Add files using upload-large-folder tool

parse/train/2UyqK45_djA/2UyqK45_djA.md

@@ -0,0 +1,498 @@
+# Plan Your Target and Learn Your Skills: State-Only Imitation Learning via Decoupled Policy Optimization
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 State-only imitation learning (SOIL) enables agents to learn from massive demon  
+2 strations without explicit action or reward information. However, previous methods   
+3 attempt to learn the implicit state-to-action mapping policy directly from state-only   
+4 data, which results in ambiguity and inefficiency. In this paper, we overcome this   
+5 issue by introducing hyper-policy as sets of policies that share the same state tran  
+6 sition to characterize the optimality in SOIL. Accordingly, we propose Decoupled   
+7 Policy Optimization (DPO) via explicitly decoupling the state-to-action mapping   
+8 policy as a state transition predictor and an inverse dynamics model. Intuitively,   
+9 we teach the agent to plan the target to go and then learn its own skills to reach.   
+10 Experiments on standard benchmarks and a real-world driving dataset demonstrate   
+11 the effectiveness of DPO and its potential of bridging the gap between reality and   
+12 simulations of reinforcement learning.
+
+# 13 1 Introduction
+
+14 Imitation learning offers a way to train an intelligent agent from demonstrations by mimicking the   
+15 expert’s behaviors without constructing hand-crafted reward functions [13, 17]. The corresponding   
+16 methods normally require the expert demonstrations include information of both states and actions.   
+17 Unfortunately, the action information is not always accessible from many real-world demonstration   
+18 resources, e.g., online video recordings of car driving or sports. Thus a natural desire to take advantage   
+19 of these massive and valuable resources motivates the study of state-only imitation learning (SOIL),   
+20 also known as learning from observations (LfO) [24]. Analogy to human beings, SOIL is a more   
+21 intuitive way to approach imitation by only matching the expert’s state sequences without having   
+22 explicit knowledge of the exact actions.   
+23 A wide range of algorithms have been proposed to solve SOIL by matching the state sequence of the   
+24 expert [22, 23, 25]. However, the action agnostic setting in SOIL makes it challenging to determine   
+25 the optimal action because of the partial observability of the expert demonstrations that multiple   
+26 policies could be chosen to match the same expert state sequence. Thus learning a state-to-action   
+27 policy is implicit, leading to a less efficient modeling of the explicit information from demonstrations,   
+28 and in result could cause suboptimality.   
+29 To this end, in this paper, we introduce the concept of hyper-policy denoting a family of policies that   
+30 share the same state transition. Based on that, instead of recovering the expert policy, we characterize   
+31 the optimality in SOIL by finding the expert hyper-policy. The proposed method is called decoupled   
+32 policy optimization (DPO), which separates the policy into two modules: an expert state transition   
+33 predictor that finds the optimal hyper-policy, followed by an inverse dynamics model that builds the   
+34 executable policy to deliver actions. Intuitively, the expert state transition predictor predicts the target,   
+35 while the inverse dynamics model enables the agent to learn its own skills to reach the target. DPO   
+36 takes the advantage of such a decoupled structure by separately learning two kinds of data: (1) the   
+37 expert state transition that is directly accessible in the demonstration; (2) the action to be performed   
+38 which should be obtained by interacting with the environment.   
+39 To ensure the benefit of DPO, these two modules should work coherently to provide accurate foresight   
+40 for targets and corresponding skills. To achieve this, we regularize the state transition predictor to   
+41 prevent the model from predicting non-neighboring states via multi-step and cycle training style.   
+42 Further, to improve the learning efficiency by encouraging the agent to reach the expert states, we   
+43 augment reward and apply policy gradient to DPO with additional generative adversarial objective.   
+44 Experiments on standard benchmarking tasks show the advantage of the decoupled structure and   
+45 the higher efficiency of DPO. We also evaluate DPO on a real-world driving dataset with state-only   
+46 demonstrations, and the result shows that DPO can learn driving behaviors closer to human drivers   
+47 when compared with baseline methods.
+
+# 48 2 Preliminaries
+
+49 Markov Decision Process. We consider a $\gamma$ -discounted infinite horizon Markov decision process   
+50 (MDP) as a tuple $\mathcal { M } = \langle \mathcal { S } , \mathcal { A } , \mathcal { T } , \rho _ { 0 } , r , \gamma \rangle$ , where $s$ is the set of states, $\mathcal { A }$ represents the action space,   
+51 $\mathcal { T } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  [ 0 , 1 ]$ is environment dynamics distribution, $\rho _ { 0 } : { \cal S }  [ 0 , 1 ]$ is the distribution of   
+52 the initial state $s _ { 0 }$ , and $\dot { \gamma } \in [ 0 , 1 ]$ is the discount factor. The agent makes decisions through a policy   
+53 $\pi ( a | s ) : S \times \mathcal { A } \to [ 0 , 1 ]$ and receives rewards $r : S \times \mathcal { A }  \mathbb { R }$ .   
+54 Occupancy Measure. The concept of occupancy measure (OM) [10] is proposed to characterize   
+55 the statistical properties of a certain policy interacting with an MDP. Specifically, the state OM is   
+56 defined as the time-discounted cumulative stationary density over the states under a given policy $\pi$ :   
+57 $\begin{array} { r } { \rho _ { \pi } ( s ) = \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } P ( s _ { t } = s | \pi ) } \end{array}$ . Following such a definition we can define different OM:
+
+58 a) State-action OM: $\rho _ { \pi } ( s , a ) = \pi ( a | s ) \rho _ { \pi } ( s )$ b) State transition OM: 59 $\begin{array} { r } { \rho _ { \pi } ( s , s ^ { \prime } ) = \int _ { \mathcal { A } } \rho _ { \pi } ( s , a ) \mathcal { T } ( s ^ { \prime } | s , a ) \mathrm { d } a } \end{array}$ c) Joint OM: 60 $\rho _ { \pi } ( s , a , s ^ { \prime } ) = \rho _ { \pi } ( s , a ) \mathcal { T } ( s ^ { \prime } | s , a )$
+
+61 Imitation Learning from State-Only Demonstrations. Imitation learning $\left( \operatorname { I L } \right)$ [13] studies the   
+62 task of learning from demonstrations (LfD), which aims to learn a policy from expert demonstrations   
+63 without getting access to the reward signals. The expert demonstrations typically consist of expert   
+64 state-action pairs. General $\mathrm { I L }$ objective minimizes the state-action OM discrepancy:
+
+$$
+\pi ^ { * } = \underset { \pi } { \arg \operatorname* { m i n } } \mathbb { E } _ { s \sim \rho _ { \pi } ^ { s } } \left[ \ell \left( \pi _ { E } ( \cdot | s ) , \pi ( \cdot | s ) \right) \right] \Rightarrow \underset { \pi } { \arg \operatorname* { m i n } } \ell \left( \rho _ { \pi _ { E } } ( s , a ) , \rho _ { \pi } ( s , a ) \right) ,
+$$
+
+65 where $\ell$ denotes some distance metric. For example, GAIL [10] chooses to minimize the JS divergence   
+66 $\mathrm { D } _ { \mathrm { J S } } \big ( \rho _ { \pi _ { E } } ( s , a ) \| \rho _ { \pi } ( s , a ) \big )$ , and AIRL [5] utilizes the KL divergence $\mathrm { D } _ { \mathrm { K L } } \big ( \rho _ { \pi _ { E } } ( s , a ) \| \rho _ { \pi } ( s , a ) \big )$ instead,   
+67 which corresponds to a maximum entropy solution with the recovered reward [17]. However, for the   
+68 scenario studied in this paper, the action information is absent in state-only demonstrations, known as   
+69 state-only imitation learning (SOIL) or learning from observations (LfO) problems, where the action   
+70 spaces between the expert and the agent can even be different. Such challenges prevent applying   
+71 typical $\mathrm { I L }$ solutions. An existing method for this problem is to instead optimize the discrepancy of   
+72 the state transition OM with the state-to-action policy $\pi ( a | s )$ [23]:
+
+$$
+\pi ^ { * } = \arg \operatorname* { m i n } _ { \pi } [ \ell \left( \rho _ { \pi _ { E } } ( s , s ^ { \prime } ) , \rho _ { \pi } ( s , s ^ { \prime } ) \right) ] .
+$$
+
+73 However, the solution to this problem is ambiguous since there is no one-to-one correspondence   
+74 between $\rho ( s , s ^ { \prime } )$ and $\pi$ as we will show in the following section. As such, the optimality of SOIL   
+75 should be reconsidered.
+
+# 76 3 Methodology
+
+# 77 3.1 Characterizing the Optimality in SOIL
+
+78 In standard $\mathrm { I L }$ tasks, when the expert actions are accessible in demonstrations, perfectly imitating the   
+79 expert policy corresponds to matching the state-action OM due to the one-to-one correspondence   
+80 between $\pi$ and $\rho _ { \pi } ( s , a )$ [10, 21]. However, such correspondence is not applicable for the state   
+81 transition OM matching in SOIL.   
+82 Proposition 1. Suppose Π is the policy space and $\mathcal { P } = \{ \rho : \rho \geq 0 \}$ is a feasible set of OM, then a   
+83 policy $\pi \in \Pi$ corresponds to one state transition O $\textit { 1 } \rho _ { \pi } \in \mathcal { P }$ . However, a state transition OM $\rho \in \mathcal P$   
+84 can correspond to more than one policy in $\Pi$ .   
+85 The proof can be found in Appendix B. As a result, if we choose to optimize a state-to-action mapping   
+86 policy, then the optimal solution to Eq. (2) is ambiguous. The ambiguity also comes from the fact   
+87 that Eq. (2) does not correspond to a maximum policy entropy solution as in normal $\mathrm { I L }$ tasks (see   
+88 Appendix C for details). Therefore, a state-to-action mapping function may be too implicit for   
+89 matching the state sequence, which could cause training instability and lead to sub-optimal policies.   
+90 In that case, we must find a one-to-one corresponding solution to solve SOIL explicitly and efficiently.   
+91 Before continuing, we introduce the definition of hyper-policy.
+
+Definition 1. A hyper-policy $\Omega$ is a set of policies such that for any $\pi _ { 1 } , \pi _ { 2 } \in \Omega$ , we have $\rho _ { \pi _ { 1 } } ( s , s ^ { \prime } ) =$ $\rho _ { \pi _ { 2 } } ( s , s ^ { \prime } )$ .
+
+94 Then by definition, there is a one-to-one correspondence between the hyper-policy $\Omega$ and the state   
+95 transition OM $\rho _ { \Omega } ( s , s ^ { \prime } )$ . Similar to the normal state-to-action mapping policy, a hyper-policy $\Omega$ can   
+96 be regarded as a state-to-state mapping function $h _ { \Omega } ( s ^ { \prime } | s )$ which predicts the state transition such that   
+97 for any $\pi \in \Omega$ :
+
+$$
+h _ { \Omega } ( s ^ { \prime } | s ) = \frac { \rho _ { \Omega } ( s , s ^ { \prime } ) } { \int _ { \tilde { s } } \rho _ { \Omega } ( s , \tilde { s } ) \mathrm { d } \tilde { s } } = \int _ { a } \pi ( a | s ) \mathcal { T } ( s ^ { \prime } | s , a ) \mathrm { d } a .
+$$
+
+99 Proposition 2. Suppose the state transition predictor is defined as in Eq. (3) and $\Gamma$ is its space,   
+100 $\mathcal { P } = \{ \rho : \rho \geq 0 \}$ , then a hyper-policy state transition predictor $h _ { \Omega } \in \Gamma$ corresponds to one state   
+101 transition OM ${ \boldsymbol \rho } _ { \Omega } \in \mathcal { P }$ ; and a state transition OM $\rho \in \mathcal { P }$ only corresponds to one hyper-policy state   
+102 transition predictor such that $h _ { \rho } = \rho ( s , s ^ { \prime } ) / \int _ { \tilde { s } } \rho ( s , \tilde { s } ) \mathrm { d } \tilde { s }$ .   
+103 Therefore, we find a one-to-one correspondence between the optimization term $\rho ( s , s ^ { \prime } )$ and a practical   
+104 target $h _ { \Omega } ( s ^ { \prime } | s )$ , which indicates that we do not have to infer the expert actions under state-only   
+105 demonstrations but only need to recover the state transition predictor of the hyper-policy $\Omega _ { E }$ :
+
+$$
+\underset { \Omega } { \arg \operatorname* { m i n } } [ \ell \left( \rho _ { \Omega _ { E } } ( s , s ^ { \prime } ) , \rho _ { \Omega } ( s , s ^ { \prime } ) \right) ] \Rightarrow \underset { h _ { \Omega } } { \arg \operatorname* { m i n } } \mathbb { E } _ { s \sim \Omega } [ \ell \left( h _ { \Omega _ { E } } ( s ^ { \prime } | s ) , h _ { \Omega } ( s ^ { \prime } | s ) \right) ] .
+$$
+
+106 However, SOIL still requires to learn a policy to interact with the MDP environment to match the   
+107 state transition OM of the expert. This is achievable since we do not have to recover the expert policy   
+108 $\pi _ { E }$ exactly but can learn any policy $\pi \in \Omega _ { E }$ according to Eq. (4).
+
+# 3.2 Policy Decoupling
+
+110 To construct an unambiguous objective for SOIL, we define hyper-policy and solve the problem by   
+111 finding the state transition predictor of the expert hyper-policy. Intuitively, this tells the agent the   
+112 target that the expert will reach without informing any feasible skill that require the agent to learn   
+113 itself. Therefore, to recover a $\pi \in \Omega _ { E }$ , we can construct an inverse dynamics such that
+
+$$
+\pi = \mathrm { ~  ~ \underbrace ~ { ~ \mathcal { T } _ { \pi } ^ { - 1 } ~ } ~ } \left( \begin{array} { l l l l l l l } { ~ } & { ~ } & { ~ } & { ~ \mathcal { T } ( \pi _ { E } ) } & { ~ } & { ~ } & { ~ } \end{array} \right) .
+$$
+
+|{z}Inverse dynamics | {z }Expert state transition predictor
+
+114 Formally, the expert policy can be decoupled as
+
+$$
+\begin{array} { l } { { \displaystyle \pi _ { E } ( a | s ) = \int _ { s ^ { \prime } } { \mathcal T } ( s ^ { \prime } | s , a ) \pi _ { E } ( a | s ) \mathrm { d } s ^ { \prime } = \int _ { s ^ { \prime } } { \frac { \rho _ { \pi _ { E } } ( s , a , s ^ { \prime } ) } { \rho _ { \pi _ { E } } ( s ) } } \mathrm { d } s ^ { \prime } = \int _ { s ^ { \prime } } { \frac { \rho _ { \pi _ { E } } ( s , s ^ { \prime } ) I _ { \pi _ { E } } ( a | s , s ^ { \prime } ) } { \rho _ { \pi _ { E } } ( s ) } } \mathrm { d } s ^ { \prime } } } \\ { { \displaystyle \qquad = \int _ { s ^ { \prime } } h _ { \pi _ { E } } ( s ^ { \prime } | s ) I _ { \pi _ { E } } ( a | s , s ^ { \prime } ) \mathrm { d } s ^ { \prime } } . } \end{array}
+$$
+
+115
+
+116 Notice that both the state transition predictor $h$ and the inverse dynamics model $I$ is policy dependent.   
+117 Nevertheless, recall that the optimality in SOIL only requires us to recover $\pi \in \Omega _ { E }$ , we do not have   
+118 to learn about $I _ { \pi _ { E } }$ but just one feasible skill $I ( a | s , s ^ { \prime } )$ . Then a policy can be recovered by
+
+$$
+\pi = \mathbb { E } \underbrace { _ { \mathcal { S } ^ { ' } \sim h _ { \Omega _ { E } } \left( \boldsymbol { s } ^ { \prime } | \boldsymbol { s } \right) } } _ { \mathrm { t a r g e t } } \Big [ \underbrace { I ( a | \boldsymbol { s } , \boldsymbol { s } ^ { \prime } ) } _ { \mathrm { s k i l l } } \Big ] .
+$$
+
+119 Here the inverse dynamics model $I$ offers an   
+120 arbitrary $s k i l l$ to reach the expected target state   
+121 provided by the state transition predictor $h$ . In   
+122 fact, it does not depend on the hyper-policy   
+123 $\Omega _ { E }$ but a sampling policy $\pi _ { B }$ to construct $I =$   
+124 $I _ { \pi _ { B } }$ . We only need a mild requirement for $\pi _ { B }$   
+125 that it covers the support of $\bar { \rho } _ { \Omega _ { E } } ( s , s ^ { \prime } )$ so that   
+126 the learned $I$ can provide a possible action to   
+127 achieve the target state. In both experiments and   
+128 theoretical analysis we show that this require  
+129 ment alleviates the dependence on the inverse   
+130 dynamics. Furthermore, if the environment and   
+131 the expert policy are both deterministic (which   
+132 is usually the case in real-world scenarios such   
+133 as robotics), the state transition is a single-point distribution (or known as the Dirac delta func  
+134 tion), and we can simply model $h$ as a deterministic function. By decoupling the policy, which is   
+135 a state-to-action mapping function, as a state-to-state mapping function (the transition predictor)   
+136 and a state-pair-to-action mapping function (the inverse dynamics model), we can mimic the expert   
+137 policy from state-only demonstrations by optimizing these two modules. The whole architecture is   
+138 illustrated in Fig. 1.
+
+![](images/0c6189272a8df27c81d57ba8da0bd7d0f9151c524d81ee23041c8b60dd9471b3.jpg)  
+Figure 1: The architecture of Decoupled Policy Optimization (DPO), which consists of an expert state transition predictor (to plan where to go) followed by an inverse dynamics model (to decide how to reach).
+
+State Transition Predictor. In practice, we construct a parameterized expert state transition predictor $h _ { \psi }$ which predicts the subsequent state of the expert taking the input as a current state $\hat { s ^ { \prime } } = h _ { \psi } ( s )$
+
+141 The state transition predictor models the explicit information of the expert, and it can be learned from   
+142 the demonstration data only. Thence, we implement Eq. (4) as a KL divergence minimization:
+
+$$
+\operatorname* { m i n } _ { \psi } \mathbb { E } _ { ( s , s ^ { \prime } ) \sim \Omega _ { E } } \big [ \mathrm { D } _ { \mathrm { K L } } \big ( h _ { \Omega _ { E } } ( s ^ { \prime } | s ) \| h _ { \psi } ( s ^ { \prime } | s ) \big ) \big ] ,
+$$
+
+which can be optimized in a supervised manner. Specifically, we sample state transitions 143 $( s , s ^ { \prime } )$ from 144 the expert demonstrations $\mathcal { D }$ and optimize the L2 loss:
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \psi } ^ { h } = \mathbb { E } _ { ( s , s ^ { \prime } ) \sim \mathcal { D } } \left[ \| s ^ { \prime } - h _ { \psi } ( s ) \| ^ { 2 } \right] . } \end{array}
+$$
+
+145 Inverse Dynamics Model. Knowing where to go is not enough since the agent has to interact with   
+146 the environment to reach the target. This can be achieved via an inverse dynamics model, which   
+147 predicts the action given two consecutive states. Formally, let the $\phi$ -parameterized inverse dynamics   
+148 model $I _ { \phi }$ take input the state pair and predict the feasible action to achieve the state transition:   
+149 $\hat { a } = I _ { \phi } ( \overset { \cdot } { s } , s ^ { \prime } )$ . Intuitively, we want the inverse dynamics to learn from possible transitions sampled   
+150 by the agent. Recall that we only need the support of learned $I ( a | s , s ^ { \prime } )$ of the sampling policy covers   
+151 the support of the expert state transition OM, from which we can infer at least one possible action.   
+152 Hence, we can optimize the KL divergence between the inverse dynamics of a sampling policy $\pi _ { B }$   
+153 and $I _ { \phi }$ :
+
+$$
+\displaystyle \operatorname* { m i n } _ { \phi } \mathbb { E } _ { ( s , s ^ { \prime } ) \sim \pi _ { \mathcal { B } } } [ \mathrm { D } _ { \mathrm { K L } } ( I _ { \pi _ { \mathcal { B } } } ( a | s , s ^ { \prime } ) \| I _ { \phi } ( a | s , s ^ { \prime } ) ) ] ,
+$$
+
+154 and we can choose to optimize L2 loss in a supervised manner by sampling from the replay buffer $\boldsymbol { B }$ :
+
+$$
+\mathcal { L } _ { \phi } ^ { I } = \mathbb { E } _ { ( s , a , s ^ { \prime } ) \sim \mathcal { B } } \left[ \Vert a - I _ { \phi } ( s , s ^ { \prime } ) \Vert ^ { 2 } \right] .
+$$
+
+55 In our implementation, both the state predictor and the inverse dynamics can be constructed as   
+56 Gaussian distributions similar to a normal stochastic policy, thus encouraging exploration.
+
+# 3.3 Tackling Compounding Error Challenges
+
+158 In our formulation, we have decoupled the state-to-action mapping policy as a state-to-state mapping   
+159 function and a state-pair-to-action mapping function. Unfortunately, the compounding error problem   
+160 exists such that the agent cannot reach where it plans due to the fitting errors of these two parts.   
+61 Theorem 1 (Error Bound of DPO). Consider a deterministic environment whose dynamics transition   
+62 function ${ \mathcal { T } } ( s , a )$ is deterministic and $L$ -Lipschitz. Assume the ground-truth state transition $h _ { \Omega _ { E } } ( s )$   
+63 is deterministic, and for each policy $\pi \in \Pi$ , its inverse dynamics $I _ { \pi }$ is also deterministic and
+
+$$
+\| s ^ { \prime } - s _ { E } ^ { \prime } \| \leq L C \| h _ { \Omega _ { E } } ( s ) - h _ { \psi } ( s ) \| + L \| I _ { \pi _ { B } } ( s , \hat { s } ^ { \prime } ) - I _ { \phi } ( s , \hat { s } ^ { \prime } ) \| ,
+$$
+
+166 where $\pi _ { B }$ is a sampling policy that covers the state transition support of the expert hyper-policy and   
+67 $\hat { s } ^ { \prime } = h _ { \psi } ( s )$ is the predicted next state.   
+168 The proof can be found in Appendix B, where we also   
+169 induce a similar error bound for rollout with a state  
+170 to-action policy as BCO [22] to show the advantage   
+171 of the decoupled structure. From Theorem 1 we   
+172 know that the compounding error can be enlarged   
+173 due to each part’s fitting error, where the first term   
+174 corresponds to the error of predicted states and the   
+175 second term indicates whether the agent can reach
+
+![](images/6ae8d1a939debab71510cc4ece3f7a2d4ed6199704bd819f9427ead6c805f5e6.jpg)  
+Figure 2: Multi-step optimization. Given an expert state $_ { s _ { E } }$ , $h _ { \psi }$ predicts the next possible state $\hat { \boldsymbol s } ^ { \prime 1 }$ , which is further fed to a target network $h _ { \psi ^ { \prime } }$ to predict the following sequence. The total loss computes the MSE loss along the state sequence.
+
+76 where it plans to. To alleviate the error, we further propose regularization on these two modules.
+
+# 3.3.1 Regularization on Target Planning
+
+178 One major problem is that the state transition predictor may suggest non-neighboring states instead   
+179 of predicting one-step reachable states. To overcome this, we draw inspiration from Asadi et al. [2]   
+180 and Edwards et al. [3], and regularize state transition predictor to prevent the model from predicting   
+181 non-neighboring states via multi-step and cycle training style.   
+182 Multi-Step Optimization. We first explain the details of the multi-step optimization objective.   
+183 This idea is motivated by Asadi et al. [2], which optimizes a multi-step outcome by executing a   
+184 sequence of actions in the dynamics model. Here we optimize the state sequence instead. As shown   
+185 in Fig. 2, given an expert state $s _ { E }$ , $h _ { \psi }$ predicts the next possible state $\hat { s } ^ { \prime }$ that the expert will reach; the   
+186 predicted state is then fed into the predictor to output the predicted two-step state $\hat { s } ^ { \prime \prime }$ . As such, the   
+187 multi-step training loss is the L2 loss computed along the $k$ -step outcome sequence:
+
+$$
+\mathcal { L } _ { \psi } ^ { h , \mathrm { m s } } = \underset { ( s , \{ s _ { E } ^ { \prime } \} _ { i = 1 } ^ { k } ) \sim \mathcal { D } } { \mathbb { E } } \left[ \| s _ { E } ^ { \prime 1 } - h _ { \psi } ( s ) \| ^ { 2 } + \sum _ { i = 2 } ^ { k } \| s _ { E } ^ { \prime i } - h _ { \psi ^ { \prime } } ( s ^ { \prime i - 1 } ) \| ^ { 2 } \right] .
+$$
+
+188 Intuitively, such a regularization makes the state prediction   
+189 $\hat { s } ^ { \prime }$ close to the expert state distribution in order to make   
+190 accurate long step predictions. It is worth noting that the   
+191 gradient of the cascading state transition predictors should   
+192 be dropped since we already have an accurate input at each   
+193 time step, and for each training step, we only update the   
+194 first one. We use a target network $h _ { \psi ^ { \prime } }$ in practice.   
+195 Cycle Training Style. Another way to regularize the   
+196 transition predictor’s output to a neighboring state is to   
+197 keep an additional function to ensure the cycle consistency,   
+198 which is also an important technique in [3]. In particular,   
+199 as illustrated in Fig. 3, given an expert state $s _ { E }$ , we take   
+200 the predicted state $\hat { s } ^ { \prime }$ and $s _ { E }$ into the inverse dynamics and
+
+![](images/826bccc6205f5766d83182c068c280c2914759b75a8454a7bd35bdc9cb1f920a.jpg)  
+Figure 3: Cycle training style. Given an expert state $s _ { E }$ , $I _ { \phi } ( s , s ^ { \prime } )$ takes input the predicted state $\hat { s } ^ { \prime }$ and $_ { s _ { E } }$ to get the execution action $a$ , then an additional forward dynamics model $M _ { \omega }$ is used to simulated one step rollout using $( s _ { E } , a )$ and get a forward next state $\tilde { s } ^ { \prime }$ . The total loss computes the MSE loss between the two predicted states.
+
+201 get the action $a$ , then we train an additional forward dynamics model $M _ { \omega }$ to simulate one step rollout that takes the input 202 $( s _ { E } , a )$ and gets a forward next state $\widetilde { s } ^ { \prime }$ :
+
+$$
+\begin{array} { r l } & { \mathcal { L } _ { \omega } ^ { M } = \mathbb { E } _ { ( s , a , s ^ { \prime } ) \sim \mathcal { B } } \left[ \| s ^ { \prime } - M _ { \omega } ( s , a ) \| ^ { 2 } \right] } \\ & { \mathcal { L } _ { \psi } ^ { h , \mathrm { c y c l e } } = \mathbb { E } _ { ( s , s ^ { \prime } ) \sim \mathcal { D } } \left[ \| s ^ { \prime } - h _ { \psi } ( s ) \| ^ { 2 } + \| h ( s ) - M _ { \omega } ( s , I _ { \phi } ( s , s ^ { \prime } ) ) \| ^ { 2 } \right] . } \end{array}
+$$
+
+203 In other words, the cycle training scheme provides a regularization on $h _ { \psi }$ to make predictions   
+204 consistent with the forward dynamics model.
+
+In previous sections, we have mentioned that learning to reach a specific place requires the datacollecting policy to cover the support of the expert hyper-policy. This is easy to achieve on simple low-dimensional tasks, but may not be satisfied in high-dimensional continuous environments. To this end, we encourage the agent to approach those state transitions from the expert’s hyper-policy $\Omega _ { E }$ by minimizing the JS divergence of the state transition occupancy using a state-to-action mapping policy $\mathrm { D } _ { \mathrm { J S } } \left( \rho _ { \pi _ { E } } ( { \bar { s } } , s ^ { \prime } ) , \rho _ { \pi } ( s , s ^ { \prime } ) \right)$ . This can be done by producing informative rewards via GAN-like methods [10, 23], and updating the decoupled policy with policy gradients (PG).
+
+213 In detail, we construct a parameterized discriminator $D _ { \omega } ( s , s ^ { \prime } )$ to compute the reward $r ( s , a ) \triangleq$   
+214 $r ( s , s ^ { \prime } )$ as $\log D _ { \omega } ( s , s ^ { \prime } )$ and the decoupled policy served as the generator. In addition, since we   
+215 decouple the policy as two parameterized modules, i.e., a state transition predictor and an inverse   
+216 dynamics model, then by chain rule, the PG for the decoupled policy can be accomplished by
+
+$$
+\begin{array} { r l } & { \nabla \mathcal { L } _ { \phi , \psi } ^ { \pi } = \mathbb { E } _ { \pi } \left[ Q ( s , a ) \nabla _ { \phi , \psi } \log \pi _ { \phi , \psi } ( a | s ) \right] } \\ & { \qquad = \mathbb { E } _ { \pi } \left[ Q ( s , a ) \displaystyle \int _ { s ^ { \prime } } \left( \nabla _ { \psi } \log h _ { \psi } ( s ^ { \prime } | s ) + \nabla _ { \phi } \log I _ { \phi } ( a | s , s ^ { \prime } ) \right) \mathrm { d } s ^ { \prime } \right] , } \end{array}
+$$
+
+217 where $Q$ is the state-action value function; the first term is the gradient for updating the state transition   
+218 predictor; and the second term is for the inverse dynamics model. Thus, the optimization for both   
+219 the state transition predictor and the inverse dynamics model can augment the supervised learning   
+220 objectives with any PG-based learning algorithms (e.g., TRPO, PPO, SAC). As the training proceeds,   
+221 the agent will sample more transition data around $\Omega _ { E }$ , and thus the support of the sampling policy   
+222 will progressively cover the support of $\rho _ { \Omega _ { E } } ( s , s ^ { \prime } )$ .
+
+# 3.4 Overall Algorithm
+
+By combining the idea of generative adversarial training, we obtain our final algorithm, composed with three essential parts: the state transition predictor $h$ used for predicting the possible future states sampled by the expert; the inverse dynamics model $I$ used for inferring the possible actions conditioned on two adjacent states; and the discriminator $D$ used for offering intermediate reward signals for training the decoupled policy $\pi = I ( h )$ . The overall objective of DPO is
+
+$$
+\begin{array} { r } { \mathcal { L } _ { \phi , \psi } ^ { \pi , h , I } = \lambda _ { G } \mathcal { L } _ { \phi , \psi } ^ { \pi } + \lambda _ { h } \mathcal { L } _ { \psi } ^ { h } + \lambda _ { I } \mathcal { L } _ { \phi } ^ { I } , } \end{array}
+$$
+
+229 where $\lambda _ { G } , \lambda _ { h }$ and $\lambda _ { I }$ are hyperparameters for trading off the training among each loss. In practice,   
+230 we try less than ten combinations for these parameters as shown in Appendix D.3, and we directly   
+231 optimize $\mathcal { L } _ { \phi , \psi } ^ { \pi }$ instead of iterative training. The detailed algorithm is summarized in Appendix A.   
+232 Besides, it is worth noting that both the inverse dynamics model and the state transition predictor   
+233 can be pre-trained, where we optimize $\mathcal { L } _ { \psi } ^ { h }$ using the state-only demonstration and optimize $\dot { \mathcal { L } } _ { \phi } ^ { I }$ using   
+234 samples collected by a randomized agent.
+
+# 5 4 Related Work
+
+Table 1: Comparison between different methods.   
+
+<table><tr><td>Method</td><td>Inverse Dynamics</td><td>State Predictor</td><td>Decoupled Policy</td><td>Task</td></tr><tr><td>BCO[22]</td><td>√</td><td>X</td><td>X</td><td>SOIL</td></tr><tr><td>GAIfO [23]</td><td>X</td><td>X</td><td>X</td><td>SOIL</td></tr><tr><td>IDDM[25]</td><td></td><td>X</td><td>X</td><td>SOIL</td></tr><tr><td>OPOLO [27]</td><td>X</td><td></td><td>X</td><td>SOIL</td></tr><tr><td>PID-GAIL [11]</td><td>X</td><td>xx</td><td>√</td><td>IL</td></tr><tr><td>QSS[3]</td><td>√</td><td>√</td><td>√</td><td>RL</td></tr><tr><td>SAIL[16]</td><td>1</td><td>√</td><td>X</td><td>IL</td></tr><tr><td>DPO (Ours)</td><td>√</td><td>√</td><td>√</td><td>SOIL</td></tr></table>
+
+State-only imitation learning (SOIL) endows the agent with the ability to learn from expert states. Although lacking the expert decision information, most of the previous works still optimize a state-to-action mapping policy to match the expert state transition distribution. For example, Torabi et al. [22] used a model-based approach to apply behavioral cloning to state-only demonstrations, while Torabi et al. [23] employed a similar structure to GAIL to match the state transition distribution. Yang et al. [25] analyzed the optimization gap between SOIL and naive IL and introduced a mutual information term to narrow it. Huang et al. [11] applied SOIL on autonomous driving tasks by decoupling the policy into a neural decision module and a non-differentiable execution module in a hierarchical way.
+
+247 Our work decouples the state-to-action policy into two modules. However, both the inverse dynamics   
+248 model and the state transition predictor have been widely used by many previous works on RL and IL   
+249 tasks. For instance, Torabi et al. [22] and Guo et al. [6] trained an inverse dynamics model to label the   
+250   
+251   
+252   
+253   
+254   
+255   
+256   
+257   
+258   
+259   
+260
+
+Table 2: Eventual performance against different methods on 6 easy-to-hard continuous control benchmarks. The means and the standard deviations are evaluated over more than 5 random seeds.   
+
+<table><tr><td></td><td>InvertedPendulum</td><td>InvertedDoublePendulum</td><td>Hopper</td><td>Walker2d</td><td>HalfCheetah</td><td>Ant</td></tr><tr><td>Random</td><td>25.28 ± 5.53</td><td>78.28 ± 10.73</td><td>13.09 ±0.10</td><td>7.07 ± 0.13</td><td>74.48 ± 12.39</td><td>713.59 ± 203.92</td></tr><tr><td>BCO</td><td>1000.00±0.00</td><td>415.04± 148.46</td><td>1430.16± 398.81</td><td>261.36 ± 25.17</td><td>-13.66± 149.94</td><td>397.79± 239.16</td></tr><tr><td>GAIfO</td><td>1000.00 ±0.00</td><td>7818.07 ± 1778.67</td><td>3068.10± 26.32</td><td>3865.20 ± 341.90</td><td>8953.35 ± 1079.41</td><td>5122.29 ± 807.19</td></tr><tr><td>GAIfO-DP</td><td>1000.00±0.00</td><td>7305.01 ± 1591.23</td><td>3031.84 ± 152.13</td><td>4003.06 ± 241.34</td><td>8675.42 ± 807.29</td><td>5535.9± 62.74</td></tr><tr><td>DPO (w/o PG)</td><td>1000.00±0.00</td><td>3545.70± 738.16</td><td>629.84 ± 344.07</td><td>334.23 ±85.42</td><td>-472.00 ± 132.81</td><td>-196.96 ± 124.26</td></tr><tr><td>DPO (w PG)</td><td>1000.00±0.00</td><td>7846.40 ± 1541.20</td><td>3165.72 ± 68.44</td><td>4407.53 ± 266.72</td><td>10501.96 ± 438.01</td><td>5338.48 ± 107.2</td></tr><tr><td>Expert (SAC)</td><td>1000.00±0.00</td><td>9358.87 ± 0.10</td><td>3402.94 ± 446.48</td><td>5639.32 ± 29.97</td><td>13711.64 ± 111.47</td><td>5404.55±1520.49</td></tr></table>
+
+state-only demonstrations with inferred actions. Nair et al. [19] proposed a method for manipulating ropes to match a single human-specified image sequence, in which an inverse dynamics model is trained in a self-supervised manner and used to generate control signals. Kimura et al. [14] utilized a state transition predictor to fit the state transition probability in the expert data, which is further used to compute a predefined reward function. Liu et al. [16] constructed a policy prior using the inverse dynamics and the state transition predictor, but the policy prior was only used for regularizing the policy network. However, as shown in this paper, the policy can be exactly decoupled as these two parts, which can be uniformly optimized through policy gradient without keeping an extra policy. Edwards et al. [3] estimated $Q ( s , s ^ { \prime } )$ for RL tasks which employs a similar policy form as Eq. (6) and updates the state transition predictor through a deterministic policy gradient similar to DDPG [15]. To sort out the difference between these methods and ours, we summarize the key factors in Tab. 1.
+
+# 5 Experiments
+
+We conduct four sets of experiments to investigate the following research questions:
+
+RQ1 Is decoupled learning structure superior than state-to-action structure on SOIL tasks? RQ2 Does DPO achieve higher efficiency or better performance than baselines on SOIL tasks? RQ3 Can agent reach where it plans and does the proposed regularization help mitigate the compounding error?
+
+RQ4 How can DPO be applied on real-world data?
+
+To answer RQ1, we conduct toy experiments with a simple 2D grid world environment and compare both qualitative and quantitative results of DPO against BCO and GAIfO. Regarding RQ2, we empirically evaluate DPO on easy-to-hard continuous control benchmarking tasks. And for RQ3, we evaluate the difference between the predicted states that the agent plans to reach and the consecutive state that the agent actually reaches in the environment for the proposed regularization. Finally, we try to imitate real-world traffic surveillance recordings in a simulated environment to investigate RQ4, which shows the potential of using real-world data for human behavior simulation. Due to the space limit, we leave experiment details, additional results and ablation study in Appendix.
+
+![](images/2572c840bae1adcb26bf42c08dee3fbfad4863b16f4814902fe7dadff55452a7.jpg)
+
+(a) Rollout density.
+
+# 79 5.1 Understanding the Decoupled Structure
+
+In this paper we design decoupled policy optimization (DPO) to perform SOIL tasks, and in previous sections we propose that the key technical contribution of DPO is the decoupled structure of policy that models the explicit state transition information and the latent action information from demonstrations, which solves the ambiguity and enhances the learning efficiency. Therefore, in this set of experiments, we aim to demonstrate how DPO is superior than state-to-action policy methods (RQ1). We first generate expert demonstrations in a 2D $6 \times 6$ grid world environment, in which the agent starts at the upper left corner and aims to reach the upper right corner. In each grid the agent has $k \times 4$ actions, which means that the agent has $k$ possible actions to reach the neighboring block and in our experiment we choose $k = 5$ to enlarge the action space.
+
+![](images/09c21ffa2883a18d3adfd01b8a7b9fbb89d2dfdfc00ad1028ba611931eb3384d.jpg)  
+Figure 4: Toy example.
+
+291 The density of the expert trajectories and the trajectories sampled by different methods are shown in   
+292 Fig. 4(a). We show that both BCO and GAIfO have troubles in directly learning the implicit action   
+293 from state-only behaviors. Notably, GAIfO only imitates the major trajectory and omit the other   
+294 choice and BCO also stucks in the middle right. By contrast, DPO recovers the expert demonstrations   
+295 much better, benefiting from the decoupled structure that first determining the target and then taking   
+296 the action to achieve it. To further illustrate the learning efficiency advantage of DPO, we illustrate the   
+297 JS divergence curves of DPO and BCO during training in Fig. 4(b). Besides, we show the policy loss   
+298 for BCO, the state predictor (SP) loss for DPO, and the inverse dynamics (ID) loss for both methods.   
+299 Except that the JS divergence of DPO decreases more quickly than BCO, it is also observable that   
+300 DPO relies less on the inverse dynamics than BCO, since the inverse dynamics loss of DPO converges   
+301 to a higher level. We further provide a theoretic analysis of the dependence on inverse dynamics with   
+302 BCO and DPO in Appendix B.
+
+![](images/06153042034d10056cca54a6053af2d92b12f25747a0f57889baf0ae15319178.jpg)  
+Figure 5: Learning curves on 6 easy-to-hard continuous control benchmarks, where the solid line and the shade represent the mean and the standard deviation of the averaged return over more than 5 random seeds. We pre-train BCO and DPO for 50k steps and show it in figures.
+
+# 5.2 Comparative Evaluations
+
+We compare the qualitative results of DPO against other baseline methods on easy-to-hard continuous control benchmarking environments (RQ2), including InvertedPendulum, InvertedDoublePendulum, Hopper, Walker2d, HalfCheetah and Ant. In each environment, besides GAIfO and BCO, we also evaluate GAIfO with decoupled policy (denoted as GAIfO-DP). For DPO we compare the reward augmented version of DPO (denoted as DPO w PG)1 with the supervised learning version of DPO, i.e., $\lambda _ { G } = 0$ (denoted as DPO w/o PG). For fairness, we re-implement all the algorithms based on a Pytorch code framework2 and adopt Soft Actor-Critic (SAC) [7] as the RL learning algorithm for GAIfO and DPO. For all environments, we first train an SAC agent to collect 4 state-only expert trajectories and then train agents with such data. All algorithms are evaluated by a deterministic policy. The eventual results are summarized in Tab. 2, and the learning curves are shown in Fig. 5. It is worth noting that for DPO, we choose the best performance among the experiments that use multi-step or cycle regularization, and we put the full experiment results in Appendix D.
+
+One can easily observe that on simple environments, BCO is able to achieve a good performance, and GAIfO also does well on harder tasks. Even so, DPO can still gain the best or comparable performance against its counterparts. Particularly, without augmented reward, DPO is able to reach the optimality with the highest sample efficiency on simple tasks like InvertedPendulum. By contrast, on higher-dimensional tasks such as Hopper, Walker2d, HalfCheetah and Ant, it is difficult to construct accurate inverse dynamics that covers the support of the expert hyper-policy from scratch. However, by combining generative adversarial policy gradients, the agent finally recovers a good policy from the expert hyper-policy. This is particularly evident on HalfCheetah where DPO behaves poorly at the beginning but improves fast as the training proceeds. Besides, as illustrated in Fig. 5, DPO benefits from better sample efficiency in most of the environments, but the improvements are limited on the hardest tasks. We think that this may be due to larger state spaces (111 dimensions for Ant) that makes it difficult to recover a good state predictor or an inverse dynamics model. In all
+
+![](images/30082c9b9985186853bef3245eff5773245686f44639de77e4b5f2e45ec4e44a.jpg)  
+Figure 6: Compounding error of the predicted consecutive states and the real states the agent reaches when rollout in the environments.
+
+28 experiments, GAIfO-DP achieves similar results as GAIfO, indicating that the network structure does   
+29 not count much for the performance.
+
+# 5.3 Compounding Error Reduction
+
+In this section, we aim to study whether the agent can effectively reach the target as it plans (RQ3). Therefore, we analyze the distance of the reaching states and the predicted consecutive states, and draw the mean square error (MSE) along the training procedure in Fig. 6. We also compare our regularization including multi-step optimization (denoted as ${ \bf M } . { \bf S } . - k$ , where $k$ is the number of rollout steps) and cycle training style (denoted as Cycle). Note that DPO needs at least 1-step rollout for training the state transition predictor. As shown in Fig. 6, the agent still has gaps to get to where it plans to, and the mismatch always deteriorates on harder tasks. Combining regularization can always achieve lower compounding error, and the cycle training is effective in most of the environments. In Appendix D.6, we further illustrate the correlation between the final performance and the distance.
+
+# 5.4 Learn to Drive from Real-World Traffic Data
+
+The rapid development of autonomous driving has brought a lot of demand for simulating and training an RL agent in the simulator, which requires realistic interactions with various social vehicles [26]. However, driver’s detailed actions are not easily to obtain yet we adopt SOIL from a traffic surveillance recording dataset (NGSIM I-80 [8]) that contains kinds of recorded human driving trajectories. We wish to further examine the potential of DPO for decreasing the gap between the real world and simulation (RQ4). We utilize the simulator provided by Henaff et al. [9] as our simulation platform and learn to imitate real-world driving behaviors. We compare DPO against GAIfO and BCO, and choose Success Rate, Mean Distance and KL Divergence as evaluation metrics. Specifically, Success Rate is the percentage of driving across the entire area without crashing into other vehicles or driving off the road, Mean Distance is the distance traveled before the episode ends, and KL Divergence measures the position distribution distance between the expert and the agent.
+
+As shown in Tab. 3, DPO outperforms baseline methods in all three metrics while possessing higher stability. The decoupled policy allows the state predictor to focus on matching the distribution of expert trajectories, thus achieving smaller deviations from the expert position distribution. Furthermore, since the policy gradient can be computed with non-differentiable inverse dynamics, we can generate 9 stable action sequences [12, 11] by replacing the inverse dynamics model with classical controllers, which can be generalized to realistic applications.
+
+Table 3: Performance on NGSIM I-80 driving task over 5 random seeds.   
+
+<table><tr><td>Method</td><td>Success Rate (%)</td><td>Mean Distance (m)</td><td>KL Divergence</td></tr><tr><td>BCO</td><td>27.4 ± 1.1</td><td>129.8 ± 2.0</td><td>24.4 ± 2.2</td></tr><tr><td>GAIfO</td><td>77.5 ± 0.8</td><td>188.3 ± 1.1</td><td>11.5 ± 3.9</td></tr><tr><td>DPO</td><td>80.3± 0.5</td><td>192.7 ± 0.6</td><td>9.5 ± 1.8</td></tr><tr><td>Expert</td><td>100</td><td>210.0</td><td>0</td></tr></table>
+
+# 6 Conclusion
+
+In this paper, we characterize the optimality and investigate the ambiguity problem in state-only imitation learning, and accordingly propose Decoupled Policy Optimization (DPO), which splits the state-to-action mapping policy into a state-to-state mapping state transition predictor and a state-pair-to-action mapping inverse dynamics model. Furthermore, we employ regularization and generative adversarial methods to mitigate the compounding error caused by the decoupled modules. The flexibility of the decoupled architecture allows a wide range of interesting future works, such as replacing the inverse dynamics with a classic control module to produce stable control signals, learning specific skills with shared state transition and multi-task target learning with shared pretrained skills.
+
+References   
+[1] Pieter Abbeel and Andrew Y. Ng. Apprenticeship learning via inverse reinforcement learning. In Machine Learning, Proceedings of the Twenty-first International Conference (ICML 2004), 2004.   
+[2] Kavosh Asadi, Dipendra Misra, Seungchan Kim, and Michel L Littman. Combating the compounding-error problem with a multi-step model. arXiv preprint arXiv:1905.13320, 2019.   
+[3] Ashley D. Edwards, Himanshu Sahni, Rosanne Liu, Jane Hung, Ankit Jain, Rui Wang, Adrien Ecoffet, Thomas Miconi, Charles Isbell, and Jason Yosinski. Estimating q(s,s’) with deep deterministic dynamics gradients. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 2020.   
+[4] Chelsea Finn, Sergey Levine, and Pieter Abbeel. Guided cost learning: Deep inverse optimal control via policy optimization. In International Conference on Machine Learning, pages 49–58, 2016.   
+[5] Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adverserial inverse reinforcement learning. In 6th International Conference on Learning Representations, ICLR 2018, 2018.   
+[6] Xiaoxiao Guo, Shiyu Chang, Mo Yu, Gerald Tesauro, and Murray Campbell. Hybrid reinforcement learning with expert state sequences. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, pages 3739–3746, 2019.   
+[7] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, pages 1856–1865, 2018.   
+[8] John Halkias and James Colyar. Next generation simulation fact sheet. US Department of Transportation: Federal Highway Administration, 2006.   
+[9] Mikael Henaff, Alfredo Canziani, and Yann LeCun. Model-predictive policy learning with uncertainty regularization for driving in dense traffic. In 7th International Conference on Learning Representations, ICLR 2019, 2019.   
+[10] Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems 29, pages 4565–4573, 2016.   
+[11] Junning Huang, Sirui Xie, Jiankai Sun, Qiurui Ma, Chunxiao Liu, Dahua Lin, and Bolei Zhou. Learning a decision module by imitating driver’s control behaviors. In Proceedings of the Conference on Robot Learning (CoRL) 2020, 2020.   
+[12] Sandy H. Huang, Nicolas Papernot, Ian J. Goodfellow, Yan Duan, and Pieter Abbeel. Adversarial attacks on neural network policies. In 5th International Conference on Learning Representations, ICLR 2017, 2017.   
+[13] Ahmed Hussein, Mohamed Medhat Gaber, Eyad Elyan, and Chrisina Jayne. Imitation learning: A survey of learning methods. ACM Computing Surveys (CSUR), 50(2):1–35, 2017.   
+[14] Daiki Kimura, Subhajit Chaudhury, Ryuki Tachibana, and Sakyasingha Dasgupta. Internal model from observations for reward shaping. arXiv preprint arXiv:1806.01267, 2018.   
+[15] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. In 4th International Conference on Learning Representations, ICLR 2016,, 2016.   
+[16] Fangchen Liu, Zhan Ling, Tongzhou Mu, and Hao Su. State alignment-based imitation learning. In 8th International Conference on Learning Representations, ICLR 2020, 2020.   
+[17] Minghuan Liu, Tairan He, Minkai Xu, and Weinan Zhang. Energy-based imitation learning. In 20th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2021, 2021.   
+[18] Leland McInnes and John Healy. UMAP: uniform manifold approximation and projection for dimension reduction. CoRR, abs/1802.03426, 2018.   
+[19] Ashvin Nair, Dian Chen, Pulkit Agrawal, Phillip Isola, Pieter Abbeel, Jitendra Malik, and Sergey Levine. Combining self-supervised learning and imitation for vision-based rope manipulation. In 2017 IEEE International Conference on Robotics and Automation, ICRA 2017, pages 2146– 2153, 2017.   
+[20] Murray Rosenblatt. Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics, pages 832–837, 1956.   
+[21] Umar Syed, Michael H. Bowling, and Robert E. Schapire. Apprenticeship learning using linear programming. In Machine Learning, Proceedings of the Twenty-Fifth International Conference (ICML 2008), pages 1032–1039, 2008.   
+[22] Faraz Torabi, Garrett Warnell, and Peter Stone. Behavioral cloning from observation. In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, pages 4950–4957, 2018.   
+[23] Faraz Torabi, Garrett Warnell, and Peter Stone. Adversarial imitation learning from state-only demonstrations. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ’19, pages 2229–2231, 2019.   
+[24] Faraz Torabi, Garrett Warnell, and Peter Stone. Recent advances in imitation learning from observation. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, pages 6325–6331, 2019.   
+[25] Chao Yang, Xiaojian Ma, Wenbing Huang, Fuchun Sun, Huaping Liu, Junzhou Huang, and Chuang Gan. Imitation learning from observations by minimizing inverse dynamics disagreement. In Advances in Neural Information Processing Systems 32, pages 239–249, 2019.   
+[26] Ming Zhou, Jun Luo, Julian Villela, Yaodong Yang, David Rusu, Jiayu Miao, Weinan Zhang, et al. Smarts: Scalable multi-agent reinforcement learning training school for autonomous driving. In Conference on Robot Learning, 2020.   
+[27] Zhuangdi Zhu, Kaixiang Lin, Bo Dai, and Jiayu Zhou. Off-policy imitation learning from observations. In Advances in Neural Information Processing Systems 33, 2020.
+
+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 3.3.   
+(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] See Theorem 1.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See Section B.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix D.3.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We ran our results with more than 5 random seeds as said in Section 5.2.   
+(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)? [No]
+
+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 re-implement all algorithms based on an existed code base, as said in Section 5.2.   
+(b) Did you mention the license of the assets? [No]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] We will public our codes after publication.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/2UyqK45_djA/2UyqK45_djA_content_list.json

@@ -0,0 +1,1511 @@
+[
+    {
+        "type": "text",
+        "text": "Plan Your Target and Learn Your Skills: State-Only Imitation Learning via Decoupled Policy Optimization ",
+        "text_level": 1,
+        "bbox": [
+            186,
+            122,
+            813,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            251,
+            578,
+            306
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            343,
+            535,
+            359
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 State-only imitation learning (SOIL) enables agents to learn from massive demon  \n2 strations without explicit action or reward information. However, previous methods   \n3 attempt to learn the implicit state-to-action mapping policy directly from state-only   \n4 data, which results in ambiguity and inefficiency. In this paper, we overcome this   \n5 issue by introducing hyper-policy as sets of policies that share the same state tran  \n6 sition to characterize the optimality in SOIL. Accordingly, we propose Decoupled   \n7 Policy Optimization (DPO) via explicitly decoupling the state-to-action mapping   \n8 policy as a state transition predictor and an inverse dynamics model. Intuitively,   \n9 we teach the agent to plan the target to go and then learn its own skills to reach.   \n10 Experiments on standard benchmarks and a real-world driving dataset demonstrate   \n11 the effectiveness of DPO and its potential of bridging the gap between reality and   \n12 simulations of reinforcement learning. ",
+        "bbox": [
+            148,
+            373,
+            767,
+            541
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "13 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            565,
+            312,
+            583
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "14 Imitation learning offers a way to train an intelligent agent from demonstrations by mimicking the   \n15 expert’s behaviors without constructing hand-crafted reward functions [13, 17]. The corresponding   \n16 methods normally require the expert demonstrations include information of both states and actions.   \n17 Unfortunately, the action information is not always accessible from many real-world demonstration   \n18 resources, e.g., online video recordings of car driving or sports. Thus a natural desire to take advantage   \n19 of these massive and valuable resources motivates the study of state-only imitation learning (SOIL),   \n20 also known as learning from observations (LfO) [24]. Analogy to human beings, SOIL is a more   \n21 intuitive way to approach imitation by only matching the expert’s state sequences without having   \n22 explicit knowledge of the exact actions.   \n23 A wide range of algorithms have been proposed to solve SOIL by matching the state sequence of the   \n24 expert [22, 23, 25]. However, the action agnostic setting in SOIL makes it challenging to determine   \n25 the optimal action because of the partial observability of the expert demonstrations that multiple   \n26 policies could be chosen to match the same expert state sequence. Thus learning a state-to-action   \n27 policy is implicit, leading to a less efficient modeling of the explicit information from demonstrations,   \n28 and in result could cause suboptimality.   \n29 To this end, in this paper, we introduce the concept of hyper-policy denoting a family of policies that   \n30 share the same state transition. Based on that, instead of recovering the expert policy, we characterize   \n31 the optimality in SOIL by finding the expert hyper-policy. The proposed method is called decoupled   \n32 policy optimization (DPO), which separates the policy into two modules: an expert state transition   \n33 predictor that finds the optimal hyper-policy, followed by an inverse dynamics model that builds the   \n34 executable policy to deliver actions. Intuitively, the expert state transition predictor predicts the target,   \n35 while the inverse dynamics model enables the agent to learn its own skills to reach the target. DPO   \n36 takes the advantage of such a decoupled structure by separately learning two kinds of data: (1) the   \n37 expert state transition that is directly accessible in the demonstration; (2) the action to be performed   \n38 which should be obtained by interacting with the environment.   \n39 To ensure the benefit of DPO, these two modules should work coherently to provide accurate foresight   \n40 for targets and corresponding skills. To achieve this, we regularize the state transition predictor to   \n41 prevent the model from predicting non-neighboring states via multi-step and cycle training style.   \n42 Further, to improve the learning efficiency by encouraging the agent to reach the expert states, we   \n43 augment reward and apply policy gradient to DPO with additional generative adversarial objective.   \n44 Experiments on standard benchmarking tasks show the advantage of the decoupled structure and   \n45 the higher efficiency of DPO. We also evaluate DPO on a real-world driving dataset with state-only   \n46 demonstrations, and the result shows that DPO can learn driving behaviors closer to human drivers   \n47 when compared with baseline methods. ",
+        "bbox": [
+            147,
+            597,
+            825,
+            722
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            728,
+            825,
+            811
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            818,
+            825,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            152,
+            825,
+            223
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            229,
+            825,
+            285
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "48 2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            303,
+            318,
+            320
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "49 Markov Decision Process. We consider a $\\gamma$ -discounted infinite horizon Markov decision process   \n50 (MDP) as a tuple $\\mathcal { M } = \\langle \\mathcal { S } , \\mathcal { A } , \\mathcal { T } , \\rho _ { 0 } , r , \\gamma \\rangle$ , where $s$ is the set of states, $\\mathcal { A }$ represents the action space,   \n51 $\\mathcal { T } : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  [ 0 , 1 ]$ is environment dynamics distribution, $\\rho _ { 0 } : { \\cal S }  [ 0 , 1 ]$ is the distribution of   \n52 the initial state $s _ { 0 }$ , and $\\dot { \\gamma } \\in [ 0 , 1 ]$ is the discount factor. The agent makes decisions through a policy   \n53 $\\pi ( a | s ) : S \\times \\mathcal { A } \\to [ 0 , 1 ]$ and receives rewards $r : S \\times \\mathcal { A }  \\mathbb { R }$ .   \n54 Occupancy Measure. The concept of occupancy measure (OM) [10] is proposed to characterize   \n55 the statistical properties of a certain policy interacting with an MDP. Specifically, the state OM is   \n56 defined as the time-discounted cumulative stationary density over the states under a given policy $\\pi$ :   \n57 $\\begin{array} { r } { \\rho _ { \\pi } ( s ) = \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } P ( s _ { t } = s | \\pi ) } \\end{array}$ . Following such a definition we can define different OM: ",
+        "bbox": [
+            145,
+            333,
+            825,
+            404
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            416,
+            825,
+            474
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "58 a) State-action OM: $\\rho _ { \\pi } ( s , a ) = \\pi ( a | s ) \\rho _ { \\pi } ( s )$ b) State transition OM: 59 $\\begin{array} { r } { \\rho _ { \\pi } ( s , s ^ { \\prime } ) = \\int _ { \\mathcal { A } } \\rho _ { \\pi } ( s , a ) \\mathcal { T } ( s ^ { \\prime } | s , a ) \\mathrm { d } a } \\end{array}$ c) Joint OM: 60 $\\rho _ { \\pi } ( s , a , s ^ { \\prime } ) = \\rho _ { \\pi } ( s , a ) \\mathcal { T } ( s ^ { \\prime } | s , a )$ ",
+        "bbox": [
+            145,
+            481,
+            578,
+            535
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "61 Imitation Learning from State-Only Demonstrations. Imitation learning $\\left( \\operatorname { I L } \\right)$ [13] studies the   \n62 task of learning from demonstrations (LfD), which aims to learn a policy from expert demonstrations   \n63 without getting access to the reward signals. The expert demonstrations typically consist of expert   \n64 state-action pairs. General $\\mathrm { I L }$ objective minimizes the state-action OM discrepancy: ",
+        "bbox": [
+            145,
+            546,
+            825,
+            602
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/af640c262fe9e6c7a79a7ee0ebed8241b3746061b68111e6ed42fea73543d7b3.jpg",
+        "text": "$$\n\\pi ^ { * } = \\underset { \\pi } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { s \\sim \\rho _ { \\pi } ^ { s } } \\left[ \\ell \\left( \\pi _ { E } ( \\cdot | s ) , \\pi ( \\cdot | s ) \\right) \\right] \\Rightarrow \\underset { \\pi } { \\arg \\operatorname* { m i n } } \\ell \\left( \\rho _ { \\pi _ { E } } ( s , a ) , \\rho _ { \\pi } ( s , a ) \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            603,
+            751,
+            628
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "65 where $\\ell$ denotes some distance metric. For example, GAIL [10] chooses to minimize the JS divergence   \n66 $\\mathrm { D } _ { \\mathrm { J S } } \\big ( \\rho _ { \\pi _ { E } } ( s , a ) \\| \\rho _ { \\pi } ( s , a ) \\big )$ , and AIRL [5] utilizes the KL divergence $\\mathrm { D } _ { \\mathrm { K L } } \\big ( \\rho _ { \\pi _ { E } } ( s , a ) \\| \\rho _ { \\pi } ( s , a ) \\big )$ instead,   \n67 which corresponds to a maximum entropy solution with the recovered reward [17]. However, for the   \n68 scenario studied in this paper, the action information is absent in state-only demonstrations, known as   \n69 state-only imitation learning (SOIL) or learning from observations (LfO) problems, where the action   \n70 spaces between the expert and the agent can even be different. Such challenges prevent applying   \n71 typical $\\mathrm { I L }$ solutions. An existing method for this problem is to instead optimize the discrepancy of   \n72 the state transition OM with the state-to-action policy $\\pi ( a | s )$ [23]: ",
+        "bbox": [
+            147,
+            630,
+            825,
+            741
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ba5613bb777d3fb507f1d248615825246e79e52e36f343c85d25a724d19ea86.jpg",
+        "text": "$$\n\\pi ^ { * } = \\arg \\operatorname* { m i n } _ { \\pi } [ \\ell \\left( \\rho _ { \\pi _ { E } } ( s , s ^ { \\prime } ) , \\rho _ { \\pi } ( s , s ^ { \\prime } ) \\right) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            741,
+            632,
+            766
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "73 However, the solution to this problem is ambiguous since there is no one-to-one correspondence   \n74 between $\\rho ( s , s ^ { \\prime } )$ and $\\pi$ as we will show in the following section. As such, the optimality of SOIL   \n75 should be reconsidered. ",
+        "bbox": [
+            147,
+            767,
+            826,
+            809
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "76 3 Methodology ",
+        "text_level": 1,
+        "bbox": [
+            155,
+            827,
+            313,
+            844
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "77 3.1 Characterizing the Optimality in SOIL ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            857,
+            483,
+            873
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "78 In standard $\\mathrm { I L }$ tasks, when the expert actions are accessible in demonstrations, perfectly imitating the   \n79 expert policy corresponds to matching the state-action OM due to the one-to-one correspondence   \n80 between $\\pi$ and $\\rho _ { \\pi } ( s , a )$ [10, 21]. However, such correspondence is not applicable for the state   \n81 transition OM matching in SOIL.   \n82 Proposition 1. Suppose Π is the policy space and $\\mathcal { P } = \\{ \\rho : \\rho \\geq 0 \\}$ is a feasible set of OM, then a   \n83 policy $\\pi \\in \\Pi$ corresponds to one state transition O $\\textit { 1 } \\rho _ { \\pi } \\in \\mathcal { P }$ . However, a state transition OM $\\rho \\in \\mathcal P$   \n84 can correspond to more than one policy in $\\Pi$ .   \n85 The proof can be found in Appendix B. As a result, if we choose to optimize a state-to-action mapping   \n86 policy, then the optimal solution to Eq. (2) is ambiguous. The ambiguity also comes from the fact   \n87 that Eq. (2) does not correspond to a maximum policy entropy solution as in normal $\\mathrm { I L }$ tasks (see   \n88 Appendix C for details). Therefore, a state-to-action mapping function may be too implicit for   \n89 matching the state sequence, which could cause training instability and lead to sub-optimal policies.   \n90 In that case, we must find a one-to-one corresponding solution to solve SOIL explicitly and efficiently.   \n91 Before continuing, we introduce the definition of hyper-policy. ",
+        "bbox": [
+            148,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            90,
+            823,
+            119
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            121,
+            825,
+            165
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            172,
+            826,
+            271
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. A hyper-policy $\\Omega$ is a set of policies such that for any $\\pi _ { 1 } , \\pi _ { 2 } \\in \\Omega$ , we have $\\rho _ { \\pi _ { 1 } } ( s , s ^ { \\prime } ) =$ $\\rho _ { \\pi _ { 2 } } ( s , s ^ { \\prime } )$ . ",
+        "bbox": [
+            160,
+            272,
+            825,
+            301
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "94 Then by definition, there is a one-to-one correspondence between the hyper-policy $\\Omega$ and the state   \n95 transition OM $\\rho _ { \\Omega } ( s , s ^ { \\prime } )$ . Similar to the normal state-to-action mapping policy, a hyper-policy $\\Omega$ can   \n96 be regarded as a state-to-state mapping function $h _ { \\Omega } ( s ^ { \\prime } | s )$ which predicts the state transition such that   \n97 for any $\\pi \\in \\Omega$ : ",
+        "bbox": [
+            147,
+            309,
+            825,
+            364
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5f11a68f805a9c082588a70ab56682d68ca2815b9c26c57e00fdc571df448c8.jpg",
+        "text": "$$\nh _ { \\Omega } ( s ^ { \\prime } | s ) = \\frac { \\rho _ { \\Omega } ( s , s ^ { \\prime } ) } { \\int _ { \\tilde { s } } \\rho _ { \\Omega } ( s , \\tilde { s } ) \\mathrm { d } \\tilde { s } } = \\int _ { a } \\pi ( a | s ) \\mathcal { T } ( s ^ { \\prime } | s , a ) \\mathrm { d } a .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            363,
+            676,
+            400
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "99 Proposition 2. Suppose the state transition predictor is defined as in Eq. (3) and $\\Gamma$ is its space,   \n100 $\\mathcal { P } = \\{ \\rho : \\rho \\geq 0 \\}$ , then a hyper-policy state transition predictor $h _ { \\Omega } \\in \\Gamma$ corresponds to one state   \n101 transition OM ${ \\boldsymbol \\rho } _ { \\Omega } \\in \\mathcal { P }$ ; and a state transition OM $\\rho \\in \\mathcal { P }$ only corresponds to one hyper-policy state   \n102 transition predictor such that $h _ { \\rho } = \\rho ( s , s ^ { \\prime } ) / \\int _ { \\tilde { s } } \\rho ( s , \\tilde { s } ) \\mathrm { d } \\tilde { s }$ .   \n103 Therefore, we find a one-to-one correspondence between the optimization term $\\rho ( s , s ^ { \\prime } )$ and a practical   \n104 target $h _ { \\Omega } ( s ^ { \\prime } | s )$ , which indicates that we do not have to infer the expert actions under state-only   \n105 demonstrations but only need to recover the state transition predictor of the hyper-policy $\\Omega _ { E }$ : ",
+        "bbox": [
+            142,
+            406,
+            825,
+            464
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            473,
+            826,
+            515
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b80d85debf9bb2d1a9b753f179011ffc42f90013c634667796805e92a1f4bf5d.jpg",
+        "text": "$$\n\\underset { \\Omega } { \\arg \\operatorname* { m i n } } [ \\ell \\left( \\rho _ { \\Omega _ { E } } ( s , s ^ { \\prime } ) , \\rho _ { \\Omega } ( s , s ^ { \\prime } ) \\right) ] \\Rightarrow \\underset { h _ { \\Omega } } { \\arg \\operatorname* { m i n } } \\mathbb { E } _ { s \\sim \\Omega } [ \\ell \\left( h _ { \\Omega _ { E } } ( s ^ { \\prime } | s ) , h _ { \\Omega } ( s ^ { \\prime } | s ) \\right) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            243,
+            517,
+            753,
+            544
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "106 However, SOIL still requires to learn a policy to interact with the MDP environment to match the   \n107 state transition OM of the expert. This is achievable since we do not have to recover the expert policy   \n108 $\\pi _ { E }$ exactly but can learn any policy $\\pi \\in \\Omega _ { E }$ according to Eq. (4). ",
+        "bbox": [
+            142,
+            553,
+            825,
+            595
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Policy Decoupling ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            609,
+            338,
+            626
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "110 To construct an unambiguous objective for SOIL, we define hyper-policy and solve the problem by   \n111 finding the state transition predictor of the expert hyper-policy. Intuitively, this tells the agent the   \n112 target that the expert will reach without informing any feasible skill that require the agent to learn   \n113 itself. Therefore, to recover a $\\pi \\in \\Omega _ { E }$ , we can construct an inverse dynamics such that ",
+        "bbox": [
+            140,
+            635,
+            825,
+            691
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c08ddf9a1e5afcd7620d6d7b4db8793fbfc5d3cada68172e3f1359edd07297cd.jpg",
+        "text": "$$\n\\pi = \\mathrm { ~  ~ \\underbrace ~ { ~ \\mathcal { T } _ { \\pi } ^ { - 1 } ~ } ~ } \\left( \\begin{array} { l l l l l l l } { ~ } & { ~ } & { ~ } & { ~ \\mathcal { T } ( \\pi _ { E } ) } & { ~ } & { ~ } & { ~ } \\end{array} \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            693,
+            640,
+            715
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "|{z}Inverse dynamics | {z }Expert state transition predictor ",
+        "bbox": [
+            392,
+            717,
+            624,
+            728
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "114 Formally, the expert policy can be decoupled as ",
+        "bbox": [
+            142,
+            731,
+            488,
+            746
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d851a43fdceab22a1747f57da54026b0facab9741702ab4eedbb3c0eecdd394.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle \\pi _ { E } ( a | s ) = \\int _ { s ^ { \\prime } } { \\mathcal T } ( s ^ { \\prime } | s , a ) \\pi _ { E } ( a | s ) \\mathrm { d } s ^ { \\prime } = \\int _ { s ^ { \\prime } } { \\frac { \\rho _ { \\pi _ { E } } ( s , a , s ^ { \\prime } ) } { \\rho _ { \\pi _ { E } } ( s ) } } \\mathrm { d } s ^ { \\prime } = \\int _ { s ^ { \\prime } } { \\frac { \\rho _ { \\pi _ { E } } ( s , s ^ { \\prime } ) I _ { \\pi _ { E } } ( a | s , s ^ { \\prime } ) } { \\rho _ { \\pi _ { E } } ( s ) } } \\mathrm { d } s ^ { \\prime } } } \\\\ { { \\displaystyle \\qquad = \\int _ { s ^ { \\prime } } h _ { \\pi _ { E } } ( s ^ { \\prime } | s ) I _ { \\pi _ { E } } ( a | s , s ^ { \\prime } ) \\mathrm { d } s ^ { \\prime } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            747,
+            787,
+            810
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "115 ",
+        "bbox": [
+            142,
+            804,
+            161,
+            815
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "116 Notice that both the state transition predictor $h$ and the inverse dynamics model $I$ is policy dependent.   \n117 Nevertheless, recall that the optimality in SOIL only requires us to recover $\\pi \\in \\Omega _ { E }$ , we do not have   \n118 to learn about $I _ { \\pi _ { E } }$ but just one feasible skill $I ( a | s , s ^ { \\prime } )$ . Then a policy can be recovered by ",
+        "bbox": [
+            140,
+            823,
+            826,
+            864
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab2f3a769999356eeeff5395ded80a44bd3fc6135d8302875ab7851b013b40fe.jpg",
+        "text": "$$\n\\pi = \\mathbb { E } \\underbrace { _ { \\mathcal { S } ^ { ' } \\sim h _ { \\Omega _ { E } } \\left( \\boldsymbol { s } ^ { \\prime } | \\boldsymbol { s } \\right) } } _ { \\mathrm { t a r g e t } } \\Big [ \\underbrace { I ( a | \\boldsymbol { s } , \\boldsymbol { s } ^ { \\prime } ) } _ { \\mathrm { s k i l l } } \\Big ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            380,
+            867,
+            617,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "119 Here the inverse dynamics model $I$ offers an   \n120 arbitrary $s k i l l$ to reach the expected target state   \n121 provided by the state transition predictor $h$ . In   \n122 fact, it does not depend on the hyper-policy   \n123 $\\Omega _ { E }$ but a sampling policy $\\pi _ { B }$ to construct $I =$   \n124 $I _ { \\pi _ { B } }$ . We only need a mild requirement for $\\pi _ { B }$   \n125 that it covers the support of $\\bar { \\rho } _ { \\Omega _ { E } } ( s , s ^ { \\prime } )$ so that   \n126 the learned $I$ can provide a possible action to   \n127 achieve the target state. In both experiments and   \n128 theoretical analysis we show that this require  \n129 ment alleviates the dependence on the inverse   \n130 dynamics. Furthermore, if the environment and   \n131 the expert policy are both deterministic (which   \n132 is usually the case in real-world scenarios such   \n133 as robotics), the state transition is a single-point distribution (or known as the Dirac delta func  \n134 tion), and we can simply model $h$ as a deterministic function. By decoupling the policy, which is   \n135 a state-to-action mapping function, as a state-to-state mapping function (the transition predictor)   \n136 and a state-pair-to-action mapping function (the inverse dynamics model), we can mimic the expert   \n137 policy from state-only demonstrations by optimizing these two modules. The whole architecture is   \n138 illustrated in Fig. 1. ",
+        "bbox": [
+            140,
+            92,
+            485,
+            284
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c6189272a8df27c81d57ba8da0bd7d0f9151c524d81ee23041c8b60dd9471b3.jpg",
+        "image_caption": [
+            "Figure 1: The architecture of Decoupled Policy Optimization (DPO), which consists of an expert state transition predictor (to plan where to go) followed by an inverse dynamics model (to decide how to reach). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
+            94,
+            820,
+            215
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            285,
+            826,
+            368
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "State Transition Predictor. In practice, we construct a parameterized expert state transition predictor $h _ { \\psi }$ which predicts the subsequent state of the expert taking the input as a current state $\\hat { s ^ { \\prime } } = h _ { \\psi } ( s )$ ",
+        "bbox": [
+            160,
+            382,
+            823,
+            412
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "141 The state transition predictor models the explicit information of the expert, and it can be learned from   \n142 the demonstration data only. Thence, we implement Eq. (4) as a KL divergence minimization: ",
+        "bbox": [
+            147,
+            419,
+            823,
+            448
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0a0b7617a6716ad3ea875572026b4d596ec953730888d621d3c218d44fb4c072.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\psi } \\mathbb { E } _ { ( s , s ^ { \\prime } ) \\sim \\Omega _ { E } } \\big [ \\mathrm { D } _ { \\mathrm { K L } } \\big ( h _ { \\Omega _ { E } } ( s ^ { \\prime } | s ) \\| h _ { \\psi } ( s ^ { \\prime } | s ) \\big ) \\big ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            349,
+            454,
+            643,
+            478
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "which can be optimized in a supervised manner. Specifically, we sample state transitions 143 $( s , s ^ { \\prime } )$ from 144 the expert demonstrations $\\mathcal { D }$ and optimize the L2 loss: ",
+        "bbox": [
+            138,
+            486,
+            828,
+            515
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/593547017770a1e2076bad09ad8757ee77795b2aa384af1845cff58f5dcbe89a.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\psi } ^ { h } = \\mathbb { E } _ { ( s , s ^ { \\prime } ) \\sim \\mathcal { D } } \\left[ \\| s ^ { \\prime } - h _ { \\psi } ( s ) \\| ^ { 2 } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            521,
+            616,
+            542
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "145 Inverse Dynamics Model. Knowing where to go is not enough since the agent has to interact with   \n146 the environment to reach the target. This can be achieved via an inverse dynamics model, which   \n147 predicts the action given two consecutive states. Formally, let the $\\phi$ -parameterized inverse dynamics   \n148 model $I _ { \\phi }$ take input the state pair and predict the feasible action to achieve the state transition:   \n149 $\\hat { a } = I _ { \\phi } ( \\overset { \\cdot } { s } , s ^ { \\prime } )$ . Intuitively, we want the inverse dynamics to learn from possible transitions sampled   \n150 by the agent. Recall that we only need the support of learned $I ( a | s , s ^ { \\prime } )$ of the sampling policy covers   \n151 the support of the expert state transition OM, from which we can infer at least one possible action.   \n152 Hence, we can optimize the KL divergence between the inverse dynamics of a sampling policy $\\pi _ { B }$   \n153 and $I _ { \\phi }$ : ",
+        "bbox": [
+            140,
+            554,
+            825,
+            679
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1e5ea9fd22698d86af186aac4739e2e412fde193627f5ba92d289a343c554ce9.jpg",
+        "text": "$$\n\\displaystyle \\operatorname* { m i n } _ { \\phi } \\mathbb { E } _ { ( s , s ^ { \\prime } ) \\sim \\pi _ { \\mathcal { B } } } [ \\mathrm { D } _ { \\mathrm { K L } } ( I _ { \\pi _ { \\mathcal { B } } } ( a | s , s ^ { \\prime } ) \\| I _ { \\phi } ( a | s , s ^ { \\prime } ) ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            676,
+            655,
+            702
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "154 and we can choose to optimize L2 loss in a supervised manner by sampling from the replay buffer $\\boldsymbol { B }$ : ",
+        "bbox": [
+            148,
+            705,
+            820,
+            720
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1c651e9744c70d5f18ffc8e838c8300e3a341e4750b34244454b6340acaf3b06.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\phi } ^ { I } = \\mathbb { E } _ { ( s , a , s ^ { \\prime } ) \\sim \\mathcal { B } } \\left[ \\Vert a - I _ { \\phi } ( s , s ^ { \\prime } ) \\Vert ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            727,
+            625,
+            747
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "55 In our implementation, both the state predictor and the inverse dynamics can be constructed as   \n56 Gaussian distributions similar to a normal stochastic policy, thus encouraging exploration. ",
+        "bbox": [
+            151,
+            752,
+            825,
+            781
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.3 Tackling Compounding Error Challenges ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            796,
+            501,
+            813
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "158 In our formulation, we have decoupled the state-to-action mapping policy as a state-to-state mapping   \n159 function and a state-pair-to-action mapping function. Unfortunately, the compounding error problem   \n160 exists such that the agent cannot reach where it plans due to the fitting errors of these two parts.   \n61 Theorem 1 (Error Bound of DPO). Consider a deterministic environment whose dynamics transition   \n62 function ${ \\mathcal { T } } ( s , a )$ is deterministic and $L$ -Lipschitz. Assume the ground-truth state transition $h _ { \\Omega _ { E } } ( s )$   \n63 is deterministic, and for each policy $\\pi \\in \\Pi$ , its inverse dynamics $I _ { \\pi }$ is also deterministic and ",
+        "bbox": [
+            143,
+            821,
+            825,
+            864
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            150,
+            868,
+            825,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9fcc36a4e66ad591146d55ddd602eb00a617e08473a7db733d70219090de0db5.jpg",
+        "text": "$$\n\\| s ^ { \\prime } - s _ { E } ^ { \\prime } \\| \\leq L C \\| h _ { \\Omega _ { E } } ( s ) - h _ { \\psi } ( s ) \\| + L \\| I _ { \\pi _ { B } } ( s , \\hat { s } ^ { \\prime } ) - I _ { \\phi } ( s , \\hat { s } ^ { \\prime } ) \\| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            281,
+            128,
+            715,
+            147
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "166 where $\\pi _ { B }$ is a sampling policy that covers the state transition support of the expert hyper-policy and   \n67 $\\hat { s } ^ { \\prime } = h _ { \\psi } ( s )$ is the predicted next state.   \n168 The proof can be found in Appendix B, where we also   \n169 induce a similar error bound for rollout with a state  \n170 to-action policy as BCO [22] to show the advantage   \n171 of the decoupled structure. From Theorem 1 we   \n172 know that the compounding error can be enlarged   \n173 due to each part’s fitting error, where the first term   \n174 corresponds to the error of predicted states and the   \n175 second term indicates whether the agent can reach ",
+        "bbox": [
+            151,
+            156,
+            825,
+            185
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            198,
+            517,
+            310
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/6ae8d1a939debab71510cc4ece3f7a2d4ed6199704bd819f9427ead6c805f5e6.jpg",
+        "image_caption": [
+            "Figure 2: Multi-step optimization. Given an expert state $_ { s _ { E } }$ , $h _ { \\psi }$ predicts the next possible state $\\hat { \\boldsymbol s } ^ { \\prime 1 }$ , which is further fed to a target network $h _ { \\psi ^ { \\prime } }$ to predict the following sequence. The total loss computes the MSE loss along the state sequence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            532,
+            176,
+            820,
+            239
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "76 where it plans to. To alleviate the error, we further propose regularization on these two modules. ",
+        "bbox": [
+            153,
+            309,
+            802,
+            323
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3.1 Regularization on Target Planning ",
+        "text_level": 1,
+        "bbox": [
+            158,
+            340,
+            465,
+            356
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "178 One major problem is that the state transition predictor may suggest non-neighboring states instead   \n179 of predicting one-step reachable states. To overcome this, we draw inspiration from Asadi et al. [2]   \n180 and Edwards et al. [3], and regularize state transition predictor to prevent the model from predicting   \n181 non-neighboring states via multi-step and cycle training style.   \n182 Multi-Step Optimization. We first explain the details of the multi-step optimization objective.   \n183 This idea is motivated by Asadi et al. [2], which optimizes a multi-step outcome by executing a   \n184 sequence of actions in the dynamics model. Here we optimize the state sequence instead. As shown   \n185 in Fig. 2, given an expert state $s _ { E }$ , $h _ { \\psi }$ predicts the next possible state $\\hat { s } ^ { \\prime }$ that the expert will reach; the   \n186 predicted state is then fed into the predictor to output the predicted two-step state $\\hat { s } ^ { \\prime \\prime }$ . As such, the   \n187 multi-step training loss is the L2 loss computed along the $k$ -step outcome sequence: ",
+        "bbox": [
+            140,
+            366,
+            825,
+            422
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            440,
+            826,
+            525
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/15a08d5cb04b9fc8253ad7214162ca061583b5be51ceec5873ae9c1ee4e2bcca.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\psi } ^ { h , \\mathrm { m s } } = \\underset { ( s , \\{ s _ { E } ^ { \\prime } \\} _ { i = 1 } ^ { k } ) \\sim \\mathcal { D } } { \\mathbb { E } } \\left[ \\| s _ { E } ^ { \\prime 1 } - h _ { \\psi } ( s ) \\| ^ { 2 } + \\sum _ { i = 2 } ^ { k } \\| s _ { E } ^ { \\prime i } - h _ { \\psi ^ { \\prime } } ( s ^ { \\prime i - 1 } ) \\| ^ { 2 } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            534,
+            728,
+            578
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "188 Intuitively, such a regularization makes the state prediction   \n189 $\\hat { s } ^ { \\prime }$ close to the expert state distribution in order to make   \n190 accurate long step predictions. It is worth noting that the   \n191 gradient of the cascading state transition predictors should   \n192 be dropped since we already have an accurate input at each   \n193 time step, and for each training step, we only update the   \n194 first one. We use a target network $h _ { \\psi ^ { \\prime } }$ in practice.   \n195 Cycle Training Style. Another way to regularize the   \n196 transition predictor’s output to a neighboring state is to   \n197 keep an additional function to ensure the cycle consistency,   \n198 which is also an important technique in [3]. In particular,   \n199 as illustrated in Fig. 3, given an expert state $s _ { E }$ , we take   \n200 the predicted state $\\hat { s } ^ { \\prime }$ and $s _ { E }$ into the inverse dynamics and ",
+        "bbox": [
+            140,
+            593,
+            549,
+            691
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            709,
+            550,
+            792
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/826bccc6205f5766d83182c068c280c2914759b75a8454a7bd35bdc9cb1f920a.jpg",
+        "image_caption": [
+            "Figure 3: Cycle training style. Given an expert state $s _ { E }$ , $I _ { \\phi } ( s , s ^ { \\prime } )$ takes input the predicted state $\\hat { s } ^ { \\prime }$ and $_ { s _ { E } }$ to get the execution action $a$ , then an additional forward dynamics model $M _ { \\omega }$ is used to simulated one step rollout using $( s _ { E } , a )$ and get a forward next state $\\tilde { s } ^ { \\prime }$ . The total loss computes the MSE loss between the two predicted states. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            565,
+            584,
+            820,
+            674
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "201 get the action $a$ , then we train an additional forward dynamics model $M _ { \\omega }$ to simulate one step rollout that takes the input 202 $( s _ { E } , a )$ and gets a forward next state $\\widetilde { s } ^ { \\prime }$ : ",
+        "bbox": [
+            133,
+            792,
+            826,
+            820
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d46cc7d30d466d488ce56e825d672c1fd16fd5056f97a8b0794c47df8db99a06.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { L } _ { \\omega } ^ { M } = \\mathbb { E } _ { ( s , a , s ^ { \\prime } ) \\sim \\mathcal { B } } \\left[ \\| s ^ { \\prime } - M _ { \\omega } ( s , a ) \\| ^ { 2 } \\right] } \\\\ & { \\mathcal { L } _ { \\psi } ^ { h , \\mathrm { c y c l e } } = \\mathbb { E } _ { ( s , s ^ { \\prime } ) \\sim \\mathcal { D } } \\left[ \\| s ^ { \\prime } - h _ { \\psi } ( s ) \\| ^ { 2 } + \\| h ( s ) - M _ { \\omega } ( s , I _ { \\phi } ( s , s ^ { \\prime } ) ) \\| ^ { 2 } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            829,
+            730,
+            876
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "203 In other words, the cycle training scheme provides a regularization on $h _ { \\psi }$ to make predictions   \n204 consistent with the forward dynamics model. ",
+        "bbox": [
+            137,
+            882,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In previous sections, we have mentioned that learning to reach a specific place requires the datacollecting policy to cover the support of the expert hyper-policy. This is easy to achieve on simple low-dimensional tasks, but may not be satisfied in high-dimensional continuous environments. To this end, we encourage the agent to approach those state transitions from the expert’s hyper-policy $\\Omega _ { E }$ by minimizing the JS divergence of the state transition occupancy using a state-to-action mapping policy $\\mathrm { D } _ { \\mathrm { J S } } \\left( \\rho _ { \\pi _ { E } } ( { \\bar { s } } , s ^ { \\prime } ) , \\rho _ { \\pi } ( s , s ^ { \\prime } ) \\right)$ . This can be done by producing informative rewards via GAN-like methods [10, 23], and updating the decoupled policy with policy gradients (PG). ",
+        "bbox": [
+            161,
+            114,
+            825,
+            213
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "213 In detail, we construct a parameterized discriminator $D _ { \\omega } ( s , s ^ { \\prime } )$ to compute the reward $r ( s , a ) \\triangleq$   \n214 $r ( s , s ^ { \\prime } )$ as $\\log D _ { \\omega } ( s , s ^ { \\prime } )$ and the decoupled policy served as the generator. In addition, since we   \n215 decouple the policy as two parameterized modules, i.e., a state transition predictor and an inverse   \n216 dynamics model, then by chain rule, the PG for the decoupled policy can be accomplished by ",
+        "bbox": [
+            142,
+            219,
+            825,
+            276
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0039627ef8f73cf68e1f42c911de7ee23f7b1ca89d88c92f1b133c193104c0e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\nabla \\mathcal { L } _ { \\phi , \\psi } ^ { \\pi } = \\mathbb { E } _ { \\pi } \\left[ Q ( s , a ) \\nabla _ { \\phi , \\psi } \\log \\pi _ { \\phi , \\psi } ( a | s ) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { \\pi } \\left[ Q ( s , a ) \\displaystyle \\int _ { s ^ { \\prime } } \\left( \\nabla _ { \\psi } \\log h _ { \\psi } ( s ^ { \\prime } | s ) + \\nabla _ { \\phi } \\log I _ { \\phi } ( a | s , s ^ { \\prime } ) \\right) \\mathrm { d } s ^ { \\prime } \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            258,
+            281,
+            736,
+            338
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "217 where $Q$ is the state-action value function; the first term is the gradient for updating the state transition   \n218 predictor; and the second term is for the inverse dynamics model. Thus, the optimization for both   \n219 the state transition predictor and the inverse dynamics model can augment the supervised learning   \n220 objectives with any PG-based learning algorithms (e.g., TRPO, PPO, SAC). As the training proceeds,   \n221 the agent will sample more transition data around $\\Omega _ { E }$ , and thus the support of the sampling policy   \n222 will progressively cover the support of $\\rho _ { \\Omega _ { E } } ( s , s ^ { \\prime } )$ . ",
+        "bbox": [
+            140,
+            343,
+            826,
+            428
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 Overall Algorithm ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            443,
+            341,
+            458
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "By combining the idea of generative adversarial training, we obtain our final algorithm, composed with three essential parts: the state transition predictor $h$ used for predicting the possible future states sampled by the expert; the inverse dynamics model $I$ used for inferring the possible actions conditioned on two adjacent states; and the discriminator $D$ used for offering intermediate reward signals for training the decoupled policy $\\pi = I ( h )$ . The overall objective of DPO is ",
+        "bbox": [
+            161,
+            468,
+            823,
+            539
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57a4af09ca3a15b7a2d938045855505704b1796c21bc1fc2e94513a67c727035.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { L } _ { \\phi , \\psi } ^ { \\pi , h , I } = \\lambda _ { G } \\mathcal { L } _ { \\phi , \\psi } ^ { \\pi } + \\lambda _ { h } \\mathcal { L } _ { \\psi } ^ { h } + \\lambda _ { I } \\mathcal { L } _ { \\phi } ^ { I } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            545,
+            620,
+            568
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "229 where $\\lambda _ { G } , \\lambda _ { h }$ and $\\lambda _ { I }$ are hyperparameters for trading off the training among each loss. In practice,   \n230 we try less than ten combinations for these parameters as shown in Appendix D.3, and we directly   \n231 optimize $\\mathcal { L } _ { \\phi , \\psi } ^ { \\pi }$ instead of iterative training. The detailed algorithm is summarized in Appendix A.   \n232 Besides, it is worth noting that both the inverse dynamics model and the state transition predictor   \n233 can be pre-trained, where we optimize $\\mathcal { L } _ { \\psi } ^ { h }$ using the state-only demonstration and optimize $\\dot { \\mathcal { L } } _ { \\phi } ^ { I }$ using   \n234 samples collected by a randomized agent. ",
+        "bbox": [
+            140,
+            573,
+            825,
+            660
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 4 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            158,
+            679,
+            320,
+            695
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c86ca5dfb21497ccb00da499b9e4357a449782649ba78ed46378a42b27b16052.jpg",
+        "table_caption": [
+            "Table 1: Comparison between different methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Inverse Dynamics</td><td>State Predictor</td><td>Decoupled Policy</td><td>Task</td></tr><tr><td>BCO[22]</td><td>√</td><td>X</td><td>X</td><td>SOIL</td></tr><tr><td>GAIfO [23]</td><td>X</td><td>X</td><td>X</td><td>SOIL</td></tr><tr><td>IDDM[25]</td><td></td><td>X</td><td>X</td><td>SOIL</td></tr><tr><td>OPOLO [27]</td><td>X</td><td></td><td>X</td><td>SOIL</td></tr><tr><td>PID-GAIL [11]</td><td>X</td><td>xx</td><td>√</td><td>IL</td></tr><tr><td>QSS[3]</td><td>√</td><td>√</td><td>√</td><td>RL</td></tr><tr><td>SAIL[16]</td><td>1</td><td>√</td><td>X</td><td>IL</td></tr><tr><td>DPO (Ours)</td><td>√</td><td>√</td><td>√</td><td>SOIL</td></tr></table>",
+        "bbox": [
+            532,
+            669,
+            821,
+            776
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "State-only imitation learning (SOIL) endows the agent with the ability to learn from expert states. Although lacking the expert decision information, most of the previous works still optimize a state-to-action mapping policy to match the expert state transition distribution. For example, Torabi et al. [22] used a model-based approach to apply behavioral cloning to state-only demonstrations, while Torabi et al. [23] employed a similar structure to GAIL to match the state transition distribution. Yang et al. [25] analyzed the optimization gap between SOIL and naive IL and introduced a mutual information term to narrow it. Huang et al. [11] applied SOIL on autonomous driving tasks by decoupling the policy into a neural decision module and a non-differentiable execution module in a hierarchical way. ",
+        "bbox": [
+            168,
+            710,
+            519,
+            780
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
+            780,
+            825,
+            863
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "247 Our work decouples the state-to-action policy into two modules. However, both the inverse dynamics   \n248 model and the state transition predictor have been widely used by many previous works on RL and IL   \n249 tasks. For instance, Torabi et al. [22] and Guo et al. [6] trained an inverse dynamics model to label the   \n250   \n251   \n252   \n253   \n254   \n255   \n256   \n257   \n258   \n259   \n260 ",
+        "bbox": [
+            147,
+            869,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/688e92dd57cc5894d24a6ce50c5bd55f7548a7783f7ac0541a4ecf7f6626846f.jpg",
+        "table_caption": [
+            "Table 2: Eventual performance against different methods on 6 easy-to-hard continuous control benchmarks. The means and the standard deviations are evaluated over more than 5 random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>InvertedPendulum</td><td>InvertedDoublePendulum</td><td>Hopper</td><td>Walker2d</td><td>HalfCheetah</td><td>Ant</td></tr><tr><td>Random</td><td>25.28 ± 5.53</td><td>78.28 ± 10.73</td><td>13.09 ±0.10</td><td>7.07 ± 0.13</td><td>74.48 ± 12.39</td><td>713.59 ± 203.92</td></tr><tr><td>BCO</td><td>1000.00±0.00</td><td>415.04± 148.46</td><td>1430.16± 398.81</td><td>261.36 ± 25.17</td><td>-13.66± 149.94</td><td>397.79± 239.16</td></tr><tr><td>GAIfO</td><td>1000.00 ±0.00</td><td>7818.07 ± 1778.67</td><td>3068.10± 26.32</td><td>3865.20 ± 341.90</td><td>8953.35 ± 1079.41</td><td>5122.29 ± 807.19</td></tr><tr><td>GAIfO-DP</td><td>1000.00±0.00</td><td>7305.01 ± 1591.23</td><td>3031.84 ± 152.13</td><td>4003.06 ± 241.34</td><td>8675.42 ± 807.29</td><td>5535.9± 62.74</td></tr><tr><td>DPO (w/o PG)</td><td>1000.00±0.00</td><td>3545.70± 738.16</td><td>629.84 ± 344.07</td><td>334.23 ±85.42</td><td>-472.00 ± 132.81</td><td>-196.96 ± 124.26</td></tr><tr><td>DPO (w PG)</td><td>1000.00±0.00</td><td>7846.40 ± 1541.20</td><td>3165.72 ± 68.44</td><td>4407.53 ± 266.72</td><td>10501.96 ± 438.01</td><td>5338.48 ± 107.2</td></tr><tr><td>Expert (SAC)</td><td>1000.00±0.00</td><td>9358.87 ± 0.10</td><td>3402.94 ± 446.48</td><td>5639.32 ± 29.97</td><td>13711.64 ± 111.47</td><td>5404.55±1520.49</td></tr></table>",
+        "bbox": [
+            178,
+            119,
+            820,
+            200
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            138,
+            210,
+            161,
+            364
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "state-only demonstrations with inferred actions. Nair et al. [19] proposed a method for manipulating ropes to match a single human-specified image sequence, in which an inverse dynamics model is trained in a self-supervised manner and used to generate control signals. Kimura et al. [14] utilized a state transition predictor to fit the state transition probability in the expert data, which is further used to compute a predefined reward function. Liu et al. [16] constructed a policy prior using the inverse dynamics and the state transition predictor, but the policy prior was only used for regularizing the policy network. However, as shown in this paper, the policy can be exactly decoupled as these two parts, which can be uniformly optimized through policy gradient without keeping an extra policy. Edwards et al. [3] estimated $Q ( s , s ^ { \\prime } )$ for RL tasks which employs a similar policy form as Eq. (6) and updates the state transition predictor through a deterministic policy gradient similar to DDPG [15]. To sort out the difference between these methods and ours, we summarize the key factors in Tab. 1. ",
+        "bbox": [
+            169,
+            210,
+            825,
+            362
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            381,
+            312,
+            398
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We conduct four sets of experiments to investigate the following research questions: ",
+        "bbox": [
+            176,
+            412,
+            722,
+            428
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "RQ1 Is decoupled learning structure superior than state-to-action structure on SOIL tasks? RQ2 Does DPO achieve higher efficiency or better performance than baselines on SOIL tasks? RQ3 Can agent reach where it plans and does the proposed regularization help mitigate the compounding error? ",
+        "bbox": [
+            184,
+            433,
+            823,
+            488
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "RQ4 How can DPO be applied on real-world data? ",
+        "bbox": [
+            191,
+            488,
+            531,
+            502
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To answer RQ1, we conduct toy experiments with a simple 2D grid world environment and compare both qualitative and quantitative results of DPO against BCO and GAIfO. Regarding RQ2, we empirically evaluate DPO on easy-to-hard continuous control benchmarking tasks. And for RQ3, we evaluate the difference between the predicted states that the agent plans to reach and the consecutive state that the agent actually reaches in the environment for the proposed regularization. Finally, we try to imitate real-world traffic surveillance recordings in a simulated environment to investigate RQ4, which shows the potential of using real-world data for human behavior simulation. Due to the space limit, we leave experiment details, additional results and ablation study in Appendix. ",
+        "bbox": [
+            173,
+            503,
+            647,
+            655
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2572c840bae1adcb26bf42c08dee3fbfad4863b16f4814902fe7dadff55452a7.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            663,
+            478,
+            820,
+            606
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "(a) Rollout density. ",
+        "bbox": [
+            684,
+            611,
+            799,
+            622
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "79 5.1 Understanding the Decoupled Structure ",
+        "text_level": 1,
+        "bbox": [
+            153,
+            671,
+            491,
+            686
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this paper we design decoupled policy optimization (DPO) to perform SOIL tasks, and in previous sections we propose that the key technical contribution of DPO is the decoupled structure of policy that models the explicit state transition information and the latent action information from demonstrations, which solves the ambiguity and enhances the learning efficiency. Therefore, in this set of experiments, we aim to demonstrate how DPO is superior than state-to-action policy methods (RQ1). We first generate expert demonstrations in a 2D $6 \\times 6$ grid world environment, in which the agent starts at the upper left corner and aims to reach the upper right corner. In each grid the agent has $k \\times 4$ actions, which means that the agent has $k$ possible actions to reach the neighboring block and in our experiment we choose $k = 5$ to enlarge the action space. ",
+        "bbox": [
+            173,
+            696,
+            647,
+            794
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/09c21ffa2883a18d3adfd01b8a7b9fbb89d2dfdfc00ad1028ba611931eb3384d.jpg",
+        "image_caption": [
+            "Figure 4: Toy example. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            661,
+            627,
+            820,
+            767
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            795,
+            825,
+            849
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "291 The density of the expert trajectories and the trajectories sampled by different methods are shown in   \n292 Fig. 4(a). We show that both BCO and GAIfO have troubles in directly learning the implicit action   \n293 from state-only behaviors. Notably, GAIfO only imitates the major trajectory and omit the other   \n294 choice and BCO also stucks in the middle right. By contrast, DPO recovers the expert demonstrations   \n295 much better, benefiting from the decoupled structure that first determining the target and then taking   \n296 the action to achieve it. To further illustrate the learning efficiency advantage of DPO, we illustrate the   \n297 JS divergence curves of DPO and BCO during training in Fig. 4(b). Besides, we show the policy loss   \n298 for BCO, the state predictor (SP) loss for DPO, and the inverse dynamics (ID) loss for both methods.   \n299 Except that the JS divergence of DPO decreases more quickly than BCO, it is also observable that   \n300 DPO relies less on the inverse dynamics than BCO, since the inverse dynamics loss of DPO converges   \n301 to a higher level. We further provide a theoretic analysis of the dependence on inverse dynamics with   \n302 BCO and DPO in Appendix B. ",
+        "bbox": [
+            142,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/06153042034d10056cca54a6053af2d92b12f25747a0f57889baf0ae15319178.jpg",
+        "image_caption": [
+            "Figure 5: Learning curves on 6 easy-to-hard continuous control benchmarks, where the solid line and the shade represent the mean and the standard deviation of the averaged return over more than 5 random seeds. We pre-train BCO and DPO for 50k steps and show it in figures. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            89,
+            787,
+            327
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            372,
+            825,
+            483
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 Comparative Evaluations ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            502,
+            390,
+            517
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We compare the qualitative results of DPO against other baseline methods on easy-to-hard continuous control benchmarking environments (RQ2), including InvertedPendulum, InvertedDoublePendulum, Hopper, Walker2d, HalfCheetah and Ant. In each environment, besides GAIfO and BCO, we also evaluate GAIfO with decoupled policy (denoted as GAIfO-DP). For DPO we compare the reward augmented version of DPO (denoted as DPO w PG)1 with the supervised learning version of DPO, i.e., $\\lambda _ { G } = 0$ (denoted as DPO w/o PG). For fairness, we re-implement all the algorithms based on a Pytorch code framework2 and adopt Soft Actor-Critic (SAC) [7] as the RL learning algorithm for GAIfO and DPO. For all environments, we first train an SAC agent to collect 4 state-only expert trajectories and then train agents with such data. All algorithms are evaluated by a deterministic policy. The eventual results are summarized in Tab. 2, and the learning curves are shown in Fig. 5. It is worth noting that for DPO, we choose the best performance among the experiments that use multi-step or cycle regularization, and we put the full experiment results in Appendix D. ",
+        "bbox": [
+            163,
+            529,
+            825,
+            695
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "One can easily observe that on simple environments, BCO is able to achieve a good performance, and GAIfO also does well on harder tasks. Even so, DPO can still gain the best or comparable performance against its counterparts. Particularly, without augmented reward, DPO is able to reach the optimality with the highest sample efficiency on simple tasks like InvertedPendulum. By contrast, on higher-dimensional tasks such as Hopper, Walker2d, HalfCheetah and Ant, it is difficult to construct accurate inverse dynamics that covers the support of the expert hyper-policy from scratch. However, by combining generative adversarial policy gradients, the agent finally recovers a good policy from the expert hyper-policy. This is particularly evident on HalfCheetah where DPO behaves poorly at the beginning but improves fast as the training proceeds. Besides, as illustrated in Fig. 5, DPO benefits from better sample efficiency in most of the environments, but the improvements are limited on the hardest tasks. We think that this may be due to larger state spaces (111 dimensions for Ant) that makes it difficult to recover a good state predictor or an inverse dynamics model. In all ",
+        "bbox": [
+            165,
+            702,
+            825,
+            867
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/30082c9b9985186853bef3245eff5773245686f44639de77e4b5f2e45ec4e44a.jpg",
+        "image_caption": [
+            "Figure 6: Compounding error of the predicted consecutive states and the real states the agent reaches when rollout in the environments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            199,
+            88,
+            790,
+            190
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "28 experiments, GAIfO-DP achieves similar results as GAIfO, indicating that the network structure does   \n29 not count much for the performance. ",
+        "bbox": [
+            156,
+            218,
+            823,
+            247
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.3 Compounding Error Reduction ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            262,
+            433,
+            277
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this section, we aim to study whether the agent can effectively reach the target as it plans (RQ3). Therefore, we analyze the distance of the reaching states and the predicted consecutive states, and draw the mean square error (MSE) along the training procedure in Fig. 6. We also compare our regularization including multi-step optimization (denoted as ${ \\bf M } . { \\bf S } . - k$ , where $k$ is the number of rollout steps) and cycle training style (denoted as Cycle). Note that DPO needs at least 1-step rollout for training the state transition predictor. As shown in Fig. 6, the agent still has gaps to get to where it plans to, and the mismatch always deteriorates on harder tasks. Combining regularization can always achieve lower compounding error, and the cycle training is effective in most of the environments. In Appendix D.6, we further illustrate the correlation between the final performance and the distance. ",
+        "bbox": [
+            173,
+            287,
+            825,
+            412
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.4 Learn to Drive from Real-World Traffic Data ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            428,
+            527,
+            443
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The rapid development of autonomous driving has brought a lot of demand for simulating and training an RL agent in the simulator, which requires realistic interactions with various social vehicles [26]. However, driver’s detailed actions are not easily to obtain yet we adopt SOIL from a traffic surveillance recording dataset (NGSIM I-80 [8]) that contains kinds of recorded human driving trajectories. We wish to further examine the potential of DPO for decreasing the gap between the real world and simulation (RQ4). We utilize the simulator provided by Henaff et al. [9] as our simulation platform and learn to imitate real-world driving behaviors. We compare DPO against GAIfO and BCO, and choose Success Rate, Mean Distance and KL Divergence as evaluation metrics. Specifically, Success Rate is the percentage of driving across the entire area without crashing into other vehicles or driving off the road, Mean Distance is the distance traveled before the episode ends, and KL Divergence measures the position distribution distance between the expert and the agent. ",
+        "bbox": [
+            173,
+            454,
+            825,
+            606
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As shown in Tab. 3, DPO outperforms baseline methods in all three metrics while possessing higher stability. The decoupled policy allows the state predictor to focus on matching the distribution of expert trajectories, thus achieving smaller deviations from the expert position distribution. Furthermore, since the policy gradient can be computed with non-differentiable inverse dynamics, we can generate 9 stable action sequences [12, 11] by replacing the inverse dynamics model with classical controllers, which can be generalized to realistic applications. ",
+        "bbox": [
+            156,
+            612,
+            550,
+            727
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/a39cbe1cda6983230b36c6e97194971a990c4759409a714faad6fe9c85c532b4.jpg",
+        "table_caption": [
+            "Table 3: Performance on NGSIM I-80 driving task over 5 random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Success Rate (%)</td><td>Mean Distance (m)</td><td>KL Divergence</td></tr><tr><td>BCO</td><td>27.4 ± 1.1</td><td>129.8 ± 2.0</td><td>24.4 ± 2.2</td></tr><tr><td>GAIfO</td><td>77.5 ± 0.8</td><td>188.3 ± 1.1</td><td>11.5 ± 3.9</td></tr><tr><td>DPO</td><td>80.3± 0.5</td><td>192.7 ± 0.6</td><td>9.5 ± 1.8</td></tr><tr><td>Expert</td><td>100</td><td>210.0</td><td>0</td></tr></table>",
+        "bbox": [
+            563,
+            637,
+            816,
+            710
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            161,
+            723,
+            779,
+            738
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            163,
+            756,
+            299,
+            772
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we characterize the optimality and investigate the ambiguity problem in state-only imitation learning, and accordingly propose Decoupled Policy Optimization (DPO), which splits the state-to-action mapping policy into a state-to-state mapping state transition predictor and a state-pair-to-action mapping inverse dynamics model. Furthermore, we employ regularization and generative adversarial methods to mitigate the compounding error caused by the decoupled modules. The flexibility of the decoupled architecture allows a wide range of interesting future works, such as replacing the inverse dynamics with a classic control module to produce stable control signals, learning specific skills with shared state transition and multi-task target learning with shared pretrained skills. ",
+        "bbox": [
+            171,
+            786,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Pieter Abbeel and Andrew Y. Ng. Apprenticeship learning via inverse reinforcement learning. In Machine Learning, Proceedings of the Twenty-first International Conference (ICML 2004), 2004.   \n[2] Kavosh Asadi, Dipendra Misra, Seungchan Kim, and Michel L Littman. Combating the compounding-error problem with a multi-step model. arXiv preprint arXiv:1905.13320, 2019.   \n[3] Ashley D. Edwards, Himanshu Sahni, Rosanne Liu, Jane Hung, Ankit Jain, Rui Wang, Adrien Ecoffet, Thomas Miconi, Charles Isbell, and Jason Yosinski. Estimating q(s,s’) with deep deterministic dynamics gradients. In Proceedings of the 37th International Conference on Machine Learning, ICML 2020, 2020.   \n[4] Chelsea Finn, Sergey Levine, and Pieter Abbeel. Guided cost learning: Deep inverse optimal control via policy optimization. In International Conference on Machine Learning, pages 49–58, 2016.   \n[5] Justin Fu, Katie Luo, and Sergey Levine. Learning robust rewards with adverserial inverse reinforcement learning. In 6th International Conference on Learning Representations, ICLR 2018, 2018.   \n[6] Xiaoxiao Guo, Shiyu Chang, Mo Yu, Gerald Tesauro, and Murray Campbell. Hybrid reinforcement learning with expert state sequences. In The Thirty-Third AAAI Conference on Artificial Intelligence, AAAI 2019, pages 3739–3746, 2019.   \n[7] Tuomas Haarnoja, Aurick Zhou, Pieter Abbeel, and Sergey Levine. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, pages 1856–1865, 2018.   \n[8] John Halkias and James Colyar. Next generation simulation fact sheet. US Department of Transportation: Federal Highway Administration, 2006.   \n[9] Mikael Henaff, Alfredo Canziani, and Yann LeCun. Model-predictive policy learning with uncertainty regularization for driving in dense traffic. In 7th International Conference on Learning Representations, ICLR 2019, 2019.   \n[10] Jonathan Ho and Stefano Ermon. Generative adversarial imitation learning. In Advances in Neural Information Processing Systems 29, pages 4565–4573, 2016.   \n[11] Junning Huang, Sirui Xie, Jiankai Sun, Qiurui Ma, Chunxiao Liu, Dahua Lin, and Bolei Zhou. Learning a decision module by imitating driver’s control behaviors. In Proceedings of the Conference on Robot Learning (CoRL) 2020, 2020.   \n[12] Sandy H. Huang, Nicolas Papernot, Ian J. Goodfellow, Yan Duan, and Pieter Abbeel. Adversarial attacks on neural network policies. In 5th International Conference on Learning Representations, ICLR 2017, 2017.   \n[13] Ahmed Hussein, Mohamed Medhat Gaber, Eyad Elyan, and Chrisina Jayne. Imitation learning: A survey of learning methods. ACM Computing Surveys (CSUR), 50(2):1–35, 2017.   \n[14] Daiki Kimura, Subhajit Chaudhury, Ryuki Tachibana, and Sakyasingha Dasgupta. Internal model from observations for reward shaping. arXiv preprint arXiv:1806.01267, 2018.   \n[15] Timothy P. Lillicrap, Jonathan J. Hunt, Alexander Pritzel, Nicolas Heess, Tom Erez, Yuval Tassa, David Silver, and Daan Wierstra. Continuous control with deep reinforcement learning. In 4th International Conference on Learning Representations, ICLR 2016,, 2016.   \n[16] Fangchen Liu, Zhan Ling, Tongzhou Mu, and Hao Su. State alignment-based imitation learning. In 8th International Conference on Learning Representations, ICLR 2020, 2020.   \n[17] Minghuan Liu, Tairan He, Minkai Xu, and Weinan Zhang. Energy-based imitation learning. In 20th International Conference on Autonomous Agents and Multiagent Systems, AAMAS 2021, 2021.   \n[18] Leland McInnes and John Healy. UMAP: uniform manifold approximation and projection for dimension reduction. CoRR, abs/1802.03426, 2018.   \n[19] Ashvin Nair, Dian Chen, Pulkit Agrawal, Phillip Isola, Pieter Abbeel, Jitendra Malik, and Sergey Levine. Combining self-supervised learning and imitation for vision-based rope manipulation. In 2017 IEEE International Conference on Robotics and Automation, ICRA 2017, pages 2146– 2153, 2017.   \n[20] Murray Rosenblatt. Remarks on some nonparametric estimates of a density function. The Annals of Mathematical Statistics, pages 832–837, 1956.   \n[21] Umar Syed, Michael H. Bowling, and Robert E. Schapire. Apprenticeship learning using linear programming. In Machine Learning, Proceedings of the Twenty-Fifth International Conference (ICML 2008), pages 1032–1039, 2008.   \n[22] Faraz Torabi, Garrett Warnell, and Peter Stone. Behavioral cloning from observation. In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI 2018, pages 4950–4957, 2018.   \n[23] Faraz Torabi, Garrett Warnell, and Peter Stone. Adversarial imitation learning from state-only demonstrations. In Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems, AAMAS ’19, pages 2229–2231, 2019.   \n[24] Faraz Torabi, Garrett Warnell, and Peter Stone. Recent advances in imitation learning from observation. In Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence, IJCAI 2019, pages 6325–6331, 2019.   \n[25] Chao Yang, Xiaojian Ma, Wenbing Huang, Fuchun Sun, Huaping Liu, Junzhou Huang, and Chuang Gan. Imitation learning from observations by minimizing inverse dynamics disagreement. In Advances in Neural Information Processing Systems 32, pages 239–249, 2019.   \n[26] Ming Zhou, Jun Luo, Julian Villela, Yaodong Yang, David Rusu, Jiayu Miao, Weinan Zhang, et al. Smarts: Scalable multi-agent reinforcement learning training school for autonomous driving. In Conference on Robot Learning, 2020.   \n[27] Zhuangdi Zhu, Kaixiang Lin, Bo Dai, and Jiayu Zhou. Off-policy imitation learning from observations. In Advances in Neural Information Processing Systems 33, 2020. ",
+        "bbox": [
+            143,
+            73,
+            828,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            82,
+            828,
+            574
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            116,
+            339,
+            131
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See Section 3.3.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            135,
+            825,
+            241
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            243,
+            493,
+            258
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Theorem 1.   \n(b) Did you include complete proofs of all theoretical results? [Yes] See Section B. ",
+        "bbox": [
+            236,
+            262,
+            825,
+            295
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            212,
+            299,
+            393,
+            313
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix D.3.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] We ran our results with more than 5 random seeds as said in Section 5.2.   \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)? [No] ",
+        "bbox": [
+            238,
+            316,
+            825,
+            449
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            217,
+            453,
+            823,
+            468
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We re-implement all algorithms based on an existed code base, as said in Section 5.2.   \n(b) Did you mention the license of the assets? [No]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No] We will public our codes after publication.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            472,
+            825,
+            608
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            212,
+            611,
+            705,
+            626
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            630,
+            825,
+            719
+        ],
+        "page_idx": 11
+    }
+]

parse/train/NlB8_hXkbby/NlB8_hXkbby.md

@@ -0,0 +1,332 @@
+# Multi-view Contrastive Graph Clustering
+
+Erlin Pan, Zhao Kang∗ School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China wujisixsix6@gmail.com zkang@uestc.edu.cn
+
+# Abstract
+
+With the explosive growth of information technology, multi-view graph data have become increasingly prevalent and valuable. Most existing multi-view clustering techniques either focus on the scenario of multiple graphs or multi-view attributes. In this paper, we propose a generic framework to cluster multi-view attributed graph data. Specifcally, inspired by the success of contrastive learning, we propose multi-view contrastive graph clustering (MCGC) method to learn a consensus graph since the original graph could be noisy or incomplete and is not directly applicable. Our method composes of two key steps: we frst flter out the undesirable highfrequency noise while preserving the graph geometric features via graph fltering and obtain a smooth representation of nodes; we then learn a consensus graph regularized by graph contrastive loss. Results on several benchmark datasets show the superiority of our method with respect to state-of-the-art approaches. In particular, our simple approach outperforms existing deep learning-based methods.
+
+# 1 Introduction
+
+An attributed graph contains of node features and edges characterizing the pairwise relations between nodes. It is a natural and effcient representation for many real-world data [Liu et al., 2021]. For example, social network users have their own profles and the topological graph refects their social relationships. Different from most classical clustering methods like K-means and hierarchical clustering which only focus on Euclidean data, graph clustering divides unlabeled nodes of graph into clusters. Typical graph clustering methods frst learn a good representation of graph and then apply a classical clustering method upon the embeddings. For example, large-scale information network embedding (LINE) [Tang et al., 2015] is a popular graph representation learning method, which can preserve both local and global information and scale up easily to large-scale networks. To incorporate node features and graph structure information, graph autoencoder (GAE) [Kipf and Welling, 2016] employs a graph convolution network (GCN) encoder and achieves signifcant performance improvement. The real-life data are often collected from various sources or obtained from different extractors, thus are naturally represented by different features or views [Kang et al., 2021, 2020c]. Each view could be noisy and incomplete, but important factors, such as geometry and semantics, tend to be shared among all views. Therefore, features and graphs of different views are complementary, which implies that it’s paramount to integrate all features and graphs of diverse views to improve the performance of clustering task.
+
+Numerous graph-based multi-view clustering methods have been developed to capture the consensus information shared by different views in the literature. Graph-based multi-view clustering constructs a graph for each view and fuses them based on a weighting mechanism [Wang et al., 2019]. Multi-view spectral clustering network [Huang et al., 2019] learns a discriminative representation by using a deep metric learning network. These methods are developed for feature matrix and can not handle graph data. To directly process graph data, some representative methods have also been proposed.
+
+Scalable multiplex network embedding (MNE) [Zhang et al., 2018] is a scalable multi-view network embedding model, which learns multiple relations by a unifed network embedding framework. Principled multilayer network embedding (PMNE) [Liu et al., 2017] proposes three strategies (“network aggregation”, “results aggregation”, and “layer co-analysis”) to project a multilayer network into a continuous vector space. Nevertheless, they fail to explore the feature information [Lin and Kang, 2021].
+
+Recently, based on GCN, One2Multi graph autoencoder clustering (O2MA) framework [Fan et al., 2020] and multi-view attribute GCNs for clustering (MAGCN) [Cheng et al., 2020] achieve superior performance on graph clustering. O2MA introduces a graph autoencoder to learn node embeddings based on one informative graph and reconstruct multiple graphs. However, the shared feature representation of multiple graphs could be incomplete because O2MA only takes into account the informative view selected by modularity. MAGCN exploits the abundant information of all views and adopts a cross-view consensus learning by enforcing the representations of different views to be as similar as possible. Nevertheless, O2MA targets for multiple graphs while MAGCN mainly solves graph structured data of multi-view attributes. They are not directly applicable to multiple graphs data with multi-view attributes. Therefore, the research of multi-view graph clustering is at the initial stage and more dedicated efforts are pressingly needed.
+
+In this paper, we propose a generic framework of clustering on attributed graph data with multiview features and multiple topological graphs, denoted by Multi-view Contrastive Graph Clustering (MCGC). To be exact, MCGC learns a new consensus graph by exploring the holistic information among various attributes and graphs rather than utilizing the initial graph. The reason of introducing graph learning is that the initial graph is often noisy or incomplete, which leads to suboptimal solutions [Chen et al., 2020b, Kang et al., 2020b]. A contrastive loss is adopted as regularization to make the consensus graph clustering-friendly. Moreover, we implement on the smooth representation rather than raw data. The contributions of this work could be summarized as follows:
+
+• To boost the quality of learned graph, we propose a novel contrastive loss at graph-level. It is capable of drawing similar nodes close and pushing those dissimilar ones apart.   
+• We propose a generic clustering framework to handle multilayer graphs with multi-view attributes, which contains graph fltering, graph learning, and graph contrastive components. The graph fltering is simple and effcient to obtain a smoothed representation; the graph learning is utilized to generate the consensus graph with an adaptive weighting mechanism for different views.   
+• Our method achieves state-of-the-art performance compared with shallow methods and deep methods on fve benchmark datasets.
+
+# 2 Related Work
+
+# 2.1 Multi-view Clustering
+
+Large quantities of multi-view clustering methods have been proposed in the last decades. Multi-view low-rank sparse subspace clustering [Brbic and Kopriva, 2018] obtains a joint subspace representation´ across all views by learning an affnity matrix constrained by sparsity and low-rank constraint. Cross-view matching clustering (COMIC) [Peng et al., 2019] enforces the graphs to be as similar as possible instead of the representation in the latent space. Robust multi-view spectral clustering (RMSC) [Xia et al., 2014] uses a shared low-rank transition probability matrix derived from each single view as input to the standard Markov chain method for clustering. These methods are designed for feature matrix and try to learn a graph from data. To directly cluster multiple graphs, self-weighted multi-view clustering (SwMC) [Nie et al., 2017] method learns a shared graph from multiple graphs by using a novel weighting strategy. Above methods are suitable for graph or feature data only, and can not simultaneously explore attributes and graph structure. As previously discussed, O2MA [Fan et al., 2020] and MAGCN [Cheng et al., 2020] can handle attributed graph, but they are not direct applicable to generic multi-view graph data.
+
+# 2.2 Contrastive Clustering
+
+Due to its impressive performance in many tasks, contrastive learning has become the most hot topic in unsupervised learning. Its motivation is to maximize the similarity of positive pairs and distance of negative pairs [Hadsell et al., 2006]. Generally, the positive pair are composed of data augmentations of the same instance while those of different instances are regarded as negatives. Several loss functions have been proposed, such as the triplet loss [Chopra et al., 2005], the noise contrastive estimation (NCE) loss [Gutmann and Hyvärinen, 2010], the normalized temperature-scaled cross entropy loss (NT-Xent) [Chen et al., 2020a]. Deep robust clustering turns maximizing mutual information into minimizing contrastive loss and achieves signifcant improvement after applying contrastive learning to decrease intra-class variance [Zhong et al., 2020]. Contrastive clustering develops a dual contrastive learning framework, which conducts contrastive learning at instance-level as well as cluster-level [Li et al., 2021]. As a result, it produces a representation that facilitates the downstream clustering task. Unfortunately, theses method can only handle single-view data.
+
+Recently, by combining reconstruction, cross-view contrastive learning, and cross-view dual prediction, incomplete multi-view clustering via contrastive prediction (COMPLETER) [Lin et al., 2021a] performs data recovery and consistency learning of incomplete multi-view data simultaneously. It also obtains promising performance on complete multi-view data. However, it can not deal with graph data. On the other hand, contrastive multi-view representation learning on graphs (MVGRL) [Hassani and Khasahmadi, 2020] method is proposed, which performs representation learning by contrasting two diffusion matrices transformed from the adjacency matrix. It reports better performance than variational GAE (VGAE) [Kipf and Welling, 2016], marginalized GAE (MGAE) [Wang et al., 2017], adversarially regularized GAE (ARGA) and VGAE (ARVGA) [Pan et al., 2018], and GALA [Park et al., 2019]. Different from MVGRL, our contrastive regularizer is directly applied on learned graph.
+
+# 3 Methodology
+
+# 3.1 Notation
+
+Defne the multi-view graph data as $G = \{ \mathcal { V } , E _ { 1 } , . . . , E _ { V } , X ^ { 1 } , . . . , X ^ { V } \}$ , where $\nu$ represents the sets of $N$ nodes, $e _ { i j } \in E _ { v }$ denotes the relationship between node $i$ and node $j$ in the $v$ -th view, $X ^ { v } = \{ x _ { 1 } ^ { v } , . . . , x _ { N } ^ { v } \} ^ { \top }$ he feature matrix. Adjacency matrices represent the degree matrices in various $\{ \widetilde { A } ^ { v } \} _ { v = 1 } ^ { V }$ characterize the initial e normalized adjacency $\{ D ^ { v } \} _ { v = 1 } ^ { V }$ matrix $A ^ { v } = ( D ^ { v } ) ^ { - \frac { 1 } { 2 } } ( \widetilde { A } ^ { v } + I ) ( D ^ { v } ) ^ { - \frac { 1 } { 2 } }$ and the corresponding graph laplacian $L ^ { v } = I - A ^ { v }$ .
+
+# 3.2 Graph Filtering
+
+A feature matrix $\boldsymbol { X } \in \mathbb { R } ^ { N \times d }$ of $N$ nodes can be treated as $d N$ -dimensional graph signals. A natural signal should be smooth on nearby nodes in term of the underlying graph. The smoothed signals $H$ can be achieved by solving the following optimization problem [Zhu et al., 2021, Lin et al., 2021b]
+
+$$
+\operatorname* { m i n } _ { H } \| H - X \| _ { F } ^ { 2 } + s \operatorname { T r } \left( \mathrm { H } ^ { \top } \mathrm { L H } \right) ,
+$$
+
+where $s > 0$ is a balance parameter and $L$ is the laplacian matrix associated with $X$ . $H$ can be obtained by taking the derivative of Eq. (1) w.r.t. $H$ and setting it to zero, which yields
+
+$$
+H = ( I + s L ) ^ { - 1 } X .
+$$
+
+To get rid of matrix inversion, we approximate $H$ by its frst-order Taylor series expansion, i.e., $H = ( I - s L ) X$ . Generally, $m$ -th order graph fltering can be written as
+
+$$
+H = ( I - s L ) ^ { m } X ,
+$$
+
+where $m$ is a non-negative integer. Graph fltering can flter out undesirable high-frequency noise while preserving the graph geometric features.
+
+# 3.3 Graph Learning
+
+Since real-world graph is often noisy or incomplete, which will degrade the downstream task performance if it is directly applied. Thus we learn an optimized graph $S$ from the smoothed representation $H$ . This can be realized based on the self-expression property of data, i.e., each data point can be represented by a linear combination of other data samples [Lv et al., 2021, Ma et al., 2020]. And the combination coeffcients represent the relationships among data points. The objective function on single-view data can be mathematically formulated as
+
+$$
+\operatorname* { m i n } _ { \boldsymbol { S } } \left. \boldsymbol { H } ^ { \intercal } - \boldsymbol { H } ^ { \intercal } \boldsymbol { S } \right. _ { \boldsymbol { F } } ^ { 2 } + \alpha \left\| \boldsymbol { S } \right\| _ { \boldsymbol { F } } ^ { 2 } ,
+$$
+
+where $S \in \mathbb { R } ^ { N \times N }$ is the graph matrix and $\alpha > 0$ is the trade-off parameter. The frst term is the reconstruction loss and the second term serves as a regularizer to avoid trivial solution. Many other regularizers could also be applied, such as the nuclear norm, sparse $\ell _ { 1 }$ norm [Kang et al., 2020a]. To tackle multi-view data, we can compute a smooth representation $H ^ { v }$ for each view and extend Eq. (4) by introducing a weighting factor to distinguish the contributions of different views
+
+$$
+\operatorname* { m i n } _ { S , \lambda ^ { v } } \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( \begin{array} { l } { H ^ { v \top } - H ^ { v \top } S _ { \textbf { \textit { F } } } ^ { 2 } + \alpha \| S \| _ { F } ^ { 2 } } \end{array} \right) + \sum _ { v = 1 } ^ { V } ( \lambda ^ { v } ) ^ { \gamma } ,
+$$
+
+where $\lambda ^ { v }$ is the weight of $v$ -th view and $\gamma$ is a smooth parameter. Eq. (5) learns a consensus graph $S$ shared by all views. To learn a more discriminative $S$ , we introduce a novel regularizer in this work.
+
+# 3.4 Graph Contrastive Regularizer
+
+Generally, contrastive learning is performed at instance-level and positive/negative pairs are constructed by data augmentation. Most graph contrastive learning methods conduct random corruption on nodes and edges to learn a good node representation. Different from them, each node and its $k$ -nearest neighbors $( k \mathrm { N N } )$ are regarded as positive pairs in this paper. Then, we perform contrastive learning at graph-level by applying a contrastive regularizer on the graph matrix $S$ instead of node features. It can be expressed as
+
+$$
+\mathcal { T } = \sum _ { i = 1 } ^ { N } \sum _ { j \in \mathbb { N } _ { i } ^ { v } } - \log \frac { \exp { ( S _ { i j } ) } } { \sum _ { p \neq i } ^ { N } \exp { ( S _ { i p } ) } } ,
+$$
+
+where $\mathbb { N } _ { i } ^ { v }$ represents the $k$ -nearest neighbors of node $i$ in $v$ -th view. The introduction of Eq. (6) is to draw neighbors close and push non-neighbors apart, so as to boost the quality of graph.
+
+Eventually, our proposed multi-view contrastive graph clustering (MCGC) model can be formulated as
+
+$$
+\operatorname* { m i n } _ { S , \lambda ^ { \nu } } \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( \begin{array} { l } { H ^ { v \top } - H ^ { v \top } S _ { F } ^ { \enspace 2 } + \alpha \sum _ { i = 1 } ^ { N } \sum _ { j \in \mathbb { N } _ { i } ^ { v } } - \log \frac { \exp { ( S _ { i j } ) } } { \sum _ { p \ne i } ^ { N } \exp { ( S _ { i p } ) } } } \end{array} \right) + \sum _ { v = 1 } ^ { V } ( \lambda ^ { v } ) ^ { \top } .
+$$
+
+Different from existing multi-view clustering methods, MCGC explores the holistic information from both multi-view attributes and multiple structural graphs. Furthermore, it constructs a consensus graph from the smooth signal rather than the raw data.
+
+# 3.5 Optimization
+
+There are two groups of variables in Eq. (7) and it’s diffcult to solve them directly. To optimize them, we adopt an alternating optimization strategy, in which we update one variable and fx all others at each time.
+
+# Fix $\lambda ^ { v }$ , Update $S$
+
+Because $\lambda ^ { v }$ is fxed, our objective function can be expressed as
+
+$$
+\operatorname* { m i n } _ { S } \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( \begin{array} { l } { H ^ { v \top } - H ^ { v \top } S _ { \mathrm { \bf ~ { \cal ~ F } ~ } } ^ { \mathrm { \bf ~ 2 } } + \alpha \sum _ { i = 1 } ^ { N } \sum _ { j \in \mathbb { N } _ { i } ^ { v } } - \log \frac { \exp { ( S _ { i j } ) } } { \sum _ { p \ne i } ^ { N } \exp { ( S _ { i p } ) } } } \end{array} \right) .
+$$
+
+$S$ can be elemently solved by gradient descent and its derivative at epoch $t$ can be denoted as
+
+$$
+\nabla _ { 1 } ^ { ( \mathrm { t } ) } + \alpha \nabla _ { 2 } ^ { ( t ) } .
+$$
+
+The frst term is
+
+$$
+\nabla _ { 1 } ^ { ( \mathrm { t } ) } = 2 \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( - \left[ H ^ { v } H ^ { v ^ { \top } } \right] _ { i j } + \left[ H ^ { v } H ^ { v ^ { \top } } S ^ { ( t - 1 ) } \right] _ { i j } \right) .
+$$
+
+Defne $\begin{array} { r } { K ^ { ( \mathrm { t } - 1 ) } = \sum _ { p \neq i } ^ { N } \exp \left( S _ { i p } ^ { ( t - 1 ) } \right) } \end{array}$ and let $n$ be the total number of neighbors (i.e., the neighbors from each graph are all incorporated). Consequently, the second term becomes
+
+$$
+\begin{array} { r } { \nabla _ { 2 } ^ { ( t ) } = \left\{ \begin{array} { l l } { \displaystyle \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( - 1 + \frac { n \exp \left( S _ { i j } ^ { ( t - 1 ) } \right) } { K ^ { ( t - 1 ) } } \right) , \mathrm { ~ i f ~ } j \mathrm { ~ i n ~ } \mathbb { N } _ { i } ^ { v } , } \\ { \displaystyle \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( \frac { n \exp \left( S _ { i j } ^ { ( t - 1 ) } \right) } { K ^ { ( t - 1 ) } } \right) , \mathrm { ~ o t h e r w i s e ~ . } } \end{array} \right. } \end{array}
+$$
+
+Then we adopt Adam optimization strategy [Kingma and Ba, 2015] to update $S$ . To increase the speed of convergence, we initialize $S$ with $S ^ { * }$ , where $S ^ { * }$ is the solution of Eq. (5). Fix $S$ , Update $\lambda ^ { v }$
+
+For each view $v$ , we defne $M ^ { v } = \ H ^ { v \top } - H ^ { v \top } S \ \mathsf { \Pi } _ { F } ^ { 2 } + \alpha \mathcal { I }$ . Then, the loss function is simplifed as
+
+$$
+\operatorname* { m i n } _ { \lambda ^ { v } } \sum _ { v = 1 } ^ { V } \lambda ^ { v } M ^ { v } + \sum _ { v } ^ { V } ( \lambda ^ { v } ) ^ { \gamma } .
+$$
+
+By setting its derivation to zero, we get
+
+$$
+\lambda ^ { v } = \left( \frac { - M ^ { v } } { \gamma } \right) ^ { \frac { 1 } { \gamma - 1 } } .
+$$
+
+We alternatively update $S$ and $\lambda ^ { v }$ until convergence. The complete procedures are outlined in Algorithm 1.
+
+# Algorithm 1 MCGC
+
+Require: adjacency matrix $\widetilde { A } ^ { 1 } , . . . , \widetilde { A } ^ { V }$ , feature $X ^ { 1 } , . . . , X ^ { V }$ , the order of graph fltering $m$ , parameter $\alpha$ , $s$ and $\gamma$ , the number of clusters $c$ .
+
+Ensure: $c$ clusters.   
+1: $\lambda ^ { v } = 1$ ;   
+2: $A ^ { v } = ( D ^ { v } ) ^ { - \frac { 1 } { 2 } } ( \widetilde { A } ^ { v } + I ) ( D ^ { v } ) ^ { - \frac { 1 } { 2 } }$ ;   
+3: $L ^ { v } = I - A ^ { v }$ ;   
+4: Graph fltering by Eq. (4) for each view;   
+5: while convergence condition does not meet do   
+6: Update $S$ in Eq. (9) via Adam;   
+7: for each view do   
+8: Update $\lambda ^ { v }$ in Eq. (13);   
+9: end for   
+10: end while   
+(|S|+|S| >) 11: C = ;   
+12: Clustering on $C$ .
+
+# 4 Experiments
+
+# 4.1 Datasets and Metrics
+
+We evaluate MCGC on fve benchmark datasets, ACM, DBLP, IMDB [Fan et al., 2020], Amazon photos and Amazon computers [Shchur et al., 2018]. The statistical information is shown in Table 1.
+
+ACM: It is a paper network from the ACM dataset. Node attribute features are the elements of a bag-of-words representing of each paper’s keywords. The two graphs are constructed by two types of relationships: "Co-Author" means that two papers are written by the same author and "Co-Subject" suggests that they focus on the same fled.
+
+DBLP: It is an author network from the DBLP dataset. Node attribute features are the elements of a bag-of-words representing of each author’s keywords. Three graphs are derived from the relationships: "Co-Author", "Co-Conference", and "Co-Term", which indicate that two authors have worked together on papers, published papers at the same conference, and published papers with the same terms.
+
+IMDB: It is a movie network from the IMDB dataset. Node attribute features correspond to elements of a bag-of-words representing of each movie. The relationships of being acted by the same actor (Co-actor) and directed by the same director (Co-director) are exploited to construct two graphs.
+
+Amazon photos and Amazon computers: They are segments of the Amazon co-purchase network dataset, in which nodes represent goods and features of each good are bag-of-words of product reviews, the edges means that two goods are purchased together. To have multi-view attributes, the second feature matrix is constructed via cartesian product by following [Cheng et al., 2020].
+
+We adopt four popular clustering metrics: Accuracy (ACC), normalized Mutual Information (NMI), Adjusted Rand Index (ARI), F1 score.
+
+Table 1: The statistical information of datasets.   
+
+<table><tr><td>Dataset</td><td>Nodes</td><td>Features</td><td>Graph and Edges</td><td>Clusters</td></tr><tr><td rowspan="2">ACM</td><td rowspan="2">3.025</td><td rowspan="2">1,830</td><td>Co-Subject (29,281)</td><td rowspan="2">3</td></tr><tr><td>Co-Author (2,210,761)</td></tr><tr><td rowspan="3">DBLP</td><td rowspan="3">4,057</td><td rowspan="3">334</td><td>Co-Author (11,113)</td><td rowspan="3">4</td></tr><tr><td>Co-Conference (5,000,495)</td></tr><tr><td>Co-Term (6,776,335)</td></tr><tr><td rowspan="2">IMDB</td><td rowspan="2">4,780</td><td rowspan="2">1,232</td><td>Co-Actor (98,010)</td><td rowspan="2">3</td></tr><tr><td>Co-Director (21,018)</td></tr><tr><td rowspan="2">Amazon photos</td><td rowspan="2">7,487</td><td>745</td><td rowspan="2">Co-Purchase(119,043)</td><td rowspan="2">8</td></tr><tr><td>7,487</td></tr><tr><td rowspan="2">Amazon computers</td><td rowspan="2">13,381</td><td>767</td><td rowspan="2">Co-Purchase(245,778)</td><td rowspan="2">10</td></tr><tr><td>13,381</td></tr></table>
+
+# 4.2 Experiment Setup
+
+We compare MCGC with multi-view methods as well as single-view methods. LINE [Tang et al., 2015] and GAE [Kipf and Welling, 2016] have been chosen as representatives of single-view methods, and we report the best one among the results from all views. Compared multi-view clustering methods include: PMNE [Liu et al., 2017], RMSC [Xia et al., 2014], SwMC [Nie et al., 2017]. PMNE and SwMC only use structural information while RMSC only exploits attribute features. PMNE uses three strategies to project a multilayer network into a continuous vector space, so we select the best result. MCGC is also compared with other methods that not only explore attribute features but also structural information, i.e., O2MA and O2MAC [Fan et al., 2020], MAGCN [Cheng et al., 2020]. In addition, MCGC is compared with COMPLETER [Lin et al., 2021a] and MVGRL [Hassani and Khasahmadi, 2020] that conduct contrastive learning to learn a common representation shared across features of different views and multiple graphs respectively. For an unbiased comparison, we copy part of the results from [Fan et al., 2020]. Since the neighbors of each node on different views could be different, we also examine another strategy: only use the shared neighbors in the contrastive loss $\begin{array} { r } { \mathbb { N } _ { i } = \bigcap _ { v = 1 } ^ { V } \mathbb { N } _ { i } ^ { v } } \end{array}$ . And our methodect neighbors and ith this appis fxed as ch is marked assince we fnd t $\mathbf { M } \mathbf { C } \mathbf { G } \mathbf { C } ^ { * }$ . During experiments,  little infuence to the $k = 1 0$ $\gamma$ $- 4$ result. According to parameter analysis, we set $m = 2$ , $s = 0 . 5$ , and tune $\alpha$ . All experiments are conducted on the same machine with the Intel(R) Core(TM) i7-8700 3.20GHz CPU, two GeForce GTX 1080 Ti GPUs and 64GB RAM. The implementation of MCGC is public available 1.
+
+# 4.3 Results
+
+All results are shown in Table 2 and Table 3. Compared with single-view method GAE, MCGC improves ACC by more than $9 \%$ , $4 \%$ , $1 9 \%$ on ACM, DBLP, IMDB, respectively. Though using deep neural networks, GAE can not explore the complementarity of views. Compared with PMNE, the ACC, NMI, ARI, F1 are boosted by $1 6 \%$ , $2 0 \%$ , $2 0 \%$ , $1 2 \%$ on average. With respect to LINE, RMSC, SwMC, the improvement is more signifcant. This can be attributed to the exploration of both feature and structure information in MCGC. Although O2MA, O2MAC, and MAGCN capture attributes and structure information, MCGC still outperforms them considerably. Specifcally, MCGC improves O2MAC on average by almost $6 \%$ , $9 \%$ , $1 \bar { 1 } \%$ on ACC, NMI, F1, respectively. With respect to MAGCN, the improvement is more than $2 0 \%$ for all metrics. Compared with contrastive learningbased approaches, our improvement is also impressive. In particular, compared with COMPLETER, the improvement is more than $3 0 \%$ on Amazon datasets, which illustrates that MCGC benefts from the graph structure information. MCGC also enhances the performance of MVGRL by $2 0 \%$ . By comparing the results of MCGC and $\mathbf { M } \mathbf { C } \mathbf { G } \mathbf { C } ^ { * }$ , we can see that the strategy of choosing neighbors does have impact on performance.
+
+Table 2: Results on ACM, DBLP, IMDB.   
+
+<table><tr><td>Method</td><td></td><td>LINE</td><td>GAE</td><td>PMNE</td><td>RMSC</td><td>SwMC</td><td>02MA</td><td>02MAC</td><td>MCGC</td><td>MCGC*</td></tr><tr><td rowspan="4">ACM</td><td>ACC</td><td>0.6479</td><td>0.8216</td><td>0.6936</td><td>0.6315</td><td>0.3831</td><td>0.888</td><td>0.9042</td><td>0.9147</td><td>0.9055</td></tr><tr><td>NMI</td><td>0.3941</td><td>0.4914</td><td>0.4648</td><td>0.3973</td><td>0.4709</td><td>0.6515</td><td>0.6923</td><td>0.7126</td><td>0.6823</td></tr><tr><td>ARI</td><td>0.3433</td><td>0.5444</td><td>0.4302</td><td>0.3312</td><td>0.0838</td><td>0.6987</td><td>0.7394</td><td>0.7627</td><td>0.7385</td></tr><tr><td>F1</td><td>0.6594</td><td>0.8225</td><td>0.6955</td><td>0.5746</td><td>0.018</td><td>0.8894</td><td>0.9053</td><td>0.9155</td><td>0.9062</td></tr><tr><td rowspan="4">DBLP</td><td>ACC</td><td>0.8689</td><td>0.8859</td><td>0.7925</td><td>0.8994</td><td>0.3253</td><td>0.904</td><td>0.9074</td><td>0.9298</td><td>0.9162</td></tr><tr><td>NMI</td><td>0.6676</td><td>0.6925</td><td>0.5914</td><td>0.7111</td><td>0.019</td><td>0.7257</td><td>0.7287</td><td>0.8302</td><td>0.7490</td></tr><tr><td>ARI</td><td>0.6988</td><td>0.741</td><td>0.5265</td><td>0.7647</td><td>0.0159</td><td>0.7705</td><td>0.778</td><td>0.7746</td><td>0.7995</td></tr><tr><td>F1</td><td>0.8546</td><td>0.8743</td><td>0.7966</td><td>0.8248</td><td>0.2808</td><td>0.8976</td><td>0.9013</td><td>0.9252</td><td>0.9112</td></tr><tr><td rowspan="4">IMDB</td><td>ACC</td><td>0.4268</td><td>0.4298</td><td>0.4958</td><td>0.2702</td><td>0.2453</td><td>0.4697</td><td>0.4502</td><td>0.6182</td><td>0.6113</td></tr><tr><td>NMI</td><td>0.0031</td><td>0.0402</td><td>0.0359</td><td>0.3775</td><td>0.0023</td><td>0.0524</td><td>0.0421</td><td>0.1149</td><td>0.1225</td></tr><tr><td>ARI</td><td>-0.009</td><td>0.0473</td><td>0.0366</td><td>50.0054</td><td>0.0017</td><td>0.0753</td><td>0.0564</td><td>0.1833</td><td>0.1811</td></tr><tr><td>F1</td><td>0.287</td><td>0.4062</td><td>0.3906</td><td>0.0018</td><td>0.3164</td><td>0.4229</td><td>0.1459</td><td>0.4401</td><td>0.4512</td></tr></table>
+
+Table 3: Results on Amazon photos and Amazon computers. The ‘-’ means that the method raises out-of-memory problem.   
+
+<table><tr><td>Dataset</td><td colspan="4"> Amazon photos</td><td colspan="4"> Amazon computers</td></tr><tr><td></td><td>ACC</td><td>NMI</td><td>ARI</td><td>F1</td><td>ACC</td><td>NMI</td><td>ARI</td><td>F1</td></tr><tr><td>COMPLETER</td><td>0.3678</td><td>0.2606</td><td>0.0759</td><td>0.3067</td><td>0.2417</td><td>0.1562</td><td>0.0536</td><td>0.1601</td></tr><tr><td>MVGRL</td><td>0.5054</td><td>0.4331</td><td>0.2379</td><td>0.4599</td><td>0.2450</td><td>0.1012</td><td>0.0553</td><td>0.1706</td></tr><tr><td>MAGCN</td><td>0.5167</td><td>0.3897</td><td>0.2401</td><td>0.4736</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>MCGC</td><td>0.7164</td><td>0.6154</td><td>0.4323</td><td>0.6864</td><td>0.5967</td><td>0.5317</td><td>0.3902</td><td>0.5204</td></tr></table>
+
+# 5 Ablation Study
+
+# 5.1 The Effect of Contrastive Loss
+
+By employing the contrastive regularizer, our method pulls neighbors into the same cluster, which decreases intra-cluster variance. To see its effect, we replace $\mathcal { I }$ with a Frobenius term, i.e. Eq. (5). As can be seen from Table 4, the performance falls precipitously without contrastive loss on all datasets. MCGC achieves ACC improvements by $1 6 \%$ , $8 \%$ , $5 \%$ , $i 2 \%$ on DBLP, ACM, IMDB, Amazon datasets, respectively. For other metrics, the contrastive regularizer also enhances the performance signifcantly. Above facts validate that MCGC benefts from the graph contrastive loss.
+
+Table 4: Results of MCGC without contrastive loss.   
+
+<table><tr><td></td><td>Datasets</td><td>ACM</td><td>DBLP</td><td>IMDB</td><td>Amazon photos</td><td>Amazon computers</td></tr><tr><td rowspan="2">ACC</td><td>MCGC</td><td>0.9147</td><td>0.9298</td><td>0.6182</td><td>0.7164</td><td>0.5967</td></tr><tr><td>MCGC w/o J</td><td>0.8334</td><td>0.7658</td><td>0.5636</td><td>0.5882</td><td>0.4662</td></tr><tr><td rowspan="2">NMI</td><td>MCGC</td><td>0.7126</td><td>0.8302</td><td>0.1149</td><td>0.6154</td><td>0.5317</td></tr><tr><td>MCGC w/o J</td><td>0.5264</td><td>0.4621</td><td>0.0707</td><td>0.5372</td><td>0.3988</td></tr><tr><td rowspan="2">ARI</td><td>MCGC</td><td>0.7627</td><td>0.7746</td><td>0.1833</td><td>0.4323</td><td>0.3902</td></tr><tr><td>MCGC w/o J</td><td>0.5779</td><td>0.4949</td><td>0.1451</td><td>0.2640</td><td>0.1745</td></tr><tr><td rowspan="2">F1</td><td>MCGC</td><td>0.9155</td><td>0.9252</td><td>0.4401</td><td>0.6864</td><td>0.5204</td></tr><tr><td>MCGC w/o J</td><td>0.8313</td><td>0.7601</td><td>0.4444</td><td>0.5437</td><td>0.3678</td></tr></table>
+
+Table 5: Results in various views of MCGC on ACM and Amazon photos. $G _ { 1 }$ and $G _ { 2 }$ denote the graphs in different views.   
+
+<table><tr><td>Dataset</td><td colspan="3">ACM</td><td colspan="3">Amazon photos</td></tr><tr><td></td><td>G1,X</td><td>G2,X</td><td>G1,G2, X</td><td>X1,G</td><td>X²,G</td><td>X1, X²,G</td></tr><tr><td>ACC</td><td>0.9088</td><td>0.8152</td><td>0.9147</td><td>0.4433</td><td>0.6935</td><td>0.7164</td></tr><tr><td>NMI</td><td>0.6929</td><td>0.4656</td><td>0.7126</td><td>0.3519</td><td>0.5976</td><td>0.6154</td></tr><tr><td>ARI</td><td>0.7470</td><td>0.5229</td><td>0.7627</td><td>0.1572</td><td>0.4291</td><td>0.4323</td></tr><tr><td>F1</td><td>0.9097</td><td>0.8184</td><td>0.9155</td><td>0.3675</td><td>0.6734</td><td>0.6864</td></tr></table>
+
+# 5.2 The Effect of Multi-View Learning
+
+To demonstrate the effect of multi-view learning, we evaluate the performance of the following single-view model
+
+$$
+\operatorname* { m i n } _ { S } ~ H ^ { \top } - H ^ { \top } S ~ _ { F } ^ { 2 } + \alpha \sum _ { i = 1 } ^ { N } \sum _ { j \in \mathbb { N } _ { i } } - \log \frac { \exp ( S _ { i j } ) } { \displaystyle \sum _ { p \neq i } ^ { N } \exp ( S _ { i p } ) } .
+$$
+
+Taking ACM and Amazon photos as examples, we report the clustering performance of various scenarios in Table 5. We can observe that the best performance is always achieved when all views are incorporated. In addition, we also see that the performance varies a lot for different views. This justifes the necessity of $\lambda ^ { v }$ in Eq. (7). Therefore, it is benefcial to explore the complementarity of multi-view information.
+
+# 5.3 The Effect of Graph Filtering
+
+To understand the contribution of graph fltering, we conduct another group of experiments. Without graph fltering, our objective function becomes
+
+$$
+\operatorname* { m i n } _ { S , \lambda ^ { \nu } } \sum _ { v = 1 } ^ { V } \lambda ^ { v } \left( \begin{array} { l } { { X ^ { v \top } - X ^ { v \top } S _ { F } ^ { \enspace 2 } + \alpha \sum _ { i = 1 } ^ { N } \sum _ { j \in \mathbb { N } _ { i } ^ { v } } - \log \frac { \exp { ( S _ { i j } ) } } { \sum _ { p \neq i } ^ { N } \exp { ( S _ { i p } ) } } } } \end{array} \right) + \sum _ { v = 1 } ^ { V } ( \lambda ^ { v } ) ^ { \top } .
+$$
+
+We denote this model as MCGC-. The results of MCGC- are shown in Table 6. With respect to MCGC, ACC on ACM, DBLP, IMDB drops by $0 . 8 \%$ , $1 . 3 \%$ , $0 . 8 \%$ , respectively. This indicates that graph fltering makes a positive impact on our model. For other metrics, MCGC also outperforms MCGC- in most cases.
+
+Table 6: The results of MCGC- (without graph fltering).   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">ACM</td><td colspan="2">DBLP</td><td colspan="2">IMDB</td></tr><tr><td>MCGC</td><td>MCGC-</td><td>MCGC</td><td>MCGC-</td><td>MCGC</td><td>MCGC-</td></tr><tr><td>ACC</td><td>0.9147</td><td>0.9061</td><td>0.9298</td><td>0.9162</td><td>0.6182</td><td>0.6109</td></tr><tr><td>NMI</td><td>0.7126</td><td>0.6974</td><td>0.8302</td><td>0.7490</td><td>0.1149</td><td>0.1219</td></tr><tr><td>ARI</td><td>0.7627</td><td>0.7439</td><td>0.7746</td><td>0.7995</td><td>0.1833</td><td>0.1804</td></tr><tr><td>F1</td><td>0.9155</td><td>0.9057</td><td>0.9252</td><td>0.9112</td><td>0.4401</td><td>0.4509</td></tr></table>
+
+# 6 Parameter Analysis
+
+Firstly, two parameters $m$ and $s$ are applied in graph fltering. Taking ACM as an example, we show their infuence on performance by setting $m = [ 1 , 2 , 3 , 4 , 5 ]$ , $s = [ 0 . 0 1 , 0 . 1 , 0 . 3 , 0 . 5 , 1 , 3 , 5 , 1 0 ]$ in Fig. 1. It can be seen that MCGC achieves a reasonable performance for a small $m$ and $s$ .
+
+![](images/7da9718be016c75436ecafa088b545a6d5e6f125dce330ec913b2a359d0191ef.jpg)  
+Figure 1: Sensitivity analysis of parameters $m$ and $s$ on ACM.
+
+Therefore, we set $m = 2$ and $s = 0 . 5$ in all experiments. Afterwards, we tune the trade-off parameter $\alpha = [ 1 0 ^ { - 3 } , 0 . 1 , 1 , 1 0 , 1 0 ^ { 2 } , 1 0 ^ { 3 } ]$ . As shown in Fig. 2, our method is not sensitive to $\alpha$ , which enhances the practicality in real-world applications. In addition, we plot the objective variation of Eq. (7) in Fig. 3. As observed from this fgure, our method converges quickly.
+
+![](images/8d8a25b5eb837b1a8d4db5b5c1cb855e520623dd5c7d7c3a12d414126acc0a88.jpg)  
+Figure 2: Sensitivity analysis of parameter $\alpha$ on ACM (a-d), DBLP (e-h), IMDB (i-l).
+
+![](images/35cbf232f72df6ebb073fa49def909e8d0d4b9f9a8c1efd4a76338e314b32358.jpg)  
+Figure 3: The evolution of objective function.
+
+# 7 Conclusion
+
+Multi-view graph clustering is till at a nascent stage with many challenges remained unsolved. In this paper, we propose a novel method (MCGC) to learn a consensus graph by exploiting not only attribute content but also graph structure information. Particularly, graph fltering is introduced to flter out noisy components and a contrastive regularizer is employed to further enhance the quality of learned graph. Experimental results on multi-view attributed graph datasets have shown the superior performance of our method. This study demonstrates that it is possible for shallow model to beat deep learning methods facing the systematic use of complex deep neural networks. Graph learning is crucial to more and more tasks and applications. Just like other methods that learn from data, brings the risk of learning biases and perpetuating them in the form of decisions. Thus our method should be deployed with careful consideration of any potential underlying biases in the data. One potential limitation of our approach is that it could take a lot of memory if the data contain too many nodes. This is because the size of learned graph is $N \times N$ . Research on large scale network is left for future work.
+
+# Acknowledgments and Disclosure of Funding
+
+This paper was in part supported by the Natural Science Foundation of China (Nos.61806045, U19A2059).
+
+# References
+
+Maria Brbic and Ivica Kopriva. Multi-view low-rank sparse subspace clustering. ´ Pattern Recognition, 73:247–258, 2018.
+
+Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pages 1597–1607. PMLR, 2020a.
+
+Yu Chen, Lingfei Wu, and Mohammed Zaki. Iterative deep graph learning for graph neural networks: Better and robust node embeddings. Advances in Neural Information Processing Systems, 33, 2020b.
+
+Jiafeng Cheng, Qianqian Wang, Zhiqiang Tao, Deyan Xie, and Quanxue Gao. Multi-view attribute graph convolution networks for clustering. IJCAI, 2020.
+
+Sumit Chopra, Raia Hadsell, and Yann LeCun. Learning a similarity metric discriminatively, with application to face verifcation. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), volume 1, pages 539–546. IEEE, 2005.
+
+Shaohua Fan, Xiao Wang, Chuan Shi, Emiao Lu, Ken Lin, and Bai Wang. One2multi graph autoencoder for multi-view graph clustering. In Proceedings of The Web Conference 2020, pages 3070–3076, 2020.
+
+Michael Gutmann and Aapo Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artifcial Intelligence and Statistics, pages 297–304. JMLR Workshop and Conference Proceedings, 2010.
+
+Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pages 1735–1742. IEEE, 2006.
+
+Kaveh Hassani and Amir Hosein Khasahmadi. Contrastive multi-view representation learning on graphs. In International Conference on Machine Learning, pages 4116–4126. PMLR, 2020.
+
+Zhenyu Huang, Joey Tianyi Zhou, Xi Peng, Changqing Zhang, Hongyuan Zhu, and Jiancheng Lv. Multi-view spectral clustering network. In IJCAI, pages 2563–2569, 2019.
+
+Zhao Kang, Xiao Lu, Yiwei Lu, chong Peng, Wenyu Chen, and Zenglin Xu. Structure learning with similarity preserving. Neural Networks, 129:138–148, 2020a.
+
+Zhao Kang, Haiqi Pan, Steven C.H. Hoi, and Zenglin Xu. Robust graph learning from noisy data. IEEE Transactions on Cybernetics, 50(5):1833–1843, 2020b.
+
+Zhao Kang, Xinjia Zhao, Shi, chong Peng, Hongyuan Zhu, Joey Tianyi Zhou, Xi Peng, Wenyu Chen, and Zenglin Xu. Partition level multiview subspace clustering. Neural Networks, 122:279–288, 2020c.
+
+Zhao Kang, Zhiping Lin, Xiaofeng Zhu, and Wenbo Xu. Structured graph learning for scalable subspace clustering: From single-view to multi-view. IEEE Transactions on Cybernetics, 2021. doi: 10.1109/TCYB.2021.3061660.
+
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015.
+
+Thomas N Kipf and Max Welling. Variational graph auto-encoders. In NIPS Bayesian Deep Learning Workshop, 2016.
+
+Yunfan Li, Peng Hu, Zitao Liu, Dezhong Peng, Joey Tianyi Zhou, and Xi Peng. Contrastive clustering. In AAAI 2021, volume 35, Feb. 2021.
+
+Yijie Lin, Yuanbiao Gou, Zitao Liu, Boyun Li, Jiancheng Lv, and Xi Peng. Completer: Incomplete multi-view clustering via contrastive prediction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2021a.
+
+Zhiping Lin and Zhao Kang. Graph flter-based multi-view attributed graph clustering. In Proceedings of the 30th International Joint Conference on Artifcial Intelligence, IJCAI, pages 19–26, 2021.
+
+Zhiping Lin, Zhao Kang, Lizong Zhang, and Ling Tian. Multi-view attributed graph clustering. IEEE Transactions on Knowledge and Data Engineering, 2021b. doi: 10.1109/TKDE.2021.3101227.
+
+Changshu Liu, Liangjian Wen, Zhao Kang, Guangchun Luo, and Ling Tian. Self-supervised consensus representation learning for attributed graph. In Proceedings of the 29th ACM International Conference on Multimedia, 2021.
+
+Weiyi Liu, Pin-Yu Chen, Sailung Yeung, Toyotaro Suzumura, and Lingli Chen. Principled multilayer network embedding. In 2017 IEEE International Conference on Data Mining Workshops (ICDMW), pages 134–141. IEEE, 2017.
+
+Juncheng Lv, Zhao Kang, Xiao Lu, and Zenglin Xu. Pseudo-supervised deep subspace clustering. IEEE Transactions on Image Processing, 30:5252–5263, 2021.
+
+Zhengrui Ma, Zhao Kang, Guangchun Luo, Ling Tian, and Wenyu Chen. Towards clusteringfriendly representations: Subspace clustering via graph fltering. In Proceedings of the 28th ACM International Conference on Multimedia, pages 3081–3089, 2020.
+
+Feiping Nie, Jing Li, Xuelong Li, et al. Self-weighted multiview clustering with multiple graphs. In IJCAI, pages 2564–2570, 2017.
+
+Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, Lina Yao, and Chengqi Zhang. Adversarially regularized graph autoencoder for graph embedding. In Proceedings of the 27th International Joint Conference on Artifcial Intelligence, pages 2609–2615, 2018.
+
+Jiwoong Park, Minsik Lee, Hyung Jin Chang, Kyuewang Lee, and Jin Young Choi. Symmetric graph convolutional autoencoder for unsupervised graph representation learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6519–6528, 2019.
+
+Xi Peng, Zhenyu Huang, Jiancheng Lv, Hongyuan Zhu, and Joey Tianyi Zhou. Comic: Multi-view clustering without parameter selection. In International Conference on Machine Learning, pages 5092–5101. PMLR, 2019.
+
+Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Günnemann. Pitfalls of graph neural network evaluation. arXiv preprint arXiv:1811.05868, 2018.
+
+Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large-scale information network embedding. In Proceedings of the 24th international conference on world wide web, pages 1067–1077, 2015.
+
+Chun Wang, Shirui Pan, Guodong Long, Xingquan Zhu, and Jing Jiang. Mgae: Marginalized graph autoencoder for graph clustering. In Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, pages 889–898, 2017.
+
+Hao Wang, Yan Yang, and Bing Liu. Gmc: Graph-based multi-view clustering. IEEE Transactions on Knowledge and Data Engineering, 32(6):1116–1129, 2019.
+
+Rongkai Xia, Yan Pan, Lei Du, and Jian Yin. Robust multi-view spectral clustering via low-rank and sparse decomposition. In Proceedings of the AAAI conference on artifcial intelligence, volume 28, 2014.
+
+Hongming Zhang, Liwei Qiu, Lingling Yi, and Yangqiu Song. Scalable multiplex network embedding. In IJCAI, volume 18, pages 3082–3088, 2018.
+
+Huasong Zhong, Chong Chen, Zhongming Jin, and Xian-Sheng Hua. Deep robust clustering by contrastive learning. arXiv preprint arXiv:2008.03030, 2020.
+
+Meiqi Zhu, Xiao Wang, Chuan Shi, Houye Ji, and Peng Cui. Interpreting and unifying graph neural networks with an optimization framework. In Proceedings of the Web Conference 2021, pages 1215–1226, 2021.

parse/train/NlB8_hXkbby/NlB8_hXkbby_content_list.json

@@ -0,0 +1,1629 @@
+[
+    {
+        "type": "text",
+        "text": "Multi-view Contrastive Graph Clustering ",
+        "text_level": 1,
+        "bbox": [
+            245,
+            122,
+            753,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Erlin Pan, Zhao Kang∗ School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu, China wujisixsix6@gmail.com zkang@uestc.edu.cn ",
+        "bbox": [
+            250,
+            200,
+            748,
+            257
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            292,
+            535,
+            309
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "With the explosive growth of information technology, multi-view graph data have become increasingly prevalent and valuable. Most existing multi-view clustering techniques either focus on the scenario of multiple graphs or multi-view attributes. In this paper, we propose a generic framework to cluster multi-view attributed graph data. Specifcally, inspired by the success of contrastive learning, we propose multi-view contrastive graph clustering (MCGC) method to learn a consensus graph since the original graph could be noisy or incomplete and is not directly applicable. Our method composes of two key steps: we frst flter out the undesirable highfrequency noise while preserving the graph geometric features via graph fltering and obtain a smooth representation of nodes; we then learn a consensus graph regularized by graph contrastive loss. Results on several benchmark datasets show the superiority of our method with respect to state-of-the-art approaches. In particular, our simple approach outperforms existing deep learning-based methods. ",
+        "bbox": [
+            233,
+            324,
+            766,
+            503
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            529,
+            310,
+            545
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "An attributed graph contains of node features and edges characterizing the pairwise relations between nodes. It is a natural and effcient representation for many real-world data [Liu et al., 2021]. For example, social network users have their own profles and the topological graph refects their social relationships. Different from most classical clustering methods like K-means and hierarchical clustering which only focus on Euclidean data, graph clustering divides unlabeled nodes of graph into clusters. Typical graph clustering methods frst learn a good representation of graph and then apply a classical clustering method upon the embeddings. For example, large-scale information network embedding (LINE) [Tang et al., 2015] is a popular graph representation learning method, which can preserve both local and global information and scale up easily to large-scale networks. To incorporate node features and graph structure information, graph autoencoder (GAE) [Kipf and Welling, 2016] employs a graph convolution network (GCN) encoder and achieves signifcant performance improvement. The real-life data are often collected from various sources or obtained from different extractors, thus are naturally represented by different features or views [Kang et al., 2021, 2020c]. Each view could be noisy and incomplete, but important factors, such as geometry and semantics, tend to be shared among all views. Therefore, features and graphs of different views are complementary, which implies that it’s paramount to integrate all features and graphs of diverse views to improve the performance of clustering task. ",
+        "bbox": [
+            174,
+            560,
+            825,
+            795
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Numerous graph-based multi-view clustering methods have been developed to capture the consensus information shared by different views in the literature. Graph-based multi-view clustering constructs a graph for each view and fuses them based on a weighting mechanism [Wang et al., 2019]. Multi-view spectral clustering network [Huang et al., 2019] learns a discriminative representation by using a deep metric learning network. These methods are developed for feature matrix and can not handle graph data. To directly process graph data, some representative methods have also been proposed. ",
+        "bbox": [
+            174,
+            795,
+            825,
+            877
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Scalable multiplex network embedding (MNE) [Zhang et al., 2018] is a scalable multi-view network embedding model, which learns multiple relations by a unifed network embedding framework. Principled multilayer network embedding (PMNE) [Liu et al., 2017] proposes three strategies (“network aggregation”, “results aggregation”, and “layer co-analysis”) to project a multilayer network into a continuous vector space. Nevertheless, they fail to explore the feature information [Lin and Kang, 2021]. ",
+        "bbox": [
+            174,
+            92,
+            825,
+            174
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recently, based on GCN, One2Multi graph autoencoder clustering (O2MA) framework [Fan et al., 2020] and multi-view attribute GCNs for clustering (MAGCN) [Cheng et al., 2020] achieve superior performance on graph clustering. O2MA introduces a graph autoencoder to learn node embeddings based on one informative graph and reconstruct multiple graphs. However, the shared feature representation of multiple graphs could be incomplete because O2MA only takes into account the informative view selected by modularity. MAGCN exploits the abundant information of all views and adopts a cross-view consensus learning by enforcing the representations of different views to be as similar as possible. Nevertheless, O2MA targets for multiple graphs while MAGCN mainly solves graph structured data of multi-view attributes. They are not directly applicable to multiple graphs data with multi-view attributes. Therefore, the research of multi-view graph clustering is at the initial stage and more dedicated efforts are pressingly needed. ",
+        "bbox": [
+            174,
+            174,
+            825,
+            325
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a generic framework of clustering on attributed graph data with multiview features and multiple topological graphs, denoted by Multi-view Contrastive Graph Clustering (MCGC). To be exact, MCGC learns a new consensus graph by exploring the holistic information among various attributes and graphs rather than utilizing the initial graph. The reason of introducing graph learning is that the initial graph is often noisy or incomplete, which leads to suboptimal solutions [Chen et al., 2020b, Kang et al., 2020b]. A contrastive loss is adopted as regularization to make the consensus graph clustering-friendly. Moreover, we implement on the smooth representation rather than raw data. The contributions of this work could be summarized as follows: ",
+        "bbox": [
+            174,
+            327,
+            825,
+            435
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• To boost the quality of learned graph, we propose a novel contrastive loss at graph-level. It is capable of drawing similar nodes close and pushing those dissimilar ones apart.   \n• We propose a generic clustering framework to handle multilayer graphs with multi-view attributes, which contains graph fltering, graph learning, and graph contrastive components. The graph fltering is simple and effcient to obtain a smoothed representation; the graph learning is utilized to generate the consensus graph with an adaptive weighting mechanism for different views.   \n• Our method achieves state-of-the-art performance compared with shallow methods and deep methods on fve benchmark datasets. ",
+        "bbox": [
+            217,
+            448,
+            825,
+            583
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            603,
+            321,
+            621
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 Multi-view Clustering ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            635,
+            367,
+            650
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Large quantities of multi-view clustering methods have been proposed in the last decades. Multi-view low-rank sparse subspace clustering [Brbic and Kopriva, 2018] obtains a joint subspace representation´ across all views by learning an affnity matrix constrained by sparsity and low-rank constraint. Cross-view matching clustering (COMIC) [Peng et al., 2019] enforces the graphs to be as similar as possible instead of the representation in the latent space. Robust multi-view spectral clustering (RMSC) [Xia et al., 2014] uses a shared low-rank transition probability matrix derived from each single view as input to the standard Markov chain method for clustering. These methods are designed for feature matrix and try to learn a graph from data. To directly cluster multiple graphs, self-weighted multi-view clustering (SwMC) [Nie et al., 2017] method learns a shared graph from multiple graphs by using a novel weighting strategy. Above methods are suitable for graph or feature data only, and can not simultaneously explore attributes and graph structure. As previously discussed, O2MA [Fan et al., 2020] and MAGCN [Cheng et al., 2020] can handle attributed graph, but they are not direct applicable to generic multi-view graph data. ",
+        "bbox": [
+            173,
+            661,
+            825,
+            840
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.2 Contrastive Clustering ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
+            372,
+            872
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Due to its impressive performance in many tasks, contrastive learning has become the most hot topic in unsupervised learning. Its motivation is to maximize the similarity of positive pairs and distance of negative pairs [Hadsell et al., 2006]. Generally, the positive pair are composed of data augmentations of the same instance while those of different instances are regarded as negatives. Several loss functions have been proposed, such as the triplet loss [Chopra et al., 2005], the noise contrastive estimation (NCE) loss [Gutmann and Hyvärinen, 2010], the normalized temperature-scaled cross entropy loss (NT-Xent) [Chen et al., 2020a]. Deep robust clustering turns maximizing mutual information into minimizing contrastive loss and achieves signifcant improvement after applying contrastive learning to decrease intra-class variance [Zhong et al., 2020]. Contrastive clustering develops a dual contrastive learning framework, which conducts contrastive learning at instance-level as well as cluster-level [Li et al., 2021]. As a result, it produces a representation that facilitates the downstream clustering task. Unfortunately, theses method can only handle single-view data. ",
+        "bbox": [
+            173,
+            883,
+            821,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            90,
+            825,
+            229
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, by combining reconstruction, cross-view contrastive learning, and cross-view dual prediction, incomplete multi-view clustering via contrastive prediction (COMPLETER) [Lin et al., 2021a] performs data recovery and consistency learning of incomplete multi-view data simultaneously. It also obtains promising performance on complete multi-view data. However, it can not deal with graph data. On the other hand, contrastive multi-view representation learning on graphs (MVGRL) [Hassani and Khasahmadi, 2020] method is proposed, which performs representation learning by contrasting two diffusion matrices transformed from the adjacency matrix. It reports better performance than variational GAE (VGAE) [Kipf and Welling, 2016], marginalized GAE (MGAE) [Wang et al., 2017], adversarially regularized GAE (ARGA) and VGAE (ARVGA) [Pan et al., 2018], and GALA [Park et al., 2019]. Different from MVGRL, our contrastive regularizer is directly applied on learned graph. ",
+        "bbox": [
+            173,
+            231,
+            826,
+            367
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Methodology ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            386,
+            313,
+            404
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Notation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            416,
+            274,
+            431
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Defne the multi-view graph data as $G = \\{ \\mathcal { V } , E _ { 1 } , . . . , E _ { V } , X ^ { 1 } , . . . , X ^ { V } \\}$ , where $\\nu$ represents the sets of $N$ nodes, $e _ { i j } \\in E _ { v }$ denotes the relationship between node $i$ and node $j$ in the $v$ -th view, $X ^ { v } = \\{ x _ { 1 } ^ { v } , . . . , x _ { N } ^ { v } \\} ^ { \\top }$ he feature matrix. Adjacency matrices represent the degree matrices in various $\\{ \\widetilde { A } ^ { v } \\} _ { v = 1 } ^ { V }$ characterize the initial e normalized adjacency $\\{ D ^ { v } \\} _ { v = 1 } ^ { V }$ matrix $A ^ { v } = ( D ^ { v } ) ^ { - \\frac { 1 } { 2 } } ( \\widetilde { A } ^ { v } + I ) ( D ^ { v } ) ^ { - \\frac { 1 } { 2 } }$ and the corresponding graph laplacian $L ^ { v } = I - A ^ { v }$ . ",
+        "bbox": [
+            173,
+            440,
+            826,
+            520
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 Graph Filtering ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            534,
+            325,
+            549
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A feature matrix $\\boldsymbol { X } \\in \\mathbb { R } ^ { N \\times d }$ of $N$ nodes can be treated as $d N$ -dimensional graph signals. A natural signal should be smooth on nearby nodes in term of the underlying graph. The smoothed signals $H$ can be achieved by solving the following optimization problem [Zhu et al., 2021, Lin et al., 2021b] ",
+        "bbox": [
+            174,
+            559,
+            825,
+            603
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8bd2abfbe3e405ad72ed1da927b4187d88cb33044d36e960f4c9da882b92128d.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { H } \\| H - X \\| _ { F } ^ { 2 } + s \\operatorname { T r } \\left( \\mathrm { H } ^ { \\top } \\mathrm { L H } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            382,
+            604,
+            611,
+            628
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $s > 0$ is a balance parameter and $L$ is the laplacian matrix associated with $X$ . $H$ can be obtained by taking the derivative of Eq. (1) w.r.t. $H$ and setting it to zero, which yields ",
+        "bbox": [
+            173,
+            631,
+            823,
+            660
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a5995c047849110614ac7cbdc661583e3796390fee249de387368c64b6f74ea.jpg",
+        "text": "$$\nH = ( I + s L ) ^ { - 1 } X .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            431,
+            661,
+            566,
+            680
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To get rid of matrix inversion, we approximate $H$ by its frst-order Taylor series expansion, i.e., $H = ( I - s L ) X$ . Generally, $m$ -th order graph fltering can be written as ",
+        "bbox": [
+            173,
+            683,
+            823,
+            710
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6eda8ce22f17412edd14bf68b8f746916b28bece8a9b21729f0aedb9d567c543.jpg",
+        "text": "$$\nH = ( I - s L ) ^ { m } X ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            433,
+            713,
+            563,
+            731
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $m$ is a non-negative integer. Graph fltering can flter out undesirable high-frequency noise while preserving the graph geometric features. ",
+        "bbox": [
+            173,
+            733,
+            823,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.3 Graph Learning ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            776,
+            328,
+            792
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Since real-world graph is often noisy or incomplete, which will degrade the downstream task performance if it is directly applied. Thus we learn an optimized graph $S$ from the smoothed representation $H$ . This can be realized based on the self-expression property of data, i.e., each data point can be represented by a linear combination of other data samples [Lv et al., 2021, Ma et al., 2020]. And the combination coeffcients represent the relationships among data points. The objective function on single-view data can be mathematically formulated as ",
+        "bbox": [
+            173,
+            801,
+            825,
+            886
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dd5418b191b05a988da2f47e7d2dc38140941e27e530c52c4b3c016a52989118.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\boldsymbol { S } } \\left. \\boldsymbol { H } ^ { \\intercal } - \\boldsymbol { H } ^ { \\intercal } \\boldsymbol { S } \\right. _ { \\boldsymbol { F } } ^ { 2 } + \\alpha \\left\\| \\boldsymbol { S } \\right\\| _ { \\boldsymbol { F } } ^ { 2 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            385,
+            888,
+            609,
+            915
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $S \\in \\mathbb { R } ^ { N \\times N }$ is the graph matrix and $\\alpha > 0$ is the trade-off parameter. The frst term is the reconstruction loss and the second term serves as a regularizer to avoid trivial solution. Many other regularizers could also be applied, such as the nuclear norm, sparse $\\ell _ { 1 }$ norm [Kang et al., 2020a]. To tackle multi-view data, we can compute a smooth representation $H ^ { v }$ for each view and extend Eq. (4) by introducing a weighting factor to distinguish the contributions of different views ",
+        "bbox": [
+            173,
+            89,
+            826,
+            161
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/684d4efe19f9cbfb299f36b33338922d2871f27a9664b5cba14eeca3dab51fc2.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { S , \\lambda ^ { v } } \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( \\begin{array} { l } { H ^ { v \\top } - H ^ { v \\top } S _ { \\textbf { \\textit { F } } } ^ { 2 } + \\alpha \\| S \\| _ { F } ^ { 2 } } \\end{array} \\right) + \\sum _ { v = 1 } ^ { V } ( \\lambda ^ { v } ) ^ { \\gamma } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            161,
+            691,
+            205
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda ^ { v }$ is the weight of $v$ -th view and $\\gamma$ is a smooth parameter. Eq. (5) learns a consensus graph $S$ shared by all views. To learn a more discriminative $S$ , we introduce a novel regularizer in this work. ",
+        "bbox": [
+            173,
+            207,
+            823,
+            234
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.4 Graph Contrastive Regularizer ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            250,
+            429,
+            265
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Generally, contrastive learning is performed at instance-level and positive/negative pairs are constructed by data augmentation. Most graph contrastive learning methods conduct random corruption on nodes and edges to learn a good node representation. Different from them, each node and its $k$ -nearest neighbors $( k \\mathrm { N N } )$ are regarded as positive pairs in this paper. Then, we perform contrastive learning at graph-level by applying a contrastive regularizer on the graph matrix $S$ instead of node features. It can be expressed as ",
+        "bbox": [
+            173,
+            273,
+            825,
+            358
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ccb3c393c7041a171f9bf0326222b1d8f54be35f5e7ff55996fbb9f8dfe73ca9.jpg",
+        "text": "$$\n\\mathcal { T } = \\sum _ { i = 1 } ^ { N } \\sum _ { j \\in \\mathbb { N } _ { i } ^ { v } } - \\log \\frac { \\exp { ( S _ { i j } ) } } { \\sum _ { p \\neq i } ^ { N } \\exp { ( S _ { i p } ) } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            359,
+            622,
+            405
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { N } _ { i } ^ { v }$ represents the $k$ -nearest neighbors of node $i$ in $v$ -th view. The introduction of Eq. (6) is to draw neighbors close and push non-neighbors apart, so as to boost the quality of graph. ",
+        "bbox": [
+            173,
+            406,
+            823,
+            434
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Eventually, our proposed multi-view contrastive graph clustering (MCGC) model can be formulated as ",
+        "bbox": [
+            176,
+            435,
+            823,
+            460
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4ddb38e8d549d3f13e0d80707c561371a426b6dfee27678409a2318d94c183be.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { S , \\lambda ^ { \\nu } } \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( \\begin{array} { l } { H ^ { v \\top } - H ^ { v \\top } S _ { F } ^ { \\enspace 2 } + \\alpha \\sum _ { i = 1 } ^ { N } \\sum _ { j \\in \\mathbb { N } _ { i } ^ { v } } - \\log \\frac { \\exp { ( S _ { i j } ) } } { \\sum _ { p \\ne i } ^ { N } \\exp { ( S _ { i p } ) } } } \\end{array} \\right) + \\sum _ { v = 1 } ^ { V } ( \\lambda ^ { v } ) ^ { \\top } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            212,
+            460,
+            782,
+            511
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Different from existing multi-view clustering methods, MCGC explores the holistic information from both multi-view attributes and multiple structural graphs. Furthermore, it constructs a consensus graph from the smooth signal rather than the raw data. ",
+        "bbox": [
+            173,
+            511,
+            825,
+            553
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.5 Optimization ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            568,
+            305,
+            583
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "There are two groups of variables in Eq. (7) and it’s diffcult to solve them directly. To optimize them, we adopt an alternating optimization strategy, in which we update one variable and fx all others at each time. ",
+        "bbox": [
+            173,
+            593,
+            826,
+            635
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Fix $\\lambda ^ { v }$ , Update $S$ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            636,
+            294,
+            648
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Because $\\lambda ^ { v }$ is fxed, our objective function can be expressed as ",
+        "bbox": [
+            174,
+            648,
+            588,
+            662
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc0467eb902d9df919e20923c988c8a538cd2402ddf2ce1f6bf96159d7726912.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { S } \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( \\begin{array} { l } { H ^ { v \\top } - H ^ { v \\top } S _ { \\mathrm { \\bf ~ { \\cal ~ F } ~ } } ^ { \\mathrm { \\bf ~ 2 } } + \\alpha \\sum _ { i = 1 } ^ { N } \\sum _ { j \\in \\mathbb { N } _ { i } ^ { v } } - \\log \\frac { \\exp { ( S _ { i j } ) } } { \\sum _ { p \\ne i } ^ { N } \\exp { ( S _ { i p } ) } } } \\end{array} \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            664,
+            740,
+            713
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "$S$ can be elemently solved by gradient descent and its derivative at epoch $t$ can be denoted as ",
+        "bbox": [
+            176,
+            714,
+            782,
+            729
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3fc838e73de5eaf3891bea3de9d5a174b17cf582c7e18b7234d7fc1520aebef0.jpg",
+        "text": "$$\n\\nabla _ { 1 } ^ { ( \\mathrm { t } ) } + \\alpha \\nabla _ { 2 } ^ { ( t ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            449,
+            729,
+            547,
+            751
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The frst term is ",
+        "bbox": [
+            173,
+            751,
+            281,
+            765
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffaeea7c1b989e22e3ba7c84b96171f4e890f34a8c007f0063380631df67488b.jpg",
+        "text": "$$\n\\nabla _ { 1 } ^ { ( \\mathrm { t } ) } = 2 \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( - \\left[ H ^ { v } H ^ { v ^ { \\top } } \\right] _ { i j } + \\left[ H ^ { v } H ^ { v ^ { \\top } } S ^ { ( t - 1 ) } \\right] _ { i j } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            763,
+            692,
+            808
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Defne $\\begin{array} { r } { K ^ { ( \\mathrm { t } - 1 ) } = \\sum _ { p \\neq i } ^ { N } \\exp \\left( S _ { i p } ^ { ( t - 1 ) } \\right) } \\end{array}$ and let $n$ be the total number of neighbors (i.e., the neighbors from each graph are all incorporated). Consequently, the second term becomes ",
+        "bbox": [
+            173,
+            809,
+            825,
+            847
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fffff56607d588afe40d9f4a451402f296fa7a413671fcf03ebb7580908f7226.jpg",
+        "text": "$$\n\\begin{array} { r } { \\nabla _ { 2 } ^ { ( t ) } = \\left\\{ \\begin{array} { l l } { \\displaystyle \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( - 1 + \\frac { n \\exp \\left( S _ { i j } ^ { ( t - 1 ) } \\right) } { K ^ { ( t - 1 ) } } \\right) , \\mathrm { ~ i f ~ } j \\mathrm { ~ i n ~ } \\mathbb { N } _ { i } ^ { v } , } \\\\ { \\displaystyle \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( \\frac { n \\exp \\left( S _ { i j } ^ { ( t - 1 ) } \\right) } { K ^ { ( t - 1 ) } } \\right) , \\mathrm { ~ o t h e r w i s e ~ . } } \\end{array} \\right. } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            847,
+            671,
+            916
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then we adopt Adam optimization strategy [Kingma and Ba, 2015] to update $S$ . To increase the speed of convergence, we initialize $S$ with $S ^ { * }$ , where $S ^ { * }$ is the solution of Eq. (5). Fix $S$ , Update $\\lambda ^ { v }$ ",
+        "bbox": [
+            174,
+            90,
+            823,
+            132
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For each view $v$ , we defne $M ^ { v } = \\ H ^ { v \\top } - H ^ { v \\top } S \\ \\mathsf { \\Pi } _ { F } ^ { 2 } + \\alpha \\mathcal { I }$ . Then, the loss function is simplifed as ",
+        "bbox": [
+            173,
+            132,
+            818,
+            152
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/474ce07d276ea3a5c631e25a20af56ea3f25cdda838970c08a7a69b88471e570.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\lambda ^ { v } } \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } M ^ { v } + \\sum _ { v } ^ { V } ( \\lambda ^ { v } ) ^ { \\gamma } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            156,
+            596,
+            200
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By setting its derivation to zero, we get ",
+        "bbox": [
+            174,
+            204,
+            433,
+            219
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a4d31276da4be2cf470b9b1e4703da7fee26c739aaffd2751bc804cab42e80d.jpg",
+        "text": "$$\n\\lambda ^ { v } = \\left( \\frac { - M ^ { v } } { \\gamma } \\right) ^ { \\frac { 1 } { \\gamma - 1 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            428,
+            223,
+            570,
+            263
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We alternatively update $S$ and $\\lambda ^ { v }$ until convergence. The complete procedures are outlined in Algorithm 1. ",
+        "bbox": [
+            171,
+            266,
+            825,
+            295
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 MCGC ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            309,
+            313,
+            324
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Require: adjacency matrix $\\widetilde { A } ^ { 1 } , . . . , \\widetilde { A } ^ { V }$ , feature $X ^ { 1 } , . . . , X ^ { V }$ , the order of graph fltering $m$ , parameter $\\alpha$ , $s$ and $\\gamma$ , the number of clusters $c$ . ",
+        "bbox": [
+            174,
+            328,
+            823,
+            358
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Ensure: $c$ clusters.   \n1: $\\lambda ^ { v } = 1$ ;   \n2: $A ^ { v } = ( D ^ { v } ) ^ { - \\frac { 1 } { 2 } } ( \\widetilde { A } ^ { v } + I ) ( D ^ { v } ) ^ { - \\frac { 1 } { 2 } }$ ;   \n3: $L ^ { v } = I - A ^ { v }$ ;   \n4: Graph fltering by Eq. (4) for each view;   \n5: while convergence condition does not meet do   \n6: Update $S$ in Eq. (9) via Adam;   \n7: for each view do   \n8: Update $\\lambda ^ { v }$ in Eq. (13);   \n9: end for   \n10: end while   \n(|S|+|S| >) 11: C = ;   \n12: Clustering on $C$ . ",
+        "bbox": [
+            176,
+            357,
+            508,
+            542
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            579,
+            312,
+            595
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.1 Datasets and Metrics ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            609,
+            359,
+            625
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We evaluate MCGC on fve benchmark datasets, ACM, DBLP, IMDB [Fan et al., 2020], Amazon photos and Amazon computers [Shchur et al., 2018]. The statistical information is shown in Table 1. ",
+        "bbox": [
+            173,
+            635,
+            825,
+            662
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "ACM: It is a paper network from the ACM dataset. Node attribute features are the elements of a bag-of-words representing of each paper’s keywords. The two graphs are constructed by two types of relationships: \"Co-Author\" means that two papers are written by the same author and \"Co-Subject\" suggests that they focus on the same fled. ",
+        "bbox": [
+            173,
+            664,
+            825,
+            718
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "DBLP: It is an author network from the DBLP dataset. Node attribute features are the elements of a bag-of-words representing of each author’s keywords. Three graphs are derived from the relationships: \"Co-Author\", \"Co-Conference\", and \"Co-Term\", which indicate that two authors have worked together on papers, published papers at the same conference, and published papers with the same terms. ",
+        "bbox": [
+            173,
+            718,
+            825,
+            786
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "IMDB: It is a movie network from the IMDB dataset. Node attribute features correspond to elements of a bag-of-words representing of each movie. The relationships of being acted by the same actor (Co-actor) and directed by the same director (Co-director) are exploited to construct two graphs. ",
+        "bbox": [
+            174,
+            787,
+            825,
+            828
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Amazon photos and Amazon computers: They are segments of the Amazon co-purchase network dataset, in which nodes represent goods and features of each good are bag-of-words of product reviews, the edges means that two goods are purchased together. To have multi-view attributes, the second feature matrix is constructed via cartesian product by following [Cheng et al., 2020]. ",
+        "bbox": [
+            174,
+            829,
+            826,
+            882
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We adopt four popular clustering metrics: Accuracy (ACC), normalized Mutual Information (NMI), Adjusted Rand Index (ARI), F1 score. ",
+        "bbox": [
+            173,
+            883,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/11f014a5f9a12c6f234408f68b5c2032551e6d382a2d0fbd2a0c4ec221bb117a.jpg",
+        "table_caption": [
+            "Table 1: The statistical information of datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Nodes</td><td>Features</td><td>Graph and Edges</td><td>Clusters</td></tr><tr><td rowspan=\"2\">ACM</td><td rowspan=\"2\">3.025</td><td rowspan=\"2\">1,830</td><td>Co-Subject (29,281)</td><td rowspan=\"2\">3</td></tr><tr><td>Co-Author (2,210,761)</td></tr><tr><td rowspan=\"3\">DBLP</td><td rowspan=\"3\">4,057</td><td rowspan=\"3\">334</td><td>Co-Author (11,113)</td><td rowspan=\"3\">4</td></tr><tr><td>Co-Conference (5,000,495)</td></tr><tr><td>Co-Term (6,776,335)</td></tr><tr><td rowspan=\"2\">IMDB</td><td rowspan=\"2\">4,780</td><td rowspan=\"2\">1,232</td><td>Co-Actor (98,010)</td><td rowspan=\"2\">3</td></tr><tr><td>Co-Director (21,018)</td></tr><tr><td rowspan=\"2\">Amazon photos</td><td rowspan=\"2\">7,487</td><td>745</td><td rowspan=\"2\">Co-Purchase(119,043)</td><td rowspan=\"2\">8</td></tr><tr><td>7,487</td></tr><tr><td rowspan=\"2\">Amazon computers</td><td rowspan=\"2\">13,381</td><td>767</td><td rowspan=\"2\">Co-Purchase(245,778)</td><td rowspan=\"2\">10</td></tr><tr><td>13,381</td></tr></table>",
+        "bbox": [
+            240,
+            107,
+            758,
+            315
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Experiment Setup ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            347,
+            339,
+            363
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We compare MCGC with multi-view methods as well as single-view methods. LINE [Tang et al., 2015] and GAE [Kipf and Welling, 2016] have been chosen as representatives of single-view methods, and we report the best one among the results from all views. Compared multi-view clustering methods include: PMNE [Liu et al., 2017], RMSC [Xia et al., 2014], SwMC [Nie et al., 2017]. PMNE and SwMC only use structural information while RMSC only exploits attribute features. PMNE uses three strategies to project a multilayer network into a continuous vector space, so we select the best result. MCGC is also compared with other methods that not only explore attribute features but also structural information, i.e., O2MA and O2MAC [Fan et al., 2020], MAGCN [Cheng et al., 2020]. In addition, MCGC is compared with COMPLETER [Lin et al., 2021a] and MVGRL [Hassani and Khasahmadi, 2020] that conduct contrastive learning to learn a common representation shared across features of different views and multiple graphs respectively. For an unbiased comparison, we copy part of the results from [Fan et al., 2020]. Since the neighbors of each node on different views could be different, we also examine another strategy: only use the shared neighbors in the contrastive loss $\\begin{array} { r } { \\mathbb { N } _ { i } = \\bigcap _ { v = 1 } ^ { V } \\mathbb { N } _ { i } ^ { v } } \\end{array}$ . And our methodect neighbors and ith this appis fxed as ch is marked assince we fnd t $\\mathbf { M } \\mathbf { C } \\mathbf { G } \\mathbf { C } ^ { * }$ . During experiments,  little infuence to the $k = 1 0$ $\\gamma$ $- 4$ result. According to parameter analysis, we set $m = 2$ , $s = 0 . 5$ , and tune $\\alpha$ . All experiments are conducted on the same machine with the Intel(R) Core(TM) i7-8700 3.20GHz CPU, two GeForce GTX 1080 Ti GPUs and 64GB RAM. The implementation of MCGC is public available 1. ",
+        "bbox": [
+            173,
+            376,
+            826,
+            627
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 Results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            651,
+            263,
+            665
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "All results are shown in Table 2 and Table 3. Compared with single-view method GAE, MCGC improves ACC by more than $9 \\%$ , $4 \\%$ , $1 9 \\%$ on ACM, DBLP, IMDB, respectively. Though using deep neural networks, GAE can not explore the complementarity of views. Compared with PMNE, the ACC, NMI, ARI, F1 are boosted by $1 6 \\%$ , $2 0 \\%$ , $2 0 \\%$ , $1 2 \\%$ on average. With respect to LINE, RMSC, SwMC, the improvement is more signifcant. This can be attributed to the exploration of both feature and structure information in MCGC. Although O2MA, O2MAC, and MAGCN capture attributes and structure information, MCGC still outperforms them considerably. Specifcally, MCGC improves O2MAC on average by almost $6 \\%$ , $9 \\%$ , $1 \\bar { 1 } \\%$ on ACC, NMI, F1, respectively. With respect to MAGCN, the improvement is more than $2 0 \\%$ for all metrics. Compared with contrastive learningbased approaches, our improvement is also impressive. In particular, compared with COMPLETER, the improvement is more than $3 0 \\%$ on Amazon datasets, which illustrates that MCGC benefts from the graph structure information. MCGC also enhances the performance of MVGRL by $2 0 \\%$ . By comparing the results of MCGC and $\\mathbf { M } \\mathbf { C } \\mathbf { G } \\mathbf { C } ^ { * }$ , we can see that the strategy of choosing neighbors does have impact on performance. ",
+        "bbox": [
+            174,
+            680,
+            826,
+            873
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/1252c4a280afd9af050f3ea348fec2ee13efd1911c6cbcaedd7f96e4bdb531cb.jpg",
+        "table_caption": [
+            "Table 2: Results on ACM, DBLP, IMDB. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td></td><td>LINE</td><td>GAE</td><td>PMNE</td><td>RMSC</td><td>SwMC</td><td>02MA</td><td>02MAC</td><td>MCGC</td><td>MCGC*</td></tr><tr><td rowspan=\"4\">ACM</td><td>ACC</td><td>0.6479</td><td>0.8216</td><td>0.6936</td><td>0.6315</td><td>0.3831</td><td>0.888</td><td>0.9042</td><td>0.9147</td><td>0.9055</td></tr><tr><td>NMI</td><td>0.3941</td><td>0.4914</td><td>0.4648</td><td>0.3973</td><td>0.4709</td><td>0.6515</td><td>0.6923</td><td>0.7126</td><td>0.6823</td></tr><tr><td>ARI</td><td>0.3433</td><td>0.5444</td><td>0.4302</td><td>0.3312</td><td>0.0838</td><td>0.6987</td><td>0.7394</td><td>0.7627</td><td>0.7385</td></tr><tr><td>F1</td><td>0.6594</td><td>0.8225</td><td>0.6955</td><td>0.5746</td><td>0.018</td><td>0.8894</td><td>0.9053</td><td>0.9155</td><td>0.9062</td></tr><tr><td rowspan=\"4\">DBLP</td><td>ACC</td><td>0.8689</td><td>0.8859</td><td>0.7925</td><td>0.8994</td><td>0.3253</td><td>0.904</td><td>0.9074</td><td>0.9298</td><td>0.9162</td></tr><tr><td>NMI</td><td>0.6676</td><td>0.6925</td><td>0.5914</td><td>0.7111</td><td>0.019</td><td>0.7257</td><td>0.7287</td><td>0.8302</td><td>0.7490</td></tr><tr><td>ARI</td><td>0.6988</td><td>0.741</td><td>0.5265</td><td>0.7647</td><td>0.0159</td><td>0.7705</td><td>0.778</td><td>0.7746</td><td>0.7995</td></tr><tr><td>F1</td><td>0.8546</td><td>0.8743</td><td>0.7966</td><td>0.8248</td><td>0.2808</td><td>0.8976</td><td>0.9013</td><td>0.9252</td><td>0.9112</td></tr><tr><td rowspan=\"4\">IMDB</td><td>ACC</td><td>0.4268</td><td>0.4298</td><td>0.4958</td><td>0.2702</td><td>0.2453</td><td>0.4697</td><td>0.4502</td><td>0.6182</td><td>0.6113</td></tr><tr><td>NMI</td><td>0.0031</td><td>0.0402</td><td>0.0359</td><td>0.3775</td><td>0.0023</td><td>0.0524</td><td>0.0421</td><td>0.1149</td><td>0.1225</td></tr><tr><td>ARI</td><td>-0.009</td><td>0.0473</td><td>0.0366</td><td>50.0054</td><td>0.0017</td><td>0.0753</td><td>0.0564</td><td>0.1833</td><td>0.1811</td></tr><tr><td>F1</td><td>0.287</td><td>0.4062</td><td>0.3906</td><td>0.0018</td><td>0.3164</td><td>0.4229</td><td>0.1459</td><td>0.4401</td><td>0.4512</td></tr></table>",
+        "bbox": [
+            189,
+            111,
+            808,
+            318
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/b54ba61e222100a02f858c009fd7be4ab5be130332b0801ef01f3465ba219681.jpg",
+        "table_caption": [
+            "Table 3: Results on Amazon photos and Amazon computers. The ‘-’ means that the method raises out-of-memory problem. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"4\"> Amazon photos</td><td colspan=\"4\"> Amazon computers</td></tr><tr><td></td><td>ACC</td><td>NMI</td><td>ARI</td><td>F1</td><td>ACC</td><td>NMI</td><td>ARI</td><td>F1</td></tr><tr><td>COMPLETER</td><td>0.3678</td><td>0.2606</td><td>0.0759</td><td>0.3067</td><td>0.2417</td><td>0.1562</td><td>0.0536</td><td>0.1601</td></tr><tr><td>MVGRL</td><td>0.5054</td><td>0.4331</td><td>0.2379</td><td>0.4599</td><td>0.2450</td><td>0.1012</td><td>0.0553</td><td>0.1706</td></tr><tr><td>MAGCN</td><td>0.5167</td><td>0.3897</td><td>0.2401</td><td>0.4736</td><td>1</td><td>1</td><td>1</td><td>1</td></tr><tr><td>MCGC</td><td>0.7164</td><td>0.6154</td><td>0.4323</td><td>0.6864</td><td>0.5967</td><td>0.5317</td><td>0.3902</td><td>0.5204</td></tr></table>",
+        "bbox": [
+            181,
+            380,
+            816,
+            487
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 Ablation Study ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            523,
+            331,
+            542
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 The Effect of Contrastive Loss ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            565,
+            424,
+            580
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "By employing the contrastive regularizer, our method pulls neighbors into the same cluster, which decreases intra-cluster variance. To see its effect, we replace $\\mathcal { I }$ with a Frobenius term, i.e. Eq. (5). As can be seen from Table 4, the performance falls precipitously without contrastive loss on all datasets. MCGC achieves ACC improvements by $1 6 \\%$ , $8 \\%$ , $5 \\%$ , $i 2 \\%$ on DBLP, ACM, IMDB, Amazon datasets, respectively. For other metrics, the contrastive regularizer also enhances the performance signifcantly. Above facts validate that MCGC benefts from the graph contrastive loss. ",
+        "bbox": [
+            173,
+            597,
+            826,
+            681
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/ac5a3f6c5f2100fcd85c04058a3bc43cec3013b199cc74bc510411fcf2bf144b.jpg",
+        "table_caption": [
+            "Table 4: Results of MCGC without contrastive loss. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Datasets</td><td>ACM</td><td>DBLP</td><td>IMDB</td><td>Amazon photos</td><td>Amazon computers</td></tr><tr><td rowspan=\"2\">ACC</td><td>MCGC</td><td>0.9147</td><td>0.9298</td><td>0.6182</td><td>0.7164</td><td>0.5967</td></tr><tr><td>MCGC w/o J</td><td>0.8334</td><td>0.7658</td><td>0.5636</td><td>0.5882</td><td>0.4662</td></tr><tr><td rowspan=\"2\">NMI</td><td>MCGC</td><td>0.7126</td><td>0.8302</td><td>0.1149</td><td>0.6154</td><td>0.5317</td></tr><tr><td>MCGC w/o J</td><td>0.5264</td><td>0.4621</td><td>0.0707</td><td>0.5372</td><td>0.3988</td></tr><tr><td rowspan=\"2\">ARI</td><td>MCGC</td><td>0.7627</td><td>0.7746</td><td>0.1833</td><td>0.4323</td><td>0.3902</td></tr><tr><td>MCGC w/o J</td><td>0.5779</td><td>0.4949</td><td>0.1451</td><td>0.2640</td><td>0.1745</td></tr><tr><td rowspan=\"2\">F1</td><td>MCGC</td><td>0.9155</td><td>0.9252</td><td>0.4401</td><td>0.6864</td><td>0.5204</td></tr><tr><td>MCGC w/o J</td><td>0.8313</td><td>0.7601</td><td>0.4444</td><td>0.5437</td><td>0.3678</td></tr></table>",
+        "bbox": [
+            179,
+            748,
+            818,
+            909
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/fdf8a1754aa540e187221a69181dd81dd2c70a38c30414a5e10a471c3e90351b.jpg",
+        "table_caption": [
+            "Table 5: Results in various views of MCGC on ACM and Amazon photos. $G _ { 1 }$ and $G _ { 2 }$ denote the graphs in different views. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"3\">ACM</td><td colspan=\"3\">Amazon photos</td></tr><tr><td></td><td>G1,X</td><td>G2,X</td><td>G1,G2, X</td><td>X1,G</td><td>X²,G</td><td>X1, X²,G</td></tr><tr><td>ACC</td><td>0.9088</td><td>0.8152</td><td>0.9147</td><td>0.4433</td><td>0.6935</td><td>0.7164</td></tr><tr><td>NMI</td><td>0.6929</td><td>0.4656</td><td>0.7126</td><td>0.3519</td><td>0.5976</td><td>0.6154</td></tr><tr><td>ARI</td><td>0.7470</td><td>0.5229</td><td>0.7627</td><td>0.1572</td><td>0.4291</td><td>0.4323</td></tr><tr><td>F1</td><td>0.9097</td><td>0.8184</td><td>0.9155</td><td>0.3675</td><td>0.6734</td><td>0.6864</td></tr></table>",
+        "bbox": [
+            241,
+            123,
+            756,
+            231
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 The Effect of Multi-View Learning ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            255,
+            455,
+            271
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effect of multi-view learning, we evaluate the performance of the following single-view model ",
+        "bbox": [
+            169,
+            281,
+            825,
+            310
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46153beec80a446bc559712fa873f25c93d5c81667e98e4b40e880885e8cb21b.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { S } ~ H ^ { \\top } - H ^ { \\top } S ~ _ { F } ^ { 2 } + \\alpha \\sum _ { i = 1 } ^ { N } \\sum _ { j \\in \\mathbb { N } _ { i } } - \\log \\frac { \\exp ( S _ { i j } ) } { \\displaystyle \\sum _ { p \\neq i } ^ { N } \\exp ( S _ { i p } ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            315,
+            686,
+            385
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Taking ACM and Amazon photos as examples, we report the clustering performance of various scenarios in Table 5. We can observe that the best performance is always achieved when all views are incorporated. In addition, we also see that the performance varies a lot for different views. This justifes the necessity of $\\lambda ^ { v }$ in Eq. (7). Therefore, it is benefcial to explore the complementarity of multi-view information. ",
+        "bbox": [
+            173,
+            390,
+            826,
+            460
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 The Effect of Graph Filtering ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            476,
+            419,
+            491
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To understand the contribution of graph fltering, we conduct another group of experiments. Without graph fltering, our objective function becomes ",
+        "bbox": [
+            174,
+            501,
+            825,
+            530
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/dbd5a23c8814ef10e26552653e85834ad5585ddf8be7c8f2adad05cec00831c4.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { S , \\lambda ^ { \\nu } } \\sum _ { v = 1 } ^ { V } \\lambda ^ { v } \\left( \\begin{array} { l } { { X ^ { v \\top } - X ^ { v \\top } S _ { F } ^ { \\enspace 2 } + \\alpha \\sum _ { i = 1 } ^ { N } \\sum _ { j \\in \\mathbb { N } _ { i } ^ { v } } - \\log \\frac { \\exp { ( S _ { i j } ) } } { \\sum _ { p \\neq i } ^ { N } \\exp { ( S _ { i p } ) } } } } \\end{array} \\right) + \\sum _ { v = 1 } ^ { V } ( \\lambda ^ { v } ) ^ { \\top } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            200,
+            536,
+            767,
+            587
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We denote this model as MCGC-. The results of MCGC- are shown in Table 6. With respect to MCGC, ACC on ACM, DBLP, IMDB drops by $0 . 8 \\%$ , $1 . 3 \\%$ , $0 . 8 \\%$ , respectively. This indicates that graph fltering makes a positive impact on our model. For other metrics, MCGC also outperforms MCGC- in most cases. ",
+        "bbox": [
+            173,
+            592,
+            825,
+            648
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/c8b0031987c71fb2b252ae26c2da117f345c1c443c24c992cd93b8a334c35e6e.jpg",
+        "table_caption": [
+            "Table 6: The results of MCGC- (without graph fltering). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td colspan=\"2\">ACM</td><td colspan=\"2\">DBLP</td><td colspan=\"2\">IMDB</td></tr><tr><td>MCGC</td><td>MCGC-</td><td>MCGC</td><td>MCGC-</td><td>MCGC</td><td>MCGC-</td></tr><tr><td>ACC</td><td>0.9147</td><td>0.9061</td><td>0.9298</td><td>0.9162</td><td>0.6182</td><td>0.6109</td></tr><tr><td>NMI</td><td>0.7126</td><td>0.6974</td><td>0.8302</td><td>0.7490</td><td>0.1149</td><td>0.1219</td></tr><tr><td>ARI</td><td>0.7627</td><td>0.7439</td><td>0.7746</td><td>0.7995</td><td>0.1833</td><td>0.1804</td></tr><tr><td>F1</td><td>0.9155</td><td>0.9057</td><td>0.9252</td><td>0.9112</td><td>0.4401</td><td>0.4509</td></tr></table>",
+        "bbox": [
+            251,
+            694,
+            746,
+            801
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 Parameter Analysis ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            838,
+            370,
+            856
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Firstly, two parameters $m$ and $s$ are applied in graph fltering. Taking ACM as an example, we show their infuence on performance by setting $m = [ 1 , 2 , 3 , 4 , 5 ]$ , $s = [ 0 . 0 1 , 0 . 1 , 0 . 3 , 0 . 5 , 1 , 3 , 5 , 1 0 ]$ in Fig. 1. It can be seen that MCGC achieves a reasonable performance for a small $m$ and $s$ . ",
+        "bbox": [
+            174,
+            868,
+            826,
+            912
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/7da9718be016c75436ecafa088b545a6d5e6f125dce330ec913b2a359d0191ef.jpg",
+        "image_caption": [
+            "Figure 1: Sensitivity analysis of parameters $m$ and $s$ on ACM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            189,
+            88,
+            805,
+            203
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Therefore, we set $m = 2$ and $s = 0 . 5$ in all experiments. Afterwards, we tune the trade-off parameter $\\alpha = [ 1 0 ^ { - 3 } , 0 . 1 , 1 , 1 0 , 1 0 ^ { 2 } , 1 0 ^ { 3 } ]$ . As shown in Fig. 2, our method is not sensitive to $\\alpha$ , which enhances the practicality in real-world applications. In addition, we plot the objective variation of Eq. (7) in Fig. 3. As observed from this fgure, our method converges quickly. ",
+        "bbox": [
+            173,
+            252,
+            826,
+            309
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/8d8a25b5eb837b1a8d4db5b5c1cb855e520623dd5c7d7c3a12d414126acc0a88.jpg",
+        "image_caption": [
+            "Figure 2: Sensitivity analysis of parameter $\\alpha$ on ACM (a-d), DBLP (e-h), IMDB (i-l). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            324,
+            820,
+            614
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/35cbf232f72df6ebb073fa49def909e8d0d4b9f9a8c1efd4a76338e314b32358.jpg",
+        "image_caption": [
+            "Figure 3: The evolution of objective function. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            192,
+            671,
+            807,
+            785
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            837,
+            299,
+            854
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Multi-view graph clustering is till at a nascent stage with many challenges remained unsolved. In this paper, we propose a novel method (MCGC) to learn a consensus graph by exploiting not only attribute content but also graph structure information. Particularly, graph fltering is introduced to flter out noisy components and a contrastive regularizer is employed to further enhance the quality of learned graph. Experimental results on multi-view attributed graph datasets have shown the superior performance of our method. This study demonstrates that it is possible for shallow model to beat deep learning methods facing the systematic use of complex deep neural networks. Graph learning is crucial to more and more tasks and applications. Just like other methods that learn from data, brings the risk of learning biases and perpetuating them in the form of decisions. Thus our method should be deployed with careful consideration of any potential underlying biases in the data. One potential limitation of our approach is that it could take a lot of memory if the data contain too many nodes. This is because the size of learned graph is $N \\times N$ . Research on large scale network is left for future work. ",
+        "bbox": [
+            174,
+            869,
+            825,
+            912
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            92,
+            825,
+            229
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            250,
+            553,
+            267
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "This paper was in part supported by the Natural Science Foundation of China (Nos.61806045, U19A2059). ",
+        "bbox": [
+            176,
+            281,
+            823,
+            310
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            330,
+            266,
+            347
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Maria Brbic and Ivica Kopriva. Multi-view low-rank sparse subspace clustering. ´ Pattern Recognition, 73:247–258, 2018. ",
+        "bbox": [
+            173,
+            354,
+            826,
+            382
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. In International conference on machine learning, pages 1597–1607. PMLR, 2020a. ",
+        "bbox": [
+            174,
+            393,
+            826,
+            436
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Yu Chen, Lingfei Wu, and Mohammed Zaki. Iterative deep graph learning for graph neural networks: Better and robust node embeddings. Advances in Neural Information Processing Systems, 33, 2020b. ",
+        "bbox": [
+            174,
+            446,
+            826,
+            489
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Jiafeng Cheng, Qianqian Wang, Zhiqiang Tao, Deyan Xie, and Quanxue Gao. Multi-view attribute graph convolution networks for clustering. IJCAI, 2020. ",
+        "bbox": [
+            171,
+            500,
+            823,
+            529
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Sumit Chopra, Raia Hadsell, and Yann LeCun. Learning a similarity metric discriminatively, with application to face verifcation. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), volume 1, pages 539–546. IEEE, 2005. ",
+        "bbox": [
+            174,
+            539,
+            823,
+            582
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Shaohua Fan, Xiao Wang, Chuan Shi, Emiao Lu, Ken Lin, and Bai Wang. One2multi graph autoencoder for multi-view graph clustering. In Proceedings of The Web Conference 2020, pages 3070–3076, 2020. ",
+        "bbox": [
+            173,
+            592,
+            823,
+            635
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Michael Gutmann and Aapo Hyvärinen. Noise-contrastive estimation: A new estimation principle for unnormalized statistical models. In Proceedings of the Thirteenth International Conference on Artifcial Intelligence and Statistics, pages 297–304. JMLR Workshop and Conference Proceedings, 2010. ",
+        "bbox": [
+            173,
+            645,
+            825,
+            702
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), volume 2, pages 1735–1742. IEEE, 2006. ",
+        "bbox": [
+            173,
+            712,
+            825,
+            756
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Kaveh Hassani and Amir Hosein Khasahmadi. Contrastive multi-view representation learning on graphs. In International Conference on Machine Learning, pages 4116–4126. PMLR, 2020. ",
+        "bbox": [
+            171,
+            765,
+            823,
+            795
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhenyu Huang, Joey Tianyi Zhou, Xi Peng, Changqing Zhang, Hongyuan Zhu, and Jiancheng Lv. Multi-view spectral clustering network. In IJCAI, pages 2563–2569, 2019. ",
+        "bbox": [
+            171,
+            804,
+            823,
+            833
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhao Kang, Xiao Lu, Yiwei Lu, chong Peng, Wenyu Chen, and Zenglin Xu. Structure learning with similarity preserving. Neural Networks, 129:138–148, 2020a. ",
+        "bbox": [
+            169,
+            843,
+            823,
+            872
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhao Kang, Haiqi Pan, Steven C.H. Hoi, and Zenglin Xu. Robust graph learning from noisy data. IEEE Transactions on Cybernetics, 50(5):1833–1843, 2020b. ",
+        "bbox": [
+            174,
+            882,
+            823,
+            912
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Zhao Kang, Xinjia Zhao, Shi, chong Peng, Hongyuan Zhu, Joey Tianyi Zhou, Xi Peng, Wenyu Chen, and Zenglin Xu. Partition level multiview subspace clustering. Neural Networks, 122:279–288, 2020c. ",
+        "bbox": [
+            174,
+            90,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhao Kang, Zhiping Lin, Xiaofeng Zhu, and Wenbo Xu. Structured graph learning for scalable subspace clustering: From single-view to multi-view. IEEE Transactions on Cybernetics, 2021. doi: 10.1109/TCYB.2021.3061660. ",
+        "bbox": [
+            173,
+            142,
+            825,
+            185
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. ",
+        "bbox": [
+            174,
+            194,
+            823,
+            237
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Thomas N Kipf and Max Welling. Variational graph auto-encoders. In NIPS Bayesian Deep Learning Workshop, 2016. ",
+        "bbox": [
+            173,
+            244,
+            823,
+            273
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yunfan Li, Peng Hu, Zitao Liu, Dezhong Peng, Joey Tianyi Zhou, and Xi Peng. Contrastive clustering. In AAAI 2021, volume 35, Feb. 2021. ",
+        "bbox": [
+            173,
+            282,
+            823,
+            311
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yijie Lin, Yuanbiao Gou, Zitao Liu, Boyun Li, Jiancheng Lv, and Xi Peng. Completer: Incomplete multi-view clustering via contrastive prediction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2021a. ",
+        "bbox": [
+            174,
+            320,
+            821,
+            363
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhiping Lin and Zhao Kang. Graph flter-based multi-view attributed graph clustering. In Proceedings of the 30th International Joint Conference on Artifcial Intelligence, IJCAI, pages 19–26, 2021. ",
+        "bbox": [
+            173,
+            371,
+            823,
+            401
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhiping Lin, Zhao Kang, Lizong Zhang, and Ling Tian. Multi-view attributed graph clustering. IEEE Transactions on Knowledge and Data Engineering, 2021b. doi: 10.1109/TKDE.2021.3101227. ",
+        "bbox": [
+            174,
+            409,
+            825,
+            439
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Changshu Liu, Liangjian Wen, Zhao Kang, Guangchun Luo, and Ling Tian. Self-supervised consensus representation learning for attributed graph. In Proceedings of the 29th ACM International Conference on Multimedia, 2021. ",
+        "bbox": [
+            174,
+            446,
+            823,
+            489
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Weiyi Liu, Pin-Yu Chen, Sailung Yeung, Toyotaro Suzumura, and Lingli Chen. Principled multilayer network embedding. In 2017 IEEE International Conference on Data Mining Workshops (ICDMW), pages 134–141. IEEE, 2017. ",
+        "bbox": [
+            173,
+            498,
+            823,
+            541
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Juncheng Lv, Zhao Kang, Xiao Lu, and Zenglin Xu. Pseudo-supervised deep subspace clustering. IEEE Transactions on Image Processing, 30:5252–5263, 2021. ",
+        "bbox": [
+            171,
+            549,
+            823,
+            579
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Zhengrui Ma, Zhao Kang, Guangchun Luo, Ling Tian, and Wenyu Chen. Towards clusteringfriendly representations: Subspace clustering via graph fltering. In Proceedings of the 28th ACM International Conference on Multimedia, pages 3081–3089, 2020. ",
+        "bbox": [
+            173,
+            587,
+            825,
+            631
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Feiping Nie, Jing Li, Xuelong Li, et al. Self-weighted multiview clustering with multiple graphs. In IJCAI, pages 2564–2570, 2017. ",
+        "bbox": [
+            169,
+            638,
+            825,
+            667
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Shirui Pan, Ruiqi Hu, Guodong Long, Jing Jiang, Lina Yao, and Chengqi Zhang. Adversarially regularized graph autoencoder for graph embedding. In Proceedings of the 27th International Joint Conference on Artifcial Intelligence, pages 2609–2615, 2018. ",
+        "bbox": [
+            176,
+            676,
+            823,
+            720
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jiwoong Park, Minsik Lee, Hyung Jin Chang, Kyuewang Lee, and Jin Young Choi. Symmetric graph convolutional autoencoder for unsupervised graph representation learning. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6519–6528, 2019. ",
+        "bbox": [
+            174,
+            728,
+            823,
+            772
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Xi Peng, Zhenyu Huang, Jiancheng Lv, Hongyuan Zhu, and Joey Tianyi Zhou. Comic: Multi-view clustering without parameter selection. In International Conference on Machine Learning, pages 5092–5101. PMLR, 2019. ",
+        "bbox": [
+            174,
+            779,
+            823,
+            823
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Oleksandr Shchur, Maximilian Mumme, Aleksandar Bojchevski, and Stephan Günnemann. Pitfalls of graph neural network evaluation. arXiv preprint arXiv:1811.05868, 2018. ",
+        "bbox": [
+            169,
+            830,
+            823,
+            861
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Jian Tang, Meng Qu, Mingzhe Wang, Ming Zhang, Jun Yan, and Qiaozhu Mei. Line: Large-scale information network embedding. In Proceedings of the 24th international conference on world wide web, pages 1067–1077, 2015. ",
+        "bbox": [
+            173,
+            869,
+            826,
+            911
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chun Wang, Shirui Pan, Guodong Long, Xingquan Zhu, and Jing Jiang. Mgae: Marginalized graph autoencoder for graph clustering. In Proceedings of the 2017 ACM on Conference on Information and Knowledge Management, pages 889–898, 2017. ",
+        "bbox": [
+            176,
+            90,
+            823,
+            133
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hao Wang, Yan Yang, and Bing Liu. Gmc: Graph-based multi-view clustering. IEEE Transactions on Knowledge and Data Engineering, 32(6):1116–1129, 2019. ",
+        "bbox": [
+            171,
+            142,
+            823,
+            171
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Rongkai Xia, Yan Pan, Lei Du, and Jian Yin. Robust multi-view spectral clustering via low-rank and sparse decomposition. In Proceedings of the AAAI conference on artifcial intelligence, volume 28, 2014. ",
+        "bbox": [
+            174,
+            179,
+            823,
+            222
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Hongming Zhang, Liwei Qiu, Lingling Yi, and Yangqiu Song. Scalable multiplex network embedding. In IJCAI, volume 18, pages 3082–3088, 2018. ",
+        "bbox": [
+            168,
+            231,
+            825,
+            261
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Huasong Zhong, Chong Chen, Zhongming Jin, and Xian-Sheng Hua. Deep robust clustering by contrastive learning. arXiv preprint arXiv:2008.03030, 2020. ",
+        "bbox": [
+            168,
+            268,
+            823,
+            297
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Meiqi Zhu, Xiao Wang, Chuan Shi, Houye Ji, and Peng Cui. Interpreting and unifying graph neural networks with an optimization framework. In Proceedings of the Web Conference 2021, pages 1215–1226, 2021. ",
+        "bbox": [
+            173,
+            306,
+            825,
+            349
+        ],
+        "page_idx": 11
+    }
+]

parse/train/SJlOq34Kwr/SJlOq34Kwr.md

@@ -0,0 +1,299 @@
+# UNSUPERVISED INTUITIVE PHYSICS FROM PAST EXPERIENCES
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We consider the problem of learning models of intuitive physics from raw, unlabelled visual input. Differently from prior work, in addition to learning general physical principles, we are also interested in learning “on the fly” physical properties specific to new environments, based on a small number of environment-specific experiences. We do all this in an unsupervised manner, using a meta-learning formulation where the goal is to predict videos containing demonstrations of physical phenomena, such as objects moving and colliding with a complex background. We introduce the idea of summarizing past experiences in a very compact manner, in our case using dynamic images, and show that this can be used to solve the problem well and efficiently. Empirically, we show, via extensive experiments and ablation studies, that our model learns to perform physical predictions that generalize well in time and space, as well as to a variable number of interacting physical objects.
+
+# 1 INTRODUCTION
+
+Many animals possess an intuitive understanding of the physical world. They use this understanding to accurately and rapidly predict events from sparse sensory inputs. In addition to general physical principles, many animals also learn specific models of new environments as they experience them over time. For example, they can explore an environment to determine which parts of it can be navigated safely and remember this knowledge for later reuse.
+
+Authors have looked at equipping artificial intelligences (AIs) with analogous capabilities, but focusing mostly on performing predictions from instantaneous observations of an environment, such as a few frames in a video. However, such predictions can be successful only if observations are combined with sufficient prior knowledge about the environment. For example, consider predicting the motion of a bouncing ball. Unless key parameters such as the ball’s elasticity are known a priori, it is impossible to predict the ball’s trajectory accurately. However, after observing at least one bounce, it is possible to infer some of the parameters and eventually perform much better predictions.
+
+In this paper, we are interested in learning intuitive physics in an entirely unsupervised manner, by passively watching videos. We consider situations in which objects interact with scenarios that can only be partially inferred from their appearance, but that also contain objects whose parameters cannot be confidently predicted from appearance alone (fig. 1). Then, we consider learning a system that can observe a few physical experiments to infer such parameters, and use this knowledge to perform better predictions in the future.
+
+Our model has three goals. First, it must learn without the use of any external or ad-hoc supervision.   
+We achieve this by training our model from raw videos, using a video prediction error as a loss.
+
+Second, our model must be able to extract information about a new scenario by observing a few experiments, which we formulate as meta-learning. We also propose a simple representation of the experiments based on the concept of “dynamic image” that allows to process long experiments more efficiently than using a conventional recurrent network.
+
+Third, our model must learn a good representation of physics without access to any explicit or external supervision. Instead, we propose three tests to support this hypothesis. (i) We show that the model can predict far in the future, which is a proxy to temporal invariance. (ii) We further show that the model can extend to scenarios that are geometrically much larger than the ones used for training, which is a proxy to spatial invariance. (iii) Finally, we show that the model can generalize to several moving objects, which is a proxy to locality. Locality and time-space invariance are of course three key properties of physical laws and thus we should expect any good intuitive model of physics to possess them.
+
+![](images/260c1b139e0592126ca5f693f20c78a00a2190f8fe352e8ddc563ff9d47e2b66.jpg)  
+Figure 1: Probing a new environment. Given the input showing a (purple) ball moving from right to left, three possible future outcomes are possible: the ball passes above (A), passes under (U), or bounces (B) off the object. This cannot be predicted from appearance alone. Instead, our model learns the property of the object by probing each new environment via a small set of experiments.
+
+In order to support these claims, we conduct extensive experiments in simulated scenarios, including testing the ability of the model to cope with non-trivial visual variations of the inputs. While the data is simpler than a real-world application, we nevertheless make substantial progress compared to previous work, as discussed in section 2. We do so by learning from passive, raw video data a good model of dynamics and collisions that generalizes well spatially, temporally, and to a variable number of objects. The scalability of our approach, via the use of the dynamic image, is also unique. Finally, we investigate the problem of learning the parameters of new scenarios on the fly via experiences and we propose an effective solution to do so.
+
+# 2 RELATED WORK
+
+A natural way to represent physics is to manually encode every object parameter and physical property (e.g., mass, velocity, positions) and use supervision to make predictions. This has been widely used to represent and propagate physics (Wu et al., 2015; 2016; Battaglia et al., 2016; Chang et al., 2017; Mrowca et al., 2018; Sanchez-Gonzalez et al., 2018). If models like Wu et al. (2015; 2016) also estimate environment parameters, these works rely on a physics engine that assumes strong priors about the scenario, while our approach does not require such constraint.
+
+Inspired by the recent successes of Convolutional Neural Networks (CNNs (Krizhevsky et al., 2012)) and their application to implicit representation of dynamics (Ondruska & Posner, 2016; Oh et al., 2015; Chiappa et al., 2017; Bhattacharyya et al., 2018), researchers (Watters et al., 2017; Ehrhardt et al., 2019; Kansky et al., 2017) have tried to base their approaches on visual inputs. They learn from several frames of a scene to regress the next physical state of a system. In general these approaches learn an implicit representation of physics (Ehrhardt et al., 2019; Watters et al., 2017) as a tensor state from recurrent deep networks.
+
+Such models are mostly supervised using ground-truth information about key physical parameters (e.g., positions, velocities, density) during training. While these approaches require an expensive annotation of data, other works have tried to learn from unsupervised data as well. Researchers have successfully learned unsupervised models either through active manipulation (Agrawal et al., 2016; Denil et al., 2016; Finn et al., 2016a), using the laws of physics (Stewart & Ermon, 2017), using dynamic clues and invariances (Greff et al., 2017; van Steenkiste et al., 2018) or features extracted from unsupervised methods (Finn et al., 2016b; Ehrhardt et al., 2018). Fragkiadaki et al. (2016) also used an unsupervised system like ours, however, they assumed the rendering system and the number of objects to be known, which we do not, which prevented the internal representation to fail over time. Perhaps most related to our approach is the work of Wang et al. (2018), where the model is learnt using future image prediction on a simple synthetic task and then transferred to real world scenarios. They also demonstrate long-term dynamic predictions, however they did not generalize to different background/number of balls.
+
+In other works, models are taught to answer simple qualitative questions about a physical setup, such as: the stability of stacks of objects (Battaglia et al., 2013; Lerer et al., 2016; Li et al., 2017; Groth et al., 2018), the likelihood of a scenario (Riochet et al., 2018), the forces acting behind a scene (Wu et al., 2017; Mottaghi et al., 2016) or properties of objects through manipulation (Agrawal et al., 2016; Denil et al., 2016). Other papers compromise between qualitative and quantitative predictions and focus on plausibility (Tompson et al., 2016; Ladický et al., 2015; Monszpart et al., 2016).
+
+![](images/4e6b55d84e3f030b72b198c2fa02083183b99be36b1ca078b0df1b13ad1450ed.jpg)  
+Figure 2: Overview of our method. $\Psi$ (bottom right block) acts as a meta-learning module. It takes as input past experiments compressed into dynamic images alongside with the median image and learn to optimise $_ { m }$ and $\textbf { \em a }$ to minimise the final loss. $\eta$ (top-left yellow block) extracts states $\mathbf { x } _ { 0 : T _ { 0 } }$ which are carried forward with the auto-regressive predictor $\Phi$ . Finally $g$ renders frame $\hat { \mathbf { I } } _ { t }$ from $\mathbf { x } _ { t }$ .
+
+Finally similar problems can be found in online learning settings (Nagabandi et al., 2019a;b). These works also use meta-learning but, differently from ours, allow their models to learn and adapt at test time from feedbacks.
+
+# 3 METHOD
+
+We describe our model, illustrated in fig. 2, starting from the input and output data.
+
+A scenario $S$ is a physical environment that supports moving objects interacting with it. In this paper, we take as scenarios 2.1D environments containing obstacles and we consider rolling balls as moving objects. Hence, interactions are in the form of bounces. Formally, a scenario is defined over a lattice $\Omega = \{ 0 , \dots , H - 1 \} \times \{ 0 , \dots , W - 1 \}$ and is specified by a list of obstacles $\boldsymbol { \mathcal { S } } = \{ ( O _ { j } , b _ { j } )$ , $j =$ $1 , \ldots , K \}$ . Here $O _ { j } \subset \Omega$ is the shape of an obstacle and $b _ { j } \in \{ B , A , U \}$ is a flag that tells whether the ball bounces against it (B), passes above it (A), or passes under it (U). Obstacles do not overlap.
+
+A run is a tuple $\mathcal { R } = ( \mathcal { S } , \mathbf { y } )$ associating a scenario $s$ with a trajectory $\mathbf { y } = ( y _ { t } \in \Omega , t = 0 , \dots , T - 1 )$ for the ball (this is trivially extended to multiple balls). Scenarios and runs are sensed visually. The symbol $\mathbf { I } ( \pmb { S } ) : \Omega  \mathbb { R } ^ { 3 \times } \mathbf { \bar { H } } \times \boldsymbol { W }$ denotes the image generated by observing a scenario with no ball and $\mathbf { I } _ { t } ( \mathcal { R } ) = \mathbf { I } ( S , y _ { t } )$ is the image generated by observing the ball at time $t$ in a run. The symbol $\mathbf { I } ( \mathcal { R } ) = ( \mathbf { I } _ { t } ( \mathcal { R } ) , t = 0 , \dots , T - 1 )$ denotes the collection of all frames in a run, which can be thought of as a video.
+
+We are interested in learning intuitive physics with no explicit supervision on the object trajectories, nor an explicit encoding of the laws of mechanics that govern the motion of the objects and their interactions and collisions with the environment. Hence, we cast this problem as predicting the video $\mathbf { I } ( \mathcal { R } )$ of the trajectory of the objects given only a few initial frames $\mathbf { I } _ { 0 : T _ { 0 } } ( \mathcal { R } ) = ( \mathbf { I } _ { 0 } ( \mathcal { R } ) , \dots , \mathbf { I } _ { T _ { 0 } - 1 } ( \mathcal { R } ) )$ , where $T _ { 0 } \ll T$ .
+
+We are not the first to consider a similar learning problem, although most prior works do require some form of external supervision, which we do not use. Here, however, we consider an additional key challenge that the images ${ \mathbf { I } } _ { 0 : T _ { 0 } } ( { \mathcal { R } } )$ do not contain sufficient information to successfully predict the long-term objects’ motion. This is because these few frames tell us nothing about the nature of the obstacles in the scenario. In fact, under the assumptions that obstacles of type $B$ , $A$ and $U$ have similar or even the identical appearance, it is not possible to predict whether the ball will bounce, move above, or move under any such obstacle. This situation is representative of an agent that needs to operate in a new complex environment and must learn more about it, from past experiences, before it can do so reliably.
+
+Thus, we consider a modification of the setup described above in which the model can experience each new scenario for a while, by observing the motion of the ball, before making its own predictions.
+
+Formally, an experience is a collection of $N$ runs ${ \mathcal { E } } = ( { \mathcal { R } } _ { 1 } , \ldots , { \mathcal { R } } _ { N } )$ all relative to the same scenario $s$ with each a different trajectories $\mathbf { y } _ { 1 } , \ldots , \mathbf { y } _ { N }$ . By observing such examples, the model must determine the nature of the obstacles and then use this information to correctly predict the motion of the ball in the future. We cast this as the problem of learning a mapping
+
+$$
+\Omega : \left( \mathrm { I } _ { 0 : T _ { 0 } } ( { \mathcal { R } } ) , \mathrm { I } ( { \mathcal { E } } ) \right) \ \longmapsto \ \mathrm { I } ( { \mathcal { R } } ) ,
+$$
+
+where $\mathbf { I } ( { \mathcal { E } } ) = ( \mathbf { I } ( { \mathcal { R } } _ { 1 } ) , \ldots , \mathbf { I } ( { \mathcal { R } } _ { N } ) )$ are the videos corresponding to the runs in the experience. We will call $\mathcal { R }$ the prediction run to distinguish it from the experience runs $\mathcal { E }$ .
+
+# 3.1 A META-LEARNING PERSPECTIVE
+
+The setup we have described above can be naturally described as meta-learning. Namely, eq. (1) can be thought of as incorporating a “local” learning rule $\mathcal { M }$ that maps the experience $\mathcal { E }$ to a scenariospecific predictor $\hat { \Omega }$ on the fly: $\hat { \Omega } ( \cdot ) = \Omega ( \cdot , \mathbf { I } ( \mathcal { E } ) ) = \mathcal { M } [ \mathbf { I } ( \mathcal { E } ) ]$ . Hence $\mathcal { M }$ must extract from the experience as much information as possible about the underlying scenario and transfer it to the scenario-specific predictor $\hat { \Omega }$ . In order to learn $\mathcal { M }$ , we consider meta-samples of the type $( \mathcal S , \mathcal R , \mathcal E )$ comprising a scenario $s$ , a prediction run $\mathcal { R }$ and $N$ experience runs $\mathcal { E }$ . Given a dataset $\mathcal { D }$ of such meta-samples, meta-learning searches for the mapping $\mathcal { M }$ that minimizes the error on the prediction runs:
+
+$$
+\mathcal { M } ^ { * } = \operatorname * { a r g m i n } _ { \mathcal { M } } \frac { 1 } { | \mathcal { D } | } \sum _ { ( \mathcal { S } , \mathcal { R } , \mathcal { E } ) \in \mathcal { D } } \ell ( \mathbf { I } ( \mathcal { R } ) , \hat { \Omega } ( \mathbf { I } _ { 0 : T _ { 0 } } ( \mathcal { R } ) ) ) , \quad \hat { \Omega } = \mathcal { M } [ \mathbf { I } ( \mathcal { E } ) ] .
+$$
+
+# 3.2 COMPRESSED DYNAMIC EXPERIENCES
+
+Concretely, we parameterise the scenario-specific predictor $\hat { \Omega } ( \cdot ; w , m )$ using parameters $\pmb { w }$ , the weights a deep neural network, which is fixed and scenario-independent, and $_ { \mathbf { \nabla } } \mathbf { m }$ , which is scenariospecific. The latter is extracted by a network ${ \pmb m } = \Psi _ { 1 } ( { \pmb I } ( { \pmb \mathcal { E } } ) )$ from the experience videos $\mathbf { I } ( \mathcal { E } )$ . Since we expect $_ { \mathbf { \nabla } } \mathbf { m } _ { \mathbf { \nabla } }$ to provide information about the nature of the obstacles in the scenario, we let $\pmb { m } \in \mathbb { R } ^ { 1 \times H ^ { \star } W }$ be a tensor with the same spatial resolution as the scenario and interpret it as an “obstacle mask”.
+
+Given that the function $\Psi _ { 1 }$ takes as input a number of video sequences, one may think to implement it as a recurrent neural network; however, recurrent networks scale badly, especially in our metalearning context — to the point that we had difficulties in even running such a model at a non-trivial scale. Instead, we propose to construct $\Psi _ { 1 }$ based on a compact representation of the experience which leads to a much more efficient design.
+
+For this, we use the concept of weighted average of its frames: $f _ { 1 } ( \mathbf { I } _ { 0 } , \dots , \mathbf { I } _ { T - 1 } ) = \sum _ { t = 0 } ^ { T - 1 } \alpha _ { t } \mathbf { I } _ { t }$ 6), wwith $\begin{array} { r } { \alpha _ { t } \ = \ \sum _ { i = t } ^ { T - 1 } \frac { 2 ( i + 1 ) - T - 1 } { i + 1 } } \end{array}$ Since the dynamic image is only sensitive to changes in the video, we complement it by computing also the median image $\bar { f } _ { 2 } ( { \bf I } _ { 0 } , \dots , { \bf I } _ { T - 1 } ) = \mathrm { m e d i a n } _ { t = 0 , \dots , T - 1 } { \bf I } _ { t }$ and combine the two functions $f _ { 1 }$ and $f _ { 2 }$ in a single representation $f$ by stacking the respective outputs along the channel dimension. With this design, we can rewrite the map $\Psi _ { 1 }$ as follows:
+
+$$
+m = \Psi _ { 1 } ( { \bf I } ( { \mathcal { E } } ) ) = \operatorname* { m a x p o o l } _ { n = 1 , \ldots , N } { \hat { \Psi } } _ { 1 } ( f ( { \bf I } ( { \mathcal { R } } _ { n } ) ) ) .
+$$
+
+Here the map $\hat { \Psi } _ { 1 } : \mathbb { R } ^ { 6 \times H \times W } \to \mathbb { R } ^ { 1 \times H \times W }$ , which can be implemented as a standard CNN, takes as input the dynamic/median image and produces as output the obstacle mask $_ { \mathbf { \nabla } } \mathbf { m } _ { \mathbf { \nabla } }$ . The pooling operator summarizes the information extracted from multiple runs into a single mask $_ { m }$ .
+
+We also consider a second similar map $\hat { \Psi } _ { 2 } : \mathbb { R } ^ { 6 \times H \times W }  \mathbb { R } ^ { C \times H \times W }$ to extract an “appearance” tensor $^ { a }$ . The latter helps the generator to render obstacles above or underneath the moving objects as needed. The response of this function is also max-pooled over runs, but channel-wise: $[ \mathrm { p o o l } _ { k = 1 , \dots , N } \mathbf { a } _ { k } ] _ { c v u } = [ \mathbf { a } _ { k ( c ) } ] _ { c v u }$ where $\begin{array} { r } { k ( c ) = \mathrm { a r g m a x } _ { k = 1 , \ldots , N } \sum _ { v u } [ { \pmb a } _ { k } ] _ { c v u } ^ { 2 } } \end{array}$ .
+
+In practice, $\hat { \Psi } _ { 1 }$ and $\hat { \Psi } _ { 2 }$ are implemented as a single neural network $\Psi : \mathbb { R } ^ { 6 \times H \times W }  \mathbb { R } ^ { ( 1 + C ) \times H \times W }$ where $\hat { \Psi } _ { 1 }$ is the first output channel and $\hat { \Psi } _ { 2 }$ the others.
+
+Optional obstacle mask supervision. In the experiments, we show that the map $\hat { \Psi }$ can be learned automatically without any external supervision. We contrast this with supervising $\hat { \Psi } _ { 1 }$ with an oracle rendition of the obstacle map. To this end, we define the tensor $m _ { \mathrm { g t } }$ for a scenario $s$ to be the indicator mask of whether a pixel contains a solid obstacles (including the perimetral walls) and then minimize the auxiliary loss $\dot { \ell } ( \pmb { m } _ { \mathrm { g t } } , \pmb { \Psi } _ { 1 } ( \mathbf { I } ( \pmb { \mathcal { E } } ) ) ) = \| \pmb { m } _ { \mathrm { g t } } - \Psi _ { 1 } ( \mathbf { I } ( \pmb { \mathcal { E } } ) ) \| ^ { \tilde { 2 } }$ .
+
+# 3.3 AUTO-REGRESSIVE PREDICTOR
+
+The predictor $\Phi$ is designed as an RNN that takes as input $T _ { 0 }$ past states $\mathbf { x } _ { t - T _ { 0 } : t } = \left( \mathbf { x } _ { t - T _ { 0 } } , \ldots , \mathbf { x } _ { t - 1 } \right)$ and outputs a new state $\mathbf { x } _ { t }$ . Each state variable is in turn a distributed representation of the physical state of the system, in the form of a “heatmap” $\mathbf { x } _ { t } \in \mathbb { R } ^ { H \times W }$ . Considering $T _ { 0 }$ past states allows the model to auto-regressively represent the dynamics of the system if so learning chooses to do.
+
+The predictor also takes as input the scenario representation $_ { m }$ given by eq. (3). The first $T _ { 0 }$ state variables are initialized from observations ${ \mathbf { I } } _ { 0 : T _ { 0 } } ( { \mathcal { R } } )$ via an initialization function $\eta : \mathbb { R } ^ { 3 \times H \times W } $ $\mathbb { R } ^ { H \times W }$ . We thus have
+
+$$
+\begin{array} { r l } { \mathbf { x } _ { 0 : T _ { 0 } } = [ \eta ( \mathbf { I } _ { t } ( \mathcal { R } ) ) ] _ { t = 0 : T _ { 0 } } \qquad } & { { } \mathrm { ( i n i t i a l i z a t i o n ) } } \\ { \mathbf { x } _ { t } = \Phi \bigl ( \mathbf { x } _ { t - T _ { 0 } : t } , m \bigr ) \qquad } & { { } \mathrm { ( a u t o - r e g r e s s i v e ~ p r e d i c t i o n ) } } \end{array}
+$$
+
+In short, this model estimates recursively the evolution of the system dynamics from the visual observations of the first $T _ { 0 }$ samples.
+
+Conditional generator. Variable $\mathbf { x } _ { t }$ contains information about the state of the moving objects (balls). This is then converted into a prediction of the corresponding video frame, combining also the appearance tensor $\textbf { \em a }$ and the median of the first $T _ { 0 }$ images in the sequence $\hat { \mathbf { I } } = \mathrm { m e d i a n } _ { t = 0 , \ldots , T _ { 0 } - 1 } \mathbf { I } _ { t }$ This is formulated as a conditional generator network $\hat { \mathbf { I } } _ { t } = g ( \mathbf { x } _ { t } , \pmb { a } , \hat { \mathbf { I } } )$ .
+
+Video reconstruction loss. Next, we discuss the nature of the loss equation 2. Owing also to the static background, the conditional video generation task is relatively simple provided that the dynamics are estimated correctly. As a consequence, the generated videos are likely to closely approximate the ground truth ones, so the loss function $\ell$ in equation 2 does not require to be complex. In our experiments, we combine the $L ^ { 2 }$ image loss with the perceptual loss of (Johnson et al., 2016). The latter is obtained by passing the ground-truth and predicted images through the first few layers of a pre-trained deep neural network $e$ , VGG-16 (Simonyan & Zisserman, 2015) in our case, and then comparing the resulting encodings in $L ^ { 2 }$ distance. The overall loss is then given by $\ell ( \mathbf { I } _ { t } , \hat { \mathbf { I } } _ { t } ) = \lambda _ { \mathbf { I } } \Vert \mathbf { I } _ { t } - \hat { \mathbf { I } } _ { t } \Vert ^ { 2 } + \lambda _ { p } \Vert e ( \mathbf { I } _ { t } ) - \bar { e } ( \hat { \mathbf { I } } _ { t } ) \Vert ^ { 2 }$ (details in the sup. mat.). The perceptual loss is robust to small shifts in the image reconstructions that may arise from imperfect physics predictions and favors reconstructing a sharp image; by comparison, the $L ^ { 2 }$ loss alone would result in blurrier images by regressing to the mean. In practice, trading-off the $L ^ { 2 }$ and perceptual losses results in qualitatively better images as well as in better trajectory prediction.
+
+State space integrity. An issue with the recursion equation 5 is that the state $\mathbf { x } _ { t }$ may, in the long run, falls outside the space of meaningful states, leading to a catastrophic failure of the predictor. This is especially true for predictions longer than the ones used for training the model. In order to encourage the recursion to maintain the integrity of the state over time, we add a self-consistency loss $\| \mathbf { x } _ { t } - \eta ( \mathbf { I } _ { t } ( \mathcal { R } ) ) \| ^ { 2 }$ . Here the network $\eta$ is the same that is used to extract the state space from the initial frames of the video sequence in equation 4 which weights are fixed. In practice, we find that, by using this additional loss, our recursion learns to maintain the integrity of the state nearly indefinitely.
+
+# 4 EXPERIMENTAL SETUP
+
+Data. A scenario $s$ (see embedded figure) is generated by sampling a certain number of obstacles of different types $m _ { i }$ and shapes $O _ { i }$ (either rectangular or custom), placing them at random locations and orientations on a 2D board. We consider boards with either two rectangular obstacles (denoted R2), a random number from 3 to 4 rectangular obstacles (R4) or two curved obstacles (C). Scenarios are rendered by painting the background and a wall around it. Then, all obstacle are painted in a solid colour, randomly picked from a fixed palette to ensure sufficient contrast. The background is also painted in solid color (general case) or by using a texture image (only curved shapes, denoted $\mathrm { C } { + } \mathrm { T }$ ). Crucially, there is no correlation between an obstacle’s type and its shape, location and color, so its type cannot be inferred by appearance alone.
+
+Runs $\mathcal { R }$ are generated by placing one or more dynamic objects (“balls”) with a random initial location and a random momentum oriented towards the obstacles, simulating the 2.1D physics, and rendering the balls as circles with a fixed color (see fig. A1 in supp. mat. for more examples).
+
+For each scenario $s$ , we sample $N + 1$ runs, using the first as a prediction run and the others as experience runs, forming triplets $( \mathcal S , \mathcal R , \mathcal E )$ . Unless otherwise stated, we set $N = 7$ and let each run evolve for 60 frames, including for the experience runs. Thanks to our compact representation, we only use 200MB for 7 experiences for backpropagation during training. Using a recurrent network requires an amount of memory larger in proportion with the length of an experience, i.e. 60-fold, so 12GB for $6 4 \times 6 4$ boards, which wouldn’t allow the reccurrent architecture to fit into memory. Using shorter runs is also not possible, as they tend to contain no collisions.
+
+For training, samples $( \mathcal { S } , \mathcal { R } , \mathcal { E } )$ are drawn in an on-line fashion and are thus unlimited. For testing, we use a fixed set of 200 samples. We generate boards of size $6 4 \times 6 4$ and $1 2 8 \times 1 2 8$ , but use the latter only for testing purposes to evaluate generalization. Unless otherwise stated, the board size is $6 4 \times 6 4$ .
+
+Evaluation metrics. We report the video prediction error as average $L _ { 2 }$ image loss. In order to assess the ability of the method to predict good trajectories, we note that blobs tend to emerge in the heatmaps $\mathbf { x } _ { t }$ in correspondence of the tracked objects. Hence, we handcrafted a simple blob detector $h ( \mathbf { x } _ { t } )$ that detects the blobs contained in the heatmaps $\mathbf { x } _ { t }$ (see sup. mat.). We then report the number of detected blobs vs the actual number of moving objects and, for each, the distance in pixel space of the predicted blob center and ground-truth object center, averaged over the different scenarios. For each experiment we report mean and standard deviation across all sampled scenarios.
+
+Baselines. We compare with three baselines. The first is a version of the Interaction Networks (Battaglia et al., 2016), trained with perfect knowledge of the object locations, background and object generation. The baseline works directly on the object positions and regresses object positions. The ground truth obstacle map is given as input and transformed into a vector thanks to a pre-trained VGG-16 architecture. We also reimplemented the RLVN network of Wang et al. (2018), which has a vector representation bottleneck, and compared video prediction error results. Note that this baseline wouldn’t adapt to bigger board. Our last baseline amounts to running the ground-truth physics simulator after removing the obstacles from the board (and thus results in perfect predictions for trajectories that do not hit solid obstacles).
+
+Implementation details. Networks $\Phi$ and $\eta$ share very similar auto-encoder type architecture, the network $\Psi$ uses a U-Net-like architecture (Ronneberger & Fischer, 2015). As stated in Wang et al. (2018) intuitively $\Psi$ would better preserve appearance while the structure of $\Phi$ would loose some appearance structure and put an inductive bias on dynamic predictions. $g$ is a fully-convolutional 6-layers stack. Our implementation uses TensorFlow 1.9 (Abadi et al., 2015) and the Adam optimizer (Kingma & Ba, 2014) with learning rate $1 0 ^ { - 4 }$ and Xavier initialization (Glorot & Bengio, 2010). We used a batch size of 10 samples $( \mathcal S , \mathcal R , \mathcal E )$ . Models are first trained for 110,000 iterations using only the $L ^ { 2 }$ and self-consistency losses for eq. (2), and then optionally fine-tuning for further 1,000 iterations using the perceptual loss as well. All models are trained on R2 with board size $6 4 ^ { 2 }$ and tested without any fine-tuning on different board types and sizes (see Tab. 1,2,3,A.3). For the $\mathrm { C } { + } \mathrm { T }$ scenarios, we first train the model using flat colour R2 scenarios and then fine-tune for 55,000 iterations using the textured data. Unless otherwise specified, models are trained in a fully unsupervised fashion. Full details can be found in the sup. mat.
+
+# 5 RESULTS
+
+Full system. Table 1 rows 1-4 report the prediction accuracy of our system on the 200 simulated test sequences using one moving object. In the table, we distinguish three durations: $T _ { 0 } = 4$ is the number of frames to bootstrap the prediction of a test runs, $T _ { \mathrm { t r a i n } } = 2 0$ is the length of prediction runs observed during meta-training and $T _ { \mathrm { t e s t } } = k T _ { \mathrm { t r a i n } }$ is the length of the runs observed during meta-testing, where $k = 1 , 3 , 5$ . We test both $6 4 \times 6 4$ boards (which is the same size used for training) and larger $1 2 8 \times 1 2 8$ boards to test spatial generalization. We also compare training using the $L ^ { 2 }$ or the $L ^ { 2 } +$ perceptual loss for video prediction supervision. We report the number of detected objects (which should be 1 in this experiment), the video prediction error, and the position prediction error.
+
+Table 1: Predicting one moving object. Obst. is the obstacle type (R2, R4, C). The test board size can be either $6 4 \times 6 4$ (same as in training) or $1 2 8 \times 1 2 8$ and we consider using either no supervision or obstacle supervision. We test the average prediction error at $T _ { \mathrm { t e s t } } = 2 0 , 6 0 , 1 0 0$ well above the duration $T _ { \mathrm { t r a i n } } = 2 0$ observed during training. Position errors were normalized by the board size diagonal. Video errors were normalized to board size $6 4 ^ { 2 }$ where larger. $L ^ { 2 }$ loss on positions is used for (14)-(15).   
+
+<table><tr><td></td><td></td><td></td><td></td><td></td><td colspan="3">Ttest =Ttrain = 20</td><td colspan="3">Ttest =3×Ttrain</td><td colspan="3">Ttest=5×Ttrain</td></tr><tr><td>No.</td><td>Obst.</td><td>Sup.</td><td>Test brd.size</td><td>Train loss</td><td># obj.</td><td>Vid. L2</td><td>Pos. err.</td><td># obj.</td><td>Vid. L2</td><td>Pos. err.</td><td># obj.</td><td>Vid. L2</td><td>Pos. err.</td></tr><tr><td>(1)</td><td>R2</td><td>None</td><td>642</td><td>L²</td><td>1.0±0.2</td><td>2.5±2.6</td><td>.036±.096</td><td>0.3±0.5</td><td>4.6±3.3</td><td>.404±.241</td><td>0.1±0.3</td><td>4.7±3.3</td><td>.526±.180</td></tr><tr><td>(2)</td><td>R2</td><td>None</td><td>1282</td><td>L²</td><td>0.7±0.4</td><td>1.7±1.3</td><td>.179±.236</td><td>0.1±0.3</td><td>1.7±1.1</td><td>.484±.228</td><td>0.0±0.1</td><td>1.7±1.1</td><td>.531±.209</td></tr><tr><td>(3)</td><td>R2</td><td>None</td><td>64</td><td>Percep.</td><td>1.0±0.2</td><td>2.5±2.9</td><td>.028±.073</td><td>1.0±0.2</td><td>5.3±4.0</td><td>.145±.148</td><td>1.0±0.2</td><td>5.5±4.0</td><td>.286±.168</td></tr><tr><td>(4)</td><td>R2</td><td>None</td><td>1282</td><td>Percep.</td><td>0.9±0.4</td><td>1.7±1.4</td><td>.104±.170</td><td>1.0±0.4</td><td>2.0±1.4</td><td>.227±.179</td><td>1.0±0.4</td><td>2.0±1.4</td><td>.320±.172</td></tr><tr><td>(5)</td><td>R2</td><td>Obst.</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>3.0±3.3</td><td>.018±.023</td><td>1.0±0.2</td><td>5.7±4.4</td><td>.112±.086</td><td>1.0±0.3</td><td>5.8±4.2</td><td>.231±.141</td></tr><tr><td>(6)</td><td>R2</td><td>Obst.</td><td>1282</td><td>Percep.</td><td>8.3±3.3</td><td>2.4±1.9</td><td>.323±.064</td><td>26.5±6.4</td><td>5.6±4.0</td><td>.355±.054</td><td>25.8±6.3</td><td>5.8±4.1</td><td>.363±.133</td></tr><tr><td>(7)</td><td>C</td><td>None</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>2.9±3.5</td><td>.040±.086</td><td>1.1±0.3</td><td>5.4±4.1</td><td>.175±.160</td><td>1.0±0.3</td><td>5.4±4.0</td><td>.265±.155</td></tr><tr><td>(8)</td><td>C</td><td>None</td><td>1282</td><td>Percep.</td><td>0.9±0.4</td><td>1.7±1.4</td><td>.104±.170</td><td>1.0±0.4</td><td>2.0±1.4</td><td>.227±.179</td><td>1.0±0.4</td><td>5.6±4.6</td><td>.320±.172</td></tr><tr><td>(9)</td><td>C</td><td>Obst</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>2.9±3.1</td><td>.024±.072</td><td>1.0±0.1</td><td>5.7±4.2</td><td>.128±.114</td><td>1.0±0.2</td><td>5.5±4.1</td><td>.247±.141</td></tr><tr><td>(10)</td><td>C+T</td><td>None</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>3.4±2.9</td><td>.017±.044</td><td>1.0±0.2</td><td>6.1±3.4</td><td>.094±.076</td><td>1.0±0.2</td><td>6.2±2.9</td><td>.222±.147</td></tr><tr><td>(11)</td><td>C+T</td><td>None</td><td>1282</td><td>Percep.</td><td>1.0±0.1</td><td>.675±.685</td><td>.012±.033</td><td>1.0±0.2</td><td></td><td>1.4±.814 .057±.065</td><td>0.9±0.3</td><td>1.4±.750</td><td>.121±.088</td></tr><tr><td></td><td></td><td></td><td></td><td>Ablation: removing the median and dynamic images,respectively (see text)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(12) (13)</td><td>R2 R2</td><td>None</td><td>642</td><td>Percep.</td><td>1.0±0.3</td><td>4.0±4.0</td><td>.074±.115 0.8±0.5</td><td></td><td>5.1±4.0</td><td>.302±.202</td><td>0.6±0.5</td><td>5.0±3.8</td><td>.357±.212</td></tr><tr><td></td><td></td><td>None</td><td>642</td><td>Percep.</td><td>1.4±0.6</td><td>3.2±3.4</td><td>.054±.062</td><td>1.8±0.8</td><td>5.5±4.2</td><td>.268±.151</td><td>1.7±0.8</td><td>5.6±4.2</td><td>.326±.136</td></tr><tr><td></td><td></td><td>Comparison: Interaction Network (14-15),RLVN(16)and Ground Truth Simulator minus Obstacles (17-18)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(14)</td><td>R2</td><td>Full</td><td>1282</td><td>L²Pos.</td><td>=</td><td>=</td><td>.038±.027</td><td></td><td></td><td>.166±.088</td><td>=</td><td>=</td><td>.331±.152</td></tr><tr><td>(15)</td><td>C</td><td>Full</td><td>1282</td><td>L²Pos.</td><td></td><td>-</td><td>.038±.031</td><td></td><td>=</td><td>.171±.095</td><td></td><td>-</td><td>.349±.203</td></tr><tr><td>(16)</td><td>R2</td><td>-</td><td>642</td><td></td><td></td><td>3.6±3.2</td><td></td><td></td><td>5.3±3.3</td><td></td><td>=</td><td>5.5±3.5</td><td></td></tr><tr><td>(17)</td><td>R2</td><td>=</td><td>64²</td><td>=</td><td></td><td>-</td><td>.060±.099</td><td></td><td>=</td><td>.229±.224</td><td>=</td><td>=</td><td>.224±.209</td></tr><tr><td>(18)</td><td>C</td><td></td><td>642</td><td></td><td></td><td>=</td><td>.055±.099</td><td></td><td></td><td>.219±.217</td><td></td><td>=</td><td>.232±.228</td></tr></table>
+
+We find that switching from the $L ^ { 2 }$ to the $L ^ { 2 }$ +perceptual loss for training (rows 1 vs 3 and 2 vs 4) reduces significantly the trajectory prediction error and prevents the object from disappearing. This, however, results to slightly larger $L _ { 2 }$ video prediction error compared to empty boards predicted in the long run by models trained with $L ^ { 2 }$ loss only. We also find that generalization through time is excellent: prediction errors accumulate over time (as it is unavoidable), but the recursion evolves without collapsing well beyond $T _ { \mathrm { t r a i n } }$ (qualitatively we found that the recursion can be applied indefinitely). This should be contrasted with prior work on intuitive physics (Fragkiadaki et al., 2016) where recursion was found to be unstable in the long term.
+
+Spatial generalization to larger boards (rows 2,4,6) is also satisfactory if the perceptual loss is used.   
+However, we did notice the emergence of a few artifacts in this case.
+
+In rows 5-6 we use external supervision to train the obstacle map predictor as suggested in section 3.2 (the rest of the system is still unsupervised). Supervision improves the trajectory prediction accuracy for boards of the same size as the ones seen in training. However, generalization to larger boards is poorer, suggesting that explicit supervision causes the model to overfit somewhat.
+
+Rows 7-11 show results for more complex shapes (C) and appearances $( \mathrm { C } \mathrm { + T } )$ . Prediction accuracy is only slightly worse than with the simpler obstacles (e.g. rows 3 vs 7 or 9 vs 10).
+
+Table 2: Importance of experience. For the unsupervised model, we report the trajectory prediction error at $T = 4 0$ (rows 1-3) and the obstacle mask $L _ { 2 }$ prediction error (rows 4-6) for different obstacle types. The number of runs $N$ in the experiences is varied from 0 to 50. For the obstacle mask prediction error, we also report the trivial baseline $\lVert \boldsymbol { m } _ { \mathrm { g t } } - \boldsymbol { m } _ { \mathrm { a l l - o n } } \rVert _ { 2 }$ as ‘All-ON’ (see text). Position errors were normalized by the board size $( 6 4 ^ { 2 } )$ diagonal. $T _ { \mathrm { t r a i n } } = 2 0$ .   
+
+<table><tr><td>No</td><td>Err.</td><td>Obst.</td><td>All-ON</td><td>N=0</td><td>1</td><td>2</td><td>5</td><td>7</td><td>10</td><td>20</td><td>50</td></tr><tr><td>(1)</td><td>Pos.</td><td>R2</td><td></td><td>.462±.136</td><td>.144±.157</td><td>.108±.136</td><td>.083±.128</td><td>.077±.120</td><td>.071±.114</td><td>.070±.116</td><td>.070±.104</td></tr><tr><td>(2)</td><td>Pos.</td><td>R4</td><td></td><td>.502±.151</td><td>.189±.169</td><td>.158±.164</td><td>.122±.155</td><td>.110±.149</td><td>.114±.130</td><td>.110±.154</td><td>.114±.152</td></tr><tr><td>(3)</td><td>Pos.</td><td>C</td><td></td><td>.459±.124</td><td>.161±.172</td><td>.125±.158</td><td>.099±.139</td><td>.099±.146</td><td>.099±.144</td><td>.101±.144</td><td>.106±.143</td></tr><tr><td>(4)</td><td>Obs.</td><td>R2</td><td>13.2±11.1</td><td>24.6±16.4</td><td>10.3±9.5</td><td>7.6±8.4</td><td>4.2±5.4</td><td>3.8±4.7</td><td>3.5±3.3</td><td>3.6±3.5</td><td>3.8±3.7</td></tr><tr><td>(6)</td><td>Obs.</td><td>R4</td><td>16.8±13.3</td><td>23.6±15.1</td><td>14.1±10.9</td><td>11.5±9.9</td><td>7.7±8.0</td><td>7.4±7.3</td><td>7.3±7.2</td><td>7.4±7.2</td><td>8.4±8.3</td></tr><tr><td>(5)</td><td>Obs.</td><td>C</td><td>9.4±8.9</td><td>27.8±15.8</td><td>10.1±8.2</td><td>7.0±7.0</td><td>5.4±5.6</td><td>5.3±5.2</td><td>5.1±5.2</td><td>5.7±5.6</td><td>6.0±5.9</td></tr></table>
+
+<table><tr><td>(U1)</td><td>(U2)</td><td>(U3)</td><td>(B1)</td><td>(B2)</td><td>(B3)</td></tr></table>
+
+Ablation. We suppress respectively the median image (row 12) and dynamic image (row 13) in the feature extractor $f$ summarizing the experiences. We notice a sensible deterioration of performance compared to row 3, suggesting that both components are important.
+
+Comparison. Compared to the fully-supervised IN (rows 14-15), our unsupervised method is highly competitive, especially for longer simulation times (rows 4,8). The method also outperforms (except for very long simulations where drift prevails) the ground-truth simulator that ignores the obstacles (rows 16,17). Finally we show that our method surpasses RLVN (Wang et al., 2018)(row 16), qualitatively, this method produces similar results to (13) and the object tends to disappear when encounting an object.
+
+The importance of experience. We investigate how much information the model extracts from the experiences $\mathcal { E }$ by ablating the number of runs $N$ in it. For the special case $N = 0$ , since there are no runs, we generate an pseudo experience run by copying the first frame $\mathbf { I } _ { 0 } ( \mathcal { R } )$ of the prediction run $\mathcal { R }$
+
+In table 2, we report the trajectory prediction accuracy for different obstacle types (row 1-3). We also test the ability to predict the obstacle mask $m _ { \mathrm { g t } }$ (defined in section 3). Since the model is unsupervised, the learned mask $_ { \mathbf { \nabla } } \mathbf { m } _ { \mathbf { \nabla } }$ will not in general be an exact match of $m _ { \mathrm { g t } }$ (e.g. the mask range is arbitrary); hence, after fixing all the parameters in the model, we train a simple regression network that maps $_ { \mathbf { \nabla } } \mathbf { m } _ { \mathbf { \nabla } }$ to $m _ { \mathrm { g t } }$ and report the prediction accuracy of the latter on the test set (see sup. mat. for details). As a point of comparison, we also report the error $\lVert m _ { \mathrm { g t } } - m _ { \mathrm { a l l - o n } } \rVert _ { 2 }$ of the trivial baseline that predict a mask $m _ { \mathrm { a l l - o n } }$ where all objects are highlighted, regardless of their flag.
+
+We note that there is a very large improvement when we move from zero to one experience runs, and a smaller but non-negligible improvement until $N = 1 0$ . Furthermore, comparing the results with predicting $m _ { \mathrm { a l l - o n } }$ shows that the system can tell from the experiences which obstacles are solid and which are not.
+
+Table 3: Predicting the obstacles: supervision vs no supervision. The table reports the obstacle mask prediction error $\left( L _ { 2 } \right)$ for a network trained with or without supervision for the obstacles (see text). The last column shows a multi-ball predictor operated with 3 balls in each run. $T _ { \mathrm { t r a i n } } = 2 0$ .   
+
+<table><tr><td>Supervision</td><td>R2</td><td>R4</td><td>C</td><td>C+T</td><td>R2 (3 balls) C+T → R2</td><td></td></tr><tr><td>Obstacle</td><td></td><td></td><td></td><td>2.7±4.4 6.3±7.8 3.5±4.9 2.6±3.9</td><td>2.6±3.9</td><td>7.1±9.2</td></tr><tr><td>None</td><td></td><td></td><td></td><td>3.8±4.7 7.4±7.3 5.3±5.2 4.0±5.2</td><td>3.9±3.5</td><td>7.6±8.9</td></tr></table>
+
+Supervised obstacle regression. Table 3 compares our unsupervised method to using full obstacle map supervision in order to predict the obstacle map $m _ { \mathrm { g t } }$ from the experiences $\mathcal { E }$ . For the unsupervised system, the obstacle map is estimated as explained in the paragraph above. As expected, supervised learning achieves a lower error, but the unsupervised method is still much better than the trivial baseline of table 2. Finally, we observe that for network fine-tuned on $\mathrm { C } { + } \mathrm { T }$ and evaluated back on R2 $( \mathbf { C } \mathbf { + } \mathbf { T } \mathbf { \xrightarrow { } } \mathbf { R } 2$ in Table 3) obstacle predictions results remain competitive with results on R2 only while object appearance and backgrounds drastically changed in the new setting.
+
+Multiple moving objects. We also test whether the system trained with a single moving object can generalize to multiple ones. In table A.3 (see sup. mat.) we show that the network can simulate the motion properly until balls collide, after which they merge (see videos in the sup. mat.). This indicates that, just as physical laws, the rules learned by the model are local. We also train a model (scenario R2) showing it from 1 to 3 balls in each run. This model is able not only to correctly handle ball collisions, but is also able to generalize correctly to several more objects on the board. Furthermore, as shown in table 3, this model still predicts correctly the obstacle masks from experiences, despite the act that the latter are much more crowded.
+
+# 6 CONCLUSIONS
+
+We have demonstrated a system that can learn an intuitive model of physics in an unsupervised manner. Differently from most prior works, our system is also able to learn on-the-fly and very efficiently some physical parameters of new scenarios from a few experiences. Prediction results are strong, competitive with fully-supervised models, and predictors generalize well over time space, and an arbitrary number of moving objects. Our next challenge is to apply the system to more difficult scenarios, including non-frontal views of the physical worlds. For this, we plan to learn a function to summarize past experiences in a more general manner than the dynamic image could.
+
+# REFERENCES
+
+Abadi et al. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.   
+P. Agrawal et al. Learning to Poke by Poking: Experiential Learning of Intuitive Physics. In Proc. NeurIPS, pp. 5074–5082, 2016.   
+P. Battaglia et al. Interaction networks for learning about objects, relations and physics. In Proc. NeurIPS, pp. 4502–4510, 2016.   
+P.W. Battaglia, J. Hamrick, and J.B. Tenenbaum. Simulation as an engine of physical scene understanding. PNAS, 110(45):18327–18332, 2013.   
+A. Bhattacharyya et al. Long-term image boundary prediction. In Thirty-Second AAAI Conference on Artificial Intelligence. AAAI, 2018.   
+H. Bilen et al. Dynamic image networks for action recognition. In IEEE CVPR, 2016.   
+M. B. Chang et al. A compositional object-based approach to learning physical dynamics. In Proc. ICLR, 2017.   
+S. Chiappa et al. Recurrent environment simulators. In Proc. ICLR, 2017.   
+M. Denil et al. Learning to perform physics experiments via deep reinforcement learning. Deep Reinforcement Learning Workshop, NIPS, 2016.   
+S. Ehrhardt et al. Unsupervised intuitive physics from visual observations. Proc. ACCV, 2018.   
+S. Ehrhardt et al. Taking visual motion prediction to new heightfields. Computer Vision and Image Understanding (CVIU), 2019.   
+C. Finn, I. Goodfellow, and S. Levine. Unsupervised learning for physical interaction through video prediction. In Advances in neural information processing systems, 2016a.   
+C. Finn et al. Deep spatial autoencoders for visuomotor learning. In Robotics and Automation (ICRA), 2016 IEEE International Conference on, pp. 512–519. IEEE, 2016b.   
+K. Fragkiadaki et al. Learning visual predictive models of physics for playing billiards. In Proc. NeurIPS, 2016.   
+X. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp. 249–256, 2010.   
+K. Greff et al. Neural expectation maximization. Proc. NeurIPS, 2017.   
+O. Groth et al. Shapestacks: Learning vision-based physical intuition for generalised object stacking. In eccv, 2018.   
+J. Johnson, A. Alahi, and L. Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. Proc. ECCV, 2016.
+
+K. Kansky et al. Schema networks: Zero-shot transfer with a generative causal model of intuitive physics. In International Conference on Machine Learning, pp. 1809–1818, 2017.
+
+D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. Proc. ICLR, 2014.
+
+A. Krizhevsky, I. Sutskever, and G. Hinton. Imagenet classification with deep convolutional neural networks. In Proc. NeurIPS, pp. 1097–1105, 2012.
+
+Ladický et al. Data-driven fluid simulations using regression forests. ACM Trans. on Graphics (TOG), 34(6):199, 2015.
+
+A. Lerer, S. Gross, and R. Fergus. Learning physical intuition of block towers by example. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, pp. 430–438, 2016.
+
+W. Li, A. Leonardis, and M. Fritz. Visual stability prediction and its application to manipulation. AAAI, 2017.
+
+A. Monszpart, N. Thuerey, and N. Mitra. SMASH: Physics-guided Reconstruction of Collisions from Videos. ACM Trans. on Graphics (TOG), 2016.
+
+R. Mottaghi et al. Newtonian scene understanding: Unfolding the dynamics of objects in static images. In IEEE CVPR, 2016.
+
+D. Mrowca et al. Flexible Neural Representation for Physics Prediction. Proc. NeurIPS, 2018.
+
+Anusha Nagabandi, Chelsea Finn, and Sergey Levine. Deep online learning via meta-learning: Continual adaptation for model-based rl. iclr, 2019a.
+
+Anusha Nagabandi et al. Learning to adapt in dynamic, real-world environments through metareinforcement learning. iclr, 2019b.
+
+J. Oh et al. Action-conditional video prediction using deep networks in atari games. In Advances in Neural Information Processing Systems, pp. 2863–2871, 2015.
+
+P. Ondruska and I. Posner. Deep tracking: Seeing beyond seeing using recurrent neural networks. In Proc. AAAI, 2016.
+
+R. Riochet et al. IntPhys: A Framework and Benchmark for Visual Intuitive Physics Reasoning. ArXiv e-prints, 2018.
+
+O. Ronneberger and T. Fischer, P.and Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, 2015.
+
+A. Sanchez-Gonzalez et al. Graph networks as learnable physics engines for inference and control. Proceedings of the 35th International Conference on International Conference on Machine Learning, 2018.
+
+K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In International Conference on Learning Representations, 2015.
+
+R. Stewart and S. Ermon. Label-free supervision of neural networks with physics and domain knowledge. In AAAI, pp. 2576–2582, 2017.
+
+J. Tompson et al. Accelerating Eulerian Fluid Simulation With Convolutional Networks. ArXiv e-print arXiv:1607.03597, 2016.
+
+S. van Steenkiste et al. Relational neural expectation maximization: Unsupervised discovery of objects and their interactions. Proc. ICLR, 2018.
+
+Z. Wang et al. Neural allocentric intuitive physics prediction from real videos. arXiv preprint arXiv:1809.03330, 2018.
+
+N. Watters et al. Visual interaction networks: Learning a physics simulator from video. In Proc. NeurIPS, 2017.
+
+J. Wu et al. Galileo: Perceiving physical object properties by integrating a physics engine with deep learning. In Proc. NeurIPS, pp. 127–135, 2015.
+
+J. Wu et al. Physics 101: Learning physical object properties from unlabeled videos. In Proc. BMVC, 2016.
+
+J. Wu et al. Learning to see physics via visual de-animation. In Proc. NeurIPS, 2017.
+
+# A1 DATA GENERATION
+
+In fig. A1 and fig. A2 we provide more details on the scene aspect. For solid background color, the objects and background colours are sampled to be different. Every rectangular object have height and width randomly sample in between 10 and 17 pixels. Custom object are loaded from template images and randomly scaled from 1 to 2.
+
+![](images/af104159ae4307f11e3728074a2d0a303c2f3ec09d6cd8af8b9fcbe9c044adf2.jpg)  
+Figure A1: Dataset samples. Pairs of sample data from left to right: R2, C, C+T.
+
+![](images/74ffdf6f57f91f4d34abc0e31310b69f4440aabd09275345e3081978f79c44f5.jpg)  
+Figure A2: Additional material used for data generation. From left to right: color palette for solid object/background color (R2,R4,C), texture used to increase background complexity in $( \mathrm { C } \mathrm { + T } )$ , set of custom objects for (C) data generation.
+
+# A2 IMPLEMENTATION DETAILS
+
+# A2.1 NETWORK ARCHITECTURE
+
+In fig. A3 we show the details of all the networks of our pipeline. The regression network mentionned in sec.5 is a simple 3 layers fully convolutional network with [3x3] kernel convolution and intermediate channel of size 64.
+
+# A2.2 TRAINING DETAILS
+
+In the overall loss $l$ is:
+
+$$
+\ell ( t ) = \lambda _ { \mathbf { I } } \Vert \mathbf { I } _ { t } - \hat { \mathbf { I } } _ { t } \Vert ^ { 2 } + \lambda _ { \mathbf { x } } \Vert \mathbf { x } _ { t } - \eta ( \mathbf { I } _ { t } ( \mathcal { R } ) ) \Vert ^ { 2 } + \lambda _ { p } \Vert e ( \mathbf { I } _ { t } ) - e ( \hat { \mathbf { I } } _ { t } ) \Vert ^ { 2 }
+$$
+
+With $t$ ranging from 0 to $T _ { t r a i n }$ . For $t = \{ 0 , . . , T _ { 0 } \}$ we used $\lambda _ { \mathbf { I } } = 1$ and $\lambda _ { \mathbf { x } } = \lambda _ { \mathbf { p } } = 0$ . In such case $\hat { I } _ { t } = g ( x _ { t } , \mathbf { a } , \hat { \mathbf { I } } ) = g ( \eta ( \mathbf { I } _ { t } ) , \mathbf { a } , \hat { \mathbf { I } } )$ .
+
+For $t > T _ { 0 }$ , when training with $L ^ { 2 }$ only we used $\lambda _ { \mathbf { I } } = \lambda _ { \mathbf { x } } = 1$ and $\lambda _ { p } = 0$ . When finetuning with perceptual we weighted the perceputal loss with coefficient $\lambda _ { p } = 1 0$ and the other losses with $\lambda _ { \mathbf { I } } = \lambda _ { \mathbf { x } } = 0 . 0 1$ . Finally, perceptual loss uses feature extracted from conv3 of VGG-16.
+
+The Interaction Network baseline is a simple Interaction Network to which state we concatenated the background feature extracted from a pre-trained VGG-16 network. The state propagator uses the last 4 states. The network is then trained over $T _ { t r a i n } = 4 0$ time steps with Adam optimizer with learning rate $1 0 ^ { - 4 }$ and batch size 50. We found that $5 0 0 0 0$ iterations were sufficient to reach convergence.
+
+All experiments are run using single NVIDIA-GPU Titan X.
+
+# A2.3 EVALUATION DETAILS
+
+To detect blob create a binary image by manually thresholding the intermediate heatmap $\mathbf { x } _ { t }$ and using a simple Hough circle detector. Every threshold coefficient is hand picked for every model but kept constant across scenarios and different data.
+
+For position errors we used an $L _ { 2 }$ loss, when more than one object was detected we took the distance to the closest detected object and applied a fix penalty to the loss for all other detection (half the board size).
+
+![](images/1fb652321621dc8fdc7664e7b623662fc03c1cb892f8b91728443e0d15089061.jpg)  
+Figure A3: Networks architecture detail. All the networks generally uses layer normalization $\eta$ and $\Phi$ are simple auto-encoder type architecture while $\Psi$ shares similarity with U-Net.
+
+Table A1: Predicting one moving object. Obstacle type is R2. The test board size is $6 4 \times 6 4$ We test the average prediction error at $T _ { \mathrm { t e s t } } = 2 0$ , 60, 100 well above the duration $T _ { \mathrm { t r a i n } } = 2 0$ observed during training. Position errors were normalized by the board size diagonal.
+
+<table><tr><td></td><td></td><td></td><td colspan="3">Ttest = Ttrain = 20</td><td colspan="3">Trest =3× Ttrain</td><td colspan="3">Ttest =5×Ttrain</td></tr><tr><td>No.</td><td>Sup.</td><td>Train loss</td><td>#obj.</td><td>Vid. L2</td><td>Pos. err.</td><td>#obj.</td><td>Vid. L2</td><td>Pos. err.</td><td>#obj.</td><td>Vid. L2</td><td>Pos. err.</td></tr><tr><td>(1)</td><td>Oracle</td><td>Perc.</td><td>1.0±0.1</td><td>2.5±2.9</td><td>.013±.012</td><td>1.0±0.1</td><td>5.8±4.4</td><td>.095±.095</td><td>1.0±0.3</td><td>6.2±4.5</td><td>.201±.136</td></tr><tr><td>(2)</td><td>None</td><td>Perc.</td><td>1.0±0.1</td><td>3.2±3.1</td><td>.030±.068</td><td>1.0±0.2</td><td>5.5±4.0</td><td>.136±.151</td><td>0.9±0.4</td><td>5.5±4.0</td><td>.262±.180</td></tr></table>
+
+# A3 ADDITIONAL RESULTS
+
+In Table A1 we report two more results on one moving object prediction. (1) is an upper bound of our system had full knowledge of $\{ B , U \}$ obstacle locations at training time. We simply replaced $_ { m }$ with the binary map of $B$ obstacles and $\textbf { \em a }$ with the binary map of $U$ obstacles where 1s indicate the presence of the obstacle type. In this scenario we note although that we obtain an upper bound of the results introduced in Table 1, our method still remains competitive with this upper bound. (2) reports results of the network trained on $\mathrm { C } { + } \mathrm { T }$ and evaluated back on R2. As in Table 3 we see that the network still manage to make accurate predictions. Table A2 is an extension of Table 2 in the paper for the supervised model and Table A3 introduces results on multiple balls prediction.
+
+Table A2: Importance of experience. For the supervised model, we report the trajectory prediction error at $T = 4 0$ (rows 1-3) and the obstacle map prediction error (rows 4-6) for different obstacle types. The number of runs $N$ in the experiences is varied from 0 to 50. Position errors were normalized by the board size diagonal. $T _ { \mathrm { t r a i n } } = 2 0$ .   
+
+<table><tr><td>No</td><td>Err.</td><td>Obst.</td><td>N=0</td><td>1</td><td>2</td><td>5</td><td>7</td><td>10</td><td>20</td><td>50</td></tr><tr><td>(1)</td><td>Pos.</td><td>R2</td><td>1.81±1.41</td><td>.94±.975</td><td>.53±.81</td><td>.15±0.41</td><td>.08±.21</td><td>.07±.16</td><td>.10±.24</td><td>.19±.38</td></tr><tr><td>(2)</td><td>Pos.</td><td>R4</td><td>3.28±1.94</td><td>2.22±1.67</td><td>1.56±1.40</td><td>.63±.84</td><td>.43±.68</td><td>.37±.60</td><td>.44±.61</td><td>.83±.97</td></tr><tr><td>(3)</td><td>Pos.</td><td>C</td><td>1.05±0.91</td><td>.58±.61</td><td>.36±.49</td><td>.15±.29</td><td>.13±.26</td><td>.12±.22</td><td>.16±.26</td><td>.24±.36</td></tr><tr><td>(4)</td><td>Obs.</td><td>R2</td><td>4.1±4.0</td><td>3.6±3.8</td><td>3.1±3.4</td><td>2.3±2.5</td><td>2.3±2.4</td><td>2.3±2.4</td><td>2.3±2.7</td><td>2.6±3.0</td></tr><tr><td>(5)</td><td>Obs.</td><td>R4</td><td>6.1±2.2</td><td>4.1±3.8</td><td>3.1±3.4</td><td>2.8±3.3</td><td>2.7±3.1</td><td>2.6±3.0</td><td>2.7±2.9</td><td>3.3±3.6</td></tr><tr><td>(6)</td><td>Obs.</td><td>C</td><td>3.9±3.9</td><td>3.3±3.7</td><td>2.8±3.1</td><td>2.4±2.7</td><td>2.4±2.8</td><td>2.3±2.6</td><td>2.4±2.6</td><td>2.5±2.9</td></tr></table>
+
+Table A3: Predicting multiple moving object. Obstacle type is R2, all network were trained on the fully unsupervised with perceptual loss. Num. is the maxmium number of balls per run during training. For instance ${ \mathrm { N u m } } { = } 3$ means that every runs sampled contained one to three balls. The test board is size $6 4 \times 6 4$ We test the average prediction error at $T _ { \mathrm { t e s t } } = 2 0 , 6 0 , 1 0 0$ well above the duration $T _ { \mathrm { t r a i n } } = 2 0$ observed during training.   
+
+<table><tr><td></td><td colspan="3">Ttest = Ttrain=20 Ttest =3×Ttrain Ttest =5×Ttrain</td></tr><tr><td></td><td></td><td></td><td>Num.#obj.Vid.L2#obj. Vid.L2 #obj. Vid. L2</td></tr><tr><td></td><td colspan="3">3 Balls</td></tr><tr><td></td><td>12.6±0.8 6.5±5.2</td><td></td><td>1.7±0.9 8.5±5.9 1.6±0.8 8.5±6.0</td></tr><tr><td></td><td></td><td>33.1±0.64.6±4.8</td><td>2.9±0.8 10.4±7.6 2.9±0.9 11.0±8.0</td></tr><tr><td></td><td></td><td>5 Balls</td><td></td></tr><tr><td>1</td><td>3.3±1.0</td><td>8.9±6.6</td><td>2.0±1.0 10.2±7.4 3.1±0.6 1.8±.775</td></tr><tr><td></td><td>35.5±0.7</td><td>6.6±6.2</td><td>5.1±1.1 13.0±9.5 5.1±1.0 13.5±10.0</td></tr></table>

parse/train/SJlOq34Kwr/SJlOq34Kwr_content_list.json

@@ -0,0 +1,1522 @@
+[
+    {
+        "type": "text",
+        "text": "UNSUPERVISED INTUITIVE PHYSICS FROM PAST EXPERIENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            826,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            181,
+            170,
+            397,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the problem of learning models of intuitive physics from raw, unlabelled visual input. Differently from prior work, in addition to learning general physical principles, we are also interested in learning “on the fly” physical properties specific to new environments, based on a small number of environment-specific experiences. We do all this in an unsupervised manner, using a meta-learning formulation where the goal is to predict videos containing demonstrations of physical phenomena, such as objects moving and colliding with a complex background. We introduce the idea of summarizing past experiences in a very compact manner, in our case using dynamic images, and show that this can be used to solve the problem well and efficiently. Empirically, we show, via extensive experiments and ablation studies, that our model learns to perform physical predictions that generalize well in time and space, as well as to a variable number of interacting physical objects. ",
+        "bbox": [
+            233,
+            270,
+            766,
+            436
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            460,
+            336,
+            477
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many animals possess an intuitive understanding of the physical world. They use this understanding to accurately and rapidly predict events from sparse sensory inputs. In addition to general physical principles, many animals also learn specific models of new environments as they experience them over time. For example, they can explore an environment to determine which parts of it can be navigated safely and remember this knowledge for later reuse. ",
+        "bbox": [
+            174,
+            484,
+            823,
+            554
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Authors have looked at equipping artificial intelligences (AIs) with analogous capabilities, but focusing mostly on performing predictions from instantaneous observations of an environment, such as a few frames in a video. However, such predictions can be successful only if observations are combined with sufficient prior knowledge about the environment. For example, consider predicting the motion of a bouncing ball. Unless key parameters such as the ball’s elasticity are known a priori, it is impossible to predict the ball’s trajectory accurately. However, after observing at least one bounce, it is possible to infer some of the parameters and eventually perform much better predictions. ",
+        "bbox": [
+            174,
+            560,
+            825,
+            659
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we are interested in learning intuitive physics in an entirely unsupervised manner, by passively watching videos. We consider situations in which objects interact with scenarios that can only be partially inferred from their appearance, but that also contain objects whose parameters cannot be confidently predicted from appearance alone (fig. 1). Then, we consider learning a system that can observe a few physical experiments to infer such parameters, and use this knowledge to perform better predictions in the future. ",
+        "bbox": [
+            174,
+            665,
+            825,
+            748
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our model has three goals. First, it must learn without the use of any external or ad-hoc supervision.   \nWe achieve this by training our model from raw videos, using a video prediction error as a loss. ",
+        "bbox": [
+            174,
+            756,
+            823,
+            784
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Second, our model must be able to extract information about a new scenario by observing a few experiments, which we formulate as meta-learning. We also propose a simple representation of the experiments based on the concept of “dynamic image” that allows to process long experiments more efficiently than using a conventional recurrent network. ",
+        "bbox": [
+            174,
+            791,
+            823,
+            847
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Third, our model must learn a good representation of physics without access to any explicit or external supervision. Instead, we propose three tests to support this hypothesis. (i) We show that the model can predict far in the future, which is a proxy to temporal invariance. (ii) We further show that the model can extend to scenarios that are geometrically much larger than the ones used for training, which is a proxy to spatial invariance. (iii) Finally, we show that the model can generalize to several moving objects, which is a proxy to locality. Locality and time-space invariance are of course three key properties of physical laws and thus we should expect any good intuitive model of physics to possess them. ",
+        "bbox": [
+            176,
+            854,
+            823,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/260c1b139e0592126ca5f693f20c78a00a2190f8fe352e8ddc563ff9d47e2b66.jpg",
+        "image_caption": [
+            "Figure 1: Probing a new environment. Given the input showing a (purple) ball moving from right to left, three possible future outcomes are possible: the ball passes above (A), passes under (U), or bounces (B) off the object. This cannot be predicted from appearance alone. Instead, our model learns the property of the object by probing each new environment via a small set of experiments. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            99,
+            790,
+            179
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            255,
+            821,
+            296
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In order to support these claims, we conduct extensive experiments in simulated scenarios, including testing the ability of the model to cope with non-trivial visual variations of the inputs. While the data is simpler than a real-world application, we nevertheless make substantial progress compared to previous work, as discussed in section 2. We do so by learning from passive, raw video data a good model of dynamics and collisions that generalizes well spatially, temporally, and to a variable number of objects. The scalability of our approach, via the use of the dynamic image, is also unique. Finally, we investigate the problem of learning the parameters of new scenarios on the fly via experiences and we propose an effective solution to do so. ",
+        "bbox": [
+            173,
+            303,
+            825,
+            415
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            435,
+            341,
+            450
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A natural way to represent physics is to manually encode every object parameter and physical property (e.g., mass, velocity, positions) and use supervision to make predictions. This has been widely used to represent and propagate physics (Wu et al., 2015; 2016; Battaglia et al., 2016; Chang et al., 2017; Mrowca et al., 2018; Sanchez-Gonzalez et al., 2018). If models like Wu et al. (2015; 2016) also estimate environment parameters, these works rely on a physics engine that assumes strong priors about the scenario, while our approach does not require such constraint. ",
+        "bbox": [
+            174,
+            458,
+            825,
+            541
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Inspired by the recent successes of Convolutional Neural Networks (CNNs (Krizhevsky et al., 2012)) and their application to implicit representation of dynamics (Ondruska & Posner, 2016; Oh et al., 2015; Chiappa et al., 2017; Bhattacharyya et al., 2018), researchers (Watters et al., 2017; Ehrhardt et al., 2019; Kansky et al., 2017) have tried to base their approaches on visual inputs. They learn from several frames of a scene to regress the next physical state of a system. In general these approaches learn an implicit representation of physics (Ehrhardt et al., 2019; Watters et al., 2017) as a tensor state from recurrent deep networks. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            645
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Such models are mostly supervised using ground-truth information about key physical parameters (e.g., positions, velocities, density) during training. While these approaches require an expensive annotation of data, other works have tried to learn from unsupervised data as well. Researchers have successfully learned unsupervised models either through active manipulation (Agrawal et al., 2016; Denil et al., 2016; Finn et al., 2016a), using the laws of physics (Stewart & Ermon, 2017), using dynamic clues and invariances (Greff et al., 2017; van Steenkiste et al., 2018) or features extracted from unsupervised methods (Finn et al., 2016b; Ehrhardt et al., 2018). Fragkiadaki et al. (2016) also used an unsupervised system like ours, however, they assumed the rendering system and the number of objects to be known, which we do not, which prevented the internal representation to fail over time. Perhaps most related to our approach is the work of Wang et al. (2018), where the model is learnt using future image prediction on a simple synthetic task and then transferred to real world scenarios. They also demonstrate long-term dynamic predictions, however they did not generalize to different background/number of balls. ",
+        "bbox": [
+            173,
+            652,
+            825,
+            833
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In other works, models are taught to answer simple qualitative questions about a physical setup, such as: the stability of stacks of objects (Battaglia et al., 2013; Lerer et al., 2016; Li et al., 2017; Groth et al., 2018), the likelihood of a scenario (Riochet et al., 2018), the forces acting behind a scene (Wu et al., 2017; Mottaghi et al., 2016) or properties of objects through manipulation (Agrawal et al., 2016; Denil et al., 2016). Other papers compromise between qualitative and quantitative predictions and focus on plausibility (Tompson et al., 2016; Ladický et al., 2015; Monszpart et al., 2016). ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/4e6b55d84e3f030b72b198c2fa02083183b99be36b1ca078b0df1b13ad1450ed.jpg",
+        "image_caption": [
+            "Figure 2: Overview of our method. $\\Psi$ (bottom right block) acts as a meta-learning module. It takes as input past experiments compressed into dynamic images alongside with the median image and learn to optimise $_ { m }$ and $\\textbf { \\em a }$ to minimise the final loss. $\\eta$ (top-left yellow block) extracts states $\\mathbf { x } _ { 0 : T _ { 0 } }$ which are carried forward with the auto-regressive predictor $\\Phi$ . Finally $g$ renders frame $\\hat { \\mathbf { I } } _ { t }$ from $\\mathbf { x } _ { t }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            103,
+            820,
+            330
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Finally similar problems can be found in online learning settings (Nagabandi et al., 2019a;b). These works also use meta-learning but, differently from ours, allow their models to learn and adapt at test time from feedbacks. ",
+        "bbox": [
+            176,
+            404,
+            821,
+            445
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHOD ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            460,
+            282,
+            477
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We describe our model, illustrated in fig. 2, starting from the input and output data. ",
+        "bbox": [
+            176,
+            483,
+            715,
+            500
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A scenario $S$ is a physical environment that supports moving objects interacting with it. In this paper, we take as scenarios 2.1D environments containing obstacles and we consider rolling balls as moving objects. Hence, interactions are in the form of bounces. Formally, a scenario is defined over a lattice $\\Omega = \\{ 0 , \\dots , H - 1 \\} \\times \\{ 0 , \\dots , W - 1 \\}$ and is specified by a list of obstacles $\\boldsymbol { \\mathcal { S } } = \\{ ( O _ { j } , b _ { j } )$ , $j =$ $1 , \\ldots , K \\}$ . Here $O _ { j } \\subset \\Omega$ is the shape of an obstacle and $b _ { j } \\in \\{ B , A , U \\}$ is a flag that tells whether the ball bounces against it (B), passes above it (A), or passes under it (U). Obstacles do not overlap. ",
+        "bbox": [
+            173,
+            505,
+            825,
+            589
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A run is a tuple $\\mathcal { R } = ( \\mathcal { S } , \\mathbf { y } )$ associating a scenario $s$ with a trajectory $\\mathbf { y } = ( y _ { t } \\in \\Omega , t = 0 , \\dots , T - 1 )$ for the ball (this is trivially extended to multiple balls). Scenarios and runs are sensed visually. The symbol $\\mathbf { I } ( \\pmb { S } ) : \\Omega  \\mathbb { R } ^ { 3 \\times } \\mathbf { \\bar { H } } \\times \\boldsymbol { W }$ denotes the image generated by observing a scenario with no ball and $\\mathbf { I } _ { t } ( \\mathcal { R } ) = \\mathbf { I } ( S , y _ { t } )$ is the image generated by observing the ball at time $t$ in a run. The symbol $\\mathbf { I } ( \\mathcal { R } ) = ( \\mathbf { I } _ { t } ( \\mathcal { R } ) , t = 0 , \\dots , T - 1 )$ denotes the collection of all frames in a run, which can be thought of as a video. ",
+        "bbox": [
+            173,
+            594,
+            825,
+            679
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are interested in learning intuitive physics with no explicit supervision on the object trajectories, nor an explicit encoding of the laws of mechanics that govern the motion of the objects and their interactions and collisions with the environment. Hence, we cast this problem as predicting the video $\\mathbf { I } ( \\mathcal { R } )$ of the trajectory of the objects given only a few initial frames $\\mathbf { I } _ { 0 : T _ { 0 } } ( \\mathcal { R } ) = ( \\mathbf { I } _ { 0 } ( \\mathcal { R } ) , \\dots , \\mathbf { I } _ { T _ { 0 } - 1 } ( \\mathcal { R } ) )$ , where $T _ { 0 } \\ll T$ . ",
+        "bbox": [
+            174,
+            686,
+            825,
+            757
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We are not the first to consider a similar learning problem, although most prior works do require some form of external supervision, which we do not use. Here, however, we consider an additional key challenge that the images ${ \\mathbf { I } } _ { 0 : T _ { 0 } } ( { \\mathcal { R } } )$ do not contain sufficient information to successfully predict the long-term objects’ motion. This is because these few frames tell us nothing about the nature of the obstacles in the scenario. In fact, under the assumptions that obstacles of type $B$ , $A$ and $U$ have similar or even the identical appearance, it is not possible to predict whether the ball will bounce, move above, or move under any such obstacle. This situation is representative of an agent that needs to operate in a new complex environment and must learn more about it, from past experiences, before it can do so reliably. ",
+        "bbox": [
+            173,
+            762,
+            825,
+            888
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Thus, we consider a modification of the setup described above in which the model can experience each new scenario for a while, by observing the motion of the ball, before making its own predictions. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Formally, an experience is a collection of $N$ runs ${ \\mathcal { E } } = ( { \\mathcal { R } } _ { 1 } , \\ldots , { \\mathcal { R } } _ { N } )$ all relative to the same scenario $s$ with each a different trajectories $\\mathbf { y } _ { 1 } , \\ldots , \\mathbf { y } _ { N }$ . By observing such examples, the model must determine the nature of the obstacles and then use this information to correctly predict the motion of the ball in the future. We cast this as the problem of learning a mapping ",
+        "bbox": [
+            173,
+            102,
+            825,
+            160
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e9dd35250c76e327cbe9e09648a78c7ad051b7c57f714f2c087c6cc5852bf6fe.jpg",
+        "text": "$$\n\\Omega : \\left( \\mathrm { I } _ { 0 : T _ { 0 } } ( { \\mathcal { R } } ) , \\mathrm { I } ( { \\mathcal { E } } ) \\right) \\ \\longmapsto \\ \\mathrm { I } ( { \\mathcal { R } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            165,
+            617,
+            183
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { I } ( { \\mathcal { E } } ) = ( \\mathbf { I } ( { \\mathcal { R } } _ { 1 } ) , \\ldots , \\mathbf { I } ( { \\mathcal { R } } _ { N } ) )$ are the videos corresponding to the runs in the experience. We will call $\\mathcal { R }$ the prediction run to distinguish it from the experience runs $\\mathcal { E }$ . ",
+        "bbox": [
+            169,
+            188,
+            823,
+            217
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 A META-LEARNING PERSPECTIVE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            228,
+            446,
+            243
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The setup we have described above can be naturally described as meta-learning. Namely, eq. (1) can be thought of as incorporating a “local” learning rule $\\mathcal { M }$ that maps the experience $\\mathcal { E }$ to a scenariospecific predictor $\\hat { \\Omega }$ on the fly: $\\hat { \\Omega } ( \\cdot ) = \\Omega ( \\cdot , \\mathbf { I } ( \\mathcal { E } ) ) = \\mathcal { M } [ \\mathbf { I } ( \\mathcal { E } ) ]$ . Hence $\\mathcal { M }$ must extract from the experience as much information as possible about the underlying scenario and transfer it to the scenario-specific predictor $\\hat { \\Omega }$ . In order to learn $\\mathcal { M }$ , we consider meta-samples of the type $( \\mathcal S , \\mathcal R , \\mathcal E )$ comprising a scenario $s$ , a prediction run $\\mathcal { R }$ and $N$ experience runs $\\mathcal { E }$ . Given a dataset $\\mathcal { D }$ of such meta-samples, meta-learning searches for the mapping $\\mathcal { M }$ that minimizes the error on the prediction runs: ",
+        "bbox": [
+            173,
+            250,
+            826,
+            364
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d60d61cb338fd8fb1f9d09e2a7d62ab65bc4f76a69777266f82965c1dd5d1b16.jpg",
+        "text": "$$\n\\mathcal { M } ^ { * } = \\operatorname * { a r g m i n } _ { \\mathcal { M } } \\frac { 1 } { | \\mathcal { D } | } \\sum _ { ( \\mathcal { S } , \\mathcal { R } , \\mathcal { E } ) \\in \\mathcal { D } } \\ell ( \\mathbf { I } ( \\mathcal { R } ) , \\hat { \\Omega } ( \\mathbf { I } _ { 0 : T _ { 0 } } ( \\mathcal { R } ) ) ) , \\quad \\hat { \\Omega } = \\mathcal { M } [ \\mathbf { I } ( \\mathcal { E } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            359,
+            735,
+            401
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 COMPRESSED DYNAMIC EXPERIENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            412,
+            480,
+            426
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Concretely, we parameterise the scenario-specific predictor $\\hat { \\Omega } ( \\cdot ; w , m )$ using parameters $\\pmb { w }$ , the weights a deep neural network, which is fixed and scenario-independent, and $_ { \\mathbf { \\nabla } } \\mathbf { m }$ , which is scenariospecific. The latter is extracted by a network ${ \\pmb m } = \\Psi _ { 1 } ( { \\pmb I } ( { \\pmb \\mathcal { E } } ) )$ from the experience videos $\\mathbf { I } ( \\mathcal { E } )$ . Since we expect $_ { \\mathbf { \\nabla } } \\mathbf { m } _ { \\mathbf { \\nabla } }$ to provide information about the nature of the obstacles in the scenario, we let $\\pmb { m } \\in \\mathbb { R } ^ { 1 \\times H ^ { \\star } W }$ be a tensor with the same spatial resolution as the scenario and interpret it as an “obstacle mask”. ",
+        "bbox": [
+            173,
+            435,
+            826,
+            520
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Given that the function $\\Psi _ { 1 }$ takes as input a number of video sequences, one may think to implement it as a recurrent neural network; however, recurrent networks scale badly, especially in our metalearning context — to the point that we had difficulties in even running such a model at a non-trivial scale. Instead, we propose to construct $\\Psi _ { 1 }$ based on a compact representation of the experience which leads to a much more efficient design. ",
+        "bbox": [
+            173,
+            527,
+            825,
+            598
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For this, we use the concept of weighted average of its frames: $f _ { 1 } ( \\mathbf { I } _ { 0 } , \\dots , \\mathbf { I } _ { T - 1 } ) = \\sum _ { t = 0 } ^ { T - 1 } \\alpha _ { t } \\mathbf { I } _ { t }$ 6), wwith $\\begin{array} { r } { \\alpha _ { t } \\ = \\ \\sum _ { i = t } ^ { T - 1 } \\frac { 2 ( i + 1 ) - T - 1 } { i + 1 } } \\end{array}$ Since the dynamic image is only sensitive to changes in the video, we complement it by computing also the median image $\\bar { f } _ { 2 } ( { \\bf I } _ { 0 } , \\dots , { \\bf I } _ { T - 1 } ) = \\mathrm { m e d i a n } _ { t = 0 , \\dots , T - 1 } { \\bf I } _ { t }$ and combine the two functions $f _ { 1 }$ and $f _ { 2 }$ in a single representation $f$ by stacking the respective outputs along the channel dimension. With this design, we can rewrite the map $\\Psi _ { 1 }$ as follows: ",
+        "bbox": [
+            173,
+            603,
+            826,
+            693
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbe5b65fc748c75e91c49709db377d0647b6e3cda28ab1293eab31446d053e09.jpg",
+        "text": "$$\nm = \\Psi _ { 1 } ( { \\bf I } ( { \\mathcal { E } } ) ) = \\operatorname* { m a x p o o l } _ { n = 1 , \\ldots , N } { \\hat { \\Psi } } _ { 1 } ( f ( { \\bf I } ( { \\mathcal { R } } _ { n } ) ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            698,
+            645,
+            727
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here the map $\\hat { \\Psi } _ { 1 } : \\mathbb { R } ^ { 6 \\times H \\times W } \\to \\mathbb { R } ^ { 1 \\times H \\times W }$ , which can be implemented as a standard CNN, takes as input the dynamic/median image and produces as output the obstacle mask $_ { \\mathbf { \\nabla } } \\mathbf { m } _ { \\mathbf { \\nabla } }$ . The pooling operator summarizes the information extracted from multiple runs into a single mask $_ { m }$ . ",
+        "bbox": [
+            173,
+            733,
+            825,
+            779
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We also consider a second similar map $\\hat { \\Psi } _ { 2 } : \\mathbb { R } ^ { 6 \\times H \\times W }  \\mathbb { R } ^ { C \\times H \\times W }$ to extract an “appearance” tensor $^ { a }$ . The latter helps the generator to render obstacles above or underneath the moving objects as needed. The response of this function is also max-pooled over runs, but channel-wise: $[ \\mathrm { p o o l } _ { k = 1 , \\dots , N } \\mathbf { a } _ { k } ] _ { c v u } = [ \\mathbf { a } _ { k ( c ) } ] _ { c v u }$ where $\\begin{array} { r } { k ( c ) = \\mathrm { a r g m a x } _ { k = 1 , \\ldots , N } \\sum _ { v u } [ { \\pmb a } _ { k } ] _ { c v u } ^ { 2 } } \\end{array}$ . ",
+        "bbox": [
+            174,
+            784,
+            826,
+            847
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In practice, $\\hat { \\Psi } _ { 1 }$ and $\\hat { \\Psi } _ { 2 }$ are implemented as a single neural network $\\Psi : \\mathbb { R } ^ { 6 \\times H \\times W }  \\mathbb { R } ^ { ( 1 + C ) \\times H \\times W }$ where $\\hat { \\Psi } _ { 1 }$ is the first output channel and $\\hat { \\Psi } _ { 2 }$ the others. ",
+        "bbox": [
+            171,
+            852,
+            821,
+            885
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Optional obstacle mask supervision. In the experiments, we show that the map $\\hat { \\Psi }$ can be learned automatically without any external supervision. We contrast this with supervising $\\hat { \\Psi } _ { 1 }$ with an oracle rendition of the obstacle map. To this end, we define the tensor $m _ { \\mathrm { g t } }$ for a scenario $s$ to be the indicator mask of whether a pixel contains a solid obstacles (including the perimetral walls) and then minimize the auxiliary loss $\\dot { \\ell } ( \\pmb { m } _ { \\mathrm { g t } } , \\pmb { \\Psi } _ { 1 } ( \\mathbf { I } ( \\pmb { \\mathcal { E } } ) ) ) = \\| \\pmb { m } _ { \\mathrm { g t } } - \\Psi _ { 1 } ( \\mathbf { I } ( \\pmb { \\mathcal { E } } ) ) \\| ^ { \\tilde { 2 } }$ . ",
+        "bbox": [
+            174,
+            892,
+            823,
+            924
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            147
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 AUTO-REGRESSIVE PREDICTOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            164,
+            431,
+            178
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The predictor $\\Phi$ is designed as an RNN that takes as input $T _ { 0 }$ past states $\\mathbf { x } _ { t - T _ { 0 } : t } = \\left( \\mathbf { x } _ { t - T _ { 0 } } , \\ldots , \\mathbf { x } _ { t - 1 } \\right)$ and outputs a new state $\\mathbf { x } _ { t }$ . Each state variable is in turn a distributed representation of the physical state of the system, in the form of a “heatmap” $\\mathbf { x } _ { t } \\in \\mathbb { R } ^ { H \\times W }$ . Considering $T _ { 0 }$ past states allows the model to auto-regressively represent the dynamics of the system if so learning chooses to do. ",
+        "bbox": [
+            174,
+            184,
+            825,
+            241
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The predictor also takes as input the scenario representation $_ { m }$ given by eq. (3). The first $T _ { 0 }$ state variables are initialized from observations ${ \\mathbf { I } } _ { 0 : T _ { 0 } } ( { \\mathcal { R } } )$ via an initialization function $\\eta : \\mathbb { R } ^ { 3 \\times H \\times W } $ $\\mathbb { R } ^ { H \\times W }$ . We thus have ",
+        "bbox": [
+            174,
+            247,
+            825,
+            290
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e4abf7ecd58d3652d8b4c2a06de1f3cf2c71413be22636fa1e01894f814dc669.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\mathbf { x } _ { 0 : T _ { 0 } } = [ \\eta ( \\mathbf { I } _ { t } ( \\mathcal { R } ) ) ] _ { t = 0 : T _ { 0 } } \\qquad } & { { } \\mathrm { ( i n i t i a l i z a t i o n ) } } \\\\ { \\mathbf { x } _ { t } = \\Phi \\bigl ( \\mathbf { x } _ { t - T _ { 0 } : t } , m \\bigr ) \\qquad } & { { } \\mathrm { ( a u t o - r e g r e s s i v e ~ p r e d i c t i o n ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            303,
+            725,
+            342
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In short, this model estimates recursively the evolution of the system dynamics from the visual observations of the first $T _ { 0 }$ samples. ",
+        "bbox": [
+            173,
+            354,
+            823,
+            383
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Conditional generator. Variable $\\mathbf { x } _ { t }$ contains information about the state of the moving objects (balls). This is then converted into a prediction of the corresponding video frame, combining also the appearance tensor $\\textbf { \\em a }$ and the median of the first $T _ { 0 }$ images in the sequence $\\hat { \\mathbf { I } } = \\mathrm { m e d i a n } _ { t = 0 , \\ldots , T _ { 0 } - 1 } \\mathbf { I } _ { t }$ This is formulated as a conditional generator network $\\hat { \\mathbf { I } } _ { t } = g ( \\mathbf { x } _ { t } , \\pmb { a } , \\hat { \\mathbf { I } } )$ . ",
+        "bbox": [
+            173,
+            390,
+            825,
+            452
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Video reconstruction loss. Next, we discuss the nature of the loss equation 2. Owing also to the static background, the conditional video generation task is relatively simple provided that the dynamics are estimated correctly. As a consequence, the generated videos are likely to closely approximate the ground truth ones, so the loss function $\\ell$ in equation 2 does not require to be complex. In our experiments, we combine the $L ^ { 2 }$ image loss with the perceptual loss of (Johnson et al., 2016). The latter is obtained by passing the ground-truth and predicted images through the first few layers of a pre-trained deep neural network $e$ , VGG-16 (Simonyan & Zisserman, 2015) in our case, and then comparing the resulting encodings in $L ^ { 2 }$ distance. The overall loss is then given by $\\ell ( \\mathbf { I } _ { t } , \\hat { \\mathbf { I } } _ { t } ) = \\lambda _ { \\mathbf { I } } \\Vert \\mathbf { I } _ { t } - \\hat { \\mathbf { I } } _ { t } \\Vert ^ { 2 } + \\lambda _ { p } \\Vert e ( \\mathbf { I } _ { t } ) - \\bar { e } ( \\hat { \\mathbf { I } } _ { t } ) \\Vert ^ { 2 }$ (details in the sup. mat.). The perceptual loss is robust to small shifts in the image reconstructions that may arise from imperfect physics predictions and favors reconstructing a sharp image; by comparison, the $L ^ { 2 }$ loss alone would result in blurrier images by regressing to the mean. In practice, trading-off the $L ^ { 2 }$ and perceptual losses results in qualitatively better images as well as in better trajectory prediction. ",
+        "bbox": [
+            174,
+            458,
+            825,
+            641
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "State space integrity. An issue with the recursion equation 5 is that the state $\\mathbf { x } _ { t }$ may, in the long run, falls outside the space of meaningful states, leading to a catastrophic failure of the predictor. This is especially true for predictions longer than the ones used for training the model. In order to encourage the recursion to maintain the integrity of the state over time, we add a self-consistency loss $\\| \\mathbf { x } _ { t } - \\eta ( \\mathbf { I } _ { t } ( \\mathcal { R } ) ) \\| ^ { 2 }$ . Here the network $\\eta$ is the same that is used to extract the state space from the initial frames of the video sequence in equation 4 which weights are fixed. In practice, we find that, by using this additional loss, our recursion learns to maintain the integrity of the state nearly indefinitely. ",
+        "bbox": [
+            173,
+            647,
+            825,
+            760
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
+            397,
+            795
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Data. A scenario $s$ (see embedded figure) is generated by sampling a certain number of obstacles of different types $m _ { i }$ and shapes $O _ { i }$ (either rectangular or custom), placing them at random locations and orientations on a 2D board. We consider boards with either two rectangular obstacles (denoted R2), a random number from 3 to 4 rectangular obstacles (R4) or two curved obstacles (C). Scenarios are rendered by painting the background and a wall around it. Then, all obstacle are painted in a solid colour, randomly picked from a fixed palette to ensure sufficient contrast. The background is also painted in solid color (general case) or by using a texture image (only curved shapes, denoted $\\mathrm { C } { + } \\mathrm { T }$ ). Crucially, there is no correlation between an obstacle’s type and its shape, location and color, so its type cannot be inferred by appearance alone. ",
+        "bbox": [
+            174,
+            803,
+            825,
+            929
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Runs $\\mathcal { R }$ are generated by placing one or more dynamic objects (“balls”) with a random initial location and a random momentum oriented towards the obstacles, simulating the 2.1D physics, and rendering the balls as circles with a fixed color (see fig. A1 in supp. mat. for more examples). ",
+        "bbox": [
+            173,
+            103,
+            549,
+            174
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each scenario $s$ , we sample $N + 1$ runs, using the first as a prediction run and the others as experience runs, forming triplets $( \\mathcal S , \\mathcal R , \\mathcal E )$ . Unless otherwise stated, we set $N = 7$ and let each run evolve for 60 frames, including for the experience runs. Thanks to our compact representation, we only use 200MB for 7 experiences for backpropagation during training. Using a recurrent network requires an amount of memory larger in proportion with the length of an experience, i.e. 60-fold, so 12GB for $6 4 \\times 6 4$ boards, which wouldn’t allow the reccurrent architecture to fit into memory. Using shorter runs is also not possible, as they tend to contain no collisions. ",
+        "bbox": [
+            174,
+            180,
+            825,
+            279
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For training, samples $( \\mathcal { S } , \\mathcal { R } , \\mathcal { E } )$ are drawn in an on-line fashion and are thus unlimited. For testing, we use a fixed set of 200 samples. We generate boards of size $6 4 \\times 6 4$ and $1 2 8 \\times 1 2 8$ , but use the latter only for testing purposes to evaluate generalization. Unless otherwise stated, the board size is $6 4 \\times 6 4$ . ",
+        "bbox": [
+            174,
+            285,
+            825,
+            340
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Evaluation metrics. We report the video prediction error as average $L _ { 2 }$ image loss. In order to assess the ability of the method to predict good trajectories, we note that blobs tend to emerge in the heatmaps $\\mathbf { x } _ { t }$ in correspondence of the tracked objects. Hence, we handcrafted a simple blob detector $h ( \\mathbf { x } _ { t } )$ that detects the blobs contained in the heatmaps $\\mathbf { x } _ { t }$ (see sup. mat.). We then report the number of detected blobs vs the actual number of moving objects and, for each, the distance in pixel space of the predicted blob center and ground-truth object center, averaged over the different scenarios. For each experiment we report mean and standard deviation across all sampled scenarios. ",
+        "bbox": [
+            174,
+            347,
+            825,
+            445
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines. We compare with three baselines. The first is a version of the Interaction Networks (Battaglia et al., 2016), trained with perfect knowledge of the object locations, background and object generation. The baseline works directly on the object positions and regresses object positions. The ground truth obstacle map is given as input and transformed into a vector thanks to a pre-trained VGG-16 architecture. We also reimplemented the RLVN network of Wang et al. (2018), which has a vector representation bottleneck, and compared video prediction error results. Note that this baseline wouldn’t adapt to bigger board. Our last baseline amounts to running the ground-truth physics simulator after removing the obstacles from the board (and thus results in perfect predictions for trajectories that do not hit solid obstacles). ",
+        "bbox": [
+            174,
+            452,
+            825,
+            577
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Implementation details. Networks $\\Phi$ and $\\eta$ share very similar auto-encoder type architecture, the network $\\Psi$ uses a U-Net-like architecture (Ronneberger & Fischer, 2015). As stated in Wang et al. (2018) intuitively $\\Psi$ would better preserve appearance while the structure of $\\Phi$ would loose some appearance structure and put an inductive bias on dynamic predictions. $g$ is a fully-convolutional 6-layers stack. Our implementation uses TensorFlow 1.9 (Abadi et al., 2015) and the Adam optimizer (Kingma & Ba, 2014) with learning rate $1 0 ^ { - 4 }$ and Xavier initialization (Glorot & Bengio, 2010). We used a batch size of 10 samples $( \\mathcal S , \\mathcal R , \\mathcal E )$ . Models are first trained for 110,000 iterations using only the $L ^ { 2 }$ and self-consistency losses for eq. (2), and then optionally fine-tuning for further 1,000 iterations using the perceptual loss as well. All models are trained on R2 with board size $6 4 ^ { 2 }$ and tested without any fine-tuning on different board types and sizes (see Tab. 1,2,3,A.3). For the $\\mathrm { C } { + } \\mathrm { T }$ scenarios, we first train the model using flat colour R2 scenarios and then fine-tune for 55,000 iterations using the textured data. Unless otherwise specified, models are trained in a fully unsupervised fashion. Full details can be found in the sup. mat. ",
+        "bbox": [
+            173,
+            584,
+            825,
+            765
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            787,
+            281,
+            804
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Full system. Table 1 rows 1-4 report the prediction accuracy of our system on the 200 simulated test sequences using one moving object. In the table, we distinguish three durations: $T _ { 0 } = 4$ is the number of frames to bootstrap the prediction of a test runs, $T _ { \\mathrm { t r a i n } } = 2 0$ is the length of prediction runs observed during meta-training and $T _ { \\mathrm { t e s t } } = k T _ { \\mathrm { t r a i n } }$ is the length of the runs observed during meta-testing, where $k = 1 , 3 , 5$ . We test both $6 4 \\times 6 4$ boards (which is the same size used for training) and larger $1 2 8 \\times 1 2 8$ boards to test spatial generalization. We also compare training using the $L ^ { 2 }$ or the $L ^ { 2 } +$ perceptual loss for video prediction supervision. We report the number of detected objects (which should be 1 in this experiment), the video prediction error, and the position prediction error. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/84f4191bf09451836dacf2555d8206d4cf94384beb92c8070943e4401fda12bc.jpg",
+        "table_caption": [
+            "Table 1: Predicting one moving object. Obst. is the obstacle type (R2, R4, C). The test board size can be either $6 4 \\times 6 4$ (same as in training) or $1 2 8 \\times 1 2 8$ and we consider using either no supervision or obstacle supervision. We test the average prediction error at $T _ { \\mathrm { t e s t } } = 2 0 , 6 0 , 1 0 0$ well above the duration $T _ { \\mathrm { t r a i n } } = 2 0$ observed during training. Position errors were normalized by the board size diagonal. Video errors were normalized to board size $6 4 ^ { 2 }$ where larger. $L ^ { 2 }$ loss on positions is used for (14)-(15). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td></td><td></td><td colspan=\"3\">Ttest =Ttrain = 20</td><td colspan=\"3\">Ttest =3×Ttrain</td><td colspan=\"3\">Ttest=5×Ttrain</td></tr><tr><td>No.</td><td>Obst.</td><td>Sup.</td><td>Test brd.size</td><td>Train loss</td><td># obj.</td><td>Vid. L2</td><td>Pos. err.</td><td># obj.</td><td>Vid. L2</td><td>Pos. err.</td><td># obj.</td><td>Vid. L2</td><td>Pos. err.</td></tr><tr><td>(1)</td><td>R2</td><td>None</td><td>642</td><td>L²</td><td>1.0±0.2</td><td>2.5±2.6</td><td>.036±.096</td><td>0.3±0.5</td><td>4.6±3.3</td><td>.404±.241</td><td>0.1±0.3</td><td>4.7±3.3</td><td>.526±.180</td></tr><tr><td>(2)</td><td>R2</td><td>None</td><td>1282</td><td>L²</td><td>0.7±0.4</td><td>1.7±1.3</td><td>.179±.236</td><td>0.1±0.3</td><td>1.7±1.1</td><td>.484±.228</td><td>0.0±0.1</td><td>1.7±1.1</td><td>.531±.209</td></tr><tr><td>(3)</td><td>R2</td><td>None</td><td>64</td><td>Percep.</td><td>1.0±0.2</td><td>2.5±2.9</td><td>.028±.073</td><td>1.0±0.2</td><td>5.3±4.0</td><td>.145±.148</td><td>1.0±0.2</td><td>5.5±4.0</td><td>.286±.168</td></tr><tr><td>(4)</td><td>R2</td><td>None</td><td>1282</td><td>Percep.</td><td>0.9±0.4</td><td>1.7±1.4</td><td>.104±.170</td><td>1.0±0.4</td><td>2.0±1.4</td><td>.227±.179</td><td>1.0±0.4</td><td>2.0±1.4</td><td>.320±.172</td></tr><tr><td>(5)</td><td>R2</td><td>Obst.</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>3.0±3.3</td><td>.018±.023</td><td>1.0±0.2</td><td>5.7±4.4</td><td>.112±.086</td><td>1.0±0.3</td><td>5.8±4.2</td><td>.231±.141</td></tr><tr><td>(6)</td><td>R2</td><td>Obst.</td><td>1282</td><td>Percep.</td><td>8.3±3.3</td><td>2.4±1.9</td><td>.323±.064</td><td>26.5±6.4</td><td>5.6±4.0</td><td>.355±.054</td><td>25.8±6.3</td><td>5.8±4.1</td><td>.363±.133</td></tr><tr><td>(7)</td><td>C</td><td>None</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>2.9±3.5</td><td>.040±.086</td><td>1.1±0.3</td><td>5.4±4.1</td><td>.175±.160</td><td>1.0±0.3</td><td>5.4±4.0</td><td>.265±.155</td></tr><tr><td>(8)</td><td>C</td><td>None</td><td>1282</td><td>Percep.</td><td>0.9±0.4</td><td>1.7±1.4</td><td>.104±.170</td><td>1.0±0.4</td><td>2.0±1.4</td><td>.227±.179</td><td>1.0±0.4</td><td>5.6±4.6</td><td>.320±.172</td></tr><tr><td>(9)</td><td>C</td><td>Obst</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>2.9±3.1</td><td>.024±.072</td><td>1.0±0.1</td><td>5.7±4.2</td><td>.128±.114</td><td>1.0±0.2</td><td>5.5±4.1</td><td>.247±.141</td></tr><tr><td>(10)</td><td>C+T</td><td>None</td><td>642</td><td>Percep.</td><td>1.0±0.1</td><td>3.4±2.9</td><td>.017±.044</td><td>1.0±0.2</td><td>6.1±3.4</td><td>.094±.076</td><td>1.0±0.2</td><td>6.2±2.9</td><td>.222±.147</td></tr><tr><td>(11)</td><td>C+T</td><td>None</td><td>1282</td><td>Percep.</td><td>1.0±0.1</td><td>.675±.685</td><td>.012±.033</td><td>1.0±0.2</td><td></td><td>1.4±.814 .057±.065</td><td>0.9±0.3</td><td>1.4±.750</td><td>.121±.088</td></tr><tr><td></td><td></td><td></td><td></td><td>Ablation: removing the median and dynamic images,respectively (see text)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(12) (13)</td><td>R2 R2</td><td>None</td><td>642</td><td>Percep.</td><td>1.0±0.3</td><td>4.0±4.0</td><td>.074±.115 0.8±0.5</td><td></td><td>5.1±4.0</td><td>.302±.202</td><td>0.6±0.5</td><td>5.0±3.8</td><td>.357±.212</td></tr><tr><td></td><td></td><td>None</td><td>642</td><td>Percep.</td><td>1.4±0.6</td><td>3.2±3.4</td><td>.054±.062</td><td>1.8±0.8</td><td>5.5±4.2</td><td>.268±.151</td><td>1.7±0.8</td><td>5.6±4.2</td><td>.326±.136</td></tr><tr><td></td><td></td><td>Comparison: Interaction Network (14-15),RLVN(16)and Ground Truth Simulator minus Obstacles (17-18)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>(14)</td><td>R2</td><td>Full</td><td>1282</td><td>L²Pos.</td><td>=</td><td>=</td><td>.038±.027</td><td></td><td></td><td>.166±.088</td><td>=</td><td>=</td><td>.331±.152</td></tr><tr><td>(15)</td><td>C</td><td>Full</td><td>1282</td><td>L²Pos.</td><td></td><td>-</td><td>.038±.031</td><td></td><td>=</td><td>.171±.095</td><td></td><td>-</td><td>.349±.203</td></tr><tr><td>(16)</td><td>R2</td><td>-</td><td>642</td><td></td><td></td><td>3.6±3.2</td><td></td><td></td><td>5.3±3.3</td><td></td><td>=</td><td>5.5±3.5</td><td></td></tr><tr><td>(17)</td><td>R2</td><td>=</td><td>64²</td><td>=</td><td></td><td>-</td><td>.060±.099</td><td></td><td>=</td><td>.229±.224</td><td>=</td><td>=</td><td>.224±.209</td></tr><tr><td>(18)</td><td>C</td><td></td><td>642</td><td></td><td></td><td>=</td><td>.055±.099</td><td></td><td></td><td>.219±.217</td><td></td><td>=</td><td>.232±.228</td></tr></table>",
+        "bbox": [
+            176,
+            184,
+            823,
+            465
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We find that switching from the $L ^ { 2 }$ to the $L ^ { 2 }$ +perceptual loss for training (rows 1 vs 3 and 2 vs 4) reduces significantly the trajectory prediction error and prevents the object from disappearing. This, however, results to slightly larger $L _ { 2 }$ video prediction error compared to empty boards predicted in the long run by models trained with $L ^ { 2 }$ loss only. We also find that generalization through time is excellent: prediction errors accumulate over time (as it is unavoidable), but the recursion evolves without collapsing well beyond $T _ { \\mathrm { t r a i n } }$ (qualitatively we found that the recursion can be applied indefinitely). This should be contrasted with prior work on intuitive physics (Fragkiadaki et al., 2016) where recursion was found to be unstable in the long term. ",
+        "bbox": [
+            174,
+            488,
+            825,
+            602
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Spatial generalization to larger boards (rows 2,4,6) is also satisfactory if the perceptual loss is used.   \nHowever, we did notice the emergence of a few artifacts in this case. ",
+        "bbox": [
+            174,
+            608,
+            820,
+            636
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In rows 5-6 we use external supervision to train the obstacle map predictor as suggested in section 3.2 (the rest of the system is still unsupervised). Supervision improves the trajectory prediction accuracy for boards of the same size as the ones seen in training. However, generalization to larger boards is poorer, suggesting that explicit supervision causes the model to overfit somewhat. ",
+        "bbox": [
+            174,
+            643,
+            825,
+            699
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Rows 7-11 show results for more complex shapes (C) and appearances $( \\mathrm { C } \\mathrm { + T } )$ . Prediction accuracy is only slightly worse than with the simpler obstacles (e.g. rows 3 vs 7 or 9 vs 10). ",
+        "bbox": [
+            173,
+            705,
+            823,
+            734
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/1678cf0ecb8baff3085d8f554a62763985a4aa8c0fd082c9d68d49c45ca3fd18.jpg",
+        "table_caption": [
+            "Table 2: Importance of experience. For the unsupervised model, we report the trajectory prediction error at $T = 4 0$ (rows 1-3) and the obstacle mask $L _ { 2 }$ prediction error (rows 4-6) for different obstacle types. The number of runs $N$ in the experiences is varied from 0 to 50. For the obstacle mask prediction error, we also report the trivial baseline $\\lVert \\boldsymbol { m } _ { \\mathrm { g t } } - \\boldsymbol { m } _ { \\mathrm { a l l - o n } } \\rVert _ { 2 }$ as ‘All-ON’ (see text). Position errors were normalized by the board size $( 6 4 ^ { 2 } )$ diagonal. $T _ { \\mathrm { t r a i n } } = 2 0$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>No</td><td>Err.</td><td>Obst.</td><td>All-ON</td><td>N=0</td><td>1</td><td>2</td><td>5</td><td>7</td><td>10</td><td>20</td><td>50</td></tr><tr><td>(1)</td><td>Pos.</td><td>R2</td><td></td><td>.462±.136</td><td>.144±.157</td><td>.108±.136</td><td>.083±.128</td><td>.077±.120</td><td>.071±.114</td><td>.070±.116</td><td>.070±.104</td></tr><tr><td>(2)</td><td>Pos.</td><td>R4</td><td></td><td>.502±.151</td><td>.189±.169</td><td>.158±.164</td><td>.122±.155</td><td>.110±.149</td><td>.114±.130</td><td>.110±.154</td><td>.114±.152</td></tr><tr><td>(3)</td><td>Pos.</td><td>C</td><td></td><td>.459±.124</td><td>.161±.172</td><td>.125±.158</td><td>.099±.139</td><td>.099±.146</td><td>.099±.144</td><td>.101±.144</td><td>.106±.143</td></tr><tr><td>(4)</td><td>Obs.</td><td>R2</td><td>13.2±11.1</td><td>24.6±16.4</td><td>10.3±9.5</td><td>7.6±8.4</td><td>4.2±5.4</td><td>3.8±4.7</td><td>3.5±3.3</td><td>3.6±3.5</td><td>3.8±3.7</td></tr><tr><td>(6)</td><td>Obs.</td><td>R4</td><td>16.8±13.3</td><td>23.6±15.1</td><td>14.1±10.9</td><td>11.5±9.9</td><td>7.7±8.0</td><td>7.4±7.3</td><td>7.3±7.2</td><td>7.4±7.2</td><td>8.4±8.3</td></tr><tr><td>(5)</td><td>Obs.</td><td>C</td><td>9.4±8.9</td><td>27.8±15.8</td><td>10.1±8.2</td><td>7.0±7.0</td><td>5.4±5.6</td><td>5.3±5.2</td><td>5.1±5.2</td><td>5.7±5.6</td><td>6.0±5.9</td></tr></table>",
+        "bbox": [
+            189,
+            829,
+            808,
+            921
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/73d91b0d0019a446ed7487df8d41ee033a16742eb26ecfabf97a4749af4fbe6d.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>(U1)</td><td>(U2)</td><td>(U3)</td><td>(B1)</td><td>(B2)</td><td>(B3)</td></tr></table>",
+        "bbox": [
+            174,
+            101,
+            823,
+            181
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Ablation. We suppress respectively the median image (row 12) and dynamic image (row 13) in the feature extractor $f$ summarizing the experiences. We notice a sensible deterioration of performance compared to row 3, suggesting that both components are important. ",
+        "bbox": [
+            176,
+            263,
+            823,
+            306
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Comparison. Compared to the fully-supervised IN (rows 14-15), our unsupervised method is highly competitive, especially for longer simulation times (rows 4,8). The method also outperforms (except for very long simulations where drift prevails) the ground-truth simulator that ignores the obstacles (rows 16,17). Finally we show that our method surpasses RLVN (Wang et al., 2018)(row 16), qualitatively, this method produces similar results to (13) and the object tends to disappear when encounting an object. ",
+        "bbox": [
+            173,
+            313,
+            825,
+            397
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The importance of experience. We investigate how much information the model extracts from the experiences $\\mathcal { E }$ by ablating the number of runs $N$ in it. For the special case $N = 0$ , since there are no runs, we generate an pseudo experience run by copying the first frame $\\mathbf { I } _ { 0 } ( \\mathcal { R } )$ of the prediction run $\\mathcal { R }$ ",
+        "bbox": [
+            174,
+            404,
+            823,
+            446
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In table 2, we report the trajectory prediction accuracy for different obstacle types (row 1-3). We also test the ability to predict the obstacle mask $m _ { \\mathrm { g t } }$ (defined in section 3). Since the model is unsupervised, the learned mask $_ { \\mathbf { \\nabla } } \\mathbf { m } _ { \\mathbf { \\nabla } }$ will not in general be an exact match of $m _ { \\mathrm { g t } }$ (e.g. the mask range is arbitrary); hence, after fixing all the parameters in the model, we train a simple regression network that maps $_ { \\mathbf { \\nabla } } \\mathbf { m } _ { \\mathbf { \\nabla } }$ to $m _ { \\mathrm { g t } }$ and report the prediction accuracy of the latter on the test set (see sup. mat. for details). As a point of comparison, we also report the error $\\lVert m _ { \\mathrm { g t } } - m _ { \\mathrm { a l l - o n } } \\rVert _ { 2 }$ of the trivial baseline that predict a mask $m _ { \\mathrm { a l l - o n } }$ where all objects are highlighted, regardless of their flag. ",
+        "bbox": [
+            173,
+            452,
+            825,
+            551
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We note that there is a very large improvement when we move from zero to one experience runs, and a smaller but non-negligible improvement until $N = 1 0$ . Furthermore, comparing the results with predicting $m _ { \\mathrm { a l l - o n } }$ shows that the system can tell from the experiences which obstacles are solid and which are not. ",
+        "bbox": [
+            174,
+            558,
+            825,
+            613
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/e06442364975740e8001958340adb8534969c69e9b84e7162fb12588471673dc.jpg",
+        "table_caption": [
+            "Table 3: Predicting the obstacles: supervision vs no supervision. The table reports the obstacle mask prediction error $\\left( L _ { 2 } \\right)$ for a network trained with or without supervision for the obstacles (see text). The last column shows a multi-ball predictor operated with 3 balls in each run. $T _ { \\mathrm { t r a i n } } = 2 0$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Supervision</td><td>R2</td><td>R4</td><td>C</td><td>C+T</td><td>R2 (3 balls) C+T → R2</td><td></td></tr><tr><td>Obstacle</td><td></td><td></td><td></td><td>2.7±4.4 6.3±7.8 3.5±4.9 2.6±3.9</td><td>2.6±3.9</td><td>7.1±9.2</td></tr><tr><td>None</td><td></td><td></td><td></td><td>3.8±4.7 7.4±7.3 5.3±5.2 4.0±5.2</td><td>3.9±3.5</td><td>7.6±8.9</td></tr></table>",
+        "bbox": [
+            274,
+            664,
+            720,
+            722
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Supervised obstacle regression. Table 3 compares our unsupervised method to using full obstacle map supervision in order to predict the obstacle map $m _ { \\mathrm { g t } }$ from the experiences $\\mathcal { E }$ . For the unsupervised system, the obstacle map is estimated as explained in the paragraph above. As expected, supervised learning achieves a lower error, but the unsupervised method is still much better than the trivial baseline of table 2. Finally, we observe that for network fine-tuned on $\\mathrm { C } { + } \\mathrm { T }$ and evaluated back on R2 $( \\mathbf { C } \\mathbf { + } \\mathbf { T } \\mathbf { \\xrightarrow { } } \\mathbf { R } 2$ in Table 3) obstacle predictions results remain competitive with results on R2 only while object appearance and backgrounds drastically changed in the new setting. ",
+        "bbox": [
+            173,
+            734,
+            825,
+            833
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Multiple moving objects. We also test whether the system trained with a single moving object can generalize to multiple ones. In table A.3 (see sup. mat.) we show that the network can simulate the motion properly until balls collide, after which they merge (see videos in the sup. mat.). This indicates that, just as physical laws, the rules learned by the model are local. We also train a model (scenario R2) showing it from 1 to 3 balls in each run. This model is able not only to correctly handle ball collisions, but is also able to generalize correctly to several more objects on the board. Furthermore, as shown in table 3, this model still predicts correctly the obstacle masks from experiences, despite the act that the latter are much more crowded. ",
+        "bbox": [
+            174,
+            840,
+            825,
+            924
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            147,
+            330,
+            162
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have demonstrated a system that can learn an intuitive model of physics in an unsupervised manner. Differently from most prior works, our system is also able to learn on-the-fly and very efficiently some physical parameters of new scenarios from a few experiences. Prediction results are strong, competitive with fully-supervised models, and predictors generalize well over time space, and an arbitrary number of moving objects. Our next challenge is to apply the system to more difficult scenarios, including non-frontal views of the physical worlds. For this, we plan to learn a function to summarize past experiences in a more general manner than the dynamic image could. ",
+        "bbox": [
+            174,
+            170,
+            825,
+            267
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            281,
+            285,
+            296
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Abadi et al. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015. Software available from tensorflow.org.   \nP. Agrawal et al. Learning to Poke by Poking: Experiential Learning of Intuitive Physics. In Proc. NeurIPS, pp. 5074–5082, 2016.   \nP. Battaglia et al. Interaction networks for learning about objects, relations and physics. In Proc. NeurIPS, pp. 4502–4510, 2016.   \nP.W. Battaglia, J. Hamrick, and J.B. Tenenbaum. Simulation as an engine of physical scene understanding. PNAS, 110(45):18327–18332, 2013.   \nA. Bhattacharyya et al. Long-term image boundary prediction. In Thirty-Second AAAI Conference on Artificial Intelligence. AAAI, 2018.   \nH. Bilen et al. Dynamic image networks for action recognition. In IEEE CVPR, 2016.   \nM. B. Chang et al. A compositional object-based approach to learning physical dynamics. In Proc. ICLR, 2017.   \nS. Chiappa et al. Recurrent environment simulators. In Proc. ICLR, 2017.   \nM. Denil et al. Learning to perform physics experiments via deep reinforcement learning. Deep Reinforcement Learning Workshop, NIPS, 2016.   \nS. Ehrhardt et al. Unsupervised intuitive physics from visual observations. Proc. ACCV, 2018.   \nS. Ehrhardt et al. Taking visual motion prediction to new heightfields. Computer Vision and Image Understanding (CVIU), 2019.   \nC. Finn, I. Goodfellow, and S. Levine. Unsupervised learning for physical interaction through video prediction. In Advances in neural information processing systems, 2016a.   \nC. Finn et al. Deep spatial autoencoders for visuomotor learning. In Robotics and Automation (ICRA), 2016 IEEE International Conference on, pp. 512–519. IEEE, 2016b.   \nK. Fragkiadaki et al. Learning visual predictive models of physics for playing billiards. In Proc. NeurIPS, 2016.   \nX. Glorot and Y. Bengio. Understanding the difficulty of training deep feedforward neural networks. In Proceedings of the thirteenth international conference on artificial intelligence and statistics, pp. 249–256, 2010.   \nK. Greff et al. Neural expectation maximization. Proc. NeurIPS, 2017.   \nO. Groth et al. Shapestacks: Learning vision-based physical intuition for generalised object stacking. In eccv, 2018.   \nJ. Johnson, A. Alahi, and L. Fei-Fei. Perceptual losses for real-time style transfer and super-resolution. Proc. ECCV, 2016. ",
+        "bbox": [
+            171,
+            301,
+            826,
+            928
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "K. Kansky et al. Schema networks: Zero-shot transfer with a generative causal model of intuitive physics. In International Conference on Machine Learning, pp. 1809–1818, 2017. ",
+        "bbox": [
+            171,
+            103,
+            823,
+            133
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. Proc. ICLR, 2014. ",
+        "bbox": [
+            171,
+            142,
+            761,
+            159
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Krizhevsky, I. Sutskever, and G. Hinton. Imagenet classification with deep convolutional neural networks. In Proc. NeurIPS, pp. 1097–1105, 2012. ",
+        "bbox": [
+            169,
+            167,
+            823,
+            196
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Ladický et al. Data-driven fluid simulations using regression forests. ACM Trans. on Graphics (TOG), 34(6):199, 2015. ",
+        "bbox": [
+            169,
+            205,
+            825,
+            236
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Lerer, S. Gross, and R. Fergus. Learning physical intuition of block towers by example. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, pp. 430–438, 2016. ",
+        "bbox": [
+            173,
+            244,
+            825,
+            287
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "W. Li, A. Leonardis, and M. Fritz. Visual stability prediction and its application to manipulation. AAAI, 2017. ",
+        "bbox": [
+            171,
+            296,
+            825,
+            327
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Monszpart, N. Thuerey, and N. Mitra. SMASH: Physics-guided Reconstruction of Collisions from Videos. ACM Trans. on Graphics (TOG), 2016. ",
+        "bbox": [
+            173,
+            337,
+            823,
+            366
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R. Mottaghi et al. Newtonian scene understanding: Unfolding the dynamics of objects in static images. In IEEE CVPR, 2016. ",
+        "bbox": [
+            174,
+            375,
+            820,
+            405
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "D. Mrowca et al. Flexible Neural Representation for Physics Prediction. Proc. NeurIPS, 2018. ",
+        "bbox": [
+            174,
+            414,
+            794,
+            430
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Anusha Nagabandi, Chelsea Finn, and Sergey Levine. Deep online learning via meta-learning: Continual adaptation for model-based rl. iclr, 2019a. ",
+        "bbox": [
+            173,
+            439,
+            825,
+            468
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Anusha Nagabandi et al. Learning to adapt in dynamic, real-world environments through metareinforcement learning. iclr, 2019b. ",
+        "bbox": [
+            171,
+            478,
+            825,
+            508
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "J. Oh et al. Action-conditional video prediction using deep networks in atari games. In Advances in Neural Information Processing Systems, pp. 2863–2871, 2015. ",
+        "bbox": [
+            171,
+            517,
+            825,
+            546
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "P. Ondruska and I. Posner. Deep tracking: Seeing beyond seeing using recurrent neural networks. In Proc. AAAI, 2016. ",
+        "bbox": [
+            176,
+            555,
+            823,
+            585
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R. Riochet et al. IntPhys: A Framework and Benchmark for Visual Intuitive Physics Reasoning. ArXiv e-prints, 2018. ",
+        "bbox": [
+            173,
+            594,
+            823,
+            625
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "O. Ronneberger and T. Fischer, P.and Brox. U-net: Convolutional networks for biomedical image segmentation. In International Conference on Medical image computing and computer-assisted intervention, 2015. ",
+        "bbox": [
+            174,
+            633,
+            825,
+            678
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A. Sanchez-Gonzalez et al. Graph networks as learnable physics engines for inference and control. Proceedings of the 35th International Conference on International Conference on Machine Learning, 2018. ",
+        "bbox": [
+            174,
+            686,
+            825,
+            729
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "K. Simonyan and A. Zisserman. Very deep convolutional networks for large-scale image recognition. In International Conference on Learning Representations, 2015. ",
+        "bbox": [
+            171,
+            739,
+            825,
+            768
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "R. Stewart and S. Ermon. Label-free supervision of neural networks with physics and domain knowledge. In AAAI, pp. 2576–2582, 2017. ",
+        "bbox": [
+            171,
+            777,
+            823,
+            808
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "J. Tompson et al. Accelerating Eulerian Fluid Simulation With Convolutional Networks. ArXiv e-print arXiv:1607.03597, 2016. ",
+        "bbox": [
+            169,
+            816,
+            825,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "S. van Steenkiste et al. Relational neural expectation maximization: Unsupervised discovery of objects and their interactions. Proc. ICLR, 2018. ",
+        "bbox": [
+            173,
+            856,
+            823,
+            886
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Z. Wang et al. Neural allocentric intuitive physics prediction from real videos. arXiv preprint arXiv:1809.03330, 2018. ",
+        "bbox": [
+            171,
+            895,
+            821,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "N. Watters et al. Visual interaction networks: Learning a physics simulator from video. In Proc. NeurIPS, 2017. ",
+        "bbox": [
+            171,
+            103,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "J. Wu et al. Galileo: Perceiving physical object properties by integrating a physics engine with deep learning. In Proc. NeurIPS, pp. 127–135, 2015. ",
+        "bbox": [
+            169,
+            141,
+            825,
+            170
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "J. Wu et al. Physics 101: Learning physical object properties from unlabeled videos. In Proc. BMVC, 2016. ",
+        "bbox": [
+            171,
+            179,
+            826,
+            208
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "J. Wu et al. Learning to see physics via visual de-animation. In Proc. NeurIPS, 2017. ",
+        "bbox": [
+            174,
+            217,
+            733,
+            232
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A1 DATA GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            382,
+            117
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In fig. A1 and fig. A2 we provide more details on the scene aspect. For solid background color, the objects and background colours are sampled to be different. Every rectangular object have height and width randomly sample in between 10 and 17 pixels. Custom object are loaded from template images and randomly scaled from 1 to 2. ",
+        "bbox": [
+            173,
+            125,
+            825,
+            180
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/af104159ae4307f11e3728074a2d0a303c2f3ec09d6cd8af8b9fcbe9c044adf2.jpg",
+        "image_caption": [
+            "Figure A1: Dataset samples. Pairs of sample data from left to right: R2, C, C+T. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            186,
+            818,
+            265
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/74ffdf6f57f91f4d34abc0e31310b69f4440aabd09275345e3081978f79c44f5.jpg",
+        "image_caption": [
+            "Figure A2: Additional material used for data generation. From left to right: color palette for solid object/background color (R2,R4,C), texture used to increase background complexity in $( \\mathrm { C } \\mathrm { + T } )$ , set of custom objects for (C) data generation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            222,
+            296,
+            761,
+            414
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            479,
+            447,
+            494
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A2.1 NETWORK ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            503,
+            415,
+            518
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In fig. A3 we show the details of all the networks of our pipeline. The regression network mentionned in sec.5 is a simple 3 layers fully convolutional network with [3x3] kernel convolution and intermediate channel of size 64. ",
+        "bbox": [
+            173,
+            525,
+            826,
+            568
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A2.2 TRAINING DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            579,
+            362,
+            593
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In the overall loss $l$ is: ",
+        "bbox": [
+            173,
+            601,
+            320,
+            614
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/caf4d8743a1564e0ef69a99782c44c679c9065597efb70c22f50d50c35e1bd1a.jpg",
+        "text": "$$\n\\ell ( t ) = \\lambda _ { \\mathbf { I } } \\Vert \\mathbf { I } _ { t } - \\hat { \\mathbf { I } } _ { t } \\Vert ^ { 2 } + \\lambda _ { \\mathbf { x } } \\Vert \\mathbf { x } _ { t } - \\eta ( \\mathbf { I } _ { t } ( \\mathcal { R } ) ) \\Vert ^ { 2 } + \\lambda _ { p } \\Vert e ( \\mathbf { I } _ { t } ) - e ( \\hat { \\mathbf { I } } _ { t } ) \\Vert ^ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            277,
+            614,
+            720,
+            633
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "With $t$ ranging from 0 to $T _ { t r a i n }$ . For $t = \\{ 0 , . . , T _ { 0 } \\}$ we used $\\lambda _ { \\mathbf { I } } = 1$ and $\\lambda _ { \\mathbf { x } } = \\lambda _ { \\mathbf { p } } = 0$ . In such case $\\hat { I } _ { t } = g ( x _ { t } , \\mathbf { a } , \\hat { \\mathbf { I } } ) = g ( \\eta ( \\mathbf { I } _ { t } ) , \\mathbf { a } , \\hat { \\mathbf { I } } )$ . ",
+        "bbox": [
+            176,
+            635,
+            823,
+            666
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For $t > T _ { 0 }$ , when training with $L ^ { 2 }$ only we used $\\lambda _ { \\mathbf { I } } = \\lambda _ { \\mathbf { x } } = 1$ and $\\lambda _ { p } = 0$ . When finetuning with perceptual we weighted the perceputal loss with coefficient $\\lambda _ { p } = 1 0$ and the other losses with $\\lambda _ { \\mathbf { I } } = \\lambda _ { \\mathbf { x } } = 0 . 0 1$ . Finally, perceptual loss uses feature extracted from conv3 of VGG-16. ",
+        "bbox": [
+            174,
+            672,
+            825,
+            715
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The Interaction Network baseline is a simple Interaction Network to which state we concatenated the background feature extracted from a pre-trained VGG-16 network. The state propagator uses the last 4 states. The network is then trained over $T _ { t r a i n } = 4 0$ time steps with Adam optimizer with learning rate $1 0 ^ { - 4 }$ and batch size 50. We found that $5 0 0 0 0$ iterations were sufficient to reach convergence. ",
+        "bbox": [
+            173,
+            722,
+            825,
+            777
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "All experiments are run using single NVIDIA-GPU Titan X. ",
+        "bbox": [
+            173,
+            784,
+            570,
+            799
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A2.3 EVALUATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            810,
+            382,
+            825
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To detect blob create a binary image by manually thresholding the intermediate heatmap $\\mathbf { x } _ { t }$ and using a simple Hough circle detector. Every threshold coefficient is hand picked for every model but kept constant across scenarios and different data. ",
+        "bbox": [
+            174,
+            833,
+            825,
+            875
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "For position errors we used an $L _ { 2 }$ loss, when more than one object was detected we took the distance to the closest detected object and applied a fix penalty to the loss for all other detection (half the board size). ",
+        "bbox": [
+            174,
+            881,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/1fb652321621dc8fdc7664e7b623662fc03c1cb892f8b91728443e0d15089061.jpg",
+        "image_caption": [
+            "Figure A3: Networks architecture detail. All the networks generally uses layer normalization $\\eta$ and $\\Phi$ are simple auto-encoder type architecture while $\\Psi$ shares similarity with U-Net. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            171,
+            101,
+            826,
+            401
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table A1: Predicting one moving object. Obstacle type is R2. The test board size is $6 4 \\times 6 4$ We test the average prediction error at $T _ { \\mathrm { t e s t } } = 2 0$ , 60, 100 well above the duration $T _ { \\mathrm { t r a i n } } = 2 0$ observed during training. Position errors were normalized by the board size diagonal. ",
+        "bbox": [
+            173,
+            440,
+            825,
+            482
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/2680f893c1259cb1f5ad14159e17a41d712960bb39b2869aa1d86ca9c5e5394e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td colspan=\"3\">Ttest = Ttrain = 20</td><td colspan=\"3\">Trest =3× Ttrain</td><td colspan=\"3\">Ttest =5×Ttrain</td></tr><tr><td>No.</td><td>Sup.</td><td>Train loss</td><td>#obj.</td><td>Vid. L2</td><td>Pos. err.</td><td>#obj.</td><td>Vid. L2</td><td>Pos. err.</td><td>#obj.</td><td>Vid. L2</td><td>Pos. err.</td></tr><tr><td>(1)</td><td>Oracle</td><td>Perc.</td><td>1.0±0.1</td><td>2.5±2.9</td><td>.013±.012</td><td>1.0±0.1</td><td>5.8±4.4</td><td>.095±.095</td><td>1.0±0.3</td><td>6.2±4.5</td><td>.201±.136</td></tr><tr><td>(2)</td><td>None</td><td>Perc.</td><td>1.0±0.1</td><td>3.2±3.1</td><td>.030±.068</td><td>1.0±0.2</td><td>5.5±4.0</td><td>.136±.151</td><td>0.9±0.4</td><td>5.5±4.0</td><td>.262±.180</td></tr></table>",
+        "bbox": [
+            232,
+            483,
+            764,
+            554
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A3 ADDITIONAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            564,
+            408,
+            579
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Table A1 we report two more results on one moving object prediction. (1) is an upper bound of our system had full knowledge of $\\{ B , U \\}$ obstacle locations at training time. We simply replaced $_ { m }$ with the binary map of $B$ obstacles and $\\textbf { \\em a }$ with the binary map of $U$ obstacles where 1s indicate the presence of the obstacle type. In this scenario we note although that we obtain an upper bound of the results introduced in Table 1, our method still remains competitive with this upper bound. (2) reports results of the network trained on $\\mathrm { C } { + } \\mathrm { T }$ and evaluated back on R2. As in Table 3 we see that the network still manage to make accurate predictions. Table A2 is an extension of Table 2 in the paper for the supervised model and Table A3 introduces results on multiple balls prediction. ",
+        "bbox": [
+            173,
+            587,
+            825,
+            699
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/cf91a7e899a7dc5329856ec4b5b0ce7f2c00880893ba926178d523ee24cf0de0.jpg",
+        "table_caption": [
+            "Table A2: Importance of experience. For the supervised model, we report the trajectory prediction error at $T = 4 0$ (rows 1-3) and the obstacle map prediction error (rows 4-6) for different obstacle types. The number of runs $N$ in the experiences is varied from 0 to 50. Position errors were normalized by the board size diagonal. $T _ { \\mathrm { t r a i n } } = 2 0$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>No</td><td>Err.</td><td>Obst.</td><td>N=0</td><td>1</td><td>2</td><td>5</td><td>7</td><td>10</td><td>20</td><td>50</td></tr><tr><td>(1)</td><td>Pos.</td><td>R2</td><td>1.81±1.41</td><td>.94±.975</td><td>.53±.81</td><td>.15±0.41</td><td>.08±.21</td><td>.07±.16</td><td>.10±.24</td><td>.19±.38</td></tr><tr><td>(2)</td><td>Pos.</td><td>R4</td><td>3.28±1.94</td><td>2.22±1.67</td><td>1.56±1.40</td><td>.63±.84</td><td>.43±.68</td><td>.37±.60</td><td>.44±.61</td><td>.83±.97</td></tr><tr><td>(3)</td><td>Pos.</td><td>C</td><td>1.05±0.91</td><td>.58±.61</td><td>.36±.49</td><td>.15±.29</td><td>.13±.26</td><td>.12±.22</td><td>.16±.26</td><td>.24±.36</td></tr><tr><td>(4)</td><td>Obs.</td><td>R2</td><td>4.1±4.0</td><td>3.6±3.8</td><td>3.1±3.4</td><td>2.3±2.5</td><td>2.3±2.4</td><td>2.3±2.4</td><td>2.3±2.7</td><td>2.6±3.0</td></tr><tr><td>(5)</td><td>Obs.</td><td>R4</td><td>6.1±2.2</td><td>4.1±3.8</td><td>3.1±3.4</td><td>2.8±3.3</td><td>2.7±3.1</td><td>2.6±3.0</td><td>2.7±2.9</td><td>3.3±3.6</td></tr><tr><td>(6)</td><td>Obs.</td><td>C</td><td>3.9±3.9</td><td>3.3±3.7</td><td>2.8±3.1</td><td>2.4±2.7</td><td>2.4±2.8</td><td>2.3±2.6</td><td>2.4±2.6</td><td>2.5±2.9</td></tr></table>",
+        "bbox": [
+            243,
+            267,
+            756,
+            359
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/be57021aa42c014fac27390a8c79f8279f16f983944c74b9d04375a9ad1be0a5.jpg",
+        "table_caption": [
+            "Table A3: Predicting multiple moving object. Obstacle type is R2, all network were trained on the fully unsupervised with perceptual loss. Num. is the maxmium number of balls per run during training. For instance ${ \\mathrm { N u m } } { = } 3$ means that every runs sampled contained one to three balls. The test board is size $6 4 \\times 6 4$ We test the average prediction error at $T _ { \\mathrm { t e s t } } = 2 0 , 6 0 , 1 0 0$ well above the duration $T _ { \\mathrm { t r a i n } } = 2 0$ observed during training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"3\">Ttest = Ttrain=20 Ttest =3×Ttrain Ttest =5×Ttrain</td></tr><tr><td></td><td></td><td></td><td>Num.#obj.Vid.L2#obj. Vid.L2 #obj. Vid. L2</td></tr><tr><td></td><td colspan=\"3\">3 Balls</td></tr><tr><td></td><td>12.6±0.8 6.5±5.2</td><td></td><td>1.7±0.9 8.5±5.9 1.6±0.8 8.5±6.0</td></tr><tr><td></td><td></td><td>33.1±0.64.6±4.8</td><td>2.9±0.8 10.4±7.6 2.9±0.9 11.0±8.0</td></tr><tr><td></td><td></td><td>5 Balls</td><td></td></tr><tr><td>1</td><td>3.3±1.0</td><td>8.9±6.6</td><td>2.0±1.0 10.2±7.4 3.1±0.6 1.8±.775</td></tr><tr><td></td><td>35.5±0.7</td><td>6.6±6.2</td><td>5.1±1.1 13.0±9.5 5.1±1.0 13.5±10.0</td></tr></table>",
+        "bbox": [
+            316,
+            659,
+            679,
+            811
+        ],
+        "page_idx": 13
+    }
+]

parse/train/il3vPAc9fiH/il3vPAc9fiH.md

@@ -0,0 +1,455 @@
+# Geometric Attention Networks for Small Point Clouds
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Much of the success of deep learning is drawn from building architectures that   
+2 properly respect underlying symmetry and structure in the data on which they   
+3 operate—a set of considerations that have been united under the banner of geo  
+4 metric deep learning. Often problems in the physical sciences deal with relatively   
+5 small sets of points in two- or three-dimensional space wherein translation, rota  
+6 tion, and permutation equivariance are important or even vital for models to be   
+7 useful in practice. In this work, we present an architecture for deep learning on   
+8 these small point clouds with rotation and permutation equivariance, composed of   
+9 a set of products of terms from the geometric algebra and reductions over those   
+10 products using an attention mechanism. The geometric algebra provides valuable   
+11 mathematical structure by which to combine vector, scalar, and other types of   
+12 geometric inputs in a systematic way to account for rotation invariance or covari  
+13 ance, while attention yields a powerful way to impose permutation equivariance.   
+14 We demonstrate the usefulness of these architectures by training models to solve   
+15 sample problems relevant to physics, chemistry, and biology.
+
+# 16 Introduction
+
+17 Deep learning has been immensely successful in solving a wide range of problems over the last   
+18 several years, driven in large part by identifying appropriate ways to embed structure of data and   
+19 symmetry of problems directly into the architecture of the network—an idea at the core of geometric   
+20 deep learning[1]. Some applications of geometric deep learning include the use of convolutional   
+21 filters in CNNs to attain translational equivariance, or graph convolutions in graph neural networks   
+22 for permutation equivariance.1 Building symmetry into the architecture of a deep neural network   
+23 can improve the data efficiency of the network and guarantee important analytical properties without   
+24 having to rely on the network to learn to approximate them from training data.
+
+In this work, we derive a family of architectures that is useful in applications from physics to biology, where problems often deal with relatively small point clouds of labeled coordinates. These could be local environments of particles assembling into a crystal[2], atoms in a molecule interacting with other atoms[3], or coarse-grained beads representing parts of a protein[4]. In many of these applications without the influence of an external field, we are interested in modeling attributes of the
+
+$$
+\begin{array} { r l r } {  { \sum _ { \theta = 0 } ^ { \infty } \sum _ { \theta = 0 } ^ { \infty } \boldsymbol { v } ( | \vec { r } _ { i } | , v _ { 0 } , v _ { i } ) f ( | \vec { r } _ { i } | , v _ { 0 } , v _ { i } ) } } \\ & { } & { \diamond \bigotimes _ { \theta = 0 } ^ { \infty } \int _ { \phi = 0 } ^ { \infty } \boldsymbol { \hat { \rho } } _ { 0 } \cdot \boldsymbol { \varrho } _ { \phi } \big ( \vec { r } _ { 0 } | , \vec { \varrho } _ { 0 } \big ) } \\ & { } & { \diamond \bigotimes _ { \phi = 0 } ^ { \infty } \sum _ { \theta = 0 } ^ { \infty } \sum _ { \rho = 1 , v _ { i } = v _ { i } } ^ { \infty } f ( | \vec { r } _ { i } | , v _ { 0 } , v _ { i } , v _ { j } ) } \\ & { } & { \cdot \bigotimes _ { \phi = 0 } ^ { \infty } \sum _ { \rho = 1 } ^ { \infty } \boldsymbol { \rho } ( | \vec { r } _ { i } | , v _ { 0 } , v _ { i } ) f ( | \vec { r } _ { i } | , v _ { 0 } , v _ { i } , v _ { j } ) \bigotimes _ { \phi } \bigotimes _ { \varrho } \bigotimes _ { \varrho } ^ { \infty } \bigotimes _ { \varrho } ^ { \infty } \int _ { \phi = 0 } ^ { \infty } \boldsymbol { \hat { \varrho } } _ { \phi } ( | \vec { r } _ { i } | , \vec { \varrho } _ { 0 } ) , } \\ & { } & { \quad \times ( | \vec { r } _ { i } | ) f ( | ) + \ldots } \end{array}
+$$
+
+Figure 1: Overall strategy for incorporating rotation and permutation equivariance into deep neural networks using attention mechanisms and geometric products. (a) At its simplest level, our proposed structure uses an attention mechanism over the bond lengths of a cloud of points, each of which carries a value as commonly used in graph neural networks. (b) Geometric products (or linear combinations thereof) can be used to combine pairs, triplets, or larger tuples of vectors in a systematic and geometrically meaningful way. Rotational equivariance can be attained by using invariant or covariant quantities, as desired. (c) An attention mechanism reduces the set of generated geometric products to enforce proper permutation equivariance, and the learned attention maps can provide insights into how models operate. For example, here, a carbon atom in a naphthalene molecule (indicated by a blue star) directs its focus broadly around the carbon atoms of the aromatic rings in which it is situated, rather than focusing exclusively on its nearest neighbors in the molecular graph. Brighter-colored bonds indicate a greater attention weight for the two atoms sharing the bond.
+
+30 system—such as the identity of a particle’s local self-assembly environment, or the potential energy   
+31 of a group of atoms—which are invariant with respect to rotation of the input coordinates, as well as   
+32 permutation in the ordering of points. Here we attain rotation invariance by constructing functions   
+33 from rotation-invariant components of geometric products of input vectors from geometric algebra,   
+34 and permutation invariance by using an attention mechanism to intelligently reduce representations   
+35 over the set of vector products.
+
+# 36 Related Work
+
+37 “Large” point clouds. Point clouds are a ubiquitous data structure and are often found in domains   
+38 outside of the physical sciences. For the purposes of this work, we focus on comparatively small sets   
+39 of points where the points are relatively information-rich—for example, carrying information about   
+40 atom identities, local environments, or other information—in contrast to point clouds commonly   
+41 found in computer vision and robotics which may represent the geometry of a mesh or be sampled   
+42 from an object, but otherwise not have as much information associated with each point. We refer   
+43 to Guo et al. for a survey of this field[5], but a few of the recently-developed notable approaches   
+44 include PointNet[6], deep sets[7], and kernel point convolutions[8].   
+45 Geometric approaches for small point clouds. Many architectures have been proposed to incor  
+46 porate rotation equivariance by augmenting graph neural networks with geometric attributes that   
+47 are known to be rotation-invariant, such as bond lengths and angles. SchNet[9] learns distance  
+48 based convolution filters which are used to propagate signals over graphs. PhysNet[10] also refines   
+49 node representations based on bond lengths, while incorporating a learnable attention mechanism.   
+50 DimeNet[11] extends the information used to calculate node-level representations to include angles   
+51 between bonds. GNNFF[12] generates rotation-covariant results by computing a weighted sum of   
+52 modulated input vectors based on a graph message passing scheme.   
+53 Group representation-based approaches. These methods take advantage of group representation   
+54 theory by first transforming inputs into a space in which rotation-equivariant maps are more eas  
+55 ily expressed. This set of methods is powerful, having been used in the past to design rotation  
+56 and permutation-equivariant models[13–15], and have even been expanded recently for arbitrary   
+57 groups[16]. Attention-based models have also been utilized in this area: SE(3) Transformers[17]   
+58 extend tensor field networks[14] with a self-attention mechanism for increased expressivity by in  
+59 corporating value- and geometry-dependent attention weights.   
+60 The approach we present here is similar to several of the ideas presented above; however, rather   
+61 than specifying particular rotation-invariant quantities to utilize or learning maps that operate on   
+62 irreducible representations, we leverage the structure provided by geometric algebra to determine   
+63 which rotation-invariant and -covariant quantities are of interest.
+
+# 64 Geometric Attention Networks
+
+65 In this work we formulate deep neural networks using learnable functions consisting of two parts:   
+66 (1) a set of geometric products of input vectors; and (2) a permutation-equivariant reduction over   
+67 these products using an attention mechanism. We describe each of these aspects below.
+
+# 68 Geometric Algebra
+
+69 The geometric algebra was developed in the 19th century and provides a consistent framework for   
+70 dealing with scalars and other geometric quantities—such as vectors, areas, and volumes in three  
+71 dimensional space—in arbitrary dimensions[18]. Here, we will describe the essential parts of geo  
+72 metric algebra as related to our proposed attention mechanism, and defer to other works for a more   
+73 thorough description[19]. The geometric algebra specifies a binary operator, the geometric product,   
+74 that works on multivectors. Multivectors can be expressed as linear combinations of terms from a   
+75 fixed basis set for a given space, such as $\mathbb { R } ^ { 2 }$ or $\mathbb { R } ^ { 3 }$ ; in three-dimensional space, this yields scalars,   
+76 vectors, bivectors (which specify signed areas within a plane and have 3 components), and trivec  
+77 tors (which specify signed volumes and have 1 component)—a total of 8 linearly independent terms   
+78 for each multivector2. When rotation invariance is desired, we can utilize the rotation-invariant   
+79 components of a multivector: scalars, trivectors, the norms of vectors, and the norms of bivectors   
+80 are rotation-invariant. As an example, the geometric product of two vectors yields a scalar plus a   
+81 bivector; the scalar component is the dot product, and the bivector component is related to the cross   
+82 product of the two.   
+83 Geometric algebra provides a general framework that can be used to build up expressive functions   
+84 as linear combinations and geometric products of multivector inputs; rotation-equivariant quantities   
+85 can then be derived from the products, depending on the application and symmetry of the problem   
+86 of interest. The types of elements produced by a geometric product of two multivectors in $\mathbb { R } ^ { 3 }$ with   
+87 the given components are listed in Table 1 below.
+
+Table 1: Terms arising from the geometric product $A B = ( A _ { s } + A _ { v } + A _ { b } + A _ { t } ) ( B _ { s } + B _ { v } + B _ { b } + B _ { t } )$ in $\mathbb { R } ^ { 3 }$ . In three dimensions, multivectors $A$ and $B$ consist of scalars $( s )$ , vectors $( \nu )$ , bivectors $( b )$ , and trivectors $\mathbf { \rho } ( t )$ .
+
+<table><tr><td></td><td>Bs</td><td>Bv</td><td>Bb</td><td>Bt</td></tr><tr><td>会</td><td>S</td><td>V</td><td>b</td><td>t</td></tr><tr><td></td><td>V</td><td>s+b</td><td>V+t</td><td>b</td></tr><tr><td></td><td>b</td><td>V+t</td><td>s+b</td><td>V</td></tr><tr><td>At</td><td>t</td><td>b</td><td>V</td><td>S</td></tr></table>
+
+88 From Table 1, we can see that successive products of vectors alternate between producing two types   
+89 of multivectors: products of even numbers of vectors yield a scalar and bivector $( ( v + t ) v = v v + t v =$   
+90 $( s + b ) + b \to s + b )$ , while products of odd numbers of vectors produce a vector and trivector   
+91 $( ( s + b ) v = s v + b v = v + ( v + t )  v + t )$ . Generating rotation-invariant quantities from these   
+92 products is the primary application of geometric algebra in this work, although in general the method   
+93 could be used to incorporate different types of scalar, vector, bivector, and trivector quantities; for   
+94 example, rotations could be input as quaternions, which are isomorphic to the scalar-and-bivector   
+95 product of even numbers of vectors.
+
+# 96 Attention from Geometric Products
+
+97 For input point clouds with $N$ points, we can construct a series of successively higher-order geo  
+98 metric products for all $N ^ { 2 }$ possible pairs, $N ^ { 3 }$ triplets, and so on; these individual points, pairs, or   
+99 triplets we will call a tuple in this context. In addition to a coordinate $\vec { r _ { i } }$ , we associate a set of values   
+100 $v _ { i }$ to each point indexed by $i$ in some space with a given working dimension (we avoid calling these   
+101 vectors to decrease the confusion with geometric vectors; these correspond to the non-geometric   
+102 attributes of the point, such as type embeddings). To create permutation-covariant functions (pro  
+103 ducing a value for each input point) or permutation-invariant functions (producing a single output   
+104 value), we make use of a simple attention mechanism based on the rotation-invariant attributes of   
+105 each tuple. Attention has been used widely in applying deep learning to a range of problem do  
+106 mains over the last few years, with particular success in the field of natural language processing[20].   
+107 Since we are already generating tuple-wise quantities, we choose to utilize a simpler mechanism   
+108 than the typical dot product self-attention. We specify four functions: a value-generating function   
+109 $\nu$ , a tuple value-merging function $\mathcal { M }$ , a joining function that summarizes the invariant and tuple   
+110 representations $\mathcal { I }$ , and a score-generating function $s$ . The functions have the following uses within   
+111 the network:
+
+• $\nu$ produces features in the working dimension of the model from the invariants associated with each tuple.   
+• $\mathcal { M }$ merges the 1, 2, 3, or more values associated with a tuple of input points into the working dimension of the model. The form of $\mathcal { M }$ could be a complex function, a learned linear projection for each tuple position, or simply taking the sum of the tuple values.   
+• $\mathcal { I }$ joins the invariant representations from $\nu$ and the tuple representations from $\mathcal { M }$ . Like $\mathcal { M }$ , it could be a learned projection or a simple sum function.   
+• $s$ generates score logits from the representation of each tuple, which incorporates invariants associated with the tuple and the values being associated with each point that is part of the tuple. The results from $s$ , passed through a softmax function, will yield the weights for the attention mechanism.
+
+123 We first calculate the multivector geometric products $p _ { i j k \ldots }$ of all combinations of input vectors   
+124 $i , j , k$ , and so on, up to a specified rank (pairwise attention would produce a two-dimensional   
+125 matrix of products $p _ { i j }$ ). We then use $\nu , \mathcal { M } , \mathcal { I }$ , and $s$ —together with a function extracting the   
+126 rotation-invariant attributes of a geometric product (the scalar component, trivector component, and   
+127 the norms of the vector and bivector components)—as follows for a network producing permutation  
+128 covariant outputs $y _ { i }$ :
+
+$$
+\begin{array} { r l } & { p _ { i j k \ldots } = \vec { r } _ { i } \vec { r } _ { j } \vec { r } _ { k } \ldots } \\ & { q _ { i j k \ldots } = \mathrm { i n v a r i a n t s } ( p _ { i j k \ldots } ) } \\ & { v _ { i j k \ldots } = \mathcal { I } ( \mathcal { V } ( q _ { i j k \ldots } ) , \mathcal { M } ( v _ { i } , v _ { j } , v _ { k } , \ldots ) ) } \\ & { w _ { i j k \ldots } = \underset { j k \ldots } { \operatorname { s o f t m a x } } ( \mathcal { S } ( v _ { i j k \ldots } ) ) } \\ & { \qquad y _ { i } = \underset { j k \ldots } { \sum } w _ { i j k \ldots } v _ { i j k \ldots } } \end{array}
+$$
+
+129 If a permutation-invariant reduction is desired, then the softmax and final sum can be performed over   
+130 all tuples simultaneously, rather than for each input point individually. While $\mathcal { I } , \mathcal { V }$ , and $\mathcal { M }$ could in   
+131 principle be used to change the working dimension as permutation-covariant layers are stacked on   
+132 top of each other, in this work we keep the working dimension constant for the sake of easily adding   
+133 residual connections.   
+134 If rotation-covariant, rather than rotation-invariant, behavior is needed for the output of the network,   
+135 the same attention structure can be used with slight modifications; here, we coerce a vector from   
+136 the product $p _ { i j k \ldots }$ (which consists of directly taking the vector component from products of odd   
+137 numbers of input vectors, or multiplying a bivector by the unit trivector to produce a vector—as   
+138 shown in the last column of Table 1—in the case of even numbers of input vectors). These vectors   
+139 can be combined with a scalar rescaling each vector—generated by a learned function $\mathcal { R }$ —and the   
+140 attention mechanism to yield
+
+$$
+\vec { r } _ { i } ^ { \prime } = \sum _ { j k \ldots } w _ { i j k \ldots } \mathcal { R } ( q _ { i j k \ldots } ) \mathrm { v e c t o r } ( p _ { i j k \ldots } ) .
+$$
+
+# 141 Results
+
+142 We demonstrate the utility of our geometric algebra attention scheme by training deep networks to   
+143 solve three problems appearing in physics, chemistry, and biology. For simplicity, all the models   
+144 presented here utilize pairwise attention with a working depth of 32 units. Value functions $\nu$ , score   
+145 functions $s$ , and rescaling functions $\mathcal { R }$ are simple multilayer perceptrons with a hidden width of   
+146 64 units, with layer normalization applied to the output of $\nu$ . The network for crystal structure   
+147 identification uses the mean function for merge functions $\mathcal { M }$ and join functions $\mathcal { I }$ , while the other   
+148 two applications use learned linear projections. Networks are trained for up to 800 epochs using   
+149 the adam optimizer[21]; the learning rate is decreased by a factor of 0.75 after the validation set   
+150 loss does not decrease for 20 epochs, and training is ended early if the validation set loss does not   
+151 decrease for 50 epochs. Numerical results are reported as the mean and standard error of the mean   
+152 over 5 samples. Python code under the MIT license implementing each experimental workflow is   
+153 included in the supplementary information.
+
+# 154 Crystal Structure Identification
+
+155 On length scales ranging from those of atoms to colloidal particles, matter often organizes itself into   
+156 ordered two- or three-dimensional structures. One of the core ideas of materials science is that struc  
+157 ture is one of the major determining factors for material behavior. With this perspective in mind,   
+158 when studying computational models of self-assembling systems we often first identify what struc  
+159 tures, if any, have formed in our simulations—a task complicated by naturally-occurring thermal   
+160 noise, crystallographic defects, and potentially the complexity of the structures themselves. Early   
+161 efforts to automatically characterize structure led to the widely-used Steinhardt order parameters[22–   
+162 24], which are rotationally-invariant sums of spherical harmonic magnitudes over local particle   
+163 environments. While the Steinhardt order parameters can be useful when studying phase transi  
+164 tions or distinguishing among a small number of phases, determining appropriate hyperparameters—   
+165 including neighborhood size to consider, spherical harmonic order $\ell$ to use, and thresholds to iden  
+166 tify behaviors of interest—can be difficult[23]. For this reason, data-driven approaches to analyzing   
+167 structure have been the subject of great interest in recent years[25].   
+168 We use geometric attention networks to identify the source structure type of small neighborhoods   
+169 of particles extracted from bulk crystals. We select 8 prototypes of single- and two-component   
+170 crystals from the AFLOW Encyclopedia of Crystallographic Prototypes[26, 27]. These structures   
+171 are chosen to demonstrate that models can learn not only geometric information ( $c F 4 – C \mathrm { u }$ and $h P 2 \cdot$ -   
+172 $\mathbf { M g }$ are similar structures but with a different stacking of their close-packed layers; the clathrates   
+173 $c P 4 6 – S \mathrm { i }$ and $c F 1 3 6 { - } \mathrm { S i }$ are also similar, with a different arrangement of many common motifs), but   
+174 also the information encoded within each point ( $c P \mathrm { 2 \mathrm { - } C s C l }$ and $c F 8 – Z \mathrm { n S }$ differ from $c I 2 – \mathrm { { W } }$ and   
+175 $c F 8 – C$ only by their particle type assignments). For each structure, we rescale the unit cell such that   
+176 the shortest nearest-neighbor distance over the structure is 1 before replicating the unit cell to consist   
+177 of at least 2048 particles. We then create three samples of each structure by adding Gaussian noise   
+178 with a standard deviation of $1 0 ^ { - 3 }$ , $5 \cdot 1 0 ^ { - 2 }$ , and 0.1 separately to the particle coordinates, in order to   
+179 emulate thermal noise. For each particle in the structure, we find the 12 nearest neighbors and their   
+180 associated types using the freud[28] python library, which we feed into the network as the pairwise   
+181 distance $\vec { r } _ { i j } = \vec { r } _ { j } - \vec { r } _ { i }$ and one-hot-encoded symmetrized type vector $\Delta t _ { i j } = [ I _ { t _ { i } } - I _ { t _ { j } } , I _ { t _ { i } } + I _ { t _ { j } } ]$   
+182 where $I$ is the identity matrix of dimension corresponding to the maximum number of types.   
+183 We train classifiers with 2 permutation-covariant attention blocks before a final reduction over the en  
+184 tire particle neighborhood—as shown in Figure 2—in order to categorize local particle environments   
+185 according to their source crystal structure type. These networks rapidly learn to identify structures   
+186 after a few epochs, with a final overall accuracy of $9 8 . 7 \% \pm 0 . 2 \%$ after training for roughly 45   
+187 minutes on an NVIDIA Titan Xp GPU.
+
+![](images/ff776cb7c3a857ef1656d4e700d60a61b4ceca7001d4cf979a73f1d5d30f8862.jpg)  
+Figure 2: (a) Network architecture for crystal structure identification. Coordinates and particle types are passed through two permutation-covariant layers before a final permutation-invariant reduction. (b) Crystal structure prototypes chosen for the structure identification benchmark. Simple and complex structure types—including two binary structures—are included.
+
+# 188 Molecule Force Regression
+
+189 One of the most dramatic contributions of deep learning to the field of chemistry lies in constructing   
+190 fast, accurate approximations of expensive physical calculations[29, 30]. Machine learning models   
+191 can be many orders of magnitude faster than the methods used to generate their training data, which   
+192 can bring vastly more detailed and longer-time simulations into the realm of possibility. Central   
+193 to the applicability of these methods are issues of symmetry and equivariance: any imperfection   
+194 in rotational invariance of a learned potential energy function could ruin the proper thermodynamic   
+195 behavior of a model, for example, so models must be carefully designed to ensure physical behavior.   
+196 In a method similar to Batzner et al.[31], we train models to predict the per-atom forces calculated   
+197 using ab initio molecular dynamics and density functional theory available in the MD17 dataset[3].   
+198 As shown in Figure 3, we first transform the raw coordinates and types of each atom in a given   
+199 molecule into the pairwise difference and symmetric sum and difference of the coordinates and one  
+200 hot type encoding for each atom with respect to each other atom, respectively, to fix translation   
+201 invariance and assign type representations to the pairwise particle bonds. We then perform a series   
+202 of geometric attention calculations, calculating new values per atom, which are finally summed to   
+203 produce a scalar energy. The gradient of this energy with respect to the input coordinates is used to   
+204 produce the force output of the network, which ensures that a conservative force field is learned.   
+205 Consistent with previous benchmarks on this dataset, we train networks using the mean squared   
+206 distance loss for each molecule using 1,000 snapshots of forces each as training, validation, and test   
+207 data sets. We also report results for models trained on all molecules’ data simultaneously, or 8,000   
+208 snapshots each for training, validation, and testing. Training a model on an individual molecule’s   
+209 data takes between 30 minutes (ethanol and malonaldehyde, with nine atoms each) to two hours   
+210 (aspirin, with twenty-one atoms) on an NVIDIA Titan Xp GPU, while the all-molecule dataset   
+211 requires roughly 16 hours to train. Test set losses, expressed as the mean absolute error over each   
+212 force component for each sample, are presented in Table 2.   
+213 Our geometric algebra attention networks produce results competitive with SchNet[9], an architec  
+214 ture using learned radial distance convolution filters. Although the models generated here do not out  
+215 perform the Neural Equivariant Interatomic Potentials by Batzner et al.[31], we note that our models   
+216 are trained for a fraction of the time (2 GPU hours and 800 epochs for our method, compared to on   
+217 the order of 8 GPU days and 2500 epochs for NequIP) and without drastic hyperparameter tuning   
+218 aside from optimizing the number of residual blocks to use in the network architecture. Notably, the   
+219 models trained on all 8,000 molecular snapshots perform significantly better than almost all of the   
+220 specialized models, indicating that additional data could likely improve the results presented here   
+221 even without careful hyperparameter optimization.
+
+![](images/2fc4c67c06ad4a6039e383b9b361000b0a01b626ee78943320c9e3377786a94e.jpg)  
+Figure 3: (a) Network architecture for molecular force regression. Coordinates and particle types for all atoms in a molecule are fed into the network as a set of pairwise distances, with the atomic representations refined through a series of geometric algebra attention layers. Six permutation-covariant layers are stacked before reducing the representations with a final, permutation-invariant geometric product attention layer. (b) Sample pairwise attention maps for four training data molecules (malonaldehyde, aspirin, benzene, and uracil) after filtering out low-attention pairs. The attention maps indicate how strongly the pair of atoms joined by the line affect the representation of the atom indicated with a star, with lighter lines indicating greater influence. Qualitatively, more complex bonding environments such as those on the right tend to have longer-range attention interactions than the simpler environments on the left.
+
+Table 2: Mean absolute error of force components (in $\frac { m e V } { \mathring { \mathsf { A } } }$ ) for geometric algebra attention networks, NequIP[31], and SchNet[9] architectures.   
+
+<table><tr><td>Molecule</td><td>This work</td><td> NequIP</td><td>SchNet</td></tr><tr><td>Aspirin</td><td>37.0 ± 1.1</td><td>15.1</td><td>58.5</td></tr><tr><td>Benzene</td><td>11.8 ± 0.5</td><td>8.1</td><td>13.4</td></tr><tr><td>Ethanol</td><td>21.4 ± 0.5</td><td>9.0</td><td>16.9</td></tr><tr><td>Malonaldehyde</td><td>30.6 ± 1.1</td><td>14.6</td><td>28.6</td></tr><tr><td>Naphthalene</td><td>23.7 ± 1.0</td><td>4.2</td><td>25.2</td></tr><tr><td>Salicylic acid</td><td>30.2 ± 1.2</td><td>10.3</td><td>36.9</td></tr><tr><td>Toluene</td><td>20.5 ± 1.3</td><td>4.4</td><td>24.7</td></tr><tr><td>Uracil</td><td>27.4 ± 0.8</td><td>7.5</td><td>24.3</td></tr><tr><td>All molecules</td><td>10.7 ± 0.2</td><td></td><td></td></tr></table>
+
+![](images/dbda3c3058ead5d71db8172159107aef6bb9a261bd655e47f63cc8c71f4481e4.jpg)  
+Figure 4: Network architecture for inverting a coarse-grained mapping of a protein. Models are trained to predict atomic-resolution coordinates from $N _ { \mathrm { c o a r s e } } = 1 2$ neighboring amino acid centers of mass using geometrically-informed scalar-to-scalar attention (blue), scalar-to-vector attention (purple), and vector-to-vector attention (red).
+
+# 222 Backmapping Coarse-Graining in Proteins
+
+223 When simulating large molecules—such as proteins or other polymers—it is common to employ   
+224 coarse graining: a process by which groups of particles are merged into (fewer) distinct beads, en  
+225 abling faster simulations by decreasing the number of degrees of freedom of the model[4]. Although   
+226 data-driven approaches have been highly successful to formulate coarse graining operations in the   
+227 forward direction (that is, from more-detailed to less-detailed systems), some problems are best   
+228 solved using the original, fine-grained system coordinates, which are not directly available in coarse  
+229 grained simulations. To demonstrate the potential for our geometric algebra attention mechanism   
+230 on this task, we train models to predict the coordinates of the heavy atoms that form an amino acid   
+231 from the centers of mass of the nearest-neighbor amino acids. We take 19 protein structures[32–50]   
+232 that have high-resolution structural refinements (with resolution error less than or equal to $1 . 0 \mathring \mathrm { A } )$ )   
+233 and were published between 2015 and 2020 from the Protein Data Bank[51]. For applications of   
+234 this method to systems at nonzero temperature, we would expect to be better-served by using an   
+235 architecture that produces distributions instead of only point values, but we disregard this here for   
+236 simplicity; in other words, here we are teaching models to memorize the results of structure refine  
+237 ment algorithms, which may be different for each PDB entry. For every amino acid in each entry,   
+238 we create a point cloud of its 12 nearest neighbor amino acid centers of mass, as well as a point   
+239 cloud of the primary amino acid’s atomic coordinates relative to its center of mass. Two layers of   
+240 permutation-covariant geometric product attention are applied to the coarse-grained amino acid co  
+241 ordinates before being passed to a layer which produces a vector output according to Equation 2 by   
+242 augmenting the tuple representation $v _ { i j k \dots }$ of Equation 1 with labels corresponding to the identity   
+243 of the atom that should be produced, so that the value is calculated as
+
+$$
+v _ { \mathrm { a t o m } , i j k \ldots } = \mathcal { I } ( v _ { \mathrm { a t o m } } , \mathcal { V } ( q _ { i j k \ldots } ) , \mathcal { M } ( v _ { i } , v _ { j } , v _ { k } , \ldots ) ) .
+$$
+
+244 Following this layer—which maps coarse-grained coordinates of amino acids to fine-grained coordi  
+245 nates of atoms—two rotation-covariant layers are applied to the atomic coordinates to further refine   
+246 them, as shown in Figure 4.   
+247 Because the resolution of the structural refinement algorithms is on the order of $0 . 5 \mathring \mathrm { A }$ or greater, we   
+248 use the training set error as a measure of the learning progress of the models instead of performing   
+249 a standard split of training, validation, and test set data. After training for roughly 3 hours on   
+250 an NVIDIA Titan $\mathrm { X p }$ GPU, models achieve a mean absolute error of $0 . 1 2 8 \mathring { \mathbf { A } } \pm 0 . 0 0 2 \mathring { \mathrm { A } }$ (down   
+251 from approximately $\bar { 0 . 5 \mathring \mathrm { A } }$ initially), indicating that they are able to learn to reconstruct atomic-scale   
+252 coordinates from coarse-grained positions.
+
+# 253 Discussion
+
+Overall, we find the architectures formulated here to be useful for a variety of tasks. Rather than being limited to operating on bond distances and angles as in SchNet[9], PhysNet[10], and DimeNet[11], geometric algebra provides a systematic way to build functions with the desired rotation- and permutation-equivariance, with the flexibility to incorporate other types of geometric objects (such as the orientation quaternion commonly used for anisotropic particles in molecular dynamics methods[52]). The attention mechanism presented here provides a simple yet powerful method to incorporate both geometric and node-level signals. The primitives of our geometric algebra attention scheme—distances, areas, angles, and volumes—and the calculated attention weights naturally lend themselves to interpretability, which we believe will prove useful in distilling insights from trained models.
+
+# Limitations
+
+Combination of terms. Although the architectures presented here work well for the problems we have selected, creating geometric products of vectors is only a subset of the valid combinations that could be generated. In these cases we have carefully chosen sums and differences of input vectors to respect symmetries we would like to impose on the system—such as using the pairwise distance of all input coordinates for the molecular force regression task to impose translation invariance—but it is possible that more powerful models could be formed by incorporating learned linear combinations of inputs or intermediate multivector quantities. We leave this as a topic of future work.
+
+Computational scaling and neighborhood definition. An obvious limitation to using higherdegree correlations lies in the computational complexity and memory scaling of generating tuples, which are both proportional to $N ^ { r }$ for neighborhoods of $N$ coordinates and tuples of length $r$ . Polynomial scaling behavior can be ameliorated by restricting which combinations of input points are considered, essentially treating the attention weights of all other combinations as 0. These combinations could be randomly sampled from all valid indices $i j k . . .$ or use more physically-relevant restrictions, such as utilizing the molecular connectivity graph for molecular force regression or edges derived from the Voronoi tessellation for other applications. If smoothness of model predictions is a concern—as may be the case for learning general N-body interaction potentials, for example—the architectures presented here could be augmented by incorporating weights that decay to 0 as bonds are broken in the Voronoi diagram graph[23].
+
+# 283 Conclusion
+
+In this work, we have presented a strategy for developing rotation- and permutation-equivariant neural network architectures by combining geometric algebra and attention mechanisms. These architectures operate directly on the vector, scalar, and other geometric quantities of interest to produce outputs which respect desirable symmetries by construction. We believe that the mathematical simplicity and the insights derived from attention maps are particularly appealing aspects of the algorithms presented here. We hope that these architectures will help a wider range of scientific disciplines reap the benefits of geometric deep learning.
+
+# References
+
+[1] Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Velickovi ˇ c. Geometric deep learn- ´ ing: Grids, groups, graphs, geodesics, and gauges. arXiv:2104.13478 [cs, stat], April 2021.
+
+[2] Julia Dshemuchadse, Pablo F. Damasceno, Carolyn L. Phillips, Michael Engel, and Sharon C. Glotzer. Moving beyond the constraints of chemistry via crystal structure discovery with isotropic multiwell pair potentials. Proceedings of the National Academy of Sciences, 118(21), May 2021.   
+[3] Stefan Chmiela, Alexandre Tkatchenko, Huziel E. Sauceda, Igor Poltavsky, Kristof T. Schütt, and Klaus-Robert Müller. Machine learning of accurate energy-conserving molecular force fields. Science Advances, 3(5):e1603015, May 2017.   
+[4] Siewert J. Marrink and D. Peter Tieleman. Perspective on the Martini model. Chemical Society Reviews, 42(16):6801–6822, July 2013.   
+[5] Yulan Guo, Hanyun Wang, Qingyong Hu, Hao Liu, Li Liu, and Mohammed Bennamoun. Deep learning for 3d point clouds: A survey. IEEE transactions on pattern analysis and machine intelligence, 2020. [6] Charles R. Qi, Hao Su, Mo Kaichun, and Leonidas J. Guibas. PointNet: Deep learning on point sets for 3D classification and segmentation. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 77–85, July 2017. [7] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R. Salakhutdinov, and Alexander J. Smola. Deep sets. Advances in Neural Information Processing Systems, 30, 2017.   
+[8] Hugues Thomas, Charles R. Qi, Jean-Emmanuel Deschaud, Beatriz Marcotegui, François Goulette, and Leonidas J. Guibas. KPConv: Flexible and deformable convolution for point clouds. Proceedings of the IEEE International Conference on Computer Vision, 2019. [9] K. T. Schütt, P.-J. Kindermans, H. E. Sauceda, S. Chmiela, A. Tkatchenko, and K.-R. Müller. SchNet: A continuous-filter convolutional neural network for modeling quantum interactions. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, pages 992–1002, Red Hook, NY, USA, December 2017. Curran Associates Inc.   
+[10] Oliver T. Unke and Markus Meuwly. PhysNet: A neural network for predicting energies, forces, dipole moments, and partial charges. Journal of Chemical Theory and Computation, 15(6):3678–3693, June 2019.   
+[11] Johannes Klicpera, Janek Groß, and Stephan Günnemann. Directional message passing for molecular graphs. In International Conference on Learning Representations, September 2019.   
+[12] Cheol Woo Park, Mordechai Kornbluth, Jonathan Vandermause, Chris Wolverton, Boris Kozinsky, and Jonathan P. Mailoa. Accurate and scalable graph neural network force field and molecular dynamics with direct force architecture. npj Computational Materials, 7(1):1–9, May 2021.   
+[13] Risi Kondor. N-body networks: A covariant hierarchical neural network architecture for learning atomic potentials. arXiv:1803.01588 [cs], March 2018.   
+[14] Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds. arXiv:1802.08219 [cs], February 2018.   
+[15] Brandon Anderson, Truong Son Hy, and Risi Kondor. Cormorant: Covariant molecular neural networks. In Advances in Neural Information Processing Systems, volume 32, 2019.   
+[16] Marc Finzi, Max Welling, and Andrew Gordon Wilson. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. arXiv:2104.09459 [cs, math, stat], April 2021.   
+339 [17] Fabian Fuchs, Daniel Worrall, Volker Fischer, and Max Welling. SE(3)-transformers: 3D roto-translation equivariant attention networks. Advances in Neural Information Processing Systems, 33:1970–1981, 2020. [18] William Clifford. Applications of Grassmann’s extensive algebra. American Journal of Mathematics, 1(4):350–358, 1878.   
+344 [19] Alan Macdonald. A survey of geometric algebra and geometric calculus. Advances in Applied Clifford Algebras, 27(1):853–891, March 2017. [20] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 5998–6008. Curran Associates, Inc., 2017. [21] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. [22] Paul J. Steinhardt, David R. Nelson, and Marco Ronchetti. Bond-orientational order in liquids and glasses. Physical Review B, 28(2):784–805, July 1983. [23] Walter Mickel, Sebastian C. Kapfer, Gerd E. Schröder-Turk, and Klaus Mecke. Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter. The Journal of Chemical Physics, 138(4):044501, January 2013. [24] Emanuele Boattini, Marjolein Dijkstra, and Laura Filion. Unsupervised learning for local structure detection in colloidal systems. The Journal of Chemical Physics, 151(15):154901, October 2019. [25] Paul S. Clegg. Characterising soft matter using machine learning. Soft Matter, 17(15):3991– 4005, April 2021. [26] Michael J. Mehl, David Hicks, Cormac Toher, Ohad Levy, Robert M. Hanson, Gus Hart, and Stefano Curtarolo. The AFLOW library of crystallographic prototypes: Part 1. Computational Materials Science, 136:S1–S828, August 2017. [27] David Hicks, Michael J. Mehl, Eric Gossett, Cormac Toher, Ohad Levy, Robert M. Hanson, Gus Hart, and Stefano Curtarolo. The AFLOW library of crystallographic prototypes: Part 2. Computational Materials Science, 161:S1–S1011, April 2019. [28] Vyas Ramasubramani, Bradley D. Dice, Eric S. Harper, Matthew P. Spellings, Joshua A. Anderson, and Sharon C. Glotzer. Freud: A software suite for high throughput analysis of particle simulation data. Computer Physics Communications, 254:107275, September 2020.   
+372 [29] Oliver T. Unke, Stefan Chmiela, Huziel E. Sauceda, Michael Gastegger, Igor Poltavsky, Kristof T. Schütt, Alexandre Tkatchenko, and Klaus-Robert Müller. Machine learning force fields. Chemical Reviews, March 2021. [30] Jörg Behler. Four generations of high-dimensional neural network potentials. Chemical Reviews, March 2021. [31] Simon Batzner, Tess E. Smidt, Lixin Sun, Jonathan P. Mailoa, Mordechai Kornbluth, Nicola Molinari, and Boris Kozinsky. SE(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. arXiv:2101.03164 [cond-mat, physics:physics], January 2021. [32] Akihiko Nakamura, Takuya Ishida, Katsuhiro Kusaka, Taro Yamada, Shinya Fushinobu, Ichiro Tanaka, Satoshi Kaneko, Kazunori Ohta, Hiroaki Tanaka, Koji Inaka, Yoshiki Higuchi, Nobuo Niimura, Masahiro Samejima, and Kiyohiko Igarashi. "Newton’s cradle" proton relay with amide–imidic acid tautomerization in inverting cellulase visualized by neutron crystallography. Science Advances, 1(7):e1500263, August 2015.
+
+385 [33] Y. Hirano, S. Kimura, and T. Tamada. High-resolution crystal structures of the solubilized   
+386 domain of porcine cytochrome b5. Acta Crystallographica Section D: Biological Crystallog  
+387 raphy, 71(7):1572–1581, July 2015.   
+388 [34] Shigeru Matsuoka, Shigeru Sugiyama, Daisuke Matsuoka, Mika Hirose, Sébastien Lethu,   
+389 Hikaru Ano, Toshiaki Hara, Osamu Ichihara, S. Roy Kimura, Satoshi Murakami, Hanako   
+390 Ishida, Eiichi Mizohata, Tsuyoshi Inoue, and Michio Murata. Water-mediated recognition   
+391 of simple alkyl chains by heart-type fatty-acid-binding protein. Angewandte Chemie Interna  
+392 tional Edition, 54(5):1508–1511, 2015.   
+393 [35] Yoshihiko Furuike, Yuka Akita, Ikuko Miyahara, and Nobuo Kamiya. ADP-ribose pyrophos  
+394 phatase reaction in crystalline state conducted by consecutive binding of two manganese(ii)   
+395 ions as cofactors. Biochemistry, 55(12):1801–1812, March 2016.   
+396 [36] Hiraku Ohno, Kazuki Takeda, Satomi Niwa, Tomotaka Tsujinaka, Yuya Hanazono, Yu Hirano,   
+397 and Kunio Miki. Crystallographic characterization of the high-potential iron-sulfur protein in   
+398 the oxidized state at $\mathbf { \bar { 0 . 8 \ : \mathring { A } } }$ resolution. PLOS ONE, 12(5):e0178183, May 2017.   
+399 [37] Harry P. Austin, Mark D. Allen, Bryon S. Donohoe, Nicholas A. Rorrer, Fiona L. Kearns, Ro  
+400 drigo L. Silveira, Benjamin C. Pollard, Graham Dominick, Ramona Duman, Kamel El Omari,   
+401 Vitaliy Mykhaylyk, Armin Wagner, William E. Michener, Antonella Amore, Munir S. Skaf,   
+402 Michael F. Crowley, Alan W. Thorne, Christopher W. Johnson, H. Lee Woodcock, John E.   
+403 McGeehan, and Gregg T. Beckham. Characterization and engineering of a plastic-degrading   
+404 aromatic polyesterase. Proceedings of the National Academy of Sciences, 115(19):E4350–   
+405 E4357, May 2018.   
+406 [38] Alessandro Vergara, Marco Caterino, and Antonello Merlino. Raman-markers of X-ray radia  
+407 tion damage of proteins. International Journal of Biological Macromolecules, 111:1194–1205,   
+408 May 2018.   
+409 [39] Alexander T. Baker, Alexander Greenshields-Watson, Lynda Coughlan, James A. Davies,   
+410 Hanni Uusi-Kerttula, David K. Cole, Pierre J. Rizkallah, and Alan L. Parker. Diversity within   
+411 the adenovirus fiber knob hypervariable loops influences primary receptor interactions. Nature   
+412 Communications, 10(1):741, February 2019.   
+413 [40] K. Takaba, Y. Tai, H. Eki, H.-A. Dao, Y. Hanazono, K. Hasegawa, K. Miki, and K. Takeda.   
+414 Subatomic resolution X-ray structures of green fluorescent protein. IUCrJ, 6(3):387–400, May   
+415 2019.   
+416 [41] Janet M. Paterson, Amy J. Shaw, Ian Burns, Alister W. Dodds, Alpana Prasad, Ken B. Reid,   
+417 Trevor J. Greenhough, and Annette K. Shrive. Atomic-resolution crystal structures of the   
+418 immune protein conglutinin from cow reveal specific interactions of its binding site with N  
+419 acetylglucosamine. Journal of Biological Chemistry, 294(45):17155–17165, November 2019.   
+420 [42] Wei Tian, Peiqiang Yan, Ning Xu, Arghya Chakravorty, Robert Liefke, Qiaoran Xi, and   
+421 Zhanxin Wang. The HRP3 PWWP domain recognizes the minor groove of double-stranded   
+422 DNA and recruits HRP3 to chromatin. Nucleic Acids Research, 47(10):5436–5448, June 2019.   
+423 [43] Matthew J Schellenberg, C Denise Appel, Amanda A Riccio, Logan R Butler, Juno M Krahn,   
+424 Jenna A Liebermann, Felipe Cortés-Ledesma, and R Scott Williams. Ubiquitin stimulated   
+425 reversal of topoisomerase 2 DNA-protein crosslinks by TDP2. Nucleic Acids Research,   
+426 48(11):6310–6325, June 2020.   
+427 [44] Marina Plaza-Garrido, Mª Carmen Salinas-García, José C. Martínez, and Ana Cámara-Artigas.   
+428 The effect of an engineered ATCUN motif on the structure and biophysical properties of   
+429 the SH3 domain of c-Src tyrosine kinase. JBIC Journal of Biological Inorganic Chemistry,   
+430 25(4):621–634, June 2020.   
+431 [45] Shanshan Wu, Tam T. T. N. Nguyen, Olga V. Moroz, Johan P. Turkenburg, Jens E. Nielsen,   
+432 Keith S. Wilson, Kasper D. Rand, and Kaare Teilum. Conformational heterogeneity of Sav  
+433 inase from NMR, HDX-MS and X-ray diffraction analysis. PeerJ, 8:e9408, June 2020.   
+434 [46] Barbara Franke, Marta Veses-Garcia, Kay Diederichs, Heather Allison, Daniel J. Rigden, and   
+435 Olga Mayans. Structural annotation of the conserved carbohydrate esterase vb_24B_21 from   
+436 Shiga toxin-encoding bacteriophage $\Phi 2 4 \mathbf { B }$ . Journal of Structural Biology, 212(1):107596,   
+437 October 2020.   
+438 [47] Matthias Barone, Matthias Müller, Slim Chiha, Jiang Ren, Dominik Albat, Arne Soicke,   
+439 Stephan Dohmen, Marco Klein, Judith Bruns, Maarten van Dinther, Robert Opitz, Peter Linde  
+440 mann, Monika Beerbaum, Kathrin Motzny, Yvette Roske, Peter Schmieder, Rudolf Volkmer,   
+441 Marc Nazaré, Udo Heinemann, Hartmut Oschkinat, Peter ten Dijke, Hans-Günther Schmalz,   
+442 and Ronald Kühne. Designed nanomolar small-molecule inhibitors of Ena/VASP EVH1 inter  
+443 action impair invasion and extravasation of breast cancer cells. Proceedings of the National   
+444 Academy of Sciences, 117(47):29684–29690, November 2020.   
+445 [48] Renee Otten, Ricardo A. P. Pádua, H. Adrian Bunzel, Vy Nguyen, Warintra Pitsawong,   
+446 MacKenzie Patterson, Shuo Sui, Sarah L. Perry, Aina E. Cohen, Donald Hilvert, and Dorothee   
+447 Kern. How directed evolution reshapes the energy landscape in an enzyme to boost catalysis.   
+448 Science, 370(6523):1442–1446, December 2020.   
+449 [49] Karla Frydenvang, Darryl S. Pickering, Giridhar U. Kshirsagar, Giulia Chemi, Sandra Gemma,   
+450 Desiree Sprogøe, Anne Mette Kærn, Simone Brogi, Giuseppe Campiani, Stefania Butini,   
+451 and Jette Sandholm Kastrup. Ionotropic glutamate receptor GluA2 in complex with bicyclic   
+452 pyrimidinedione-based compounds: When small compound modifications have distinct effects   
+453 on binding interactions. ACS Chemical Neuroscience, 11(12):1791–1800, June 2020.   
+454 [50] Steffen Glöckner, Andreas Heine, and Gerhard Klebe. A proof-of-concept fragment screening   
+455 of a hit-validated 96-compounds library against human carbonic anhydrase ii. Biomolecules,   
+456 10(4):518, April 2020.   
+457 [51] Helen M. Berman, John Westbrook, Zukang Feng, Gary Gilliland, T. N. Bhat, Helge Weissig,   
+458 Ilya N. Shindyalov, and Philip E. Bourne. The Protein Data Bank. Nucleic Acids Research,   
+459 28(1):235–242, January 2000.   
+460 [52] H. Kamberaj, R. J. Low, and M. P. Neal. Time reversible and symplectic integrators   
+461 for molecular dynamics simulations of rigid molecules. The Journal of Chemical Physics,   
+462 122(22):224114, June 2005.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes]   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] We feel that the methods presented in this paper are sufficiently agnostic to applications that it is hard to directly address any negative or positive societal impacts of the work.   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code and scripts to reproduce results to be included as SI before the appropriate deadline.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Details of training and architectures are provided in the Results section, as well as the code that will be included in the SI.   
+(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? [N/A] No existing assets were evaluated.   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Trained models and workflows will be included in the SI.   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/il3vPAc9fiH/il3vPAc9fiH_content_list.json

@@ -0,0 +1,923 @@
+[
+    {
+        "type": "text",
+        "text": "Geometric Attention Networks for Small Point Clouds ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            122,
+            821,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            196,
+            578,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            289,
+            535,
+            306
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Much of the success of deep learning is drawn from building architectures that   \n2 properly respect underlying symmetry and structure in the data on which they   \n3 operate—a set of considerations that have been united under the banner of geo  \n4 metric deep learning. Often problems in the physical sciences deal with relatively   \n5 small sets of points in two- or three-dimensional space wherein translation, rota  \n6 tion, and permutation equivariance are important or even vital for models to be   \n7 useful in practice. In this work, we present an architecture for deep learning on   \n8 these small point clouds with rotation and permutation equivariance, composed of   \n9 a set of products of terms from the geometric algebra and reductions over those   \n10 products using an attention mechanism. The geometric algebra provides valuable   \n11 mathematical structure by which to combine vector, scalar, and other types of   \n12 geometric inputs in a systematic way to account for rotation invariance or covari  \n13 ance, while attention yields a powerful way to impose permutation equivariance.   \n14 We demonstrate the usefulness of these architectures by training models to solve   \n15 sample problems relevant to physics, chemistry, and biology. ",
+        "bbox": [
+            148,
+            320,
+            767,
+            547
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "16 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            571,
+            282,
+            588
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "17 Deep learning has been immensely successful in solving a wide range of problems over the last   \n18 several years, driven in large part by identifying appropriate ways to embed structure of data and   \n19 symmetry of problems directly into the architecture of the network—an idea at the core of geometric   \n20 deep learning[1]. Some applications of geometric deep learning include the use of convolutional   \n21 filters in CNNs to attain translational equivariance, or graph convolutions in graph neural networks   \n22 for permutation equivariance.1 Building symmetry into the architecture of a deep neural network   \n23 can improve the data efficiency of the network and guarantee important analytical properties without   \n24 having to rely on the network to learn to approximate them from training data. ",
+        "bbox": [
+            147,
+            603,
+            825,
+            724
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we derive a family of architectures that is useful in applications from physics to biology, where problems often deal with relatively small point clouds of labeled coordinates. These could be local environments of particles assembling into a crystal[2], atoms in a molecule interacting with other atoms[3], or coarse-grained beads representing parts of a protein[4]. In many of these applications without the influence of an external field, we are interested in modeling attributes of the ",
+        "bbox": [
+            173,
+            731,
+            825,
+            806
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27ce48a5cf12afdc4b3d1d5f586f74bcaaefe6f8657aa238a3895c4902be649c.jpg",
+        "text": "$$\n\\begin{array} { r l r } {  { \\sum _ { \\theta = 0 } ^ { \\infty } \\sum _ { \\theta = 0 } ^ { \\infty } \\boldsymbol { v } ( | \\vec { r } _ { i } | , v _ { 0 } , v _ { i } ) f ( | \\vec { r } _ { i } | , v _ { 0 } , v _ { i } ) } } \\\\ & { } & { \\diamond \\bigotimes _ { \\theta = 0 } ^ { \\infty } \\int _ { \\phi = 0 } ^ { \\infty } \\boldsymbol { \\hat { \\rho } } _ { 0 } \\cdot \\boldsymbol { \\varrho } _ { \\phi } \\big ( \\vec { r } _ { 0 } | , \\vec { \\varrho } _ { 0 } \\big ) } \\\\ & { } & { \\diamond \\bigotimes _ { \\phi = 0 } ^ { \\infty } \\sum _ { \\theta = 0 } ^ { \\infty } \\sum _ { \\rho = 1 , v _ { i } = v _ { i } } ^ { \\infty } f ( | \\vec { r } _ { i } | , v _ { 0 } , v _ { i } , v _ { j } ) } \\\\ & { } & { \\cdot \\bigotimes _ { \\phi = 0 } ^ { \\infty } \\sum _ { \\rho = 1 } ^ { \\infty } \\boldsymbol { \\rho } ( | \\vec { r } _ { i } | , v _ { 0 } , v _ { i } ) f ( | \\vec { r } _ { i } | , v _ { 0 } , v _ { i } , v _ { j } ) \\bigotimes _ { \\phi } \\bigotimes _ { \\varrho } \\bigotimes _ { \\varrho } ^ { \\infty } \\bigotimes _ { \\varrho } ^ { \\infty } \\int _ { \\phi = 0 } ^ { \\infty } \\boldsymbol { \\hat { \\varrho } } _ { \\phi } ( | \\vec { r } _ { i } | , \\vec { \\varrho } _ { 0 } ) , } \\\\ & { } & { \\quad \\times ( | \\vec { r } _ { i } | ) f ( | ) + \\ldots } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            202,
+            88,
+            795,
+            239
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "",
+        "image_caption": [
+            "Figure 1: Overall strategy for incorporating rotation and permutation equivariance into deep neural networks using attention mechanisms and geometric products. (a) At its simplest level, our proposed structure uses an attention mechanism over the bond lengths of a cloud of points, each of which carries a value as commonly used in graph neural networks. (b) Geometric products (or linear combinations thereof) can be used to combine pairs, triplets, or larger tuples of vectors in a systematic and geometrically meaningful way. Rotational equivariance can be attained by using invariant or covariant quantities, as desired. (c) An attention mechanism reduces the set of generated geometric products to enforce proper permutation equivariance, and the learned attention maps can provide insights into how models operate. For example, here, a carbon atom in a naphthalene molecule (indicated by a blue star) directs its focus broadly around the carbon atoms of the aromatic rings in which it is situated, rather than focusing exclusively on its nearest neighbors in the molecular graph. Brighter-colored bonds indicate a greater attention weight for the two atoms sharing the bond. "
+        ],
+        "image_footnote": [],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "30 system—such as the identity of a particle’s local self-assembly environment, or the potential energy   \n31 of a group of atoms—which are invariant with respect to rotation of the input coordinates, as well as   \n32 permutation in the ordering of points. Here we attain rotation invariance by constructing functions   \n33 from rotation-invariant components of geometric products of input vectors from geometric algebra,   \n34 and permutation invariance by using an attention mechanism to intelligently reduce representations   \n35 over the set of vector products. ",
+        "bbox": [
+            147,
+            444,
+            825,
+            534
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "36 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            147,
+            553,
+            272,
+            568
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "37 “Large” point clouds. Point clouds are a ubiquitous data structure and are often found in domains   \n38 outside of the physical sciences. For the purposes of this work, we focus on comparatively small sets   \n39 of points where the points are relatively information-rich—for example, carrying information about   \n40 atom identities, local environments, or other information—in contrast to point clouds commonly   \n41 found in computer vision and robotics which may represent the geometry of a mesh or be sampled   \n42 from an object, but otherwise not have as much information associated with each point. We refer   \n43 to Guo et al. for a survey of this field[5], but a few of the recently-developed notable approaches   \n44 include PointNet[6], deep sets[7], and kernel point convolutions[8].   \n45 Geometric approaches for small point clouds. Many architectures have been proposed to incor  \n46 porate rotation equivariance by augmenting graph neural networks with geometric attributes that   \n47 are known to be rotation-invariant, such as bond lengths and angles. SchNet[9] learns distance  \n48 based convolution filters which are used to propagate signals over graphs. PhysNet[10] also refines   \n49 node representations based on bond lengths, while incorporating a learnable attention mechanism.   \n50 DimeNet[11] extends the information used to calculate node-level representations to include angles   \n51 between bonds. GNNFF[12] generates rotation-covariant results by computing a weighted sum of   \n52 modulated input vectors based on a graph message passing scheme.   \n53 Group representation-based approaches. These methods take advantage of group representation   \n54 theory by first transforming inputs into a space in which rotation-equivariant maps are more eas  \n55 ily expressed. This set of methods is powerful, having been used in the past to design rotation  \n56 and permutation-equivariant models[13–15], and have even been expanded recently for arbitrary   \n57 groups[16]. Attention-based models have also been utilized in this area: SE(3) Transformers[17]   \n58 extend tensor field networks[14] with a self-attention mechanism for increased expressivity by in  \n59 corporating value- and geometry-dependent attention weights.   \n60 The approach we present here is similar to several of the ideas presented above; however, rather   \n61 than specifying particular rotation-invariant quantities to utilize or learning maps that operate on   \n62 irreducible representations, we leverage the structure provided by geometric algebra to determine   \n63 which rotation-invariant and -covariant quantities are of interest. ",
+        "bbox": [
+            147,
+            580,
+            825,
+            702
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            708,
+            825,
+            830
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            837,
+            825,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            127,
+            825,
+            188
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "64 Geometric Attention Networks ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            207,
+            433,
+            224
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "65 In this work we formulate deep neural networks using learnable functions consisting of two parts:   \n66 (1) a set of geometric products of input vectors; and (2) a permutation-equivariant reduction over   \n67 these products using an attention mechanism. We describe each of these aspects below. ",
+        "bbox": [
+            147,
+            239,
+            825,
+            284
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "68 Geometric Algebra ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            300,
+            310,
+            314
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "69 The geometric algebra was developed in the 19th century and provides a consistent framework for   \n70 dealing with scalars and other geometric quantities—such as vectors, areas, and volumes in three  \n71 dimensional space—in arbitrary dimensions[18]. Here, we will describe the essential parts of geo  \n72 metric algebra as related to our proposed attention mechanism, and defer to other works for a more   \n73 thorough description[19]. The geometric algebra specifies a binary operator, the geometric product,   \n74 that works on multivectors. Multivectors can be expressed as linear combinations of terms from a   \n75 fixed basis set for a given space, such as $\\mathbb { R } ^ { 2 }$ or $\\mathbb { R } ^ { 3 }$ ; in three-dimensional space, this yields scalars,   \n76 vectors, bivectors (which specify signed areas within a plane and have 3 components), and trivec  \n77 tors (which specify signed volumes and have 1 component)—a total of 8 linearly independent terms   \n78 for each multivector2. When rotation invariance is desired, we can utilize the rotation-invariant   \n79 components of a multivector: scalars, trivectors, the norms of vectors, and the norms of bivectors   \n80 are rotation-invariant. As an example, the geometric product of two vectors yields a scalar plus a   \n81 bivector; the scalar component is the dot product, and the bivector component is related to the cross   \n82 product of the two.   \n83 Geometric algebra provides a general framework that can be used to build up expressive functions   \n84 as linear combinations and geometric products of multivector inputs; rotation-equivariant quantities   \n85 can then be derived from the products, depending on the application and symmetry of the problem   \n86 of interest. The types of elements produced by a geometric product of two multivectors in $\\mathbb { R } ^ { 3 }$ with   \n87 the given components are listed in Table 1 below. ",
+        "bbox": [
+            147,
+            327,
+            826,
+            537
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            545,
+            825,
+            619
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Terms arising from the geometric product $A B = ( A _ { s } + A _ { v } + A _ { b } + A _ { t } ) ( B _ { s } + B _ { v } + B _ { b } + B _ { t } )$ in $\\mathbb { R } ^ { 3 }$ . In three dimensions, multivectors $A$ and $B$ consist of scalars $( s )$ , vectors $( \\nu )$ , bivectors $( b )$ , and trivectors $\\mathbf { \\rho } ( t )$ . ",
+        "bbox": [
+            173,
+            631,
+            826,
+            674
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/9287884a30f8e943731f6a94987626b4a60e83aecd0c777962239e076d17a500.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Bs</td><td>Bv</td><td>Bb</td><td>Bt</td></tr><tr><td>会</td><td>S</td><td>V</td><td>b</td><td>t</td></tr><tr><td></td><td>V</td><td>s+b</td><td>V+t</td><td>b</td></tr><tr><td></td><td>b</td><td>V+t</td><td>s+b</td><td>V</td></tr><tr><td>At</td><td>t</td><td>b</td><td>V</td><td>S</td></tr></table>",
+        "bbox": [
+            385,
+            681,
+            612,
+            761
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "88 From Table 1, we can see that successive products of vectors alternate between producing two types   \n89 of multivectors: products of even numbers of vectors yield a scalar and bivector $( ( v + t ) v = v v + t v =$   \n90 $( s + b ) + b \\to s + b )$ , while products of odd numbers of vectors produce a vector and trivector   \n91 $( ( s + b ) v = s v + b v = v + ( v + t )  v + t )$ . Generating rotation-invariant quantities from these   \n92 products is the primary application of geometric algebra in this work, although in general the method   \n93 could be used to incorporate different types of scalar, vector, bivector, and trivector quantities; for   \n94 example, rotations could be input as quaternions, which are isomorphic to the scalar-and-bivector   \n95 product of even numbers of vectors. ",
+        "bbox": [
+            145,
+            775,
+            825,
+            852
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            90,
+            825,
+            136
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "96 Attention from Geometric Products ",
+        "text_level": 1,
+        "bbox": [
+            151,
+            152,
+            424,
+            167
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "97 For input point clouds with $N$ points, we can construct a series of successively higher-order geo  \n98 metric products for all $N ^ { 2 }$ possible pairs, $N ^ { 3 }$ triplets, and so on; these individual points, pairs, or   \n99 triplets we will call a tuple in this context. In addition to a coordinate $\\vec { r _ { i } }$ , we associate a set of values   \n100 $v _ { i }$ to each point indexed by $i$ in some space with a given working dimension (we avoid calling these   \n101 vectors to decrease the confusion with geometric vectors; these correspond to the non-geometric   \n102 attributes of the point, such as type embeddings). To create permutation-covariant functions (pro  \n103 ducing a value for each input point) or permutation-invariant functions (producing a single output   \n104 value), we make use of a simple attention mechanism based on the rotation-invariant attributes of   \n105 each tuple. Attention has been used widely in applying deep learning to a range of problem do  \n106 mains over the last few years, with particular success in the field of natural language processing[20].   \n107 Since we are already generating tuple-wise quantities, we choose to utilize a simpler mechanism   \n108 than the typical dot product self-attention. We specify four functions: a value-generating function   \n109 $\\nu$ , a tuple value-merging function $\\mathcal { M }$ , a joining function that summarizes the invariant and tuple   \n110 representations $\\mathcal { I }$ , and a score-generating function $s$ . The functions have the following uses within   \n111 the network: ",
+        "bbox": [
+            142,
+            180,
+            826,
+            405
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "• $\\nu$ produces features in the working dimension of the model from the invariants associated with each tuple.   \n• $\\mathcal { M }$ merges the 1, 2, 3, or more values associated with a tuple of input points into the working dimension of the model. The form of $\\mathcal { M }$ could be a complex function, a learned linear projection for each tuple position, or simply taking the sum of the tuple values.   \n• $\\mathcal { I }$ joins the invariant representations from $\\nu$ and the tuple representations from $\\mathcal { M }$ . Like $\\mathcal { M }$ , it could be a learned projection or a simple sum function.   \n• $s$ generates score logits from the representation of each tuple, which incorporates invariants associated with the tuple and the values being associated with each point that is part of the tuple. The results from $s$ , passed through a softmax function, will yield the weights for the attention mechanism. ",
+        "bbox": [
+            215,
+            420,
+            825,
+            603
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "123 We first calculate the multivector geometric products $p _ { i j k \\ldots }$ of all combinations of input vectors   \n124 $i , j , k$ , and so on, up to a specified rank (pairwise attention would produce a two-dimensional   \n125 matrix of products $p _ { i j }$ ). We then use $\\nu , \\mathcal { M } , \\mathcal { I }$ , and $s$ —together with a function extracting the   \n126 rotation-invariant attributes of a geometric product (the scalar component, trivector component, and   \n127 the norms of the vector and bivector components)—as follows for a network producing permutation  \n128 covariant outputs $y _ { i }$ : ",
+        "bbox": [
+            140,
+            618,
+            826,
+            709
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/772c131ca16473698e5dc17cf5141573c2aa6849dc0e5529bea3537a656a616e.jpg",
+        "text": "$$\n\\begin{array} { r l } & { p _ { i j k \\ldots } = \\vec { r } _ { i } \\vec { r } _ { j } \\vec { r } _ { k } \\ldots } \\\\ & { q _ { i j k \\ldots } = \\mathrm { i n v a r i a n t s } ( p _ { i j k \\ldots } ) } \\\\ & { v _ { i j k \\ldots } = \\mathcal { I } ( \\mathcal { V } ( q _ { i j k \\ldots } ) , \\mathcal { M } ( v _ { i } , v _ { j } , v _ { k } , \\ldots ) ) } \\\\ & { w _ { i j k \\ldots } = \\underset { j k \\ldots } { \\operatorname { s o f t m a x } } ( \\mathcal { S } ( v _ { i j k \\ldots } ) ) } \\\\ & { \\qquad y _ { i } = \\underset { j k \\ldots } { \\sum } w _ { i j k \\ldots } v _ { i j k \\ldots } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            733,
+            637,
+            854
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "129 If a permutation-invariant reduction is desired, then the softmax and final sum can be performed over   \n130 all tuples simultaneously, rather than for each input point individually. While $\\mathcal { I } , \\mathcal { V }$ , and $\\mathcal { M }$ could in   \n131 principle be used to change the working dimension as permutation-covariant layers are stacked on   \n132 top of each other, in this work we keep the working dimension constant for the sake of easily adding   \n133 residual connections.   \n134 If rotation-covariant, rather than rotation-invariant, behavior is needed for the output of the network,   \n135 the same attention structure can be used with slight modifications; here, we coerce a vector from   \n136 the product $p _ { i j k \\ldots }$ (which consists of directly taking the vector component from products of odd   \n137 numbers of input vectors, or multiplying a bivector by the unit trivector to produce a vector—as   \n138 shown in the last column of Table 1—in the case of even numbers of input vectors). These vectors   \n139 can be combined with a scalar rescaling each vector—generated by a learned function $\\mathcal { R }$ —and the   \n140 attention mechanism to yield ",
+        "bbox": [
+            140,
+            866,
+            823,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            150,
+            90,
+            825,
+            121
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            127,
+            825,
+            234
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9ea34c232c71b954848367e7038eacee323ce43eb05adf831eea16f6e4be3139.jpg",
+        "text": "$$\n\\vec { r } _ { i } ^ { \\prime } = \\sum _ { j k \\ldots } w _ { i j k \\ldots } \\mathcal { R } ( q _ { i j k \\ldots } ) \\mathrm { v e c t o r } ( p _ { i j k \\ldots } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            251,
+            633,
+            286
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "141 Results ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            303,
+            236,
+            319
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "142 We demonstrate the utility of our geometric algebra attention scheme by training deep networks to   \n143 solve three problems appearing in physics, chemistry, and biology. For simplicity, all the models   \n144 presented here utilize pairwise attention with a working depth of 32 units. Value functions $\\nu$ , score   \n145 functions $s$ , and rescaling functions $\\mathcal { R }$ are simple multilayer perceptrons with a hidden width of   \n146 64 units, with layer normalization applied to the output of $\\nu$ . The network for crystal structure   \n147 identification uses the mean function for merge functions $\\mathcal { M }$ and join functions $\\mathcal { I }$ , while the other   \n148 two applications use learned linear projections. Networks are trained for up to 800 epochs using   \n149 the adam optimizer[21]; the learning rate is decreased by a factor of 0.75 after the validation set   \n150 loss does not decrease for 20 epochs, and training is ended early if the validation set loss does not   \n151 decrease for 50 epochs. Numerical results are reported as the mean and standard error of the mean   \n152 over 5 samples. Python code under the MIT license implementing each experimental workflow is   \n153 included in the supplementary information. ",
+        "bbox": [
+            140,
+            334,
+            825,
+            515
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "154 Crystal Structure Identification ",
+        "text_level": 1,
+        "bbox": [
+            145,
+            531,
+            395,
+            546
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "155 On length scales ranging from those of atoms to colloidal particles, matter often organizes itself into   \n156 ordered two- or three-dimensional structures. One of the core ideas of materials science is that struc  \n157 ture is one of the major determining factors for material behavior. With this perspective in mind,   \n158 when studying computational models of self-assembling systems we often first identify what struc  \n159 tures, if any, have formed in our simulations—a task complicated by naturally-occurring thermal   \n160 noise, crystallographic defects, and potentially the complexity of the structures themselves. Early   \n161 efforts to automatically characterize structure led to the widely-used Steinhardt order parameters[22–   \n162 24], which are rotationally-invariant sums of spherical harmonic magnitudes over local particle   \n163 environments. While the Steinhardt order parameters can be useful when studying phase transi  \n164 tions or distinguishing among a small number of phases, determining appropriate hyperparameters—   \n165 including neighborhood size to consider, spherical harmonic order $\\ell$ to use, and thresholds to iden  \n166 tify behaviors of interest—can be difficult[23]. For this reason, data-driven approaches to analyzing   \n167 structure have been the subject of great interest in recent years[25].   \n168 We use geometric attention networks to identify the source structure type of small neighborhoods   \n169 of particles extracted from bulk crystals. We select 8 prototypes of single- and two-component   \n170 crystals from the AFLOW Encyclopedia of Crystallographic Prototypes[26, 27]. These structures   \n171 are chosen to demonstrate that models can learn not only geometric information ( $c F 4 – C \\mathrm { u }$ and $h P 2 \\cdot$ -   \n172 $\\mathbf { M g }$ are similar structures but with a different stacking of their close-packed layers; the clathrates   \n173 $c P 4 6 – S \\mathrm { i }$ and $c F 1 3 6 { - } \\mathrm { S i }$ are also similar, with a different arrangement of many common motifs), but   \n174 also the information encoded within each point ( $c P \\mathrm { 2 \\mathrm { - } C s C l }$ and $c F 8 – Z \\mathrm { n S }$ differ from $c I 2 – \\mathrm { { W } }$ and   \n175 $c F 8 – C$ only by their particle type assignments). For each structure, we rescale the unit cell such that   \n176 the shortest nearest-neighbor distance over the structure is 1 before replicating the unit cell to consist   \n177 of at least 2048 particles. We then create three samples of each structure by adding Gaussian noise   \n178 with a standard deviation of $1 0 ^ { - 3 }$ , $5 \\cdot 1 0 ^ { - 2 }$ , and 0.1 separately to the particle coordinates, in order to   \n179 emulate thermal noise. For each particle in the structure, we find the 12 nearest neighbors and their   \n180 associated types using the freud[28] python library, which we feed into the network as the pairwise   \n181 distance $\\vec { r } _ { i j } = \\vec { r } _ { j } - \\vec { r } _ { i }$ and one-hot-encoded symmetrized type vector $\\Delta t _ { i j } = [ I _ { t _ { i } } - I _ { t _ { j } } , I _ { t _ { i } } + I _ { t _ { j } } ]$   \n182 where $I$ is the identity matrix of dimension corresponding to the maximum number of types.   \n183 We train classifiers with 2 permutation-covariant attention blocks before a final reduction over the en  \n184 tire particle neighborhood—as shown in Figure 2—in order to categorize local particle environments   \n185 according to their source crystal structure type. These networks rapidly learn to identify structures   \n186 after a few epochs, with a final overall accuracy of $9 8 . 7 \\% \\pm 0 . 2 \\%$ after training for roughly 45   \n187 minutes on an NVIDIA Titan Xp GPU. ",
+        "bbox": [
+            142,
+            558,
+            826,
+            753
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            761,
+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/ff776cb7c3a857ef1656d4e700d60a61b4ceca7001d4cf979a73f1d5d30f8862.jpg",
+        "image_caption": [
+            "Figure 2: (a) Network architecture for crystal structure identification. Coordinates and particle types are passed through two permutation-covariant layers before a final permutation-invariant reduction. (b) Crystal structure prototypes chosen for the structure identification benchmark. Simple and complex structure types—including two binary structures—are included. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            254,
+            89,
+            740,
+            318
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            416,
+            825,
+            493
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            501,
+            825,
+            575
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "188 Molecule Force Regression ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            597,
+            364,
+            612
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "189 One of the most dramatic contributions of deep learning to the field of chemistry lies in constructing   \n190 fast, accurate approximations of expensive physical calculations[29, 30]. Machine learning models   \n191 can be many orders of magnitude faster than the methods used to generate their training data, which   \n192 can bring vastly more detailed and longer-time simulations into the realm of possibility. Central   \n193 to the applicability of these methods are issues of symmetry and equivariance: any imperfection   \n194 in rotational invariance of a learned potential energy function could ruin the proper thermodynamic   \n195 behavior of a model, for example, so models must be carefully designed to ensure physical behavior.   \n196 In a method similar to Batzner et al.[31], we train models to predict the per-atom forces calculated   \n197 using ab initio molecular dynamics and density functional theory available in the MD17 dataset[3].   \n198 As shown in Figure 3, we first transform the raw coordinates and types of each atom in a given   \n199 molecule into the pairwise difference and symmetric sum and difference of the coordinates and one  \n200 hot type encoding for each atom with respect to each other atom, respectively, to fix translation   \n201 invariance and assign type representations to the pairwise particle bonds. We then perform a series   \n202 of geometric attention calculations, calculating new values per atom, which are finally summed to   \n203 produce a scalar energy. The gradient of this energy with respect to the input coordinates is used to   \n204 produce the force output of the network, which ensures that a conservative force field is learned.   \n205 Consistent with previous benchmarks on this dataset, we train networks using the mean squared   \n206 distance loss for each molecule using 1,000 snapshots of forces each as training, validation, and test   \n207 data sets. We also report results for models trained on all molecules’ data simultaneously, or 8,000   \n208 snapshots each for training, validation, and testing. Training a model on an individual molecule’s   \n209 data takes between 30 minutes (ethanol and malonaldehyde, with nine atoms each) to two hours   \n210 (aspirin, with twenty-one atoms) on an NVIDIA Titan Xp GPU, while the all-molecule dataset   \n211 requires roughly 16 hours to train. Test set losses, expressed as the mean absolute error over each   \n212 force component for each sample, are presented in Table 2.   \n213 Our geometric algebra attention networks produce results competitive with SchNet[9], an architec  \n214 ture using learned radial distance convolution filters. Although the models generated here do not out  \n215 perform the Neural Equivariant Interatomic Potentials by Batzner et al.[31], we note that our models   \n216 are trained for a fraction of the time (2 GPU hours and 800 epochs for our method, compared to on   \n217 the order of 8 GPU days and 2500 epochs for NequIP) and without drastic hyperparameter tuning   \n218 aside from optimizing the number of residual blocks to use in the network architecture. Notably, the   \n219 models trained on all 8,000 molecular snapshots perform significantly better than almost all of the   \n220 specialized models, indicating that additional data could likely improve the results presented here   \n221 even without careful hyperparameter optimization. ",
+        "bbox": [
+            140,
+            626,
+            825,
+            732
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            738,
+            826,
+            875
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            882,
+            823,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/2fc4c67c06ad4a6039e383b9b361000b0a01b626ee78943320c9e3377786a94e.jpg",
+        "image_caption": [
+            "Figure 3: (a) Network architecture for molecular force regression. Coordinates and particle types for all atoms in a molecule are fed into the network as a set of pairwise distances, with the atomic representations refined through a series of geometric algebra attention layers. Six permutation-covariant layers are stacked before reducing the representations with a final, permutation-invariant geometric product attention layer. (b) Sample pairwise attention maps for four training data molecules (malonaldehyde, aspirin, benzene, and uracil) after filtering out low-attention pairs. The attention maps indicate how strongly the pair of atoms joined by the line affect the representation of the atom indicated with a star, with lighter lines indicating greater influence. Qualitatively, more complex bonding environments such as those on the right tend to have longer-range attention interactions than the simpler environments on the left. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            89,
+            754,
+            267
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            450,
+            825,
+            541
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/557f287b29592e3862598db2acc3283d6aaf0e723f20710c6fd81cdab4b7a44a.jpg",
+        "table_caption": [
+            "Table 2: Mean absolute error of force components (in $\\frac { m e V } { \\mathring { \\mathsf { A } } }$ ) for geometric algebra attention networks, NequIP[31], and SchNet[9] architectures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Molecule</td><td>This work</td><td> NequIP</td><td>SchNet</td></tr><tr><td>Aspirin</td><td>37.0 ± 1.1</td><td>15.1</td><td>58.5</td></tr><tr><td>Benzene</td><td>11.8 ± 0.5</td><td>8.1</td><td>13.4</td></tr><tr><td>Ethanol</td><td>21.4 ± 0.5</td><td>9.0</td><td>16.9</td></tr><tr><td>Malonaldehyde</td><td>30.6 ± 1.1</td><td>14.6</td><td>28.6</td></tr><tr><td>Naphthalene</td><td>23.7 ± 1.0</td><td>4.2</td><td>25.2</td></tr><tr><td>Salicylic acid</td><td>30.2 ± 1.2</td><td>10.3</td><td>36.9</td></tr><tr><td>Toluene</td><td>20.5 ± 1.3</td><td>4.4</td><td>24.7</td></tr><tr><td>Uracil</td><td>27.4 ± 0.8</td><td>7.5</td><td>24.3</td></tr><tr><td>All molecules</td><td>10.7 ± 0.2</td><td></td><td></td></tr></table>",
+        "bbox": [
+            320,
+            599,
+            678,
+            753
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            775,
+            826,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/dbda3c3058ead5d71db8172159107aef6bb9a261bd655e47f63cc8c71f4481e4.jpg",
+        "image_caption": [
+            "Figure 4: Network architecture for inverting a coarse-grained mapping of a protein. Models are trained to predict atomic-resolution coordinates from $N _ { \\mathrm { c o a r s e } } = 1 2$ neighboring amino acid centers of mass using geometrically-informed scalar-to-scalar attention (blue), scalar-to-vector attention (purple), and vector-to-vector attention (red). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            263,
+            90,
+            732,
+            314
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "222 Backmapping Coarse-Graining in Proteins ",
+        "text_level": 1,
+        "bbox": [
+            140,
+            412,
+            475,
+            429
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "223 When simulating large molecules—such as proteins or other polymers—it is common to employ   \n224 coarse graining: a process by which groups of particles are merged into (fewer) distinct beads, en  \n225 abling faster simulations by decreasing the number of degrees of freedom of the model[4]. Although   \n226 data-driven approaches have been highly successful to formulate coarse graining operations in the   \n227 forward direction (that is, from more-detailed to less-detailed systems), some problems are best   \n228 solved using the original, fine-grained system coordinates, which are not directly available in coarse  \n229 grained simulations. To demonstrate the potential for our geometric algebra attention mechanism   \n230 on this task, we train models to predict the coordinates of the heavy atoms that form an amino acid   \n231 from the centers of mass of the nearest-neighbor amino acids. We take 19 protein structures[32–50]   \n232 that have high-resolution structural refinements (with resolution error less than or equal to $1 . 0 \\mathring \\mathrm { A } )$ )   \n233 and were published between 2015 and 2020 from the Protein Data Bank[51]. For applications of   \n234 this method to systems at nonzero temperature, we would expect to be better-served by using an   \n235 architecture that produces distributions instead of only point values, but we disregard this here for   \n236 simplicity; in other words, here we are teaching models to memorize the results of structure refine  \n237 ment algorithms, which may be different for each PDB entry. For every amino acid in each entry,   \n238 we create a point cloud of its 12 nearest neighbor amino acid centers of mass, as well as a point   \n239 cloud of the primary amino acid’s atomic coordinates relative to its center of mass. Two layers of   \n240 permutation-covariant geometric product attention are applied to the coarse-grained amino acid co  \n241 ordinates before being passed to a layer which produces a vector output according to Equation 2 by   \n242 augmenting the tuple representation $v _ { i j k \\dots }$ of Equation 1 with labels corresponding to the identity   \n243 of the atom that should be produced, so that the value is calculated as ",
+        "bbox": [
+            143,
+            440,
+            825,
+            758
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f3c0759d9a167e8bfec3083e9cded2eb0d514af37d76b1cb37b9729b9e73ed43.jpg",
+        "text": "$$\nv _ { \\mathrm { a t o m } , i j k \\ldots } = \\mathcal { I } ( v _ { \\mathrm { a t o m } } , \\mathcal { V } ( q _ { i j k \\ldots } ) , \\mathcal { M } ( v _ { i } , v _ { j } , v _ { k } , \\ldots ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            328,
+            782,
+            669,
+            800
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "244 Following this layer—which maps coarse-grained coordinates of amino acids to fine-grained coordi  \n245 nates of atoms—two rotation-covariant layers are applied to the atomic coordinates to further refine   \n246 them, as shown in Figure 4.   \n247 Because the resolution of the structural refinement algorithms is on the order of $0 . 5 \\mathring \\mathrm { A }$ or greater, we   \n248 use the training set error as a measure of the learning progress of the models instead of performing   \n249 a standard split of training, validation, and test set data. After training for roughly 3 hours on   \n250 an NVIDIA Titan $\\mathrm { X p }$ GPU, models achieve a mean absolute error of $0 . 1 2 8 \\mathring { \\mathbf { A } } \\pm 0 . 0 0 2 \\mathring { \\mathrm { A } }$ (down   \n251 from approximately $\\bar { 0 . 5 \\mathring \\mathrm { A } }$ initially), indicating that they are able to learn to reconstruct atomic-scale   \n252 coordinates from coarse-grained positions. ",
+        "bbox": [
+            143,
+            814,
+            826,
+            859
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            866,
+            825,
+            912
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            90,
+            825,
+            136
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "253 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            159,
+            264,
+            175
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Overall, we find the architectures formulated here to be useful for a variety of tasks. Rather than being limited to operating on bond distances and angles as in SchNet[9], PhysNet[10], and DimeNet[11], geometric algebra provides a systematic way to build functions with the desired rotation- and permutation-equivariance, with the flexibility to incorporate other types of geometric objects (such as the orientation quaternion commonly used for anisotropic particles in molecular dynamics methods[52]). The attention mechanism presented here provides a simple yet powerful method to incorporate both geometric and node-level signals. The primitives of our geometric algebra attention scheme—distances, areas, angles, and volumes—and the calculated attention weights naturally lend themselves to interpretability, which we believe will prove useful in distilling insights from trained models. ",
+        "bbox": [
+            171,
+            193,
+            826,
+            343
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Limitations ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            363,
+            256,
+            377
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Combination of terms. Although the architectures presented here work well for the problems we have selected, creating geometric products of vectors is only a subset of the valid combinations that could be generated. In these cases we have carefully chosen sums and differences of input vectors to respect symmetries we would like to impose on the system—such as using the pairwise distance of all input coordinates for the molecular force regression task to impose translation invariance—but it is possible that more powerful models could be formed by incorporating learned linear combinations of inputs or intermediate multivector quantities. We leave this as a topic of future work. ",
+        "bbox": [
+            173,
+            391,
+            825,
+            497
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Computational scaling and neighborhood definition. An obvious limitation to using higherdegree correlations lies in the computational complexity and memory scaling of generating tuples, which are both proportional to $N ^ { r }$ for neighborhoods of $N$ coordinates and tuples of length $r$ . Polynomial scaling behavior can be ameliorated by restricting which combinations of input points are considered, essentially treating the attention weights of all other combinations as 0. These combinations could be randomly sampled from all valid indices $i j k . . .$ or use more physically-relevant restrictions, such as utilizing the molecular connectivity graph for molecular force regression or edges derived from the Voronoi tessellation for other applications. If smoothness of model predictions is a concern—as may be the case for learning general N-body interaction potentials, for example—the architectures presented here could be augmented by incorporating weights that decay to 0 as bonds are broken in the Voronoi diagram graph[23]. ",
+        "bbox": [
+            173,
+            503,
+            826,
+            670
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "283 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            691,
+            269,
+            708
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have presented a strategy for developing rotation- and permutation-equivariant neural network architectures by combining geometric algebra and attention mechanisms. These architectures operate directly on the vector, scalar, and other geometric quantities of interest to produce outputs which respect desirable symmetries by construction. We believe that the mathematical simplicity and the insights derived from attention maps are particularly appealing aspects of the algorithms presented here. We hope that these architectures will help a wider range of scientific disciplines reap the benefits of geometric deep learning. ",
+        "bbox": [
+            174,
+            727,
+            825,
+            832
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            854,
+            266,
+            869
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[1] Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Velickovi ˇ c. Geometric deep learn- ´ ing: Grids, groups, graphs, geodesics, and gauges. arXiv:2104.13478 [cs, stat], April 2021. ",
+        "bbox": [
+            179,
+            881,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "[2] Julia Dshemuchadse, Pablo F. Damasceno, Carolyn L. Phillips, Michael Engel, and Sharon C. Glotzer. Moving beyond the constraints of chemistry via crystal structure discovery with isotropic multiwell pair potentials. Proceedings of the National Academy of Sciences, 118(21), May 2021.   \n[3] Stefan Chmiela, Alexandre Tkatchenko, Huziel E. Sauceda, Igor Poltavsky, Kristof T. Schütt, and Klaus-Robert Müller. Machine learning of accurate energy-conserving molecular force fields. Science Advances, 3(5):e1603015, May 2017.   \n[4] Siewert J. Marrink and D. Peter Tieleman. Perspective on the Martini model. Chemical Society Reviews, 42(16):6801–6822, July 2013.   \n[5] Yulan Guo, Hanyun Wang, Qingyong Hu, Hao Liu, Li Liu, and Mohammed Bennamoun. Deep learning for 3d point clouds: A survey. IEEE transactions on pattern analysis and machine intelligence, 2020. [6] Charles R. Qi, Hao Su, Mo Kaichun, and Leonidas J. Guibas. PointNet: Deep learning on point sets for 3D classification and segmentation. In 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pages 77–85, July 2017. [7] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R. Salakhutdinov, and Alexander J. Smola. Deep sets. Advances in Neural Information Processing Systems, 30, 2017.   \n[8] Hugues Thomas, Charles R. Qi, Jean-Emmanuel Deschaud, Beatriz Marcotegui, François Goulette, and Leonidas J. Guibas. KPConv: Flexible and deformable convolution for point clouds. Proceedings of the IEEE International Conference on Computer Vision, 2019. [9] K. T. Schütt, P.-J. Kindermans, H. E. Sauceda, S. Chmiela, A. Tkatchenko, and K.-R. Müller. SchNet: A continuous-filter convolutional neural network for modeling quantum interactions. In Proceedings of the 31st International Conference on Neural Information Processing Systems, NIPS’17, pages 992–1002, Red Hook, NY, USA, December 2017. Curran Associates Inc.   \n[10] Oliver T. Unke and Markus Meuwly. PhysNet: A neural network for predicting energies, forces, dipole moments, and partial charges. Journal of Chemical Theory and Computation, 15(6):3678–3693, June 2019.   \n[11] Johannes Klicpera, Janek Groß, and Stephan Günnemann. Directional message passing for molecular graphs. In International Conference on Learning Representations, September 2019.   \n[12] Cheol Woo Park, Mordechai Kornbluth, Jonathan Vandermause, Chris Wolverton, Boris Kozinsky, and Jonathan P. Mailoa. Accurate and scalable graph neural network force field and molecular dynamics with direct force architecture. npj Computational Materials, 7(1):1–9, May 2021.   \n[13] Risi Kondor. N-body networks: A covariant hierarchical neural network architecture for learning atomic potentials. arXiv:1803.01588 [cs], March 2018.   \n[14] Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation- and translation-equivariant neural networks for 3d point clouds. arXiv:1802.08219 [cs], February 2018.   \n[15] Brandon Anderson, Truong Son Hy, and Risi Kondor. Cormorant: Covariant molecular neural networks. In Advances in Neural Information Processing Systems, volume 32, 2019.   \n[16] Marc Finzi, Max Welling, and Andrew Gordon Wilson. A practical method for constructing equivariant multilayer perceptrons for arbitrary matrix groups. arXiv:2104.09459 [cs, math, stat], April 2021.   \n339 [17] Fabian Fuchs, Daniel Worrall, Volker Fischer, and Max Welling. SE(3)-transformers: 3D roto-translation equivariant attention networks. Advances in Neural Information Processing Systems, 33:1970–1981, 2020. [18] William Clifford. Applications of Grassmann’s extensive algebra. American Journal of Mathematics, 1(4):350–358, 1878.   \n344 [19] Alan Macdonald. A survey of geometric algebra and geometric calculus. Advances in Applied Clifford Algebras, 27(1):853–891, March 2017. [20] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems 30, pages 5998–6008. Curran Associates, Inc., 2017. [21] Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In Yoshua Bengio and Yann LeCun, editors, 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings, 2015. [22] Paul J. Steinhardt, David R. Nelson, and Marco Ronchetti. Bond-orientational order in liquids and glasses. Physical Review B, 28(2):784–805, July 1983. [23] Walter Mickel, Sebastian C. Kapfer, Gerd E. Schröder-Turk, and Klaus Mecke. Shortcomings of the bond orientational order parameters for the analysis of disordered particulate matter. The Journal of Chemical Physics, 138(4):044501, January 2013. [24] Emanuele Boattini, Marjolein Dijkstra, and Laura Filion. Unsupervised learning for local structure detection in colloidal systems. The Journal of Chemical Physics, 151(15):154901, October 2019. [25] Paul S. Clegg. Characterising soft matter using machine learning. Soft Matter, 17(15):3991– 4005, April 2021. [26] Michael J. Mehl, David Hicks, Cormac Toher, Ohad Levy, Robert M. Hanson, Gus Hart, and Stefano Curtarolo. The AFLOW library of crystallographic prototypes: Part 1. Computational Materials Science, 136:S1–S828, August 2017. [27] David Hicks, Michael J. Mehl, Eric Gossett, Cormac Toher, Ohad Levy, Robert M. Hanson, Gus Hart, and Stefano Curtarolo. The AFLOW library of crystallographic prototypes: Part 2. Computational Materials Science, 161:S1–S1011, April 2019. [28] Vyas Ramasubramani, Bradley D. Dice, Eric S. Harper, Matthew P. Spellings, Joshua A. Anderson, and Sharon C. Glotzer. Freud: A software suite for high throughput analysis of particle simulation data. Computer Physics Communications, 254:107275, September 2020.   \n372 [29] Oliver T. Unke, Stefan Chmiela, Huziel E. Sauceda, Michael Gastegger, Igor Poltavsky, Kristof T. Schütt, Alexandre Tkatchenko, and Klaus-Robert Müller. Machine learning force fields. Chemical Reviews, March 2021. [30] Jörg Behler. Four generations of high-dimensional neural network potentials. Chemical Reviews, March 2021. [31] Simon Batzner, Tess E. Smidt, Lixin Sun, Jonathan P. Mailoa, Mordechai Kornbluth, Nicola Molinari, and Boris Kozinsky. SE(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. arXiv:2101.03164 [cond-mat, physics:physics], January 2021. [32] Akihiko Nakamura, Takuya Ishida, Katsuhiro Kusaka, Taro Yamada, Shinya Fushinobu, Ichiro Tanaka, Satoshi Kaneko, Kazunori Ohta, Hiroaki Tanaka, Koji Inaka, Yoshiki Higuchi, Nobuo Niimura, Masahiro Samejima, and Kiyohiko Igarashi. \"Newton’s cradle\" proton relay with amide–imidic acid tautomerization in inverting cellulase visualized by neutron crystallography. Science Advances, 1(7):e1500263, August 2015. ",
+        "bbox": [
+            166,
+            53,
+            830,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            150,
+            70,
+            830,
+            920
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "385 [33] Y. Hirano, S. Kimura, and T. Tamada. High-resolution crystal structures of the solubilized   \n386 domain of porcine cytochrome b5. Acta Crystallographica Section D: Biological Crystallog  \n387 raphy, 71(7):1572–1581, July 2015.   \n388 [34] Shigeru Matsuoka, Shigeru Sugiyama, Daisuke Matsuoka, Mika Hirose, Sébastien Lethu,   \n389 Hikaru Ano, Toshiaki Hara, Osamu Ichihara, S. Roy Kimura, Satoshi Murakami, Hanako   \n390 Ishida, Eiichi Mizohata, Tsuyoshi Inoue, and Michio Murata. Water-mediated recognition   \n391 of simple alkyl chains by heart-type fatty-acid-binding protein. Angewandte Chemie Interna  \n392 tional Edition, 54(5):1508–1511, 2015.   \n393 [35] Yoshihiko Furuike, Yuka Akita, Ikuko Miyahara, and Nobuo Kamiya. ADP-ribose pyrophos  \n394 phatase reaction in crystalline state conducted by consecutive binding of two manganese(ii)   \n395 ions as cofactors. Biochemistry, 55(12):1801–1812, March 2016.   \n396 [36] Hiraku Ohno, Kazuki Takeda, Satomi Niwa, Tomotaka Tsujinaka, Yuya Hanazono, Yu Hirano,   \n397 and Kunio Miki. Crystallographic characterization of the high-potential iron-sulfur protein in   \n398 the oxidized state at $\\mathbf { \\bar { 0 . 8 \\ : \\mathring { A } } }$ resolution. PLOS ONE, 12(5):e0178183, May 2017.   \n399 [37] Harry P. Austin, Mark D. Allen, Bryon S. Donohoe, Nicholas A. Rorrer, Fiona L. Kearns, Ro  \n400 drigo L. Silveira, Benjamin C. Pollard, Graham Dominick, Ramona Duman, Kamel El Omari,   \n401 Vitaliy Mykhaylyk, Armin Wagner, William E. Michener, Antonella Amore, Munir S. Skaf,   \n402 Michael F. Crowley, Alan W. Thorne, Christopher W. Johnson, H. Lee Woodcock, John E.   \n403 McGeehan, and Gregg T. Beckham. Characterization and engineering of a plastic-degrading   \n404 aromatic polyesterase. Proceedings of the National Academy of Sciences, 115(19):E4350–   \n405 E4357, May 2018.   \n406 [38] Alessandro Vergara, Marco Caterino, and Antonello Merlino. Raman-markers of X-ray radia  \n407 tion damage of proteins. International Journal of Biological Macromolecules, 111:1194–1205,   \n408 May 2018.   \n409 [39] Alexander T. Baker, Alexander Greenshields-Watson, Lynda Coughlan, James A. Davies,   \n410 Hanni Uusi-Kerttula, David K. Cole, Pierre J. Rizkallah, and Alan L. Parker. Diversity within   \n411 the adenovirus fiber knob hypervariable loops influences primary receptor interactions. Nature   \n412 Communications, 10(1):741, February 2019.   \n413 [40] K. Takaba, Y. Tai, H. Eki, H.-A. Dao, Y. Hanazono, K. Hasegawa, K. Miki, and K. Takeda.   \n414 Subatomic resolution X-ray structures of green fluorescent protein. IUCrJ, 6(3):387–400, May   \n415 2019.   \n416 [41] Janet M. Paterson, Amy J. Shaw, Ian Burns, Alister W. Dodds, Alpana Prasad, Ken B. Reid,   \n417 Trevor J. Greenhough, and Annette K. Shrive. Atomic-resolution crystal structures of the   \n418 immune protein conglutinin from cow reveal specific interactions of its binding site with N  \n419 acetylglucosamine. Journal of Biological Chemistry, 294(45):17155–17165, November 2019.   \n420 [42] Wei Tian, Peiqiang Yan, Ning Xu, Arghya Chakravorty, Robert Liefke, Qiaoran Xi, and   \n421 Zhanxin Wang. The HRP3 PWWP domain recognizes the minor groove of double-stranded   \n422 DNA and recruits HRP3 to chromatin. Nucleic Acids Research, 47(10):5436–5448, June 2019.   \n423 [43] Matthew J Schellenberg, C Denise Appel, Amanda A Riccio, Logan R Butler, Juno M Krahn,   \n424 Jenna A Liebermann, Felipe Cortés-Ledesma, and R Scott Williams. Ubiquitin stimulated   \n425 reversal of topoisomerase 2 DNA-protein crosslinks by TDP2. Nucleic Acids Research,   \n426 48(11):6310–6325, June 2020.   \n427 [44] Marina Plaza-Garrido, Mª Carmen Salinas-García, José C. Martínez, and Ana Cámara-Artigas.   \n428 The effect of an engineered ATCUN motif on the structure and biophysical properties of   \n429 the SH3 domain of c-Src tyrosine kinase. JBIC Journal of Biological Inorganic Chemistry,   \n430 25(4):621–634, June 2020.   \n431 [45] Shanshan Wu, Tam T. T. N. Nguyen, Olga V. Moroz, Johan P. Turkenburg, Jens E. Nielsen,   \n432 Keith S. Wilson, Kasper D. Rand, and Kaare Teilum. Conformational heterogeneity of Sav  \n433 inase from NMR, HDX-MS and X-ray diffraction analysis. PeerJ, 8:e9408, June 2020.   \n434 [46] Barbara Franke, Marta Veses-Garcia, Kay Diederichs, Heather Allison, Daniel J. Rigden, and   \n435 Olga Mayans. Structural annotation of the conserved carbohydrate esterase vb_24B_21 from   \n436 Shiga toxin-encoding bacteriophage $\\Phi 2 4 \\mathbf { B }$ . Journal of Structural Biology, 212(1):107596,   \n437 October 2020.   \n438 [47] Matthias Barone, Matthias Müller, Slim Chiha, Jiang Ren, Dominik Albat, Arne Soicke,   \n439 Stephan Dohmen, Marco Klein, Judith Bruns, Maarten van Dinther, Robert Opitz, Peter Linde  \n440 mann, Monika Beerbaum, Kathrin Motzny, Yvette Roske, Peter Schmieder, Rudolf Volkmer,   \n441 Marc Nazaré, Udo Heinemann, Hartmut Oschkinat, Peter ten Dijke, Hans-Günther Schmalz,   \n442 and Ronald Kühne. Designed nanomolar small-molecule inhibitors of Ena/VASP EVH1 inter  \n443 action impair invasion and extravasation of breast cancer cells. Proceedings of the National   \n444 Academy of Sciences, 117(47):29684–29690, November 2020.   \n445 [48] Renee Otten, Ricardo A. P. Pádua, H. Adrian Bunzel, Vy Nguyen, Warintra Pitsawong,   \n446 MacKenzie Patterson, Shuo Sui, Sarah L. Perry, Aina E. Cohen, Donald Hilvert, and Dorothee   \n447 Kern. How directed evolution reshapes the energy landscape in an enzyme to boost catalysis.   \n448 Science, 370(6523):1442–1446, December 2020.   \n449 [49] Karla Frydenvang, Darryl S. Pickering, Giridhar U. Kshirsagar, Giulia Chemi, Sandra Gemma,   \n450 Desiree Sprogøe, Anne Mette Kærn, Simone Brogi, Giuseppe Campiani, Stefania Butini,   \n451 and Jette Sandholm Kastrup. Ionotropic glutamate receptor GluA2 in complex with bicyclic   \n452 pyrimidinedione-based compounds: When small compound modifications have distinct effects   \n453 on binding interactions. ACS Chemical Neuroscience, 11(12):1791–1800, June 2020.   \n454 [50] Steffen Glöckner, Andreas Heine, and Gerhard Klebe. A proof-of-concept fragment screening   \n455 of a hit-validated 96-compounds library against human carbonic anhydrase ii. Biomolecules,   \n456 10(4):518, April 2020.   \n457 [51] Helen M. Berman, John Westbrook, Zukang Feng, Gary Gilliland, T. N. Bhat, Helge Weissig,   \n458 Ilya N. Shindyalov, and Philip E. Bourne. The Protein Data Bank. Nucleic Acids Research,   \n459 28(1):235–242, January 2000.   \n460 [52] H. Kamberaj, R. J. Low, and M. P. Neal. Time reversible and symplectic integrators   \n461 for molecular dynamics simulations of rigid molecules. The Journal of Chemical Physics,   \n462 122(22):224114, June 2005. ",
+        "bbox": [
+            140,
+            90,
+            830,
+            922
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            137,
+            90,
+            828,
+            646
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            117,
+            341,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes]   \n(c) Did you discuss any potential negative societal impacts of your work? [N/A] We feel that the methods presented in this paper are sufficiently agnostic to applications that it is hard to directly address any negative or positive societal impacts of the work.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            137,
+            825,
+            266
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            271,
+            493,
+            286
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
+        "bbox": [
+            238,
+            290,
+            738,
+            324
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            214,
+            329,
+            393,
+            343
+        ],
+        "page_idx": 13
+    },
+    {
+        "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] Code and scripts to reproduce results to be included as SI before the appropriate deadline.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Details of training and architectures are provided in the Results section, as well as the code that will be included in the SI.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
+        "bbox": [
+            238,
+            348,
+            825,
+            507
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            512,
+            823,
+            527
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [N/A] No existing assets were evaluated.   \n(b) Did you mention the license of the assets? [Yes]   \n(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] Trained models and workflows will be included in the SI.   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            532,
+            825,
+            679
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            683,
+            705,
+            698
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            703,
+            825,
+            800
+        ],
+        "page_idx": 13
+    }
+]