figure 9: Normalized Laplacian eigenvectors and learned positional encodings for the graph of fluorescein. (Top row) From left to right: smallest and second smallest nontrivial eigenvectors, then second largest and largest eigenvectors. (Bottom row) From left to right: first four principal components of the output $\\rho \\big ( \\mathrm { \\bar { [ } } \\phi ( v _ { i } ) \\mathrm { \\bar { + } } \\phi ( - v _ { i } ) \\mathrm { ] } _ { i = 1 , \\dots , n } \\big )$ of SignNet. Note: we will put this back in the main paper for the camera-ready. " + ], + "image_footnote": [], + "bbox": [ + 178, + 84, + 813, + 315 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "902 multiplicity for adjacency matrix eigenvalues [Babai et al., 1982], with a time complexity that is \n903 lower for graphs with lower maximum multiplicities. \n904 \n905 \n906 \n907 \n908 \n909 \n910 \n911 \n912 \n913 \n914 \n915 ", + "bbox": [ + 147, + 420, + 825, + 449 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 424, + 163, + 626 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "A recent work of Wang et al. [2022] proposes full orthogonal group invariance for functions that process positional encodings. In particular, for positional encodings $Z \\in \\mathbb { R } ^ { n \\times k }$ , they parameterize functions $f ( Z )$ such that ${ \\bar { f } } ( Z ) { \\bar { = } } f ( Z Q )$ for all $Q \\in O ( k )$ . This indeed makes sense for network embeddings like node2vec [Grover and Leskovec, 2016], as their objective functions are based on inner products and are thus orthogonally invariant. While they prove stability results when enforcing full orthogonal invariance for eigenvectors, this is a very strict constraint compared to our basis invariance. For instance, when $k = n$ and all eigenvectors are used in $V$ , the condition $f ( V ) = f ( V Q )$ implies that $f$ is a constant function on orthogonal matrices, since any orthogonal matrix $W$ can be obtained as $W = V Q$ for $Q = V ^ { \\top } W \\in O ( n )$ . In other words, for bases of eigenspaces $V _ { 1 } , \\dots , V _ { l }$ and $V = [ V _ { 1 } \\quad \\ldots \\quad V _ { l } ]$ , Wang et al. [2022] enforces $V Q \\cong V$ , while we enforce $V \\mathrm { D i a g } ( Q _ { 1 } , \\dots , Q _ { l } ) \\cong V$ . While the columns of $V \\mathrm { D i a g } ( Q _ { 1 } , \\dots , Q _ { l } )$ are still eigenvectors, the columns of $V Q$ generally are not. ", + "bbox": [ + 171, + 455, + 825, + 622 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "916 E.3 Graph Spectra and Learning on Graphs ", + "text_level": 1, + "bbox": [ + 142, + 640, + 495, + 655 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "More generally, graph spectra are widely used in analyzing graphs, and spectral graph theory [Chung, 1997] studies the connection between graph properties and graph spectra. Different graph kernels have been defined based on graph spectra, which use robust and discriminative notions of generalized spectral distance [Verma and Zhang, 2017], the spectral density of states [Huang et al., 2021], random walk return probabilities [Zhang et al., 2018b], or the trace of the heat kernel [Tsitsulin et al., 2018]. Graph signal processing relies on spectral operations to define Fourier transforms, frequencies, convolutions, and other useful concepts for processing data on graphs [Ortega et al., 2018]. The closely related spectral graph neural networks [Wu et al., 2020, Balcilar et al., 2020] parameterize neural architectures that are based on similar spectral operations. ", + "bbox": [ + 171, + 666, + 825, + 791 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "F Definitions, Notation, and Background ", + "text_level": 1, + "bbox": [ + 169, + 811, + 529, + 829 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "F.1 Basic Topology and Algebra Definitions ", + "text_level": 1, + "bbox": [ + 166, + 843, + 486, + 858 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "We will use some basic topology and algebra for our theoretical results. A topological space $( \\mathcal { X } , \\tau )$ is a set $\\mathcal { X }$ along with a family of subsets $\\tau \\subseteq 2 ^ { \\mathcal { X } }$ satisfying certain properties, which gives useful notions like continuity and compactness. From now on, we will omit mention of $\\tau$ , and refer to a ", + "bbox": [ + 171, + 869, + 825, + 911 + ], + "page_idx": 25 + }, + { + "type": "text", + "text": "931 topological space as the set $\\mathcal { X }$ itself. For topological spaces $\\mathcal { X }$ and $\\mathcal { V }$ , we write $\\chi \\cong \\mathcal { V }$ and say that \n932 $\\mathcal { X }$ is homeomorphic to $\\mathcal { V }$ if there exists a continuous bijection with continuous inverse from $\\mathcal { X }$ to \n933 $\\mathcal { V }$ . We will say $\\mathcal { X } = \\mathcal { y }$ if the underlying sets and topologies are equal as sets (we will often use this \n934 notion of equality for simplicity, even though it can generally be substituted with homeomorphism). \n935 For a function $f : \\mathcal { X } \\mathcal { Y }$ between topological spaces $\\mathcal { X }$ and $\\mathcal { V }$ , the image $\\operatorname { i m } f$ is the set of values \n936 that $f$ takes, ${ \\mathrm { i m } } f = \\{ f ( x ) : x \\in \\mathcal { X } \\}$ . This is also denoted $f ( \\mathcal X )$ . A function $f : \\mathcal { X } \\mathcal { Y }$ is called a \n937 topological embedding if it is a homeomorphism from $\\mathcal { X }$ to its image. \n938 A group $G$ is a set along with a multiplication operation $G \\times G \\to G$ , such that multiplication is \n939 associative, there is a multiplicative identity $e \\in G$ , and each $g \\in G$ has a multiplicative inverse $g ^ { - 1 }$ \n940 A topological group is a group that is also a topological space such that the multiplication and inverse \n941 operations are continuous. \n942 A group $G$ may act on a set $\\mathcal { X }$ by a function $\\cdot : G \\times \\mathcal { X } \\to \\mathcal { X }$ . We usually denote $g \\cdot x$ as $g x$ . A \n943 topological group is said to act continuously on a topological space $\\mathcal { X }$ if $\\cdot$ is continuous. For any \n944 group $G$ and topological space $\\mathcal { X }$ , we define the coset $G x = \\{ g x : g \\in G \\}$ , which can be viewed as \n945 an equivalance class of elements that can be transformed from one to another by a group element. \n946 The quotient space ${ \\mathcal { X } } / G = \\{ G x : x \\in { \\mathcal { X } } \\}$ is the set of all such equivalence classes, with a topology \n947 induced by that of $\\mathcal { X }$ . The quotient map $\\dot { \\pi } : \\mathcal { X } \\to \\mathcal { X } / G$ is a surjective continuous map that sends $x$ \n948 to its coset, $\\pi ( x ) = G x$ . ", + "bbox": [ + 140, + 90, + 825, + 189 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 194, + 825, + 251 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 256, + 825, + 354 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "For $x \\in \\mathbb { R } ^ { s }$ , $\\| x \\| _ { 2 }$ denotes the standard Euclidean norm. By the $\\infty$ norm of functions $f : \\mathcal { Z } \\to \\mathbb { R } ^ { s }$ from a compact $\\mathcal { Z }$ to a Euclidean space $\\mathbb { R } ^ { s }$ , we mean $\\| f \\| _ { \\infty } = \\operatorname* { s u p } _ { z \\in { \\mathcal { Z } } } \\| f ( z ) \\| _ { 2 }$ . ", + "bbox": [ + 171, + 359, + 821, + 390 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "F.2 Background on Eigenspace Invariances ", + "text_level": 1, + "bbox": [ + 158, + 406, + 486, + 421 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Let $V = [ v _ { 1 } \\quad \\ldots \\quad v _ { d } ]$ and $W = [ w _ { 1 } \\quad \\ldots \\quad w _ { d } ] \\in \\mathbb { R } ^ { n \\times d }$ be two orthonormal bases for the same $d$ dimensional subspace of $\\mathbb { R } ^ { n }$ . Since $V$ and $W$ span the same space, their orthogonal projectors are the same, so $\\bar { V V } ^ { \\top } = W W ^ { \\top }$ . Also, since $V$ and $W$ have orthonormal columns, we have $V ^ { \\top } V = W ^ { \\top } W = I \\in \\mathbb { R } ^ { d \\times d }$ . Define $Q = V ^ { \\top } W$ . Then $Q$ is orthogonal because ", + "bbox": [ + 173, + 431, + 825, + 488 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/7b85ac896f219beac7bbd7703ecef290d588b4382fe8d3f27c268d3eec728332.jpg", + "text": "$$\n\\begin{array} { r } { Q ^ { \\top } Q = W ^ { \\top } V V ^ { \\top } W = W ^ { \\top } W W ^ { \\top } W = I } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 352, + 496, + 645, + 516 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "956 Moreover, we have that ", + "bbox": [ + 140, + 525, + 330, + 539 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/5b2cda6cdd00816d047a7610f586ca583143220e0d176001ded6a34de6c13ff6.jpg", + "text": "$$\nV Q = V V ^ { \\top } W = W W ^ { \\top } W = W\n$$", + "text_format": "latex", + "bbox": [ + 382, + 537, + 616, + 556 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Thus, for any orthonormal bases $V$ and $W$ of the same subspace, there exists an orthogonal $Q \\in O ( d )$ such that $V Q = W$ . ", + "bbox": [ + 161, + 563, + 825, + 592 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "959 For another perspective on this, define the Grassmannian ${ \\mathrm { G r } } ( d , n )$ as the smooth manifold consisting \n960 of all $d$ dimensional subspaces of $\\mathbb { R } ^ { n }$ . Further define the Stiefel manifold $\\operatorname { S t } ( d , n )$ as the set \n961 of all orthonormal tuples $\\begin{array} { r l r } { [ v _ { 1 } } & { { } \\ldots } & { v _ { d } ] \\in \\mathbb { R } ^ { n \\times d } } \\end{array}$ of $d$ vectors in $\\mathbb { R } ^ { n }$ . Letting $O ( d )$ act by right \n962 multiplication, it holds that $\\mathrm { S t } ( d , n ) / O ( d ) \\cong \\mathrm { G r } ( d , n )$ . This implies that any $O ( d )$ invariant function \n963 on $\\operatorname { S t } ( d , n )$ can be viewed as a function on subspaces. See e.g. Gallier and Quaintance [2020] Chapter \n964 5 for more information on this. We will use this relationship in our proofs of universal representation. \n965 When we consider permutation invariance or equivariance, the permutation acts on dimensions of size \n966 $n$ . Then a tensor $\\bar { X } \\in \\mathbb { R } ^ { n ^ { k } \\times d }$ is called an order $k$ tensor with respect to this permutation symmetry, \n967 where order 0 are called scalars, order 1 tensors are called vectors, and order 2 tensors are called \n968 matrices. Note that this does not depend on $d$ ; in this work, we only ever consider vectors and scalars \n969 with respect to the $O ( d )$ action. ", + "bbox": [ + 142, + 597, + 825, + 683 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "", + "bbox": [ + 143, + 688, + 825, + 761 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "70 G Proofs of Universality ", + "text_level": 1, + "bbox": [ + 158, + 781, + 392, + 799 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "We begin by proving the two propositions for the single subspace case from Section 2.1. ", + "bbox": [ + 169, + 814, + 750, + 830 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "Proposition 1. A continuous function $h : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }$ is sign invariant if and only if ", + "bbox": [ + 161, + 833, + 717, + 849 + ], + "page_idx": 26 + }, + { + "type": "equation", + "img_path": "images/7145e66f5f1c9e895db97c9e74b94e0c4fd232537ad908a1b48c00c760d1953c.jpg", + "text": "$$\nh ( v ) = \\phi ( v ) + \\phi ( - v )\n$$", + "text_format": "latex", + "bbox": [ + 423, + 858, + 575, + 875 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "for some continuous 973 $\\phi : \\mathbb { R } ^ { n } \\mathbb { R } ^ { s }$ . A continuous $h : \\mathbb { R } ^ { n } \\mathbb { R } ^ { n }$ is sign invariant and permutation equivariant if and only974 $i f$ (3) holds for a continuous permutation equivariant $\\phi : \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { n }$ . ", + "bbox": [ + 138, + 882, + 830, + 912 + ], + "page_idx": 26 + }, + { + "type": "text", + "text": "975 Proof. If $h ( v ) = \\phi ( v ) + \\phi ( - v )$ , then $h$ is obviously sign invariant. On the other hand, if $h$ is sign \n976 invariant, then letting $\\dot { \\phi } ( v ) \\dot { = } h ( v ) / 2$ gives that $h ( v ) { \\dot { = } } { \\bar { \\phi } } ( v ) + \\phi ( - v )$ , and $\\phi$ is of course continuous. \n977 If $h ( v ) = \\phi ( v ) + \\phi ( - v )$ for a permutation equivariant $\\phi$ , then $h ( - P v ) = \\phi ( - P v ) + \\phi ( P v ) =$ \n978 $P \\phi ( - v ) + P \\phi ( v ) = P ( \\phi ( v ) + \\phi ( - v ) ) = P h ( v )$ , so $h$ is permutation equivariant and sign invariant. \n979 If $h$ is permutation equivariant and sign invariant, then define $\\phi ( v ) = h ( \\bar { v } ) / 2$ again; it is clear that $\\phi$ \n980 is continuous and permutation equivariant. $\\boxed { \\begin{array} { r l } \\end{array} }$ \n981 Proposition 2. Any continuous, $O ( d )$ invariant $h : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { s }$ is of the form $h ( V ) = \\phi ( V V ^ { \\top } )$ for \n982 a continuous $\\phi$ . For a compact domain ${ \\mathcal { Z } } \\subseteq \\mathbb { R } ^ { n \\times d }$ , maps of the form $V \\mapsto \\operatorname { I G N } ( V V ^ { \\top } )$ universally \n983 approximate continuous functions $h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n }$ that are $O ( d )$ invariant and permutation \n984 equivariant. \n85 Proof. The case without permutation equivariance holds by the First Fundamental Theorem of $O ( d )$ \n86 (Lemma 2). \n987 For the permutation equivariant case, let $\\mathcal { Z } ^ { \\prime } = \\{ V V ^ { \\top } : V \\in \\mathcal { Z } \\}$ and let $\\epsilon > 0$ . Note that $\\mathcal { Z } ^ { \\prime }$ \n988 is compact, as it is the continuous image of a compact set. Since $h$ is $O ( d )$ invariant, the first \n989 fundamental theorem of $O ( d )$ shows that there exists a continuous function $\\dot { \\phi } : \\mathcal { Z } ^ { \\prime } \\subseteq \\mathbb { R } ^ { n \\times n } \\to \\mathbb { R } ^ { n }$ \n990 such that $h ( V ) = \\phi ( V V ^ { \\top } )$ . Since $h$ is permutation equivariant, for any permutation matrix $P$ we \n991 have that ", + "bbox": [ + 145, + 90, + 826, + 121 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 125, + 826, + 181 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 189, + 825, + 250 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 153, + 265, + 825, + 295 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 297, + 825, + 371 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/5b69e0b6cc75000da269b00e4cf17b0b003e528ed6a9f35e04cc2f3ac0f0ef80.jpg", + "text": "$$\n\\begin{array} { c } { h ( P V ) = P \\cdot h ( V ) } \\\\ { \\phi ( P V V ^ { \\top } P ^ { \\top } ) = P \\cdot \\phi ( V V ^ { \\top } ) , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 393, + 376, + 602, + 417 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "992 so $\\phi$ is a continuous permutation equivariant function from matrices to vectors. Then note that Keriven \n993 and Peyré [2019] show that invariant graph networks (of generally high tensor order in hidden layers) \n994 universally approximate continuous permutation equivariant functions from matrices to vectors on \n995 compact sets of matrices. Thus, an IGN can $\\epsilon$ -approximate $\\phi$ , and hence $V \\mapsto \\operatorname { I G N } ( V V ^ { \\top } )$ can \n996 $\\epsilon$ -approximate $h$ . □ ", + "bbox": [ + 140, + 421, + 825, + 492 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "997 G.1 Proof of Decomposition Theorem ", + "text_level": 1, + "bbox": [ + 143, + 508, + 447, + 523 + ], + "page_idx": 27 + }, + { + "type": "equation", + "img_path": "images/da82823c4a512f5bb72b1fe024e46db66b48577f811da5d7be97851474b73433.jpg", + "text": "$$\n\\begin{array} { c } { { \\phi = \\psi \\circ \\pi , \\qquad \\pi = \\pi _ { 1 } \\times . . . \\times \\frac { \\chi _ { k } } { \\chi } } } \\\\ { { \\pi = \\pi _ { 1 } \\times . . . \\pi \\displaystyle \\downarrow } } \\\\ { { \\psi ^ { - 1 } \\qquad \\longleftrightarrow \\quad \\left( \\overline { { { X _ { 1 } / G _ { 1 } } } } \\right) \\times . . . \\times \\left( \\overline { { { X _ { k } / G _ { k } } } } \\right) \\displaystyle \\mathop { \\longrightarrow } \\mathbb { R } ^ { s } } } \\\\ { { \\mathcal { Z } = \\mathrm { i m } ( \\psi ) \\subseteq \\mathbb { R } ^ { a } \\longleftrightarrow \\psi _ { 1 } \\times . . . \\times \\psi _ { k } } } \\\\ { { \\psi \\qquad \\quad = \\pi _ { 1 } \\times . . . \\pi \\displaystyle - \\frac { \\Gamma } { \\rho \\rho \\psi ^ { - 1 } } } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 264, + 540, + 733, + 719 + ], + "page_idx": 27 + }, + { + "type": "image", + "img_path": "", + "image_caption": [ + "Figure 10: Commutative diagram for our proof of Theorem 3. Black arrows denote functions from topological constructions, and red dashed lines denote functions that we parameterize by neural networks $( \\phi = \\phi _ { 1 } \\times \\ldots \\times \\phi _ { k }$ and $\\rho \\mathrm { \\hbar }$ ). " + ], + "image_footnote": [], + "page_idx": 27 + }, + { + "type": "text", + "text": "998 Here, we give the formal statement of Theorem 3, which provides the necessary topological assump \n999 tions for the theorem to hold. In particular, we only require the $G _ { i }$ be a topological group that acts \n1000 continuously on $\\mathcal { X } _ { i }$ for each $i$ , and that there exists a topological embedding of each quotient space \n1001 into some Euclidean space. That the group action is continuous is a very mild assumption, and it \n1002 holds for any finite or compact matrix group, which all of the invariances we consider in this paper \n1003 can be represented as. \n1004 A topological embedding of the quotient space into a Euclidean space is desired, as we know how to \n1005 parameterize neural networks with Euclidean outputs and inputs, whereas dealing with a quotient \n1006 space is generally difficult. Many different conditions can guarantee existence of such an embedding. \n1007 For instance, if the quotient space is a smooth manifold, then the Whitney Embedding Theorem \n1008 (Lemma 5) guarantees such an embedding. Also, if the base space $\\mathcal { X } _ { i }$ is a Euclidean space and $G _ { i }$ is \n1009 a finite or compact matrix Lie group, then a map built from $G$ -invariant polynomials gives such an \n1010 embedding (González and de Salas [2003] Lemma 11.13). ", + "bbox": [ + 137, + 792, + 825, + 877 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 138, + 882, + 825, + 912 + ], + "page_idx": 27 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 90, + 825, + 161 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Figure 10 provides a commutative diagram representing the constructions in our proof. ", + "bbox": [ + 161, + 166, + 741, + 183 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "1012 Theorem 3 (Decomposition Theorem). Let $\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }$ be topological spaces, and let $G _ { i }$ be $a$ \n1013 topological group acting continuously on $\\mathcal { X } _ { i }$ for each i. Assume that there is a topological embedding \n1014 $\\psi _ { i } : \\mathcal { X } _ { i } / G _ { i } \\to \\mathbb { R } ^ { a _ { i } }$ of each quotient space into a Euclidean space $\\mathbb { R } ^ { a _ { i } }$ for some dimension $a _ { i }$ . \n1015 Then, for any continuous function $f : \\mathcal { X } = \\mathcal { X } _ { 1 } \\times . . . \\times \\mathcal { X } _ { k } \\to \\mathbb { R } ^ { s }$ that is invariant to the action of \n1016 $G = G _ { 1 } \\times \\ldots \\times G _ { k }$ , there exists continuous functions $\\phi _ { i } : \\mathcal { X } _ { i } \\mathbb { R } ^ { a _ { i } }$ and a continuous function \n1017 $\\rho : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { a } \\to \\mathbb { R } ^ { s }$ , where $a = \\textstyle \\sum _ { i } a _ { i }$ such that ", + "bbox": [ + 135, + 186, + 826, + 272 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/826747c222b972092c5d37978042b49560c40b9d6ebf0fbe52a05371dceef0a1.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } ) = \\rho ( \\phi _ { 1 } ( v _ { 1 } ) , \\dots , \\phi _ { k } ( v _ { k } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 364, + 279, + 632, + 296 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Furthermore:1018 $( l )$ each $\\phi _ { i }$ can be taken to be invariant to $G _ { i }$ , (2) the domain $\\mathcal { Z }$ is compact if each $\\mathcal { X } _ { i }$ 1019 is compact, (3) if ${ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , then $\\phi _ { i }$ can be taken to be equal to $\\phi _ { j }$ . ", + "bbox": [ + 137, + 304, + 825, + 334 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Proof. Let 1020 $\\pi _ { i } : \\mathcal { X } _ { i } \\mathcal { X } _ { i } / G _ { i }$ denote the quotient map for $\\mathcal { X } _ { i } / G _ { i }$ . Since each $G _ { i }$ acts continuously, 1021 Lemma 3 gives that the quotient of the product space is the product of the quotient spaces, i.e. that ", + "bbox": [ + 138, + 353, + 825, + 383 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/ef1227d197d396d3e24c439ceaaad258bfb651beb80ea6ca87f8d5985b674f5f.jpg", + "text": "$$\n( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots G _ { k } ) \\cong ( { \\mathcal { X } } _ { 1 } / G _ { 1 } ) \\times \\ldots \\times ( { \\mathcal { X } } _ { k } / G _ { k } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 285, + 390, + 709, + 409 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "1022 and the corresponding quotient map $\\pi : { \\mathcal { X } } / G$ is given by ", + "bbox": [ + 137, + 416, + 549, + 433 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/b18c05cbed56cd5f83ec25f4bd1d1b6e41c09353a1696e961433dcdb06caff19.jpg", + "text": "$$\n\\pi = \\pi _ { 1 } \\times \\ldots \\times \\pi _ { k } , \\qquad \\pi ( x _ { 1 } , \\ldots , x _ { k } ) = ( \\pi _ { 1 } ( x _ { 1 } ) , \\ldots , \\pi _ { k } ( x _ { k } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 284, + 440, + 714, + 458 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "1023 By passing to the quotient (Lemma 1), there exists a continuous ${ \\tilde { f } } : { \\mathcal { X } } / G \\to \\mathbb { R } ^ { s }$ on the quotient space such that 1024 $f = \\tilde { f } \\circ \\pi$ . By Lemma 4, each $\\mathcal { X } _ { i } / G _ { i }$ is compact if $\\mathcal { X } _ { i }$ is compact. Defining the 1025 image $\\mathcal { Z } _ { i } = \\psi _ { i } ( \\mathcal { X } _ { i } / G _ { i } ) \\subseteq \\mathbb { R } ^ { a _ { i } }$ , we thus know that $\\mathcal { Z } _ { i }$ is compact if $\\mathcal { X } _ { i }$ is compact. ", + "bbox": [ + 137, + 467, + 826, + 513 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Moreover, as 26 $\\psi _ { i }$ is a topological embedding, it has a continuous inverse $\\psi _ { i } ^ { - 1 }$ on its image $\\mathcal { Z } _ { i }$ . Further, 27 we have a topological embedding $\\psi : \\mathcal { X } / G \\to \\mathcal { Z } = \\mathcal { Z } _ { 1 } \\times . . . \\times \\mathcal { Z } _ { k }$ given by $\\psi = \\psi _ { 1 } \\times \\ldots \\times \\psi _ { k }$ , with continuous inverse 28 $\\psi ^ { - 1 } = \\psi _ { 1 } ^ { - 1 } \\times \\ldots \\times \\psi _ { k } ^ { - 1 }$ . ", + "bbox": [ + 156, + 520, + 826, + 565 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "Note that ", + "bbox": [ + 173, + 570, + 238, + 584 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/2c839791d2d9da809d8401d45f8649f1ee337cc7c0154017fdef228b71597202.jpg", + "text": "$$\nf = \\tilde { f } \\circ \\pi = ( \\tilde { f } \\circ \\psi ^ { - 1 } ) \\circ ( \\psi \\circ \\pi ) .\n$$", + "text_format": "latex", + "bbox": [ + 385, + 583, + 612, + 603 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "1030 So we define ", + "bbox": [ + 137, + 609, + 261, + 623 + ], + "page_idx": 28 + }, + { + "type": "equation", + "img_path": "images/cd5a35a1ff401f8c296b0ecfe3e8d795a0e688bcd9cd9833432aec4fbb2ab164.jpg", + "text": "$$\n\\begin{array} { r l r } & { \\rho = \\tilde { f } \\circ \\psi ^ { - 1 } } & { \\rho : \\mathcal { Z } \\to \\mathbb { R } ^ { s } } \\\\ & { \\phi _ { i } = \\psi _ { i } \\circ \\pi _ { i } } & { \\phi _ { i } : \\mathcal { X } _ { i } \\to \\mathcal { Z } _ { i } } \\\\ & { \\phi = \\psi \\circ \\pi = \\phi _ { 1 } \\times \\ldots \\times \\phi _ { k } } & { \\phi : \\mathcal { X } \\to \\mathcal { Z } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 294, + 628, + 704, + 686 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "1031 Thus, $f = \\rho \\circ \\phi = \\rho \\circ ( \\phi _ { 1 } \\times \\ldots \\times \\phi _ { k } )$ , so equation (9) holds. Moreover, the $\\rho$ and $\\phi _ { i }$ are continuous, \n1032 as they are compositions of continuous functions. Furthermore, (1) holds as each $\\phi _ { i }$ is invariant \n1033 to $G _ { i }$ because each $\\pi _ { i }$ is invariant to $G _ { i }$ . Since each $\\mathcal { Z } _ { i }$ is compact if $\\mathcal { X } _ { i }$ is compact, the product \n1034 $\\mathcal { Z } = \\mathcal { Z } _ { 1 } \\times \\ldots \\times \\mathcal { Z } _ { k }$ is compact if each $\\mathcal { X } _ { i }$ is compact, thus proving (2). ", + "bbox": [ + 137, + 693, + 826, + 750 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "To show the last statement (3), note simply that if ${ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , then the quotient maps are equal, i.e. $\\pi _ { i } = \\pi _ { j }$ . Moreover, we can choose the embeddings to be equal, so say $\\psi _ { i } = \\psi _ { j }$ . Then, $\\phi _ { i } = \\psi _ { i } \\circ \\pi _ { i } = \\bar { \\psi _ { j } } \\circ \\pi _ { j } = \\phi _ { j }$ , so we are done. □ ", + "bbox": [ + 166, + 753, + 825, + 799 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "G.2 Universality of SignNet and BasisNet ", + "text_level": 1, + "bbox": [ + 166, + 814, + 475, + 830 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "039 Here, we prove Corollary 1 on the universal representation and approximation capabilities of our \n040 Unconstrained-SignNets, Unconstrained-BasisNets, and Expressive-BasisNets. We proceed in sev \n041 eral steps, first proving universal representation of continuous functions when we do not require \n042 permutation equivariance, then proving universal approximation when we do require permutation \n043 equivariance. ", + "bbox": [ + 148, + 842, + 825, + 911 + ], + "page_idx": 28 + }, + { + "type": "text", + "text": "G.2.1 Sign Invariant Universal Representation ", + "text_level": 1, + "bbox": [ + 173, + 90, + 509, + 107 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Recall that $\\mathbb { S } ^ { n - 1 }$ denotes the unit sphere in $\\mathbb { R } ^ { n }$ . As we normalize eigenvectors to unit norm, the domain of our functions on $k$ eigenvectors are on the compact space $( \\bar { \\mathbb { S } } ^ { n - 1 } ) ^ { k }$ . ", + "bbox": [ + 165, + 114, + 823, + 145 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Corollary 2 (Universal Representation for SignNet). $A$ continuous function $f : ( \\mathbb { S } ^ { n - 1 } ) ^ { k } \\to \\mathbb { R } ^ { s }$ is sign invariant, i.e. $f ( s _ { 1 } v _ { 1 } , \\ldots , s _ { k } v _ { k } ) = f ( { \\bar { v _ { 1 } } } , \\ldots , v _ { k } )$ for any $s _ { i } \\in \\{ - 1 , 1 \\}$ , if and only if there exists a continuous $\\phi : \\mathbb { R } ^ { n } \\to \\mathbb { R } ^ { 2 n - 2 }$ and a continuous $\\rho : \\mathbb { R } ^ { ( 2 n - 2 ) k } \\mathbb { R } ^ { s }$ such that ", + "bbox": [ + 173, + 150, + 825, + 194 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/61b97349cd51fa11150e4910c84cc8dfc1bfec138a958d6d6b21df99af98488b.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } ) = \\rho \\left( [ \\phi ( v _ { i } ) + \\phi ( - v _ { i } ) ] _ { i = 1 } ^ { k } \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 357, + 200, + 640, + 220 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "1050 Proof. It can be directly seen that any $f$ of the above form is sign invariant. ", + "bbox": [ + 143, + 237, + 668, + 253 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Thus, we show that any sign invariant $f$ can be expressed in the above form. First, we show that we can apply the general Theorem 3. The group $\\bar { G _ { i } } = \\{ 1 , - 1 \\}$ acts continuously and satisfies that $\\mathbb { S } ^ { n - 1 } / \\{ 1 , - \\bar { 1 } \\} = \\mathbf { \\bar { \\mathbb { R } } } \\mathbb { P } ^ { n - 1 }$ , where $\\mathbb { R } \\mathbb { P } ^ { n - 1 }$ is the real projective space of dimension $n - 1$ . Since $\\mathbb { R } \\mathbb { P } ^ { n - 1 }$ is a smooth manifold of dimension $n - 1$ , Whitney’s embedding theorem states that there exists a (smooth) topological embedding $\\psi _ { i } : \\mathbb { R P } ^ { n - 1 } \\to \\mathbb { R } ^ { \\bar { 2 } n - 2 }$ (Lemma 5). ", + "bbox": [ + 173, + 257, + 825, + 329 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Thus, we can apply the general theorem to see that $f = \\rho \\circ { \\tilde { \\phi } } ^ { k }$ for some continuous $\\rho$ and $\\tilde { \\phi } ^ { k }$ . Note that each $\\tilde { \\phi } _ { i } = \\bar { \\tilde { \\phi } }$ is the same, as each $\\mathcal { X } _ { i } = \\mathbb { S } ^ { n - 1 }$ and $G _ { i } = \\{ 1 , - 1 \\}$ is the same. Also, Theorem 3 says that we may assume that $\\tilde { \\phi }$ is sign invariant, so $\\tilde { \\phi } ( x ) = \\tilde { \\phi } ( - x )$ . Letting $\\phi ( { x } ) = \\tilde { \\phi } ( { x } ) / 2$ , we are done with the proof. □ ", + "bbox": [ + 174, + 334, + 825, + 397 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "G.2.2 Sign Invariant Universal Representation with Extra Features ", + "text_level": 1, + "bbox": [ + 176, + 415, + 653, + 431 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Recall that we may want our sign invariant functions to process other data besides eigenvectors, such as eigenvalues or node features associated to a graph. Here, we show universal representation for when we have this other data that does not possess sign symmetry. The proof is a simple extension of Corollary 2, but we provide the technical details for completeness. ", + "bbox": [ + 174, + 439, + 826, + 496 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Corollary 3 (Universal Representation for SignNet with features). For a compact space of features $\\Omega \\subseteq \\mathbb { R } ^ { d }$ , let $f ( v _ { 1 } , \\ldots , v _ { k } , x _ { 1 } , \\ldots , x _ { k } )$ be a continuous function $f : ( \\mathbb { S } ^ { n - 1 } \\times \\bar { \\Omega } ) ^ { k } \\overset { \\cdot } { } \\mathbb { R } ^ { s }$ . ", + "bbox": [ + 166, + 500, + 826, + 530 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "1067 Then $f$ is sign invariant for the inputs on the sphere, i.e. ", + "bbox": [ + 137, + 535, + 540, + 550 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/5c764048aacb5b126b36754e2ae76180aaa40305912deeda39c41a92fdbb6ab1.jpg", + "text": "$$\nf ( s _ { 1 } v _ { 1 } , \\ldots , s _ { k } v _ { k } , x _ { 1 } , \\ldots , x _ { k } ) = f ( v _ { 1 } , \\ldots , v _ { k } , x _ { 1 } , \\ldots , x _ { k } ) \\qquad s _ { i } \\in \\{ 1 , - 1 \\} ,\n$$", + "text_format": "latex", + "bbox": [ + 241, + 558, + 753, + 575 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "if and only if there exists a continuous 1068 $\\psi : \\mathbb { R } ^ { n + d } \\mathbb { R } ^ { 2 n - 2 + d }$ and a continuous $\\rho : \\mathbb { R } ^ { ( 2 n - 2 + d ) k } \\mathbb { R } ^ { s }$ 1069 such that ", + "bbox": [ + 137, + 583, + 826, + 613 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/2b27ec961a2e7bc8bfa86b6bd6d972163bd48ec4d3113ff4894c2b26602b6d45.jpg", + "text": "$$\nf ( v _ { 1 } , \\ldots , v _ { k } ) = \\rho \\left( \\phi ( v _ { 1 } , x _ { 1 } ) + \\phi ( - v _ { 1 } , x _ { 1 } ) , \\ldots , \\phi ( v _ { k } , x _ { k } ) + \\phi ( - v _ { k } , x _ { k } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 246, + 622, + 751, + 640 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "1070 Proof. Once again, the sign invariance of any $f$ in the above form is clear. ", + "bbox": [ + 137, + 656, + 661, + 672 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "1071 We follow very similar steps to the proof of Corollary 2 to show that we may apply Theorem 3. We \n1072 can view $\\Omega$ as a quotient space, after quotienting by the trivial group that does nothing, $\\Omega \\cong \\Omega / \\{ 1 \\}$ . \n1073 The corresponding quotient map is $\\mathrm { i d } _ { \\Omega }$ , the identity map. Also, $\\Omega$ trivially topologically embeds in \n1074 $\\mathbb { R } ^ { d }$ by the inclusion map. ", + "bbox": [ + 135, + 678, + 826, + 734 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "1075 As $G _ { i } = \\{ - 1 , 1 \\} \\times \\{ 1 \\}$ acts continuously, by Lemma 3 we have that ", + "bbox": [ + 142, + 739, + 629, + 756 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/81d19892c20a2aebcea2d5e5ee610ccf937a58355b63b58ed3e8d351f1a4ed67.jpg", + "text": "$$\n( \\mathbb { S } ^ { n - 1 } \\times \\Omega ) / ( \\{ 1 , - 1 \\} \\times \\{ 1 \\} ) \\cong ( \\mathbb { S } ^ { n - 1 } / \\{ 1 , - 1 \\} ) \\times ( \\Omega / \\{ 1 \\} ) \\cong \\mathbb { R } \\mathbb { P } ^ { n - 1 } \\times \\Omega ,\n$$", + "text_format": "latex", + "bbox": [ + 241, + 763, + 754, + 782 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "with corresponding quotient map 1076 $\\pi \\times \\mathrm { i d } _ { \\Omega }$ , where $\\pi$ is the quotient map to $\\mathbb { R } \\mathbb { P } ^ { n - 1 }$ . ", + "bbox": [ + 135, + 790, + 715, + 805 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "Letting 1077 $\\tilde { \\psi }$ be the embedding of $\\mathbb { R } \\mathbb { P } ^ { n - 1 } \\to \\mathbb { R } ^ { 2 n - 2 }$ guaranteed by Whitney’s embedding theorem 1078 (Lemma 5), we have that $\\psi \\overset { \\cdot } { = } \\tilde { \\psi } \\times \\mathrm { i d } _ { \\Omega }$ is an embedding of $\\mathbb { R } \\mathbb { P } ^ { n - 1 } \\times \\Omega \\to \\mathbb { R } ^ { 2 n - 2 + d }$ . Thus, we can apply Theorem 3 to write 1079 $f = \\rho \\circ { \\tilde { \\phi } } ^ { k }$ for $\\tilde { \\phi } = ( \\tilde { \\psi } \\times \\mathrm { i d } _ { \\Omega } \\mathbf { \\bar { ) } } \\circ ( \\pi \\times \\mathrm { i d } _ { \\Omega } )$ , so ", + "bbox": [ + 135, + 810, + 825, + 859 + ], + "page_idx": 29 + }, + { + "type": "equation", + "img_path": "images/42b1f1d10b062172ba5bcf284f0c25f5e44d8049dca111dfa81ef070a191c231.jpg", + "text": "$$\n\\tilde { \\phi } ( v _ { i } , x _ { i } ) = ( \\tilde { \\psi } ( v _ { i } ) , x _ { i } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 419, + 867, + 578, + 886 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "where 1080 $\\tilde { \\phi } ( v _ { i } , x _ { i } ) = \\tilde { \\phi } ( - v _ { i } , x _ { i } )$ . Letting $\\phi ( v _ { i } , x _ { i } ) = \\tilde { \\phi } ( v _ { i } , x _ { i } ) / 2$ , we are done. ", + "bbox": [ + 133, + 895, + 673, + 912 + ], + "page_idx": 29 + }, + { + "type": "text", + "text": "G.2.3 Basis Invariant Universal Representation ", + "text_level": 1, + "bbox": [ + 173, + 90, + 514, + 106 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Recall that $\\operatorname { S t } ( d , n )$ is the Stiefel manifold of $d$ -tuples of vectors $( v _ { 1 } , \\ldots , v _ { d } )$ where $\\ b { v } _ { i } \\in \\mathbb { R } ^ { n }$ and $v _ { 1 } , \\ldots , v _ { d }$ are orthonormal. This is where our inputs lie, as our eigenvectors are unit norm and orthogonal. We will also make use of the Grassmannian ${ \\mathrm { G r } } ( d , n )$ , which consists of all $d$ -dimensional subspaces in $\\mathbb { R } ^ { n }$ . This is because the Grassmannian is the quotient space for the group action we want, $\\operatorname { G r } ( d , n ) \\cong \\operatorname { S t } ( d , n ) / O ( d )$ , where $Q \\in O ( d )$ acts on $\\bar { V } \\in \\mathrm { S t } ( d , \\bar { n } ) \\subseteq \\mathbb { R } ^ { n \\times d }$ by mapping $V$ to $V Q$ [Gallier and Quaintance, 2020]. ", + "bbox": [ + 173, + 113, + 825, + 198 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Corollary 4 (Universal Representation for BasisNet). For dimensions $d _ { 1 } , \\dotsc , d _ { l } \\leq n$ let $f$ be $a$ continuous function on $\\mathrm { S t } ( \\bar { d } _ { 1 } , n ) \\times \\ldots \\times \\mathrm { S t } ( d _ { l } , n )$ . Further assume that $f$ is invariant to $O ( d _ { 1 } ) \\times$ . $\\dots \\times O ( d _ { l } )$ , where $O ( d _ { i } )$ acts on $\\operatorname { S t } ( d _ { i } , n )$ by multiplication on the right. ", + "bbox": [ + 173, + 200, + 825, + 243 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Then there exist continuous $\\rho : \\mathbb { R } ^ { \\sum _ { i = 1 } ^ { l } 2 d _ { i } ( n - d _ { i } ) } \\mathbb { R } ^ { s }$ and continuous $\\phi _ { i } : \\mathrm { S t } ( d _ { i } , n ) \\to \\mathbb { R } ^ { 2 d _ { i } ( n - d _ { i } ) }$ such that ", + "bbox": [ + 169, + 248, + 821, + 279 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/9d14a0d0a3b8c59bf2f44746483ee0752b9e89058b0df5b8555a2c4876c79a22.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( \\phi _ { 1 } ( V _ { 1 } ) , \\dots , \\phi _ { l } ( V _ { l } ) \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 362, + 279, + 632, + 295 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "where the $\\phi _ { i }$ are $O ( d _ { i } )$ invariant functions, and we can take $\\phi _ { i } = \\phi _ { j }$ if $d _ { i } = d _ { j }$ . ", + "bbox": [ + 173, + 295, + 697, + 310 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Proof. Letting $\\mathcal { X } _ { i } = \\mathrm { S t } ( d _ { i } , n )$ and $G _ { i } = O ( d _ { i } )$ , it can be seen that $G _ { i }$ acts continuously on $\\mathcal { X } _ { i }$ . Also, we have that the quotient space $\\mathrm { S t } ( d _ { i } , n ) / O ( d _ { i } ) = \\mathrm { G r } ( d _ { i } , n )$ is the Grassmannian of $d _ { i }$ dimensional subspaces in $\\mathbb { R } ^ { n }$ , which is a smooth manifold of dimension $d _ { i } ( n - d _ { i } )$ . Thus, the Whitney embedding theorem (Lemma 5) gives a topological embedding $\\psi _ { i } : { \\mathrm { G r } } ( d _ { i } , n ) \\to \\mathbb { R } ^ { 2 d _ { i } ( n - d _ { i } ) }$ . ", + "bbox": [ + 173, + 321, + 825, + 382 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "Hence, we may apply Theorem 3 to obtain continuous $O ( d _ { i } )$ invariant $\\phi _ { i } : \\mathrm { S t } ( d _ { i } , n ) \\to \\mathbb { R } ^ { 2 d _ { i } ( n - d _ { i } ) }$ and continuous $\\rho : \\mathbb { R } ^ { \\sum _ { i = 1 } ^ { l } 2 d _ { i } ( n - d _ { i } ) } \\mathbb { R } ^ { s }$ , such that $f = \\rho \\circ ( \\phi _ { 1 } \\times \\ldots \\times \\phi _ { l } )$ . Also, if $d _ { i } = d _ { j }$ , then ${ \\mathcal { X } } _ { i } = { \\mathcal { X } } _ { j }$ and $G _ { i } = G _ { j }$ , so we can take $\\phi _ { i } = \\phi _ { j }$ . ", + "bbox": [ + 174, + 388, + 825, + 435 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "G.2.4 Basis Invariant and Permutation Equivariant Universal Approximation ", + "text_level": 1, + "bbox": [ + 176, + 468, + 725, + 484 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "With the restriction that $f ( V _ { 1 } , \\dots , V _ { l } ) : \\mathbb { R } ^ { n \\times \\sum _ { i } d _ { i } } \\to \\mathbb { R } ^ { n }$ be permutation equivariant and basis invariant, we need to use the impractically expensive Expressive-BasisNet to approximate $f$ . Universality of permutation invariant or equivariant functions from matrices to scalars or matrices to vectors is difficult to achieve in a computationally tractable manner [Maron et al., 2019, Keriven and Peyré, 2019, Maehara and NT, 2019]. One intuitive reason to expect this is that universally approximating such functions allows solution of the graph isomorphism problem [Chen et al., 2019b], which is a computationally difficult problem. While we have exact representation of basis invariant functions by continuous $\\rho$ and $\\phi _ { i }$ when there is no permutation equivariance constraint, we can only achieve approximation up to an arbitrary $\\epsilon > 0$ when we require permutation equivariance. ", + "bbox": [ + 168, + 491, + 826, + 618 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "1112 Corollary 5 (Universal Approximation for Expressive-BasisNets). Let $f ( V _ { 1 } , \\dots , V _ { l } ) : \\mathrm { S t } ( d _ { 1 } , n ) \\times$ \n1113 $\\dots \\times \\operatorname { S t } ( d _ { l } , n ) \\to \\mathbb { R } ^ { n }$ be continuous, $O ( d _ { 1 } ) \\times \\ldots \\times O ( d _ { l } )$ invariant, and permutation equivariant. \n1114 Then $f$ can be ϵ-approximated by an Expressive-BasisNet. ", + "bbox": [ + 137, + 619, + 825, + 662 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "1115 Proof. By invariance, Corollary 4 of the decomposition theorem shows that $f$ can be written as ", + "bbox": [ + 145, + 675, + 797, + 691 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/86912fe25fcca61f87347d3da9c8bbcf522615f5db83ed3f4c98de9786643cc8.jpg", + "text": "$$\nf ( V _ { 1 } , \\dots , V _ { l } ) = \\rho \\left( \\varphi _ { d _ { 1 } } ( V _ { 1 } ) , \\dots , \\varphi _ { d _ { l } } ( V _ { l } ) \\right)\n$$", + "text_format": "latex", + "bbox": [ + 357, + 691, + 638, + 708 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "for some continuous 1116 $O ( d _ { i } )$ invariant $\\varphi _ { d _ { i } }$ and continuous $\\rho$ . By the first fundamental theorem of $O ( d )$ (Lemma 2), each 1117 $\\varphi _ { d _ { i } }$ can be written as $\\varphi _ { d _ { i } } ( V _ { i } ) = \\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } )$ for some continuous $\\phi _ { d _ { i } }$ . Let ", + "bbox": [ + 142, + 710, + 810, + 739 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/bf667495dbb3b2d36cad4b6156f6cb2299aba431ea0e7d0d4ebd24eaaaea6446.jpg", + "text": "$$\n{ \\mathcal { Z } } = \\{ ( V _ { 1 } V _ { 1 } ^ { \\top } , \\ldots , V _ { l } V _ { l } ^ { \\top } ) : V _ { i } \\in { \\mathrm { S t } } ( d _ { i } , n ) \\} \\subseteq \\mathbb { R } ^ { n ^ { 2 } \\times l } ,\n$$", + "text_format": "latex", + "bbox": [ + 320, + 742, + 678, + 761 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "which is compact as it is the image of the compact space 18 $\\mathrm { S t } ( d _ { 1 } , n ) \\times \\ldots \\times \\mathrm { S t } ( d _ { l } , n )$ under a continuous function. Define 119 $h : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n ^ { 2 } \\times l } \\to \\mathbb { R } ^ { n }$ by ", + "bbox": [ + 151, + 762, + 825, + 792 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/cddf73eba3d761b1fa61ef79a063fb5b5ac415e473513ed703b244528b509afc.jpg", + "text": "$$\n\\begin{array} { r } { h ( V _ { 1 } V _ { 1 } ^ { \\top } , \\ldots , V _ { l } V _ { l } ^ { \\top } ) = \\rho \\left( \\phi _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \\top } ) , \\ldots , \\phi _ { d _ { l } } ( V _ { l } V _ { l } ^ { \\top } ) \\right) . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 305, + 794, + 691, + 814 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "1120 Then note that $h$ is continuous and permutation equivariant from matrices to vectors, so it can be \n1121 $\\epsilon$ -approximated by an invariant graph network [Keriven and Peyré, 2019], call it $\\widetilde { \\mathrm { I G N } }$ . If we define \n1122 $\\tilde { \\rho } = \\widetilde { \\mathrm { I G N } }$ and $\\mathrm { I G N } _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } ) = V _ { i } V _ { i } ^ { \\top }$ (this identity operation is linear and permutation equivariant, \n1123 so it can be exactly expressed by an IGN), then we have $\\epsilon$ -approximation of $f$ by ", + "bbox": [ + 135, + 814, + 825, + 876 + ], + "page_idx": 30 + }, + { + "type": "equation", + "img_path": "images/4e63104867e723be0733de902a09bbc91361b2008566f7ff91e22bf0d8e90a35.jpg", + "text": "$$\n\\widetilde { \\mathrm { I G N } } ( V _ { 1 } V _ { 1 } ^ { \\top } , \\dots , V _ { l } V _ { l } ^ { \\top } ) = \\widetilde { \\rho } \\left( \\mathrm { I G N } _ { d _ { 1 } } ( V _ { 1 } V _ { 1 } ^ { \\top } ) , \\dots , \\mathrm { I G N } _ { d _ { l } } ( V _ { l } V _ { l } ^ { \\top } ) \\right) .\n$$", + "text_format": "latex", + "bbox": [ + 274, + 877, + 722, + 897 + ], + "page_idx": 30 + }, + { + "type": "text", + "text": "1126 Theorem 4. Consider the same setup as Theorem 3, where $\\mathcal { X } _ { i }$ are also compact. Let $\\Phi _ { i }$ be a \n1127 family of $G _ { i }$ -invariant functions that universally approximate $G _ { i }$ -invariant continuous functions \n1128 $\\mathcal { X } _ { i } \\ \\to \\ \\mathbb { R } ^ { a _ { i } }$ , and let $\\mathcal { R }$ be a set of continuous function that universally approximate continuous \n1129 functions $\\mathcal { Z } \\subseteq \\mathbb { R } ^ { a } \\to \\mathbb { R } ^ { s }$ for every compact $\\mathcal { Z }$ , where $a = \\textstyle \\sum _ { i } a _ { i }$ . Then for any $\\varepsilon > 0$ and any \n1130 $G$ -invariant continuous function $f : \\mathcal { X } _ { 1 } \\times . . . \\times \\mathcal { X } _ { k } \\to \\mathbb { R } ^ { s }$ there exists $\\phi \\in \\Phi$ and $\\rho \\in \\mathcal R$ such that \n1131 $\\| f - \\rho ( \\phi _ { 1 } , \\ldots , \\phi _ { k } ) \\| _ { \\infty } < \\varepsilon$ . ", + "bbox": [ + 135, + 116, + 826, + 202 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "Proof. Consider a particular 1132 $G$ -invariant continuous function $f : \\mathcal { X } _ { 1 } \\times . . . \\times \\mathcal { X } _ { k } \\to \\mathbb { R } ^ { s }$ . By Theorem 3 there exists 1133 $G _ { i }$ -invariant continuous functions $\\phi _ { i } ^ { \\prime } : \\mathcal { X } _ { i } \\mathbb { R } ^ { a _ { i } }$ and a continuous function 1134 $\\rho ^ { \\prime } : \\mathcal { Z } \\subseteq \\mathbb { R } ^ { a } \\to \\mathbb { R } ^ { s }$ (where $a = \\textstyle \\sum _ { i } a _ { i } )$ such that ", + "bbox": [ + 135, + 213, + 825, + 258 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/5bc3c7b945bb7db266fc52e82253897ddc81175f9238d933dbdb66b3080be50f.jpg", + "text": "$$\nf ( v _ { 1 } , \\dots , v _ { k } ) = \\rho ^ { \\prime } ( \\phi _ { 1 } ^ { \\prime } ( v _ { 1 } ) , \\dots , \\phi _ { k } ^ { \\prime } ( v _ { k } ) ) .\n$$", + "text_format": "latex", + "bbox": [ + 362, + 263, + 633, + 281 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "1135 Now fix an $\\varepsilon > 0$ . For any $\\rho \\in \\mathcal R$ and any $\\phi _ { i } \\in \\Phi _ { i } ( i = 1 , \\dots k )$ we may bound the difference from \n1136 $f$ as follows (suppressing the $v _ { i }$ ’s for brevity), ", + "bbox": [ + 135, + 286, + 825, + 316 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/826eda30ae040114afe927ae5ebf90a48b2a44169c10245178b7c0b25afc67c7.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| f - \\rho ( \\phi _ { 1 } , \\ldots , \\phi _ { k } ) \\| _ { \\infty } } \\\\ & { = \\| \\rho ^ { \\prime } ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } , \\ldots , \\phi _ { k } ) \\| _ { \\infty } } \\\\ & { = \\| \\rho ^ { \\prime } ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) + \\rho ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } , \\ldots , \\phi _ { k } ) \\| _ { \\infty } } \\\\ & { \\leq \\| \\rho ^ { \\prime } ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) \\| _ { \\infty } + \\| \\rho ( \\phi _ { 1 } ^ { \\prime } , \\ldots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } , \\ldots , \\phi _ { k } ) \\| _ { \\infty } } \\\\ & { = \\mathrm { I } + \\mathrm { I I } } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 238, + 320, + 758, + 415 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "1137 Now let $\\begin{array} { r } { K ^ { \\prime } = \\prod _ { i = 1 } ^ { k } \\mathrm { i m } \\phi _ { i } ^ { \\prime } } \\end{array}$ . Since each $\\phi _ { i } ^ { \\prime }$ is continuous and defined on a compact set $\\mathcal { X } _ { i }$ we know \n1138 that $\\mathrm { i m } \\phi _ { i } ^ { \\prime }$ is compact, and so the product $K$ is also compact. Since $K ^ { \\prime }$ is compact, it is contained in a \n1139 closed ball $B ( r )$ of radius $r > 0$ centered at the origin. Let $K$ be the closed ball $\\boldsymbol { B } ( \\boldsymbol { r } + 1 )$ of radius \n1140 $r + 1$ centered at the origin, so $K$ contains $K ^ { \\prime }$ and a ball of radius 1 around each point of $K ^ { \\prime }$ . We \n1141 may extend $\\rho ^ { \\prime }$ continuously to $K$ as needed, so assume $\\rho ^ { \\prime } : K \\to \\mathbb { R } ^ { s }$ . By universality of $\\mathcal { R }$ we may \n1142 pick a particular $\\rho : K \\mathbb { R } ^ { s }$ , $\\rho \\in \\mathcal R$ such that ", + "bbox": [ + 135, + 421, + 825, + 508 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/0b571240026a80e473c73776e92e4d6c429e3c4aefefc1f35b9728885e61aafc.jpg", + "text": "$$\n\\mathrm { I } = \\operatorname* { s u p } _ { \\{ v _ { i } \\in \\mathcal { X } _ { i } \\} _ { i = 1 } ^ { k } } \\| \\rho ^ { \\prime } ( \\phi _ { 1 } ^ { \\prime } , \\dots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } ^ { \\prime } , \\dots , \\phi _ { k } ^ { \\prime } ) \\| _ { \\infty } \\leq \\operatorname* { s u p } _ { z \\in K } \\| \\rho ^ { \\prime } ( z ) - \\rho ( z ) \\| _ { 2 } < \\varepsilon / 2 .\n$$", + "text_format": "latex", + "bbox": [ + 222, + 512, + 772, + 544 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "1143 Keeping this choice of $\\rho$ , it remains only to bound II. As $\\rho$ is continuous on a compact domain, it \n1144 is in fact uniformly continuous. Thus, we can choose a $\\delta ^ { \\prime } > 0$ such that if $\\| y - \\bar { z } \\| _ { 2 } \\leq \\delta ^ { \\prime }$ , then \n1145 $\\| \\rho ( y ) - \\rho ( z ) \\| _ { \\infty } < \\epsilon / 2$ , and then we define $\\delta = \\operatorname* { m i n } ( \\delta ^ { \\prime } , 1 )$ . \n1146 Since $\\Phi _ { i }$ universally approximates $\\phi _ { i } ^ { \\prime }$ we may pick $\\phi _ { i } \\in \\Phi _ { i }$ such that $\\| \\phi _ { i } - \\phi _ { i } ^ { \\prime } \\| _ { \\infty } < \\delta / \\sqrt { k }$ , and \n1147 thus $\\| ( \\phi _ { 1 } , \\dots , \\phi _ { k } ) - ( \\phi _ { 1 } ^ { \\prime } , \\dots \\phi _ { k } ^ { \\prime } ) \\| _ { \\infty } \\leq \\delta$ . With this choice of $\\phi _ { i }$ , we know that $\\textstyle \\prod _ { i = 1 } ^ { k } \\operatorname { i m } \\phi _ { i } \\subseteq K$ \n1148 (because each $\\phi _ { i } ( x _ { i } )$ is within distance 1 of $\\phi _ { i } ^ { \\prime } ( x _ { i } ) )$ . Thus, $\\rho ( \\phi _ { 1 } ( x _ { 1 } ) , \\ldots , \\phi _ { k } ( x _ { k } ) { \\bar { ) } }$ is well-defined, \n1149 and we have ", + "bbox": [ + 135, + 549, + 826, + 593 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 598, + 823, + 659 + ], + "page_idx": 31 + }, + { + "type": "equation", + "img_path": "images/263baaee4abfb6e0b8bd6ca8c9112770ce48f16437e8a8635938540c4279c360.jpg", + "text": "$$\n\\begin{array} { r l } & { \\mathrm { I I } = \\| \\rho ( \\phi _ { 1 } ^ { \\prime } , \\dots , \\phi _ { k } ^ { \\prime } ) - \\rho ( \\phi _ { 1 } , \\dots , \\phi _ { k } ) \\| _ { \\infty } } \\\\ & { \\quad = \\underset { \\{ x _ { i } \\in \\mathcal { X } _ { i } \\} _ { i = 1 } ^ { k } } { \\operatorname* { s u p } } \\| \\rho ( \\phi _ { 1 } ^ { \\prime } ( x _ { 1 } ) , \\dots , \\phi _ { k } ^ { \\prime } ( x _ { k } ) ) - \\rho ( \\phi _ { 1 } ( x _ { 1 } ) , \\dots , \\phi _ { k } ( x _ { k } ) ) \\| _ { 2 } } \\\\ & { \\quad < \\varepsilon / 2 } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 269, + 662, + 725, + 733 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "1150 due to our choice of $\\delta$ , which completes the proof. ", + "bbox": [ + 137, + 737, + 503, + 752 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "1151 H Basis Invariance for Graph Representation Learning ", + "text_level": 1, + "bbox": [ + 143, + 771, + 653, + 790 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "152 H.1 Spectral Graph Convolution ", + "text_level": 1, + "bbox": [ + 151, + 801, + 415, + 818 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "In this section, we consider spectral graph convolutions, which for node features $\\ b X \\in \\mathbb { R } ^ { n \\times q }$ take the form $\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { n } \\dot { \\theta _ { i } v _ { i } } v _ { i } ^ { \\top } X } \\end{array}$ for some parameters $\\theta _ { i }$ . We can optionally take $\\theta _ { i } = h ( \\lambda _ { i } )$ for some continuous function $h : \\mathbb { R } \\mathbb { R }$ of the eigenvalues. This form captures most popular spectral graph convolutions in the literature [Bruna et al., 2014, Hamilton, 2020, Bronstein et al., 2017]; often, such convolutions are parameterized by taking $h$ to be some analytic function such as a simple affine function [Kipf and Welling, 2017], a linear combination in a polynomial basis [Defferrard et al., ", + "bbox": [ + 161, + 827, + 825, + 912 + ], + "page_idx": 31 + }, + { + "type": "text", + "text": "59 2016, Chien et al., 2021], or a parameterization of rational functions [Levie et al., 2018, Bianchi et al., \n60 2021]. \n161 First, it is well known and easy to see that spectral graph convolutions are permutation equivariant, as \n162 for a permutation matrix $P$ we have ", + "bbox": [ + 150, + 90, + 825, + 119 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 126, + 825, + 154 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/84c49a6fab0a7cbb717420f2368f038ac25a5418a7124cc491bf48e5963ec130.jpg", + "text": "$$\nf ( P V , \\Lambda , P X ) = \\sum _ { i } \\theta _ { i } P v _ { i } v _ { i } ^ { \\top } P ^ { \\top } P X = \\sum _ { i } \\theta _ { i } P v _ { i } v _ { i } ^ { \\top } X = P f ( V , \\Lambda , X ) .\n$$", + "text_format": "latex", + "bbox": [ + 251, + 161, + 745, + 194 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "1163 Also, it is easy to see that they are sign invariant, as $( - v _ { i } ) ( - v _ { i } ) ^ { \\top } = v _ { i } v _ { i } ^ { \\top }$ . However, if the $\\theta _ { i }$ do not \n1164 depend on the eigenvalues, then the spectral graph convolution is not necessarily basis invariant. For \n1165 instance, if $v _ { 1 }$ and $v _ { 2 }$ are in the same eigenspace, and we change basis by permuting $v _ { 1 } ^ { \\prime } = v _ { 2 }$ and \n1166 $v _ { 2 } ^ { \\prime } = v _ { 1 }$ , then if $\\theta _ { 1 } \\neq \\theta _ { 2 }$ the spectral graph convolution will generally change as well. \n1167 On the other hand, if $\\theta _ { i } = h ( \\lambda _ { i } )$ for some function $h : \\mathbb { R } \\mathbb { R }$ , then the spectral graph convolution \n1168 is basis invariant. This is because if $v _ { i }$ and $v _ { j }$ belong to the same eigenspace, then $\\lambda _ { i } = \\lambda _ { j }$ so \n1169 $h ( \\lambda _ { i } ) = h ( \\lambda _ { j } )$ . Thus, if $v _ { i _ { 1 } } , \\ldots , v _ { i _ { d } }$ are eigenvectors of the same eigenspace with eigenvalue $\\lambda$ , \n1170 we have that $\\begin{array} { r } { \\sum _ { l = 1 } ^ { d } \\underline { h } ( \\lambda _ { i _ { l } } ) v _ { i _ { l } } v _ { i _ { l } } ^ { \\top } = h ( \\lambda ) \\sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \\top } . } \\end{array}$ . Now, note that $\\scriptstyle \\sum _ { l = 1 } ^ { d } v _ { i _ { l } } v _ { i _ { l } } ^ { \\top }$ is the orthogonal \n1171 projector onto the eigenspace [Trefethen and Bau III, 1997]. A change of basis does not change this \n1172 orthogonal projector, so such spectral graph convolutions are basis invariant. \n1173 Another way to see this basis invariance is with a simple computation. Let $V _ { 1 } , \\dots , V _ { l }$ be the \n1174 eigenspaces of dimension $d _ { 1 } , \\ldots , d _ { l }$ , where $V _ { i } \\in \\mathbb { R } ^ { n \\times d _ { i } ^ { \\star } }$ . Let the corresponding eigenvalues be \n1175 $\\mu _ { 1 } , \\ldots , \\mu _ { l }$ . Then for any orthogonal matrices $Q _ { i } \\in O ( d _ { i } )$ , we have ", + "bbox": [ + 135, + 202, + 825, + 261 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 265, + 826, + 353 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 358, + 823, + 401 + ], + "page_idx": 32 + }, + { + "type": "equation", + "img_path": "images/f7186e5a54143ef94319274046c6343d0da931c5b39aebdb5444d9c1b892e9c3.jpg", + "text": "$$\n\\begin{array} { l } { { \\displaystyle \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } = \\sum _ { j = 1 } ^ { l } V _ { j } h ( \\mu _ { j } ) I _ { d _ { j } } V _ { j } ^ { \\top } } } \\\\ { ~ } \\\\ { { \\displaystyle = \\sum _ { j = 1 } ^ { l } V _ { j } h ( \\mu _ { j } ) I _ { d _ { j } } Q _ { j } Q _ { j } ^ { \\top } V _ { j } ^ { \\top } } } \\\\ { { \\displaystyle ~ = \\sum _ { j = 1 } ^ { l } ( V _ { j } Q _ { j } ) h ( \\mu _ { j } ) I _ { d _ { j } } ( V _ { j } Q _ { j } ) ^ { \\top } } , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 338, + 409, + 656, + 546 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "1176 so the spectral graph convolution is invariant to substituting $V _ { j } Q _ { j }$ for $V _ { j }$ . ", + "bbox": [ + 137, + 551, + 651, + 568 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "77 Now, we give the proof that shows SignNet and BasisNet can universally approximate spectral graph \n78 convolutions. \n1179 Theorem 1 (Learning Spectral Graph Convolutions). Suppose the node features $X \\in \\mathbb { R } ^ { n \\times q }$ take \n1180 values in compact sets. Then SignNet can universally approximate any spectral graph convolution, \n1181 and both BasisNet and Expressive-BasisNet can universally approximate any parametric spectral \n1182 graph convolution. \n1183 Proof. Note that eigenvectors and eigenvalues of normalized Laplacian matrices take values in \n1184 compact sets, since the eigenvalues are in [0, 2] and we take eigenvectors to have unit-norm. Thus, \n1185 the whole domain of the spectral graph convolution is compact. \n1186 Let $\\varepsilon > 0$ . First, consider a spectral graph convolution $\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X } \\end{array}$ . For SignNet, \n1187 let $\\phi ( v _ { i } , \\lambda _ { i } , X )$ approximate the function $\\tilde { \\phi } ( v _ { i } , \\lambda _ { i } , X ) = \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X$ to within $\\varepsilon / n$ error, which \n1188 DeepSets can do since this is a continuous permutation equivariant function from vectors to vectors \n1189 1190 [Segol and Lipman, 2019] (note1 is the all ones vector). Then $\\rho = \\textstyle \\sum _ { i = 1 } ^ { n }$ pass is a $\\lambda _ { i }$ as a vector in ear permutati $\\mathbb { R } ^ { n }$ by instead passing equivariant operati $\\lambda _ { i } \\mathbf { 1 }$ , wherehat can \n1191 be exactly expressed by DeepSets, so the total error is within . The same argument applies when \n1192 $\\theta _ { i } = h ( \\lambda _ { i } )$ for some continuous function $h$ . \n1193 For the basis invariant case, consider a parametric spectral graph convolution $f ( V , \\Lambda , X ) ~ =$ \n1194 $\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } X } \\end{array}$ . Note that if the eigenspace bases are $V _ { 1 } , \\dots , V _ { l }$ with eigenvalues $\\mu _ { 1 } , \\ldots , \\mu _ { l }$ , we \n1195 can write the $\\begin{array} { r } { f ( V , \\Lambda , X ) = \\sum _ { i = 1 } ^ { l } h ( \\mu _ { j } ) V _ { j } V _ { j } ^ { \\top } X } \\end{array}$ . Again, we will let $\\rho = \\textstyle \\sum _ { i = 1 } ^ { l }$ be a sum function, \n1196 which can be expressed exactly by DeepSets. Thus, it suffices to show that $h ( \\mu _ { j } ) V _ { j } V _ { j } ^ { \\top } X$ can be $\\epsilon / n$ \n1197 approximated by a 2-IGN (i.e. an IGN that only uses vectors and matrices). \n1198 Note that since $h$ is continuous, we can use an elementwise MLP (which IGNs can learn) to \n1199 approximate $f _ { 1 } ( \\mu { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) = ( h ( \\mu ) { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X )$ to arbitrary precision (note that we rep \n1200 resent the eigenvalue $\\mu$ as a constant matrix $\\mu \\mathbf { 1 1 } ^ { \\top }$ ). Also, since a 2-IGN can learn matrix vector \n1201 multiplication (Cai and Wang [2022] Lemma 10), we can approximate $f _ { 2 } ( h ( \\mu ) { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) =$ \n1202 $( h ( \\mu ) \\mathbf { 1 1 } ^ { \\top } , V V ^ { \\top } X )$ , as $V _ { i } V _ { i } ^ { \\top } \\in \\mathbb { R } ^ { n ^ { 2 } }$ is a matrix and $\\ b X \\in \\mathbb { R } ^ { n \\times q }$ is a vector with respect to permuta \n1203 tion symmetries. Finally, we use an elementwise MLP to approximate the scalar-vector multiplication \n1204 $f _ { 3 } ( h ( \\mu ) { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) = h ( \\mu ) V V ^ { \\top } X$ . Since $f _ { 3 } \\circ f _ { 2 } \\circ \\bar { f } _ { 1 } ( \\mu { \\bf 1 1 } ^ { \\top } , V V ^ { \\top } , X ) = h ( \\mu ) V V ^ { \\top } X$ , and \n1205 since 2-IGNs universally approximate each $f _ { i }$ , applying Lemma 6 shows that a 2-IGN can approx \n1206 imate $h ( \\mu ) V V ^ { \\top } X$ to $\\epsilon / n$ accuracy, so we are done. Since Expressive-BasisNet is stronger than \n1207 BasisNet, it can also universally approximate these functions. □ \n1208 From the proof, we can see that SignNet and BasisNet need only learn simple functions for the $\\rho$ and \n1209 $\\phi$ when $h$ is simple, or when the filter is non-parametric and we need only learn $\\theta _ { i }$ . Xu et al. [2020] \n1210 propose the principle of algorithmic alignment, and show that if separate modules of a neural network \n1211 each need only learn simple functions (that is, functions that are well-approximated by low-order \n1212 polynomials with small coefficients), then the network may be more sample efficient. If we do not \n1213 require permutation equivariance, and parameterize SignNet and BasisNet with simple MLPs, then \n1214 algorithmic alignment may suggest that our models are sample efficient. Indeed, $\\rho \\overset { \\cdot } { = } \\sum$ is a simple \n1215 linear function with coefficients 1, and $\\phi ( V , \\lambda , X ) = h ( \\lambda ) V V ^ { \\top } X$ is quadratic in $V$ and linear in $X$ \n1216 so it is simple if $h$ is simple. ", + "bbox": [ + 156, + 573, + 826, + 602 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 604, + 825, + 661 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 678, + 826, + 722 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 726, + 825, + 828 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 833, + 825, + 912 + ], + "page_idx": 32 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 90, + 826, + 238 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 251, + 825, + 378 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "Proposition 3. There exist infinitely many pairs of non-isomorphic graphs that SignNet and BasisNet can distinguish, but spectral graph convolutions or spectral GNNs cannot distinguish. ", + "bbox": [ + 158, + 381, + 823, + 410 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "1219 Proof. The idea is as follows: we will take graphs $G$ and give them the node feature matrix $X _ { G } =$ \n1220 $D ^ { 1 / 2 } \\mathbf { 1 }$ , i.e. each node has as feature the square root of its degree. Then any spectral graph convolution \n1221 (or, the first layer of any spectral GNN) will map $V \\mathrm { D i a g } ( \\theta ) V ^ { \\mathrm { ~ l ~ } } X$ to something that only depends on \n1222 the degree sequence and number of nodes. Thus, any spectral graph convolution or spectral GNN \n1223 will have the same output (up to permutation) for any such graphs $G$ with node features $X _ { G }$ and the \n1224 same number of nodes and same degree sequence. On the other hand, SignNet and BasisNet can \n1225 distinguish between infinitely many pairs of graphs $\\left( G ^ { ( 1 ) } , G ^ { ( 2 ) } \\right)$ with node features $( X _ { G ^ { ( 1 ) } } , X _ { G ^ { ( 2 ) } } )$ \n1226 and the same number of nodes and degree sequence; this is because SignNet and BasisNet can tell \n1227 when a graph is bipartite. \n1228 For each $n \\geq 5$ , we will define $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ as connected graphs with $n$ nodes, with the same \n1229 degree sequence. Also, we define $G ^ { ( 1 ) }$ to have node features $X _ { i } ^ { ( 1 ) } = \\sqrt { d _ { i } ^ { ( 1 ) } }$ , where $d _ { i } ^ { ( 1 ) }$ is the degree \n1230 of node $i$ in $G ^ { ( 1 ) }$ , and similarly $G ^ { ( 2 ) }$ has node features $X _ { i } ^ { ( 2 ) } = \\sqrt { d _ { i } ^ { ( 2 ) } }$ . Now, note that $X ^ { ( 1 ) }$ is an \n1231 eigenvector of the normalized Laplacian of $G ^ { ( 1 ) }$ , and it has eigenvalue $0$ . As we take the eigenvectors \n1232 to be orthonormal (since the normalized Laplacian is symmetric), for any spectral graph convolution \n1233 we have that ", + "bbox": [ + 133, + 424, + 825, + 553 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "", + "bbox": [ + 133, + 558, + 825, + 666 + ], + "page_idx": 33 + }, + { + "type": "equation", + "img_path": "images/73b0411817ae20399bac710f759fa286521b3064c1b5c321715b8814e56837f8.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { n } \\theta _ { i } v _ { i } v _ { i } ^ { \\top } X ^ { ( 1 ) } = \\theta _ { 1 } v _ { 1 } v _ { 1 } ^ { \\top } X ^ { ( 1 ) } = \\theta _ { 1 } D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } ( D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } ) ^ { \\top } D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } = \\theta _ { 1 } \\sum _ { j = 1 } ^ { n } ( d _ { j } ^ { ( 1 ) } ) D _ { 1 } ^ { 1 / 2 } \\mathbf { 1 } .\n$$", + "text_format": "latex", + "bbox": [ + 205, + 667, + 761, + 710 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "1234 Where $D _ { 1 }$ is the diagonal degree matrix of $G ^ { ( 1 ) }$ . Likewise, any spectral graph convolution outputs \n1235 $\\theta _ { 1 } \\sum _ { j } ( d _ { j } ^ { ( 2 ) } ) D _ { 2 } ^ { 1 / 2 } { \\bf 1 }$ for $G ^ { ( 2 ) }$ . Since $D _ { 1 }$ and $D _ { 2 }$ are the same up to a permutation, we have that any \n1236 spectral graph convolution has the same output for $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ , up to a permutation. In fact, this \n1237 also holds for spectral GNNs, as the first layer will always have the same output (up to a permutation) \n1238 on $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ , so the latter layers will also have the same output up to a permutation. \n1239 Now, we concretely define $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ . This is illustrated in Figure 11 and Figure 12. For $n = 5$ , \n1240 let $G ^ { ( 1 ) }$ contain a triangle with nodes $w _ { 1 } , w _ { 2 } , w _ { 3 }$ , and have a path of length 2 coming out of one of \n1241 the nodes in the triangle, say $w _ { 1 }$ connects to $w _ { 4 }$ , and $w _ { 4 }$ connects to $w _ { 5 }$ . This is not bipartite, as there \n1242 is a triangle. Let $G ^ { ( 2 ) }$ be a bipartite graph that has 2 nodes on the left $( v _ { 1 } , v _ { 2 } )$ and 3 nodes on the \n1243 right $( v _ { 3 } , v _ { 4 } , v _ { 5 } )$ . Connect $v _ { 1 }$ with all nodes on the right, and connect $v _ { 2 }$ with $v _ { 3 }$ and $\\boldsymbol { v } _ { 4 }$ . \n1244 Note that both $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ have the same number of nodes and the same degree sequence \n1245 $\\{ 3 , 2 , 2 , 2 , 1 \\}$ . Thus, spectral graph convolutions or spectral GNNs cannot distinguish them. How", + "bbox": [ + 133, + 715, + 825, + 796 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 801, + 825, + 876 + ], + "page_idx": 33 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 882, + 825, + 912 + ], + "page_idx": 33 + }, + { + "type": "image", + "img_path": "images/c447c5d87e24ae97195454045af4a8f9ce0acdcec39d39916c422939fc00c4e0.jpg", + "image_caption": [ + "Figure 11: Illustration of our constructed $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ for $n = 5$ , as used in the proof of Proposition 3. " + ], + "image_footnote": [], + "bbox": [ + 308, + 85, + 689, + 227 + ], + "page_idx": 34 + }, + { + "type": "image", + "img_path": "images/1dd1428f70451ee7830fca1dd260abae9fac15f7d525169066b3d20b615b8ff1.jpg", + "image_caption": [ + "Figure 12: Illustration of our constructed $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ for $n = 6$ , as used in the proof of Proposition 3. " + ], + "image_footnote": [], + "bbox": [ + 284, + 280, + 714, + 426 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "ever, SignNet and BasisNet can distinguish them, as they can tell whether a graph is bipartite by checking the highest eigenvalue of the normalized Laplacian. This is because the multiplicity of the eigenvalue 2 is the number of bipartite components. In particular, SignNet can approximate the function $\\phi ( v _ { i } , \\lambda _ { i } , X ) = \\lambda _ { i }$ and $\\rho \\approx \\mathrm { m a x } _ { i = 1 } ^ { n }$ . Likewise, BasisNet can approximate the function $\\phi _ { d _ { i } } ( V _ { i } V _ { i } ^ { \\top } , \\lambda _ { i } ) = \\lambda _ { i }$ and $\\rho \\approx \\mathrm { m a x } _ { i = 1 } ^ { l }$ . ", + "bbox": [ + 147, + 488, + 825, + 561 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "This in fact gives an infinite family of graphs that SignNet / BasisNet can distinguish, but spectral graph convolutions or spectral graph GNNs cannot. To see why, suppose we have $G ^ { ( 1 ) }$ and $G ^ { ( 2 ) }$ for some $n \\geq 5$ . Then we construct a pair of graphs on $n + 1$ nodes with the same degree sequence. To do this, we add another node to the path of $G ^ { ( 1 ) }$ , thus giving it degree sequence $\\{ 3 , 2 , \\ldots , 2 , 1 \\}$ . For $G ^ { ( 2 ) }$ , we add a node $v _ { n + 1 }$ to the side that $v _ { n }$ is not contained on (e.g. for $n = 5$ , we add $\\boldsymbol { v } _ { 6 }$ to the left side, as $\\boldsymbol { v } _ { 5 }$ was on the right), then connect $v _ { n }$ to $v _ { n + 1 }$ to also give a degree sequence $\\{ 3 , 2 , \\ldots , 2 , 1 \\}$ . Note that the non-bipartiteness of $G ^ { ( 1 ) }$ and bipartiteness of $G ^ { ( 2 ) }$ are preserved. ", + "bbox": [ + 173, + 565, + 825, + 670 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "H.2 Existing Positional Encodings ", + "text_level": 1, + "bbox": [ + 173, + 707, + 423, + 722 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Here, we show that our SignNets and BasisNets universally approximate various types of existing graph positional encodings. The key is to show that these positional encodings are related to spectral graph convolution matrices and the diagonals of these matrices, and to show that our networks can approximate these matrices and diagonals. ", + "bbox": [ + 173, + 732, + 825, + 787 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proposition 5. If the eigenvalues take values in a compact set, SignNets and BasisNets universally approximate the diagonal of any spectral graph convolution matrix $\\begin{array} { r } { \\pmb { f } ( V , \\Lambda ) = \\mathrm { d i a g } \\left( \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) \\hat { v _ { i } v _ { i } ^ { \\top } } \\right) } \\end{array}$ BasisNets can additionally universally approximate any spectral graph convolution matrix $f ( V , \\Lambda ) =$ $\\textstyle \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top }$ . ", + "bbox": [ + 173, + 792, + 825, + 853 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "Proof. Note that the $v _ { i }$ come from a compact set as they are of unit norm. The $\\lambda _ { i }$ are from a compact set by assumption; this assumption holds for the normalized Laplacian, as $\\lambda _ { i } \\in [ 0 , 2 ]$ . Also, as diag is linear, the spectral graph convolution diagonal can be written $\\begin{array} { r } { \\sum _ { i = 1 } ^ { n } h ( \\lambda _ { i } ) \\mathrm { d i a g } ( v _ { i } v _ { i } ^ { \\top } ) } \\end{array}$ . ", + "bbox": [ + 138, + 868, + 825, + 912 + ], + "page_idx": 34 + }, + { + "type": "text", + "text": "1271 Let $\\epsilon > 0$ . For SignNet, let $\\rho = \\textstyle \\sum _ { i = 1 } ^ { n }$ , which can be exactly expressed as it is a permutation \n1272 equivariant linear operation from vectors to vectors. Then $\\phi ( v _ { i } , \\lambda _ { i } )$ can approximate the function \n1273 $\\lambda _ { i } \\mathrm { d i a g } ( v _ { i } v _ { i } ^ { \\top } )$ to arbitrary precision, as it is a permutation equivariant function from vectors to \n1274 vectors [Segol and Lipman, 2019]. Thus, letting $\\phi$ approximate the function to $\\epsilon / n$ accuracy, SignNet \n1275 can approximate $f$ to $\\epsilon$ accuracy. ", + "bbox": [ + 135, + 89, + 825, + 162 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Let $l$ be the number of eigenspaces $V _ { 1 } , \\dots , V _ { l }$ , so $\\begin{array} { r } { f ( V , \\Lambda ) = \\sum _ { i = 1 } ^ { l } h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top } } \\end{array}$ . For BasisNet, we need only show that it can approximate the spectral graph convolution matrix to $\\epsilon / l$ accuracy, as a 2-IGN can exactly express the diag function in each $\\phi _ { d _ { i } }$ , since it is a linear permutation equivariant function from matrices to vectors. A 2-IGN can universally approximate the function $f _ { 1 } ( \\mu _ { i } , V _ { i } V _ { i } ^ { \\top } ) =$ $( h ( \\mu _ { i } ) , V _ { i } V _ { i } ^ { \\top } )$ , as it can express any elementwise MLP. Also, a 2-IGN can universally approximate the scalar-matrix multiplication $f _ { 2 } ( h ( \\mu _ { i } ) , V _ { i } V _ { i } ^ { \\top } ) ~ = ~ h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top }$ by another elementwise MLP. Since $h ( \\mu _ { i } ) V _ { i } V _ { i } ^ { \\top } = f _ { 2 } \\overset { \\cdot } { \\circ } f _ { 1 } ( \\mu _ { i } , \\bar { V _ { i } } { V _ { i } ^ { \\top } } )$ , Lemma 6 shows that a single 2-IGN can approximate this composition to $\\epsilon / l$ accuracy, so we are done. ", + "bbox": [ + 165, + 167, + 826, + 289 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "Proposition 4. SignNet and BasisNet universally approximate node positional encodings based on heat kernels [Feldman et al., 2022] and random walks [Dwivedi et al., 2022]. BasisNet universally approximates diffusion and $p$ -step random walk relative positional encodings [Mialon et al., 2021], and generalized PageRank and landing probability distance encodings [Li et al., 2020]. ", + "bbox": [ + 173, + 315, + 825, + 372 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "89 Proof. We will show that we can apply the above Proposition 5, by showing that all of these \n90 positional encodings are spectral graph convolutions. The heat kernel embeddings are of the form \n91 diag $\\begin{array} { r } { \\big ( \\sum _ { i = 1 } ^ { n } \\exp ( - t \\lambda _ { i } ) v _ { i } v _ { i } ^ { \\top } \\big ) } \\end{array}$ for some choices of the parameter $t$ , so they can be approximated by \n92 SignNets or BasisNets. Also, the diffusion kernel [Mialon et al., 2021] is just the matrix of this \n93 heat kernel, and the $p$ -step random walk kernel is ${ \\textstyle \\sum _ { i = 1 } ^ { n } } ( 1 - \\gamma \\lambda _ { i } ) ^ { p } v _ { i } v _ { i } ^ { \\top }$ for some parameter $\\gamma$ , so \n94 BasisNets can universally approximate both of these. ", + "bbox": [ + 151, + 386, + 825, + 470 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "For the other positional encodings, we let $v _ { i }$ be the eigenvectors of the random walk Laplacian $I - D ^ { - 1 } A$ instead of the normalized Laplacian $I - { D ^ { - 1 / 2 } A D ^ { - 1 / 2 } }$ . The eigenvalues of these two Laplacians are the same, and if $\\tilde { v } _ { i }$ is an eigenvector of the normalized Laplacian then $D ^ { - 1 / 2 } \\tilde { v } _ { i }$ is an eigenvector of the random walk Laplacian with the same eigenvalue [Von Luxburg, 2007]. ", + "bbox": [ + 171, + 476, + 825, + 535 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "299 Then with $v _ { i }$ as the eigenvectors of the random walk Laplacian, the random walk positional encodings \n300 (RWPE) in Dwivedi et al. [2022] take the form ", + "bbox": [ + 147, + 540, + 823, + 569 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/4839f9c714294b8ea73ccd640ec8269cab9ddc4b3d063e003a39bb28c937cc0f.jpg", + "text": "$$\n\\operatorname { d i a g } \\left( ( D ^ { - 1 } A ) ^ { k } \\right) = \\operatorname { d i a g } \\left( \\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top } \\right) ,\n$$", + "text_format": "latex", + "bbox": [ + 336, + 574, + 660, + 617 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "1301 for any choices of integer $k$ . ", + "bbox": [ + 138, + 621, + 359, + 636 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "1302 The distance encodings proposed in Li et al. [2020] take the form ", + "bbox": [ + 142, + 641, + 602, + 656 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/a9184b956438d64bc72d74ece9c02c68670d711733dfcd9ff1e0d279fa1ebad4.jpg", + "text": "$$\nf _ { 3 } ( A D ^ { - 1 } , ( A D ^ { - 1 } ) ^ { 2 } , ( A D ^ { - 1 } ) ^ { 3 } , \\cdot \\cdot \\cdot ) ,\n$$", + "text_format": "latex", + "bbox": [ + 374, + 660, + 620, + 679 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "1303 for some function $f _ { 3 }$ . We restrict to continuous $f _ { 3 }$ here; shortest path distances can be obtained by a \n1304 discontinuous $f _ { 3 }$ that we discuss below. Their generalized PageRank based distance encodings can \n1305 be obtained by ", + "bbox": [ + 137, + 683, + 825, + 726 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/d980f65cb177744157bf862be09dbb663a65ef87946aa9259afa1da86732a37d.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { n } \\left( \\sum _ { k \\geq 1 } \\gamma _ { k } ( 1 - \\lambda _ { i } ) ^ { k } \\right) v _ { i } v _ { i } ^ { \\top }\n$$", + "text_format": "latex", + "bbox": [ + 397, + 723, + 599, + 773 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "1306 for some $\\gamma _ { k } \\in \\mathbb { R }$ , so this is a spectral graph convolution. They also define so-called landing probability \n1307 based positional encodings, which take the form ", + "bbox": [ + 135, + 775, + 828, + 803 + ], + "page_idx": 35 + }, + { + "type": "equation", + "img_path": "images/5e9fed7a20bb8d88af668acdaf4fa25d852ac4c75f9642d676007069023581ba.jpg", + "text": "$$\n\\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top } ,\n$$", + "text_format": "latex", + "bbox": [ + 433, + 808, + 562, + 849 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "1308 for some choices of integer $k$ . Thus, BasisNets can approximate these distance encoding matrices. ", + "bbox": [ + 143, + 853, + 797, + 869 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "1309 Another powerful class of positional encodings is based on shortest path distances between nodes \n1310 in the graph [Ying et al., 2021, Li et al., 2020]. Shortest path distances can be expressed in a \n1311 form similar to the spectral graph convolution, but require a highly discontinuous function. If we \n1312 define $f _ { 3 } ( x _ { 1 } , \\dots , x _ { n } ) = \\operatorname* { m i n } _ { i : x _ { i } \\neq 0 } i$ to be the lowest index such that $x _ { i }$ is nonzero, then we can \n1313 1314 write the shortest path diselementwise to return an $n \\times n$ matrix as matrix. $f _ { 3 } ( D ^ { - 1 } A , ( D ^ { - 1 } A ) ^ { 2 } , \\dots , ( D ^ { - 1 } A ) ^ { n } )$ $\\textstyle ( D ^ { - 1 } A ) ^ { k } = \\sum _ { i = 1 } ^ { n } ( 1 - \\lambda _ { i } ) ^ { k } v _ { i } v _ { i } ^ { \\top }$ , where BasisN $f _ { 3 }$ is applied can learn \n1315 the inside arguments, but cannot learn the discontinuous function $f _ { 3 }$ . ", + "bbox": [ + 135, + 882, + 826, + 912 + ], + "page_idx": 35 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 90, + 825, + 164 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "1316 H.3 Spectral Invariants ", + "text_level": 1, + "bbox": [ + 142, + 180, + 352, + 195 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Here, we consider the graph angles $\\alpha _ { i j } = \\lVert V _ { i } V _ { i } ^ { \\top } e _ { j } \\rVert _ { 2 }$ , for $i = 1 , \\dots , l$ where $l$ is the number of eigenspaces, and $j = 1 , \\dotsc , n$ . It is clear that graph angles are permutation equivariant and basis invariant. These graph angles have been extensively studied, so we cite a number of interesting properties of them. That graph angles determine the number of length 3, 4 and 5 cycles, the connectivity of a graph, and the number of length $k$ closed walks is all shown in Chapter 4 of Cvetkovic´ et al. [1997]. Other properties may be of use for graph representation learning as well. For instance, the eigenvalues of node-deleted subgraphs of a graph $\\mathcal { G }$ are determined by the eigenvalues and graph angles of $\\mathcal { G }$ ; this may be useful in extending recent graph neural networks that are motivated by node deletion and the reconstruction conjecture [Cotta et al., 2021, Bevilacqua et al., 2022, Papp et al., 2021, Tahmasebi et al., 2020]. ", + "bbox": [ + 173, + 205, + 825, + 344 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Now, we prove that BasisNet can universally approximate the graph angles. The graph properties we consider in the theorem are all integer valued (e.g. the number of cycles of length 3 in a graph is an integer). Thus, any two graphs that differ in these properties will differ by at least 1, so as long as we have approximation to $\\varepsilon < 1 / 2$ , we can distinguish any two graphs that differ in these properties. Recall the statement of Theorem 2. ", + "bbox": [ + 174, + 351, + 825, + 421 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "Theorem 2. BasisNet can universally approximate the graph angles $\\alpha _ { i j }$ . The eigenvalues and graph angles (and thus BasisNets) can determine the number of length 3, 4, and 5 cycles, whether a graph is connected, and the number of length $k$ closed walks from any vertex to itself. ", + "bbox": [ + 173, + 425, + 825, + 468 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "1335 Proof. Note that the graph angles satisfy ", + "bbox": [ + 138, + 486, + 442, + 501 + ], + "page_idx": 36 + }, + { + "type": "equation", + "img_path": "images/8cc88578ff9cceb30f318860140740261f2b943886071db6f68630443d346b39.jpg", + "text": "$$\n\\alpha _ { i j } = \\| { V _ { i } V _ { i } } ^ { \\top } e _ { j } \\| _ { 2 } = \\sqrt { e _ { j } ^ { \\top } V _ { i } V _ { i } ^ { \\top } V _ { i } V _ { i } ^ { \\top } e _ { j } } = \\sqrt { e _ { j } ^ { \\top } V _ { i } V _ { i } ^ { \\top } e _ { j } } ,\n$$", + "text_format": "latex", + "bbox": [ + 303, + 508, + 692, + 536 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "1336 where $V _ { i }$ is a basis for the $i$ th adjacency matrix eigenspace, and $e _ { j } ^ { \\top } V _ { i } V _ { i } ^ { \\top } e _ { j }$ is the $( j , j )$ -entry of $V _ { i } V _ { i } ^ { \\top }$ . \n1337 These graph angles are just the elementwise square roots of the diagonals of the matrices $V _ { i } V _ { i } ^ { \\top }$ . \n1338 As $f _ { 1 } ( V _ { i } V _ { i } ^ { \\top } ) = \\mathrm { d i a g } ( V _ { i } V _ { i } ^ { \\top } )$ is a permutation equivariant linear function from matrices to vectors, \n1339 2-IGN on $V _ { i } V _ { i } ^ { \\top }$ can exactly compute this with 0 error. Then a 2-IGN can learn an elementwise \n1340 MLP to approximate the elementwise square root $f _ { 2 } ( \\mathrm { d i a g } ( V _ { i } V _ { i } ^ { \\top } ) ) = \\sqrt { \\mathrm { d i a g } ( V _ { i } V _ { i } ^ { \\top } ) }$ to arbitrary \n1341 precision. Finally, there may be remaining operations $f _ { 3 }$ that are permutation invariant or permutation \n1342 equivariant from vectors to vectors; for instance, the $\\alpha _ { i j }$ are typically gathered into a matrix of size \n1343 $l \\times n$ where the columns are lexicographically sorted $\\it l$ is the number of eigenspaces) [Cvetkovic´ \n1344 et al., 1997], or we may have a permutation invariant readout to compute a subgraph count. A \n1345 DeepSets can approximate $f _ { 3 }$ without any higher order tensors besides vectors [Zaheer et al., 2017, \n1346 Segol and Lipman, 2019]. ", + "bbox": [ + 135, + 542, + 826, + 707 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "As 2-IGNs can approximate each 1347 $f _ { i }$ individually, a single 2-IGN can approximate $f _ { 3 } \\circ f _ { 2 } \\circ f _ { 1 }$ by 1348 Lemma 6. Also, since the graph properties considered in the theorem are integer-valued, BasisNet 1349 can distinguish any two graphs that differ in one of these properties. □ ", + "bbox": [ + 137, + 712, + 825, + 755 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "1350 To see that message passing graph neural networks (MPNNs) cannot determine these quantities, we \n1351 use the fact that MPNNs cannot distinguish between two graphs that have the same number of nodes \n1352 and where each node (in both graphs) has the same degree. For $k \\geq 3$ , let $C _ { k }$ denote the cycle graph \n1353 of size $k$ , and $C _ { k } + C _ { k }$ denote the graph that is the union of two disjoint cycle graphs of size $k$ \n1354 MPNNs cannot distinguish between $C _ { 2 k }$ and $C _ { k } + C _ { k }$ for $k \\geq 3$ , because they have the same number \n1355 of nodes, and each node has degree 2. Thus, MPNNs cannot tell whether a graph is connected, as \n1356 $C _ { 2 k }$ is but $C _ { k } + C _ { k }$ is not. Also, it cannot count the number of 3, 4, or 5 cycles, as $C _ { k } + C _ { k }$ has two \n1357 $k$ cycles while $C _ { 2 k }$ has no $k$ cycles. Likewise, any node in $C _ { k } + C _ { k }$ has more length $k$ closed walks \n1358 than any node in $C _ { 2 k }$ . This is because any length $k$ closed walk in $C _ { 2 k }$ has an analogous closed walk \n1359 in $C _ { k } + C _ { k }$ , but the nodes in $C _ { k } + C _ { k }$ also have a closed walk that completely goes around a cycle. ", + "bbox": [ + 135, + 772, + 825, + 912 + ], + "page_idx": 36 + }, + { + "type": "text", + "text": "In this section, we collect useful lemmas for our proofs. These lemmas generally only require basic tools to prove. Our first lemma is a crucial property of quotient spaces. ", + "bbox": [ + 158, + 119, + 823, + 148 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma 1 (Passing to the quotient). Let $\\mathcal { X }$ and $\\mathcal { V }$ be topological spaces, and let $\\mathcal { X } / G$ be a quotient space, with corresponding quotient map $\\pi$ . Then for every continuous $G$ -invariant function $f : \\mathcal { X } $ $\\mathcal { V } _ { : }$ , there is a unique continuous ${ \\tilde { f } } : { \\mathcal { X } } / G \\to { \\mathcal { Y } }$ such that $f = \\tilde { f } \\circ \\pi$ . ", + "bbox": [ + 171, + 150, + 825, + 195 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. For $z \\in \\mathcal { X } / G$ , by surjectivity of $\\pi$ we can choose an $x _ { z } \\in \\mathcal { X }$ such that $\\pi ( x _ { z } ) = z$ . Define ${ \\tilde { f } } : { \\mathcal { X } } / G \\to { \\mathcal { Y } }$ by $\\tilde { f } ( z ) = f ( x _ { z } )$ . This is well-defined, since if $\\pi ( x _ { z } ) = \\pi ( x )$ for any other $x \\in \\mathcal { X }$ , then $g x _ { z } = x$ for some $g \\in G$ , so ", + "bbox": [ + 166, + 208, + 825, + 253 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/2939c4d707ebcb7b44a84ee5dee32154219ba89910c5415c812d80db3f42a20d.jpg", + "text": "$$\nf ( x ) = f ( g x _ { z } ) = f ( x _ { z } ) = \\tilde { f } ( z ) ,\n$$", + "text_format": "latex", + "bbox": [ + 387, + 256, + 609, + 275 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "1369 where the second equality uses the $G$ -invariance of $f$ . Note that $\\tilde { f }$ is continuous by the universal \n1370 property of quotient spaces. Also, $\\tilde { f }$ is the unique function such that $f = \\tilde { f } \\circ \\pi$ ; if there were another \n1371 function $h : \\mathcal { X } / G \\to \\mathcal { Y }$ with $h ( z ) \\neq { \\tilde { f } } ( z )$ , then $h ( z ) \\neq f ( x _ { z } )$ , so $h ( \\pi ( x _ { z } ) ) = h ( z ) \\neq f ( x _ { z } )$ . \n1372 Next, we give the First Fundamental Theorem of $O ( d )$ , a classical result that has been recently used \n1373 for machine learning by Villar et al. [2021]. This result shows that an orthogonally invariant $f ( V )$ \n1374 can be expressed as a function $h ( V V ^ { \\top } )$ . We give a proof that if $f$ is continuous, then $h$ is also \n1375 continuous. ", + "bbox": [ + 135, + 279, + 825, + 327 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "", + "bbox": [ + 140, + 339, + 825, + 397 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Lemma 2 (First Fundamental Theorem of $O ( d ) { \\big \\rangle }$ ). A continuous function $f : \\mathbb { R } ^ { n \\times d } \\mathbb { R } ^ { s }$ is orthogonally invariant, i.e. $f ( V Q ) = f ( V )$ for all $Q \\in O ( d )$ , if and only if $f ( V ) = h ( V V ^ { \\top } ) f o \\iota$ r some continuous $h$ . ", + "bbox": [ + 161, + 400, + 825, + 444 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Proof. If $f ( V ) = h ( V V ^ { \\top } )$ , then we have $f ( V Q ) = h ( V Q Q ^ { \\top } V ^ { \\top } ) = h ( V V ^ { \\top } )$ so $f$ is orthogonally invariant. ", + "bbox": [ + 168, + 455, + 825, + 486 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "1381 For the other direction, invariant theory shows that the $O ( d )$ invariant polynomials are generated \n1382 by the inner products $v _ { i } ^ { \\top } v _ { j }$ , where $v _ { i } \\in \\mathbb { R } ^ { d }$ are the rows of $V$ [Kraft and Procesi, 1996]. Let $p :$ \n1383 $\\mathbb { R } ^ { n \\times d } \\to \\mathbb { R } ^ { n \\times n }$ be the map $p ( V ) = V V ^ { \\top }$ . Then González and de Salas [2003] Lemma 11.13 shows \n1384 that the quotient space $\\mathbb { R } ^ { n \\times d } / O ( d )$ is homeomorphic to a closed subset $p ( \\mathbb { R } ^ { n \\times d } ) = \\mathcal { Z } \\subseteq \\mathbb { R } ^ { n \\times n }$ . \n1385 Let $\\tilde { p }$ refer to this homeomorphism, and note that ${ \\tilde { p } } \\circ \\pi = p$ by passing to the quotient (Lemma 1). \n1386 Then any continuous $O ( d )$ invariant $f$ passes to a unique continuous $\\widetilde { f } \\ : \\ \\mathbb { R } ^ { n \\times d } / O ( d ) \\ \\to \\ \\mathbb { R } ^ { s }$ \n1387 (Lemma 1), so $f = \\tilde { f } \\circ \\pi$ where $\\pi$ is the quotient map. Define $h : { \\mathcal { Z } } \\to \\mathbb { R } ^ { s }$ by $h = \\tilde { f } \\circ \\tilde { p } ^ { - 1 }$ , and \n1388 note that $h$ is a composition of continuous functions and hence continuous. Finally, we have that \n1389 $\\begin{array} { r } { h ( V V ^ { \\top } ) = h ( \\tilde { p } \\circ \\pi ( V ) ) = \\tilde { f } \\circ \\pi ( V ) = f ( V ) } \\end{array}$ , so we are done. □ \n390 The next lemma allows us to decompose a quotient of a product space into a product of smaller \n391 quotient spaces. ", + "bbox": [ + 135, + 491, + 826, + 628 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "", + "bbox": [ + 148, + 641, + 825, + 671 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "92 Lemma 3. Let $\\mathcal { X } _ { 1 } , \\ldots , \\mathcal { X } _ { k }$ be topological spaces and $G _ { 1 } , \\ldots , G _ { k }$ be topological groups such that each 93 $G _ { i }$ acts continuously on $\\mathcal { X } _ { i }$ . Denote the quotient maps by $\\pi _ { i } : \\mathcal { X } _ { i } \\mathcal { X } _ { i } / G _ { i }$ . Then the quotient 94 of the product is the product of the quotient, i.e. ", + "bbox": [ + 151, + 672, + 825, + 715 + ], + "page_idx": 37 + }, + { + "type": "equation", + "img_path": "images/d676959cc7db9abfd041355e41887bce3911a4d1b84dd02fdd59dc4e9683bf19.jpg", + "text": "$$\n( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots \\times G _ { k } ) \\cong ( { \\mathcal { X } } _ { 1 } / G _ { 1 } ) \\times \\ldots \\times ( { \\mathcal { X } } _ { k } / G _ { k } ) ,\n$$", + "text_format": "latex", + "bbox": [ + 276, + 717, + 717, + 734 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "1395 and π1 × . . . $\\times \\pi _ { k } : \\mathcal { X } _ { 1 } \\times . . . \\mathcal { X } _ { k } \\to ( \\mathcal { X } _ { 1 } / G _ { 1 } ) \\times . . . \\times ( \\mathcal { X } _ { k } / G _ { k } )$ ) is quotient map. ", + "bbox": [ + 140, + 736, + 715, + 753 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "1396 Proof. First, we show that $\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }$ is a quotient map. This is because 1. the quotient map \n1397 of any continuous group action is an open map, so each $\\pi _ { i }$ is an open map, 2. the product of open \n1398 maps is an open map, so $\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }$ is an open map and 3. a continuous surjective open map is a \n1399 quotient map, so $\\pi _ { 1 } \\times \\ldots \\times \\pi _ { k }$ , which is continuous and surjective, is a quotient map. ", + "bbox": [ + 137, + 765, + 825, + 823 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "Now, we need only apply the theorem of uniqueness of quotient spaces to show (51) (see e.g. Lee [2013], Theorem A.31). Letting $q : { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } \\to ( { \\mathcal { X } } _ { 1 } \\times \\ldots \\times { \\mathcal { X } } _ { k } ) / ( G _ { 1 } \\times \\ldots \\times G _ { k } )$ denote the quotient map for this space, it is easily seen that $q ( x _ { 1 } , \\ldots , x _ { k } ) = q ( y _ { 1 } \\ldots , y _ { k } )$ if and only if $\\pi _ { 1 } \\stackrel { \\textstyle \\ldots } { \\times } \\ldots \\times \\pi _ { k } ( \\bar { x _ { 1 } } , \\ldots , x _ { k } ) \\stackrel { \\textstyle \\ldots } { = } \\pi _ { 1 } \\times \\ldots \\times \\bar { \\pi } _ { k } ( y _ { 1 } , \\ldots , y _ { k } )$ , since either of these is true if and only if there exist $g _ { i } \\in G _ { i }$ such that $x _ { i } = g _ { i } y _ { i }$ for each $i$ . Thus, we have an isomorphism of these quotient spaces. □ ", + "bbox": [ + 173, + 827, + 825, + 911 + ], + "page_idx": 37 + }, + { + "type": "text", + "text": "406 The following lemma shows that quotients of compact spaces are also compact, which is useful for \n407 universal approximation on quotient spaces. \n08 Lemma 4 (Compactness of quotients of compact spaces). Let $\\mathcal { X }$ be a compact space. Then the \n409 quotient space $\\mathcal { X } / G$ is compact. \n1410 Proof. Denoting the quotient map by $\\pi : { \\mathcal { X } } \\to { \\mathcal { X } } / G$ and letting $\\{ U _ { \\alpha } \\} _ { \\alpha }$ be an open cover of $\\mathcal { X } / G$ , \n1411 we have that $\\{ \\check { \\pi } ^ { - 1 } ( U _ { \\alpha } ) \\} _ { \\alpha }$ is an open cover of $\\mathcal { X }$ . By compactness of $\\mathcal { X }$ , we can choose a finite \n1412 subcover $\\{ \\pi ^ { - 1 } ( U _ { \\alpha _ { i } } ) \\} _ { i = 1 , \\dots , n }$ . Then $\\{ \\pi ( \\pi ^ { - 1 } ( U _ { \\alpha _ { i } } ) ) \\} _ { i = 1 , \\dots , n } = \\{ U _ { \\alpha _ { i } } \\} _ { i = 1 , \\dots , n }$ by surjectivity, and \n1413 $\\{ U _ { \\alpha _ { i } } \\} _ { i = 1 , \\dots , n }$ is thus an open cover of $\\mathcal { X } / G$ . □ \n1414 The Whitney embedding theorem gives a nice condition that we apply to show that the quotient \n1415 spaces $\\chi / \\bar { G }$ that we deal with embed into Euclidean space. It says that when $\\mathcal { X } / G$ is a smooth \n1416 manifold, then it can be embedded into a Euclidean space of double the dimension of the manifold. \n1417 The proof is outside the scope of this paper. ", + "bbox": [ + 150, + 90, + 825, + 119 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "", + "bbox": [ + 150, + 125, + 826, + 155 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 176, + 826, + 234 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 257, + 826, + 314 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "18 Lemma 5 (Whitney Embedding Theorem [Whitney, 1944]). Every smooth manifold $\\mathcal { M }$ of dimension n > 0 can be smoothly embedded in R2n 19 . ", + "bbox": [ + 155, + 319, + 823, + 347 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "1420 Finally, we give a lemma that helps prove universal approximation results. It says that if functions \n1421 $f$ that we want to approximate can be written as compositions $f = f _ { L } \\circ \\dots \\circ f _ { 1 }$ , then it suffices \n1422 to universally approximate each $f _ { i }$ and compose the results to universally approximate the $f$ . This \n1423 is especially useful for proving universality of neural networks, as we may use some layers to \n1424 approximate each $f _ { i }$ , then compose these layers to approximate the target function $f$ . \n1425 Lemma 6 (Layer-wise universality implies universality). Let $\\mathcal { Z } \\subseteq \\mathbb { R } ^ { d _ { 0 } }$ be a compact domain, let \n1426 $\\mathcal { F } _ { 1 } , \\ldots , \\mathcal { F } _ { L }$ be families of continuous functions where ${ \\mathcal { F } } _ { i }$ consists of functions from $\\mathbb { R } ^ { d _ { i - 1 } } \\mathbb { R } ^ { d _ { i } }$ \n1427 for some $d _ { 1 } , \\ldots , d _ { L }$ . Let $\\mathcal { F }$ be the family of functions $\\{ f _ { L } \\circ . . . f _ { 1 } : \\mathcal { Z } \\mathbb { R } ^ { d _ { L } } , f _ { i } \\in \\mathcal { F } _ { i } \\}$ that are \n1428 compositions of functions $f _ { i } \\in \\mathcal { F } _ { i }$ . ", + "bbox": [ + 135, + 359, + 825, + 430 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 434, + 825, + 492 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "For each 1429 $i$ , let $\\Phi _ { i }$ be a family of continuous functions that universally approximates ${ \\mathcal { F } } _ { i }$ . Then the 1430 family of compositions $\\Phi = \\left\\{ \\phi _ { L } \\circ . . . \\circ \\phi _ { 1 } : \\phi _ { i } \\in \\Phi _ { i } \\right\\}$ universally approximates $\\mathcal { F }$ . ", + "bbox": [ + 143, + 497, + 826, + 527 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Proof. Let 1431 $f = f _ { L } \\circ . . . \\circ f _ { 1 } \\in \\mathcal { F }$ . Let $\\tilde { \\mathcal { Z } } _ { 1 } = \\mathcal { Z }$ , and then for $i \\geq 2$ let $\\tilde { \\mathcal { Z } } _ { i } = f _ { i - 1 } ( \\tilde { \\mathcal { Z } } _ { i - 1 } )$ . Then each 1432 $\\mathcal { \\tilde { Z } } _ { i }$ is compact by continuity of the $f _ { i }$ . For $1 \\leq i < L$ , let $\\mathcal { Z } _ { i } = \\tilde { \\mathcal { Z } } _ { i }$ , and for $i = L$ let $\\mathcal { Z } _ { L }$ be a compact 1433 set containing $\\tilde { \\mathcal { Z } } _ { L }$ such that every ball of radius one centered at a point in $\\tilde { \\mathcal { Z } } _ { L }$ is still contained in $\\mathcal { Z } _ { L }$ . ", + "bbox": [ + 135, + 549, + 826, + 599 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "1434 Let $\\epsilon > 0$ . We will show that there is a $\\phi \\in \\Phi$ such that $\\| f - \\phi \\| _ { \\infty } < \\epsilon$ by induction on $L$ . This holds \n435 trivially for $L = 1$ , as then $\\Phi = \\Phi _ { 1 }$ . \n1436 Now, let $L \\geq 2$ , and suppose it holds for $L - 1$ . By universality of $\\Phi _ { L }$ , we can choose a $\\phi _ { L } : \\mathcal { Z } _ { L } $ \n1437 $\\mathbb { R } ^ { d _ { L } } \\in \\Phi _ { L }$ such that $\\| \\phi _ { L } - f _ { L } \\| _ { \\infty } < \\epsilon / 2$ . As $\\phi _ { L }$ is continuous on a compact domain, it is also \n1438 uniformly continuous, so we can choose a $\\tilde { \\delta } > 0$ such that $\\| y - z \\| _ { 2 } < \\tilde { \\delta } \\implies \\| \\phi _ { L } ( y ) - \\phi _ { L } ( z ) \\| _ { 2 } <$ \n1439 $\\epsilon / 2$ . ", + "bbox": [ + 148, + 603, + 825, + 633 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 637, + 825, + 698 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "Let 1440 $\\delta = \\operatorname* { m i n } ( \\tilde { \\delta } , 1 )$ . By induction, we can choose $\\phi _ { L - 1 } \\circ . . . \\circ \\phi _ { 1 } , \\phi _ { i } \\in \\Phi _ { i }$ such that ", + "bbox": [ + 135, + 704, + 723, + 722 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/85cb932ef395a0fc013c4eae79b49503953a06f12000a4aa8583515cdcfb5dac.jpg", + "text": "$$\n\\begin{array} { r } { \\| \\phi _ { L - 1 } \\circ . . . \\circ \\phi _ { 1 } - f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } < \\delta . } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 354, + 729, + 643, + 747 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "1441 Note that $\\phi _ { L - 1 } \\circ . . . \\circ \\phi _ { 1 } ( \\mathcal { Z } ) \\subseteq \\mathcal { Z } _ { L }$ , because for each $x \\in { \\mathcal { Z } }$ , $\\phi _ { L - 1 } \\circ . . . \\circ \\phi _ { 1 } ( x )$ is within $\\delta \\leq 1$ 1442 Euclidean distance to $f _ { L - 1 } \\circ \\dots \\circ f _ { 1 } ( x ) \\in \\tilde { \\mathcal { Z } } _ { L }$ , so it is contained in $\\mathcal { Z } _ { L }$ by construction. Thus, we may define 1443 $\\phi = \\phi _ { L } \\circ . . . \\circ \\phi _ { 1 } : \\mathcal { Z } \\mathbb { R } ^ { d _ { L } }$ , and compute that ", + "bbox": [ + 135, + 755, + 825, + 803 + ], + "page_idx": 38 + }, + { + "type": "equation", + "img_path": "images/b7f0b67eb993843680cd8b3d392aa29e3f46e2ac017fd8811321b4017f5f8a13.jpg", + "text": "$$\n\\begin{array} { r l } & { \\| \\phi - f \\| _ { \\infty } \\leq \\| \\phi - \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } + \\| \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } - f \\| _ { \\infty } } \\\\ & { \\qquad < \\| \\phi - \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } + \\epsilon / 2 , } \\end{array}\n$$", + "text_format": "latex", + "bbox": [ + 246, + 810, + 750, + 848 + ], + "page_idx": 38 + }, + { + "type": "text", + "text": "1444 since $\\| \\phi _ { L } - f _ { L } \\| _ { \\infty } < \\epsilon / 2$ . To bound this other term, let $x \\in { \\mathcal { Z } }$ , and for $y = \\phi _ { L - 1 } \\circ . . . \\circ \\phi _ { 1 } ( x )$ \n1445 and $z = f _ { L - 1 } \\circ . . . \\circ f _ { 1 } ( x )$ , we know that $\\| y - z \\| _ { 2 } < \\delta$ , so $\\| \\phi _ { L } ( y ) - \\phi _ { L } ( z ) \\| _ { 2 } < \\epsilon / 2$ by uniform \n1446 continuity. As this holds for all $x$ , we have $\\| \\phi - \\phi _ { L } \\circ f _ { L - 1 } \\circ . . . \\circ f _ { 1 } \\| _ { \\infty } \\leq \\epsilon / 2$ , so $\\| \\phi - f \\| _ { \\infty } < \\epsilon$ \n1447 and we are done. □ ", + "bbox": [ + 135, + 854, + 825, + 911 + ], + "page_idx": 38 + }, + { + "type": "table", + "img_path": "images/cf9d849e58009d78da15aa2e27b951227bbf17f652c389df8aa37249168fcfd0.jpg", + "table_caption": [ + "Table 7: Results on the ZINC dataset with $5 0 0 \\mathrm { k }$ parameter budget and no edge features. Numbers are the mean and standard deviation over 4 runs each with different seeds. " + ], + "table_footnote": [], + "table_body": "
Base modelPositional encodingk#paramsTest MAE (↓)
GINNo PE16497k0.348±0.014
LapPE (flip)16498k0.341±0.011
SignNet16500k0.238±0.012
GATNo PE16501k0.464±0.011
LapPE (flip)16502k0.462±0.013
SignNet16499k0.243±0.008
", + "bbox": [ + 282, + 131, + 715, + 242 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "1448 J Further Experiments ", + "text_level": 1, + "bbox": [ + 145, + 268, + 383, + 287 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "J.1 Graph Regression with no Edge Features ", + "text_level": 1, + "bbox": [ + 166, + 299, + 498, + 315 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "All graph regression models in Table 1 use edge features for learning and inference. To show that SignNet is also useful when no edge features are available, we ran ZINC experiments without edge features as well. The results are displayed in Table 7. In this setting, SignNet still significantly improves the performance over message passing networks without positional encodings, and over Laplacian positional encodings with sign flipping data augmentation. ", + "bbox": [ + 173, + 325, + 825, + 396 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "1455 J.2 Learning Spectral Graph Convolutions ", + "text_level": 1, + "bbox": [ + 137, + 410, + 485, + 426 + ], + "page_idx": 39 + }, + { + "type": "table", + "img_path": "images/7002f9c12eabc99a50abae1c5cacb82c225ff78a619a50459a78bbd8ef247f8d.jpg", + "table_caption": [ + "Table 8: Sum of squared errors for spectral graph convolution regression (with no test set). Lower is better. Numbers are mean and standard deviation over 50 images from He et al. [2021]. " + ], + "table_footnote": [], + "table_body": "
Low-passHigh-passBand-passBand-rejectionComb
GCN.111±.0683.092±5.111.720±3.151.418±1.031.753±1.17
GAT.113±.065.954±.6961.105±.964.543±.340.638±.446
GPR-GNN.033±.032.012±.007.137±.081.256±.197.369±.460
ARMA.053±.029.042±.024.107±.039.148±.089.202±.116
ChebNet.003±.002.001±.001.005±.003.009±.006.022±.016
BernNet.001±.002.001±.001.000±.000.048±.042.027±.019
Transformer3.662±1.973.715±1.981.531±1.301.506±1.293.178±1.93
Transformer Eig Flip4.454±2.324.425±2.381.651±1.532.567±1.733.720±1.94
Transformer Eig Abs2.727±1.403.172±1.611.264±.7881.445±.9432.607±1.32
DeepSets SignNet.004±.013.086±.405.021±.115.008±.037.003±.016
Transformer SignNet.003±.016.004±.025.001±.004.006±.023.093±.641
DeepSets BasisNet.009±.018.003±.015.008±.030.004±.011.015±.060
Transformer BasisNet.079±.471.014±.038.005±.018.006±.016.014±.051
", + "bbox": [ + 197, + 479, + 797, + 684 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "1456 To numerically test the ability of our basis invariant networks for learning spectral graph convolutions, \n1457 we follow the experimental setups of Balcilar et al. [2020], He et al. [2021]. We take the dataset of 50 \n1458 images in He et al. [2021] (originally from the Image Processing Toolbox of MATLAB), and resize \n1459 them from $1 0 0 \\times 1 0 0$ to $3 2 \\times 3 2$ . Then we apply the same spectral graph convolutions on them as in \n1460 He et al. [2021], and train neural networks to learn these as regression targets. As in prior work, we \n1461 report sum of squared errors on the training set to measure expressivity. \n1462 We compare against message passing GNNs [Kipf and Welling, 2017, Velickovi ˇ c et al. ´ , 2018] and \n1463 spectral GNNs [Chien et al., 2021, Bianchi et al., 2021, Defferrard et al., 2016, He et al., 2021]. \n1464 Also, we consider standard Transformers with only node features, with eigenvectors and sign flip \n1465 augmentation, and with absolute values of eigenvectors. These models are all approximately sign \n1466 invariant (they either use eigenvectors in a sign invariant way or do not use eigenvectors). We use \n1467 DeepSets [Zaheer et al., 2017] in SignNet and 2-IGN [Maron et al., 2018] in BasisNet for $\\phi$ , use \n1468 a DeepSets for $\\rho$ in both cases, and then feed the features into another DeepSets or a standard \n1469 Transformer [Vaswani et al., 2017] to make the final predictions. That is, we are only given graph \n1470 information through the eigenvectors and eigenvalues, and we do not use message passing. \n1471 Table 8 displays the results, which validate our theoretical results in Section 3.1. Without any message \n1472 passing, SignNet and BasisNet allow DeepSets and Transformers to perform strongly, beating the \n1473 spectral GNNs GPR-GNN and ARMA on all tasks. Also, our networks outperform all other methods \n1474 on the band-rejection and comb filters, and are mostly close to the best model on the other filters. ", + "bbox": [ + 135, + 696, + 826, + 781 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 786, + 825, + 912 + ], + "page_idx": 39 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 92, + 825, + 147 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "1475 K Further Experimental Details ", + "text_level": 1, + "bbox": [ + 145, + 166, + 459, + 184 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "K.1 Hardware, Software, and Data Details ", + "text_level": 1, + "bbox": [ + 168, + 199, + 482, + 213 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "All experiments could fit on one GPU at a time. Most experiments were run on a server with 8 NVIDIA RTX 2080 Ti GPUs. We run all of our experiments in Python, using the PyTorch [Paszke et al., 2019] framework (license URL). We also make use of Deep Graph Library (DGL) [Wang et al., 2019] (Apache License 2.0), and PyTorch Geometric (PyG) [Fey and Lenssen, 2019] (MIT License) for experiments with graph data. ", + "bbox": [ + 173, + 224, + 825, + 294 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "We open source our code [redacted for anonymous review]. ", + "bbox": [ + 166, + 300, + 562, + 314 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "The data we use are all freely available online. The datasets we use are ZINC [Irwin et al., 2012], Alchemy [Chen et al., 2019a], the synthetic counting substructures dataset [Chen et al., 2020], the multi-task graph property regression synthetic dataset [Corso et al., 2020] (MIT License), the images dataset used by Balcilar et al. [2020] (GNU General Public License v3.0), the cat mesh from free3d.com/3d-model/cat-v1--522281.html (Personal Use License), and the human mesh from turbosquid.com/3d-models/water-park-slides-3d-max/1093267 (TurboSquid 3D Model License). If no license is listed, this means that we cannot find a license for the dataset. As they appear to be freely available with permissive licenses or no licenses, we do not ask for permission from the creators or hosts of the data. ", + "bbox": [ + 165, + 320, + 825, + 445 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "1492 We do not believe that any of this data contains offensive content or personally identifiable information. \n1493 The 50 images used in the spectral graph convolution experiments are mostly images of objects, with \n1494 a few low resolution images of humans that do not appear to have offensive content. The only other \n1495 human-related data appears to be the human mesh, which appears to be from a 3D scan of a human. \n1496 The human mesh does have tattoos, but they do not appear to be offensive. ", + "bbox": [ + 137, + 452, + 826, + 521 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "K.2 Graph Regression Details ", + "text_level": 1, + "bbox": [ + 174, + 537, + 395, + 553 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "1498 \n1499 \n1500 \n1501 \n1502 \n1503 \n1504 \n1505 \n1506 \n1507 \n1508 \n1509 ", + "bbox": [ + 135, + 563, + 163, + 742 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "ZINC. In Section 4.1 we study the effectiveness of SignNet for learning positional encodings to boost the expressive power, and thereby generalization, on the graph regression problem ZINC. In all cases we take our $\\phi$ encoder to be an 8 layer GIN with ReLU activation. The input eigenvector $v _ { i } \\in \\mathbb { R } ^ { n }$ , where $n$ is the number of nodes in the graph, is treated as a single scalar feature for each node. In the case of using a fixed number of eigenvectors $k$ , the aggregator $\\rho$ is taken to be an 8 layer MLP with batch normalization and ReLU activation. The aggregator $\\rho$ is applied separately to the concatenatation of the $k$ different embeddings for each node in a graph, resulting in one single embedding per node. This embedding is concatenated to the node features for that node, and the result passed as input to the base (predictor) model. We also consider using all available eigenvectors in each graph instead of a fixed number $k$ . Since the total number of eigenvectors is a variable quantity, equal to the number of nodes in the underlying graph, an MLP cannot be used for $\\rho$ . To handle the variable sized input in this case, we take $\\rho$ to be an MLP preceded by a sum over the $\\phi$ outputs. In other words, the SignNet is of the form MLP $\\begin{array} { r } { \\left( \\sum _ { i = 1 } ^ { k } \\phi ( v _ { i } ) + \\phi ( - v _ { i } ) \\right) } \\end{array}$ in this case. ", + "bbox": [ + 168, + 564, + 825, + 753 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "1511 As well as testing SignNet, we also checked whether simple transformations that resolve the sign \n1512 ambiguity of the Laplacian eigenvectors $p = ( v _ { 1 } , \\ldots , v _ { k } )$ could serve as effective positional encoding. \n1513 We considered three options. First is to randomly flip the sign of each $\\pm v _ { i }$ during training. This \n1514 is a common heuristic used in prior work on Laplacian positional encoding [Kreuzer et al., 2021, \n1515 Dwivedi et al., 2020]. Second, take the element-wise absolute value $| v _ { i } |$ . This is a non-injective \n1516 map, creating sign invariance at the cost of destroying positional information. Third is a different \n1517 canonicalization that avoids stochasticity and use of absolute values by selecting the sign of each \n1518 $v _ { i }$ so that the majority of entries are non-negative, with ties broken by comparing the $\\ell _ { 1 }$ -norm of \n1519 positive and negative parts. When the tie-break also fails, the sign is chosen randomly. Results for \n1520 GatedGCN base model on ZINC in Table 1 show that all three of these approaches are significantly \n1521 poorer positional encodings compared to SignNet. \n1522 Our training pipeline largely follows that of Dwivedi et al. [2022], and we use the GatedGCN \n1523 and PNA base models from the accompanying implementation (see https://github.com/ \n1524 vijaydwivedi75/gnn-lspe). The Sparse Transformer base model architecture we use, which \n1525 like GAT computes attention only across neighbouring nodes, is introduced by Kreuzer et al. [2021]. \n1526 Finally, the GINE implementation is based on the PyTorch Geometric implementation [Fey and \n1527 Lenssen, 2019]. For the state-of-the-art comparison, all baseline results are from their respective \n1528 papers, except for GIN, which we run. \n1529 ZINC-full. We also run our method on the full ZINC dataset, termed ZINC-full. The result we \n1530 report for SignNet is a larger version of the GatedGCN base model with a SignNet that takes in \n1531 all eigenvectors. This model has 994,113 parameters in total. All baseline results are from their \n1532 respective papers, except for GIN, which is from [Bodnar et al., 2021]. ", + "bbox": [ + 135, + 758, + 825, + 911 + ], + "page_idx": 40 + }, + { + "type": "text", + "text": "", + "bbox": [ + 135, + 92, + 825, + 188 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "", + "bbox": [ + 137, + 194, + 825, + 251 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "Alchemy. We run our method and compare with the state-of-the-art on Alchemy (with 10,000 training graphs). We use the same data split as Morris et al. [2020b]. Our base model is a GIN that takes in edge features (i.e. a GINE). The SignNet consists of GIN for $\\phi$ and a Transformer for $\\rho$ , as in the counting substructures and graph property regression experiments in Section 4.2. The model has 907,371 parameters in total. Our training setting is very similar to that of Morris et al. [2022], as we build off of their code. We train with an Adam optimizer [Kingma and Ba, 2014] with a starting learning rate of .001, and a minimum learning rate of .000001. The learning rate schedule cuts the learning rate in half with a patience of 20 epochs, and training ends when we reach the minimum learning rate. All baseline results are from their respective papers, except for GIN, which is from [Morris et al., 2022]. ", + "bbox": [ + 165, + 257, + 825, + 395 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "1543 K.3 Spectral Graph Convolution Details ", + "text_level": 1, + "bbox": [ + 137, + 414, + 467, + 429 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "In Appendix J.2, we conduct node regression experiments for learning spectral graph convolutions. The experimental setup is mostly taken from He et al. [2021]. However, we resize the $1 0 0 \\times 1 0 0$ images to $3 2 \\times 3 2$ . Thus, each image is viewed as a 1024-node graph. The node features $X \\in \\mathbb { R } ^ { n }$ are the grayscale pixel intensities of each node. Just as in He et al. [2021], we only train and evaluate on nodes that are not connected to the boundary of the grid (that is, we only evaluate on the $2 8 \\times 2 8$ middle section). For all experiments we limit each model to 50,000 parameters. We use the Adam [Kingma and Ba, 2014] optimizer for all experiments. For each of the GNN baselines (GCN, GAT, GPR-GNN, ARMA, ChebNet, BernNet), we select the best performing out of 4 hyperparameter settings: either 2 or 4 convolution layers, and a hidden dimension of size 32 or $D$ , where $D$ is just large enough to stay with 50,000 parameters (for instance, $D = 1 2 8$ for GCN, GPR-GNN, and BernNet). ", + "bbox": [ + 168, + 440, + 825, + 593 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "1555 We use DeepSets or standard Transformers as our prediction network. This takes in the output of \n1556 SignNet or BasisNet and concatenates it with the node features, then outputs a scalar prediction for \n1557 each node. We use a 3 layer output network for DeepSets SignNet, and 2 layer output networks for \n1558 all other configurations. All networks use ReLU activations. ", + "bbox": [ + 135, + 599, + 825, + 655 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "For SignNet, we use DeepSets for both $\\phi$ and $\\rho$ . Our $\\phi$ takes in eigenvectors only, then our $\\rho$ takes the outputs of $\\phi$ and the eigenvalues. We use three layers for $\\phi$ and $\\rho$ . ", + "bbox": [ + 163, + 661, + 823, + 690 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "For BasisNet, we use the same DeepSets for $\\rho$ as in SignNet, and 2-IGNs for the $\\phi _ { d _ { i } }$ . There are three distinct multiplicities for the grid graph (1, 2, and 32), so we only need 3 separate IGNs. Each IGN consists of an $\\mathbb { R } ^ { n ^ { 2 } \\times 1 } \\to \\mathbb { R } ^ { n \\times d ^ { \\prime } }$ layer and two $\\mathbb { R } ^ { n \\times d ^ { \\prime \\prime } } \\to \\mathbb { R } ^ { n \\times d ^ { \\prime \\prime \\prime } }$ layers, where the $d ^ { \\prime }$ are hidden dimensions. There are no matrix to matrix operations used, as the memory requirements are intensive for these $\\geq 1 0 0 0$ node graphs. The $\\phi _ { d _ { i } }$ only take in $V _ { i } V _ { i } ^ { \\top }$ from the eigenspaces, and the $\\rho$ takes the output of the $\\phi _ { d _ { i } }$ as well as the eigenvalues. ", + "bbox": [ + 166, + 695, + 825, + 782 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "K.4 Substructures and Graph Properties Regression Details ", + "text_level": 1, + "bbox": [ + 168, + 801, + 602, + 816 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "We use the random graph dataset from Chen et al. [2020] for counting substructures and the synthetic dataset from Corso et al. [2020] for regressing graph properties. For fair comparison we fix the base model as a 4-layer GIN model with hidden size 128. We choose $\\phi$ as 4-layer GIN (independently applied to every eigenvector) and $\\rho$ as 1-layer Transformer (independently applied to every node). Combined with proper batching and masking, we have a SignNet that takes Laplacian eigenvectors $V \\in \\mathbb { R } ^ { n \\times n }$ and outputs fixed size sign-invariant encoding node features $f ( V , \\Lambda , \\bar { \\lambda } ) \\in \\mathbb R ^ { \\bar { n } \\times d }$ , where ", + "bbox": [ + 173, + 828, + 825, + 911 + ], + "page_idx": 41 + }, + { + "type": "text", + "text": "574 $n$ varies between graphs but $d$ is fixed. We use this SignNet in our experiments and compare with \n575 other methods of handling PEs. \n1577 We closely follow the experimental setting of Koestler et al. [2022] for the texture reconstruction \n1578 experiments. In this work, we use the cotangent Laplacian [Rustamov et al., 2007] of a triangle mesh \n1579 with the lowest 1023 eigenvectors besides the trivial eigenvector of eigenvalue 0. We implemented \n1580 SignNet in the authors’ original code, which was privately shared with us. Both $\\rho$ and $\\phi$ are taken \n1581 to be MLPs. Hyperparameter settings and number of parameters are given in Table 9. We chose \n1582 hyperparameters so that the total number of parameters in the SignNet model was no larger than that \n1583 of the original model. ", + "bbox": [ + 148, + 90, + 825, + 119 + ], + "page_idx": 42 + }, + { + "type": "table", + "img_path": "images/8bf4a947b57c4d49198594f72fd2e55442db3d5823079c4c2d9e5e99729cdea1.jpg", + "table_caption": [ + "Table 9: Parameter settings for the texture reconstruction experiments. " + ], + "table_footnote": [], + "table_body": "
ParamsBase MLP widthBase MLP layers out dimp out dimp,width
Intrinsic NF328,5791286448
SignNet323,5631086
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Base modelPositional encodingk#paramTest MAE (↓)
GatedGCNNo PEN/A492k0.252±0.007
LapPE (flip)8492k0.198±0.011
LapPE (abs.)8492k0.204±0.009
LapPE (can.)8505k0.298±0.019
SignNet (𝜙(u) only)8495k0.148±0.007
SignNet8495k0.121±0.005
SignNetAll491k0.100±0.007
Sparse TransformerNo PEN/A473k0.283±0.030
LapPE (flip)16487k0.223±0.007
SignNet16479k0.115±0.008
SignNetAll486k0.102±0.005
GINENo PEN/A470k0.170±0.002
LapPE (flip)16470k0.178±0.004
SignNet16470k0.147±0.005
SignNetAll417k0.102±0.002
PNANo PEN/A474k
LapPE (flip)8474k0.133±0.011 0.132±0.010
SignNet8476k0.105±0.007
SignNetAll487k0.084±0.006
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ZINC (10K)↓ZINC-full ↓Alchemy (10k)↓
HIMP † [Fey et al., 2020].151±.006.036±.002
CIN-small † [Bodnar et al., 2021].094±.004.044±.003
CIN† [Bodnar et al., 2021].079±.006.022±.002
GIN [Xu et al., 2019].170±.002.088±.002.180±.006
δ-2-GNN[Morris et al., 2020b].374±.022.042±.003.118±.001
δ-2-LGNN[Morris etal., 2020b].306±.044.045±.006.122±.003
SpeqNet [Morris et al., 2022].115±.001
GNN-IR [Dupty and Lee, 2022].137±.010.119±.002
PF-GNN [Dupty et al., 2021].122±.01.111±.01
Recon-GNN [Cotta et al., 2021].170±.006.125±.001
SignNet (ours).084±.006.024±.003.113±.002
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CatHuman
MethodParamsPSNR↑DSSIM↓LPIPS↓PSNR↑DSSIM↓LPIPS↓
Intrinsic NF329k34.25.099.18932.29.119.330
Absolute value329k34.67.106.25232.42.132.363
Sign flip329k23.151.282.3521.521.052.71
SignNet324k34.91.090.14732.43.125.316
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Permutation equiv.×√√×√x√√
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32 × 32 image1,02451333296.9
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Base modelPositional encodingk#paramsTest MAE (↓)
GINNo PE16497k0.348±0.014
LapPE (flip)16498k0.341±0.011
SignNet16500k0.238±0.012
GATNo PE16501k0.464±0.011
LapPE (flip)16502k0.462±0.013
SignNet16499k0.243±0.008
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Low-passHigh-passBand-passBand-rejectionComb
GCN.111±.0683.092±5.111.720±3.151.418±1.031.753±1.17
GAT.113±.065.954±.6961.105±.964.543±.340.638±.446
GPR-GNN.033±.032.012±.007.137±.081.256±.197.369±.460
ARMA.053±.029.042±.024.107±.039.148±.089.202±.116
ChebNet.003±.002.001±.001.005±.003.009±.006.022±.016
BernNet.001±.002.001±.001.000±.000.048±.042.027±.019
Transformer3.662±1.973.715±1.981.531±1.301.506±1.293.178±1.93
Transformer Eig Flip4.454±2.324.425±2.381.651±1.532.567±1.733.720±1.94
Transformer Eig Abs2.727±1.403.172±1.611.264±.7881.445±.9432.607±1.32
DeepSets SignNet.004±.013.086±.405.021±.115.008±.037.003±.016
Transformer SignNet.003±.016.004±.025.001±.004.006±.023.093±.641
DeepSets BasisNet.009±.018.003±.015.008±.030.004±.011.015±.060
Transformer BasisNet.079±.471.014±.038.005±.018.006±.016.014±.051
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