Original: G RAPH EBM: M OLECULAR G RAPH G ENERATION WITH E NERGY -B ASED M ODELS
Revised: G RAPH EBM: M OLECULAR G RAPH G ENERATION WITH E NERGY -B ASED M ODELS

Original: Anonymous authors Paper under double-blind review A BSTRACT We note that most existing approaches for molecular graph generation fail to guarantee the intrinsic property of permutation invariance, resulting in unexpected bias in generative models.
Revised: We note that most existing approaches for molecular graph generation fail to guarantee the intrinsic property of permutation invariance, resulting in unexpected bias in generative models.

Original: In this work, we propose GraphEBM to generate molecular graphs using energy-based models.
Revised: In this work, we propose GraphEBM to generate molecular graphs using energy-based models.

Original: In particular, we parameterize the energy function in a permutation invariant manner, thus making GraphEBM permutation invariant.
Revised: In particular, we parameterize the energy function in a permutation invariant manner, thus making GraphEBM permutation invariant.

Original: We apply Langevin dynamics to train the energy function by approximately maximizing likelihood and generate samples with low energies.
Revised: We apply Langevin dynamics to train the energy function by approximately maximizing likelihood and generate samples with low energies.

Original: Furthermore, to generate molecules with a desirable property, we propose a simple yet effective strategy, which pushes down energies with ﬂexible degrees according to the properties of corresponding molecules.
Revised: Furthermore, to generate molecules with a desirable property, we propose a simple yet effective strategy, which pushes down energies with ﬂexible degrees according to the properties of corresponding molecules.

Original: Finally, we explore the use of
Revised: Finally, we explore the use of

Original: GraphEBM for generating molecules with multiple objectives in a compositional manner.
Revised: GraphEBM for generating molecules with multiple objectives in a compositional manner.

Original: Comprehensive experimental results on random, goal-directed and compositional generation tasks demonstrate the effectiveness of our proposed method.
Revised: Comprehensive experimental results on random, goal-directed, and compositional generation tasks demonstrate the effectiveness of our proposed method

Original:  I NTORDUCTION
Revised:  I NTRODUCTION

Original: A fundamental problem in drug discovery and material science is to ﬁnd novel molecules with desirable properties.
Revised: A fundamental problem in drug discovery and material science is to ﬁnd novel molecules with desirable properties.

Original: One way is to search in the chemical space based on molecular property prediction (Gilmer et al., 2017; Wu et al., 2018;
Revised: One way is to search in the chemical space based on molecular property prediction (Gilmer et al., 2017; Wu et al., 2018;

Original: Yang et al., 2019; Stokes et al., 2020; Wang et al., 2020).
Revised: Yang et al., 2019; Stokes et al., 2020; Wang et al., 2020).

Original: Recently, molecular graph generation has provided an alternative and promising way for this problem by directly generating desirable molecules, thus circumventing the expensive search of the chemical space.
Revised: Recently, molecular graph generation has provided an alternative and promising way for this problem by directly generating desirable molecules, thus circumventing the expensive search of the chemical space.

Original: Despite intensive efforts recently, molecular graph generation remains challenging since the underlying graphs are discrete, irregular, and permutation invariant to node order.
Revised: Despite intensive efforts recently, molecular graph generation remains challenging since the underlying graphs are discrete, irregular, and permutation invariant to node order.

Original: Existing approaches (Appendix A.1) have achieved promising success by generating molecular graphs based on various generative methods, including variational autoencoders (VAEs) (Kingma & Welling, 2013), generative adversarial networks (GANs) (Goodfellow et al., 2014), ﬂow models (Dinh et al., 2014;
Revised: Existing approaches (Appendix A.1) have achieved promising success by generating molecular graphs based on various generative methods, including variational autoencoders (VAEs) (Kingma & Welling, 2013), generative adversarial networks (GANs) (Goodfellow et al., 2014), ﬂow models (Dinh et al., 2014;

Original: Rezende & Mohamed, 2015) and recurrent neural networks (RNNs).
Revised: Rezende & Mohamed, 2015) and recurrent neural networks (RNNs).

Original: However, as analyzed in Appendix A.1, most of them fail to preserve the intrinsic property of permutation invariance, which might yield different likelihoods for different permutations of the same graph.
Revised: However, as analyzed in Appendix A.1, most of them fail to preserve the intrinsic property of permutation invariance, which might yield different likelihoods for different permutations of the same graph.

Original: Notably, EBMs (LeCun et al., 2006)also can  be naturally used as generative models since data points near the underlying data manifold are assigned lower energies than other data points, which deﬁnes an unnormalized probability distribution over the data.
Revised: Notably, energy-based models (EBMs) (LeCun et al., 2006) can also be naturally used as generative models since data points near the underlying data manifold are assigned lower energies than other data points, which deﬁnes an unnormalized probability distribution over the data.

Original: Given a data point x , let E θ ( x ) ∈ R be the corresponding energy, where θ denotes the learnable parameters of the energy function.
Revised: Given a data point x , let E θ ( x ) ∈ R be the corresponding energy, where θ denotes the learnable parameters of the energy function.

Original: Then, the energy function deﬁnes a data distribution via the Boltzmann distribution as
Revised: Then, the energy function deﬁnes a data distribution via the Boltzmann distribution as

Original: [Equation]
Revised: [Equation]

Original: where Z ( θ )
Revised: where Z ( θ )

Original: = (cid:82) e
Revised: = (cid:82) e

Original: − E θ ( x ) dx is the normalization constant and usually intractable.
Revised: − E θ ( x ) dx is the normalization constant and usually intractable.

Original: Related works on
Revised: Related works on

Original: EBMs are reviewed in Appendix A.2.
Revised: EBMs are reviewed in Appendix A.2.

Original: In this work, we propose GraphEBM to generate molecular graphs with energy-based models (EBMs) (LeCun et al., 2006).
Revised: In this work, we propose GraphEBM to generate molecular graphs with energy-based models.

Original: Since our parameterized energy function is permutation invariant, we can show that our GraphEBM preserves the permutation invariance property.
Revised: Since our parameterized energy function is permutation invariant, we can show that our GraphEBM pre- serves the permutation invariance property.

Original: We use Langevin dynamics (Welling & Teh, 2011) to train the energy function by maximizing likelihood approximately and generate samples from the trained energy function.
Revised: We use Langevin dynamics (Welling & Teh, 2011) to train the energy function by maximizing likelihood approximately and generate samples from the trained energy function.

Original: To our knowledge, our GraphEBM is the ﬁrst energy-based model that can generate attributed molecular graphs.
Revised: To our knowledge, our GraphEBM is the ﬁrst energy-based model that can generate attributed molecular graphs.

Original: Furthermore, in order to generate molecules with a speciﬁc desirable property, we propose a novel, simple, and effective strategy to train our GraphEBM for goal-directed generation by pushing down energies with ﬂexible degrees according to the property values of corresponding molecules.
Revised: Furthermore, in order to generate molecules with a speciﬁc desirable property, we propose a novel, simple, and effective strategy to train our GraphEBM for goal-directed generation by pushing down energies with ﬂexible degrees according to the property values of corresponding molecules.

Original: Signiﬁcantly, we show that GraphEBM can generate molecules with multiple objectives in a compositional manner, which cannot be achieved by any existing methods.
Revised: Signiﬁcantly, we show that GraphEBM can generate molecules with multiple objectives in a compositional manner, which cannot be achieved by any existing methods.

Original: This provides a new and promising way for multiobjective molecule generation.
Revised: This provides a new and promising way for multi-objective molecule generation.

Original: Experimental results on random, goal-directed, and compositional generation tasks show that our proposed method is effective.
Revised: Experimental results on random, goal-directed, and compositional generation tasks show that our proposed method is effective.

Original:  T HE P ROPOSED G RAPH EBM
Revised:  T HE P ROPOSED G RAPH EBM

Original: In this section, we present GraphEBM by describing the internal structure of the parameterized energy function (Section 2.1), showing that GraphEBM satisﬁes the desirable property of permutation invariance (Section 2.2), and describing the training (Section 2.3) and generation (Section 2.4) process of GraphEBM.
Revised: In this section, we present GraphEBM by describing the internal structure of the parameterized energy function (Section 2.1), showing that GraphEBM satisﬁes the desirable property of permutation invariance (Section 2.2), and describing the training (Section 2.3) and generation (Section 2.4) process of GraphEBM.

Original: Then, we introduce our proposed strategy for goal-directed generation based on GraphEBM (Section 2.5).
Revised: Then, we introduce our proposed strategy for goal-directed generation based on GraphEBM (Section 2.5).

Original: Finally, we explore the potential of compositional generation using
Revised: Finally, we explore the potential of compositional generation using

Original: GraphEBM (Section 2.6).
Revised: GraphEBM (Section 2.6).

Original: Molecules can be naturally represented as graphs by considering atoms and bonds as nodes and edges, respectively.
Revised: Molecules can be naturally represented as graphs by considering atoms and bonds as nodes and edges, respectively.

Original: We formally represent a molecular graph as G =
Revised: We formally represent a molecular graph as G =

Original: ( X, A ) , where X is the node feature matrix and A is the adjacency tensor.
Revised: ( X, A ) , where X is the node feature matrix and A is the adjacency tensor.

Original: Let k be the number of nodes in the graph.
Revised: Let k be the number of nodes in the graph.

Original: b and c denote the number of possible types of nodes and edges, respectively.
Revised: b and c denote the number of possible types of nodes and edges, respectively.

Original: Then we have X ∈ { 0 , 1 } k × b and X ( i,p )
Revised: Then we have X ∈ { 0 , 1 } k × b and X ( i,p )

Original: =
Revised: =

Original: if node i belongs to type p .
Revised: if node i belongs to type p .

Original: A ∈ { 0 , 1 } k × k × c and A ( i,j,q ) = 1 denotes that an edge with type q exists between node i and node j .
Revised: A ∈ { 0 , 1 } k × k × c and A ( i,j,q ) = 1 denotes that an edge with type q exists between node i and node j .

Original: Following Madhawa et al.
Revised: Following Madhawa et al.

Original: and Zang & Wang (2020), we let n denote the maximum number of atoms that a molecule has in a given dataset.
Revised: and Zang & Wang (2020), we let n denote the maximum number of atoms that a molecule has in a given dataset.

Original: We insert virtual nodes into molecular graphs that have less than n nodes such that the dimensions of
Revised: We insert virtual nodes into molecular graphs that have less than n nodes such that the dimensions of

Original: X and A keep the same for all molecules.
Revised: X and A keep the same for all molecules.

Original: Also, for any two nodes that are not connected in the molecule, we add a virtual edge between them.
Revised: Also, for any two nodes that are not connected in the molecule, we add a virtual edge between them.

Original: We can consider the virtual node and the virtual edge as an additional node type and edge type, respectively.
Revised: We can consider the virtual node and the virtual edge as an additional node type and edge type, respectively.

Original: Hence, for all molecules in a certain dataset, X ∈ { 0 , 1 } n × ( b +1) and A ∈ { 0 , 1 } n × n × ( c +1) .
Revised: Hence, for all molecules in a certain dataset, X ∈ { 0 , 1 } n × ( b +1) and A ∈ { 0 , 1 } n × n × ( c +1) .

Original: 2. P ARAMETERIZED E NERGY F UNCTION
Revised: 2. P ARAMETERIZED E NERGY F UNCTION

Original: Following the above notations, the energy function for molecular graphs can be denoted as
Revised: Following the above notations, the energy function for molecular graphs can be denoted as

Original: E θ
Revised: E θ

Original: ( X, A ) .
Revised: ( X, A ) .

Original: Speciﬁcally, we model E θ ( X, A ) by a graph neural network, where θ denotes parameters in the network.
Revised: Speciﬁcally, we model E θ ( X, A ) by a graph neural network, where θ denotes parameters in the network.

Original: Many deep learning methods on graphs have been proposed and have achieved great success in many tasks, such as node classiﬁcation (Kipf & Welling, 2017; Hamilton et al., 2017;
Revised: Many deep learning methods on graphs have been proposed and have achieved great success in many tasks, such as node classiﬁcation (Kipf & Welling, 2017; Hamilton et al., 2017;

Original: Monti et al., 2017;
Revised: Monti et al., 2017;

Original: Veli ˇ ckovi ´ c et al., 2018; Gao et al., 2018; Xu et al., 2018; Klicpera et al., 2019; Wu et al., 2019; Liu et al., 2020b; Jin et al., 2020a; Chen et al., 2020; Liu et al., 2020c; Hu et al., 2020), graph classiﬁcation (Zhang et al., 2018;
Revised: Veli ˇ ckovi ´ c et al., 2018; Gao et al., 2018; Xu et al., 2018; Klicpera et al., 2019; Wu et al., 2019; Liu et al., 2020b; Jin et al., 2020a; Chen et al., 2020; Liu et al., 2020c; Hu et al., 2020), graph classiﬁcation (Zhang et al., 2018;

Original: Ying et al., 2018; Maron et al., 2018; Xu et al., 2019; Gao & Ji, 2019; Ma et al., 2019; Yuan & Ji, 2020;
Revised: Ying et al., 2018; Maron et al., 2018; Xu et al., 2019; Gao & Ji, 2019; Ma et al., 2019; Yuan & Ji, 2020;

Original: You et al., 2020; Gao et al., 2020), and link prediction (Zhang & Chen, 2018;
Revised: You et al., 2020; Gao et al., 2020), and link prediction (Zhang & Chen, 2018;

Original: Cai et al., 2020).
Revised: Cai et al., 2020).

Original: In this work, we use a variant of relational graph convolutional networks (R-GCN) (Schlichtkrull et al., 2018) to learn the node representations since molecular graphs have categorical edge types.
Revised: In this work, we use a variant of relational graph convolutional networks (R-GCN) (Schlichtkrull et al., 2018) to learn the node representations since molecular graphs have categorical edge types.

Original: Formally, the layer-wise forward-propagation is deﬁned as
Revised: Formally, the layer-wise forward-propagation is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: A (: , : ,k ) is the k -th channel of the adjacency tensor.
Revised: A (: , : ,k ) is the k -th channel of the adjacency tensor.

Original: H (cid:96)  is the node representation matrix at layer (cid:96)  .
Revised: H (cid:96) ∈ R n × d (cid:96) is the node representation matrix at layer (cid:96) , where d (cid:96) denotes the hidden dimension at layer (cid:96) .

Original: W (cid:96)k  represents the trainable weight matrix for edge type k at layer (cid:96) .
Revised: W (cid:96)k ∈ R d (cid:96) × d (cid:96) +1 represents the trainable weight matrix for edge type k at layer (cid:96) .

Original: σ ( · ) denotes a non-linear activation function.
Revised: σ ( · ) denotes a non-linear activation function.

Original: The initial node representation matrix H 

Revised: The initial node representation matrix H

Original: = X .
Revised: = X .

Original: In each layer, message passing is conducted among the nodes independently for each type of edge.
Revised: In each layer, message passing is conducted among the nodes independently for each type of edge.

Original: Then, the information is integrated together by a sum operator.
Revised: Then, the information is integrated together by a sum operator.

Original: We stack L such layers.
Revised: We stack L such layers.

Original: Hence, the ﬁnal node representation matrix is H L ∈ R n × d , where d is the hidden dimension.
Revised: Hence, the ﬁnal node representation matrix is H L ∈ R n × d , where d is the hidden dimension.

Original: Then, the representation of the whole graph can be derived by a readout function.
Revised: Then, the representation of the whole graph can be derived by a readout function.

Original: In this work, we use the sum operation to compute the graph-level representation h G as
Revised: In this work, we use the sum operation to compute the graph-level representation h G as

Original: [Equation]
Revised: [Equation]

Original: Finally, the scalar energy associated with the molecular graph can be obtained by applying a transformation as
Revised: Finally, the scalar energy associated with the molecular graph can be obtained by applying a transformation as

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: W ∈ R
Revised: W ∈ R

Original: d is the trainable parameters.
Revised: d is the trainable parameters.

Original: 2. P ERMUTATION I NVARIANCE
Revised: 2. P ERMUTATION I NVARIANCE

Original: Permutation invariance is an intrinsic and desirable inductive bias for graph modeling.
Revised: Permutation invariance is an intrinsic and desirable inductive bias for graph modeling.

Original: We note that our proposed GraphEBM satisﬁes this fundamental property due to our permutation invariant energy function.
Revised: We note that our proposed GraphEBM satisﬁes this fundamental property due to our permutation invariant energy function.

Original: Speciﬁcally, each layer of our graph neural network in Eq.
Revised: Speciﬁcally, each layer of our graph neural network in Eq.

Original: (2) is permutation equivariant.
Revised: (2) is permutation equivariant.

Original: In addition, the readout operation in Eq.
Revised: In addition, the readout operation in Eq.

Original: (3) is permutation invariant.
Revised: (3) is permutation invariant.

Original: Therefore, our parameterized energy function is permutation invariant thus satisfying E θ ( X, A ) =
Revised: Therefore, our parameterized energy function is permutation invariant thus satisfying E θ ( X, A ) =

Original: E θ ( X π , A π ) , where π denotes any permutation of node order.
Revised: E θ ( X π , A π ) , where π denotes any permutation of node order.

Original: For simplicity, we use the superscript π to denote that the corresponding matrix or tensor is arranged according to the node order given by π .
Revised: For simplicity, we use the superscript π to denote that the corresponding matrix or tensor is arranged according to the node order given by π .

Original: According to Eq.
Revised: According to Eq.

Original: (1), the energy function deﬁnes a distribution over data.
Revised: (1), the energy function deﬁnes a distribution over data.

Original: Speciﬁcally, the likelihood is proportional to the negative exponential of the corresponding energy.
Revised: Speciﬁcally, the likelihood is proportional to the negative exponential of the corresponding energy.

Original: Hence, we can further obtain p θ ( X, A ) =
Revised: Hence, we can further obtain p θ ( X, A ) =

Original: p θ ( X π , A π ) .
Revised: p θ ( X π , A π ) .

Original: Thus, our GraphEBM can preserve permutation invariance by modeling graphs in a permutation invariant manner.
Revised: Thus, our GraphEBM can preserve permutation invariance by modeling graphs in a permutation invariant manner.

Original: 2. T RAINING
Revised: 2. T RAINING

Original: Intuitively, a good energy function should assign lower energies to data points that correspond to real molecular graphs and higher energies to other data points.
Revised: Intuitively, a good energy function should assign lower energies to data points that correspond to real molecular graphs and higher energies to other data points.

Original: Hence, a straightforward idea is to train the parameterized energy function by maximizing the likelihood of real data deﬁned in Eq.
Revised: Hence, a straightforward idea is to train the parameterized energy function by maximizing the likelihood of real data deﬁned in Eq.

Original: Let p D be the distribution of the real data.
Revised: Let p D be the distribution of the real data.

Original: We can achieve maximum likelihood by minimizing the negative log likelihood of real data.
Revised: We can achieve maximum likelihood by minimizing the negative log likelihood of real data.

Original: Formally,
Revised: Formally,

Revised: [Equation]

Original: [Equation]
Revised: [Equation]

Original: where − log p θ ( X, A )
Revised: where − log p θ ( X, A )

Original: =
Revised: =

Original: E θ ( X, A )
Revised: E θ ( X, A )

Original: + log Z ( θ ) according to Eq.
Revised: + log Z ( θ ) according to Eq.

Original: It has been shown (Hinton, 2002; Turner, 2005) that the objective in Eq.
Revised: It has been shown (Hinton, 2002; Turner, 2005; Song & Kingma, 2021) that the objective in Eq.

Original: (5) has the below gradient:
Revised: (5) has the below gradient:

Original: [Equation]
Revised: [Equation]

Original: As deﬁned in Eq.
Revised: As deﬁned in Eq.

Original: (1), p θ is the distribution given by the energy function.
Revised: (1), p θ is the distribution given by the energy function.

Original: Following Du et al.
Revised: Following Du et al.

Original: (2020b), we refer to ( X (cid:12) , A (cid:12) ) as hallucinated samples.
Revised: (2020b), we refer to ( X (cid:12) , A (cid:12) ) as hallucinated samples.

Original: Obviously, this gradient pushes down the energies of positive samples ( X ⊕ , A ⊕ ) and pushes up the energies of hallucinated samples ( X (cid:12) , A (cid:12) ) .
Revised: Obviously, this gradient pushes down the energies of positive samples ( X ⊕ , A ⊕ ) and pushes up the energies of hallucinated samples ( X (cid:12) , A (cid:12) ) .

Original: However, sampling ( X (cid:12) , A (cid:12) ) from p θ is challenging, since Z ( θ ) in Eq.
Revised: However, sampling ( X (cid:12) , A (cid:12) ) from p θ is challenging, since Z ( θ ) in Eq.

Original: (1) is intractable.
Revised: (1) is intractable.

Original: To overcome this issue, we follow Du & Mordatch (2019) to sample ( X (cid:12) , A (cid:12) ) from an approximated p θ using Langevin dynamics (Welling & Teh, 2011).
Revised: To overcome this issue, we follow Du & Mordatch (2019) to sample ( X (cid:12) , A (cid:12) ) from an approximated p θ using Langevin dynamics (Welling & Teh, 2011).

Original: Particularly, a sample ( X (cid:12) , A (cid:12) ) is initialized randomly and reﬁned iteratively by
Revised: Particularly, a sample ( X (cid:12) , A (cid:12) ) is initialized randomly and reﬁned iteratively by

Original: [Equation]
Revised: [Equation]

Original: where w k and η k are added noise sampled from a Gaussian distribution N (0 , σ 2 ) .
Revised: where w k and η k are added noise sampled from a Gaussian distribution N (0 , σ 2 ) .

Original: k denotes the iteration step, and λ 2 is the step size.
Revised: k denotes the iteration step, and λ 2 is the step size.

Original: As demonstrated by Welling & Teh (2011), the obtained samples ( X k , A k ) approach samples from p θ as k → ∞ and λ 2 → 0 .
Revised: As demonstrated by Welling & Teh (2011), the obtained samples ( X k , A k ) approach samples from p θ as k → ∞ and λ 2 → 0 .

Original: In practice, we let K denote the number of iteration steps of Langevin dynamics and use the resulting sample ( X K , A K ) as ( X (cid:12) , A (cid:12) ) in Eq.
Revised: In practice, we let K denote the number of iteration steps of Langevin dynamics and use the resulting sample ( X K , A K ) as ( X (cid:12) , A (cid:12) ) in Eq.

Original: We illustrate the training process of our GraphEBM in Figure 1.
Revised: We illustrate the training process of our GraphEBM in Figure 1.

Original: Since Langevin dynamics is for continuous data, we model the hallucinated samples by continuous format.
Revised: Since Langevin dynamics is for continuous data, we model the hallucinated samples by continuous format.

Original: For consistency, we can also use dequantization techniques (Dinh et al., 2016; Kingma & Dhariwal, 2018) to convert the discrete positive samples to continuous data by adding uniform noise, as shown in the left part of
Revised: For consistency, we can also use dequantization techniques (Dinh et al., 2016; Kingma & Dhariwal, 2018) to convert the discrete positive samples to continuous data by adding uniform noise, as shown in the left part of

Original: Figure 1.
Revised: Figure 1.

Original: The dequantization can be formally expressed as
Revised: The dequantization can be formally expressed as

Original: [Equation]
Revised: [Equation]

Original: t ∈ [0 , 1) is a scaling hyperparameter.
Revised: t ∈ [0 , 1) is a scaling hyperparameter.

Original: We then apply a normalization to the adjacency tensor, which is a common step in modern graph neural networks (Kipf & Welling, 2017).
Revised: We then apply a normalization to the adjacency tensor, which is a common step in modern graph neural networks (Kipf & Welling, 2017).

Original: Formally,
Revised: Formally,

Original: [Equation]
Revised: [Equation]

Original: where D is the diagonal degree matrix in which D ( i,i )
Revised: where D is the diagonal degree matrix in which D ( i,i )

Original: = (cid:80) j,k A (cid:48)
Revised: = (cid:80) j,k A (cid:48)

Original: ( i,j,k ) .
Revised: ( i,j,k ) .

Original: We treat the above A ⊕ and
Revised: We treat the above A ⊕ and

Original: X ⊕ = X (cid:48)
Revised: X ⊕ = X (cid:48)

Original: as the input for the energy function.
Revised: as the input for the energy function.

Original: In our case, each element of X ⊕ is in [0 , 1 + t ) and each element of A ⊕ is in [0 , 1) .
Revised: In our case, each element of X ⊕ is in [0 , 1 + t ) and each element of A ⊕ is in [0 , 1) .

Original: Note that the above dequantization for positive samples is optional.
Revised: Note that the above dequantization for positive samples is optional.

Original: This indicates that we can set t = 0 and keep the positive data discrete since Langevin dynamics is only required for obtaining hallucinated samples.
Revised: This indicates that we can set t = 0 and keep the positive data discrete since Langevin dynamics is only required for obtaining hallucinated samples.

Original: Applying dequantization to positive samples can be viewed as a data augmentation technique and we can easily convert the dequantized continuous data back to the original one-hot discrete data by simply applying the argmax operation.
Revised: Applying dequantization to positive samples can be viewed as a data augmentation technique and we can easily convert the dequantized continuous data back to the original one-hot discrete data by simply applying the argmax operation.

Original: To keep the same value range as ( X ⊕ , A ⊕ ) , the hallucinated sample ( X (cid:12) , A (cid:12) ) is initialized as
Revised: To keep the same value range as ( X ⊕ , A ⊕ ) , the hallucinated sample ( X (cid:12) , A (cid:12) ) is initialized as

Original: [Equation]
Revised: [Equation]

Original: Then we apply K steps of Langevin dynamics as Eq.
Revised: Then we apply K steps of Langevin dynamics as Eq.

Original: (7) to reﬁne the sample, as illustrated in the right part of Figure 1.
Revised: (7) to reﬁne the sample, as illustrated in the right part of Figure 1.

Original: After each step of reﬁnement, we clamp the data to guarantee that the values are still in the desirable ranges.
Revised: After each step of reﬁnement, we clamp the data to guarantee that the values are still in the desirable ranges.

Original: As demonstrated in Eq.
Revised: As demonstrated in Eq.

Original: (6), the energies of positive samples are expected to be pushed down and the energies of hallucinated samples should be pushed up.
Revised: (6), the energies of positive samples are expected to be pushed down and the energies of hallucinated samples should be pushed up.

Original: Hence, to shape the energy function as expected, our loss function is deﬁned as
Revised: Hence, to shape the energy function as expected, our loss function is deﬁned as

Original: [Equation]
Revised: [Equation]

Original: As shown in the middle part of Figure 1, the gradient backpropagated from L energy can update the parameters θ , thus pushing the energy function E θ ( X, A ) to approach our expected shape.
Revised: As shown in the middle part of Figure 1, the gradient backpropagated from L energy can update the parameters θ , thus pushing the energy function E θ ( X, A ) to approach our expected shape.

Original: Notably,  the gradient from L energy will not be propagated to the energy function used in Langevin dynamics.
Revised: Notably, [Equation] the gradient from L energy will not be propagated to the energy function used in Langevin dynamics.

Original: We apply parameter sharing to keep thethe energy function used in Langevin dynamics up-to-date.
Revised: We apply parameter sharing to keep the energy function used in Langevin dynamics up-to-date.

Original: To stabilize training, we also apply a regularization technique to the energy magnitudes.
Revised: To stabilize training, we also apply a regularization technique to the energy magnitudes.

Original: Speciﬁcally, we use the same regularization as Du & Mordatch (2019).
Revised: Speciﬁcally, we use the same regularization as Du & Mordatch (2019).

Original: Formally,
Revised: Formally,

Original: [Equation]
Revised: [Equation]

Original: Hence, the total loss function is L =
Revised: Hence, the total loss function is L =

Original: L energy + α L reg , where α ∈ R is a hyperparameter.
Revised: L energy + α L reg , where α ∈ R is a hyperparameter.

Original: 2. G ENERATION
Revised: 2. G ENERATION

Original: Let E θ (cid:63) ( X, A ) denote the trained energy function, where θ (cid:63) represents the obtained parameters.
Revised: Let E θ (cid:63) ( X, A ) denote the trained energy function, where θ (cid:63) represents the obtained parameters.

Original: Intuitively, if an energy function is well-shaped, the conﬁgurations with low energies should correspond to desirable molecular graphs.
Revised: Intuitively, if an energy function is well-shaped, the conﬁgurations with low energies should correspond to desirable molecular graphs.

Original: Hence, the generation process is to generate molecules based on the conﬁgurations ( X, A ) that yield low energies.
Revised: Hence, the generation process is to generate molecules based on the conﬁgurations ( X, A ) that yield low energies.

Original: An overview of the generation process is given in Figure 3 in Appendix B.
Revised: An overview of the generation process is given in Figure 5 in Appendix B.

Original: The steps are as follows.
Revised: The steps are as follows.

Original: First, we initialize a data point as in Eq.
Revised: First, we initialize a data point as in Eq.

Original: (10) and then apply K steps of Langevin dynamics as in
Revised: (10) and then apply K steps of Langevin dynamics as in

Original: Eq.
Revised: Eq.

Original: (7) to obtain data points that have low energy.
Revised: (7) to obtain data points that have low energy.

Original: We denote the obtained conﬁguration as ( X (cid:63) , A (cid:63) ) .
Revised: We denote the obtained conﬁguration as ( X (cid:63) , A (cid:63) ) .

Original: Second, since molecular graphs are undirected, we make the adjacency tensor to be symmetric by using A (cid:63) + A (cid:63)T as the new adjacency tensor.
Revised: Second, since molecular graphs are undirected, we make the adjacency tensor to be symmetric by using A (cid:63) + A (cid:63)T as the new adjacency tensor.

Original: Third, we convert the continuous data to discrete ones by applying the argmax operation in the dimensions of atom types and bond types.
Revised: Third, we convert the continuous data to discrete ones by applying the argmax operation in the dimensions of atom types and bond types.

Original: Finally, we use validity correction introduced by Zang & Wang (2020) to reﬁne the corresponding molecule so that the valency constraint is satisﬁed.
Revised: Finally, we use validity correction introduced by Zang & Wang (2020) to reﬁne the corresponding molecule so that the valency constraint is satisﬁed.

Original: 2. G OAL -D IRECTED G ENERATION
Revised: 2. G OAL -D IRECTED G ENERATION

Original: For drug discovery and material design, we also need to generate molecules with desirable chemical properties.
Revised: For drug discovery and material design, we also need to generate molecules with desirable chemical properties.

Original: This task is termed as goal-directed generation.
Revised: This task is termed as goal-directed generation.

Original: As noted in Appendix C, it is not straightforward to apply existing strageties to our GraphEBM for goal-directed generation.
Revised: As noted in Appendix C, it is not straightforward to apply existing strageties to our GraphEBM for goal-directed generation.

Original: To generate molecules with desirable chemical properties, we propose a novel, simple, and effective strategy to achieve goal-directed generation based on our GraphEBM.
Revised: To generate molecules with desirable chemical properties, we propose a novel, simple, and effective strategy to achieve goal-directed generation based on our GraphEBM.

Original: Our basic idea is to push down energies with ﬂexible degrees according to the property values of corresponding molecules.
Revised: Our basic idea is to push down energies with ﬂexible degrees according to the property values of corresponding molecules.

Original: If a molecule has a higher value of desirable property, we push down the corresponding energy harder.
Revised: If a molecule has a higher value of desirable property, we push down the corresponding energy harder.

Original: Formally, in goal-directed generation, the loss function deﬁned in Eq.
Revised: Formally, in goal-directed generation, the loss function deﬁned in Eq.

Original: (11) becomes
Revised: (11) becomes

Original: [Equation]
Revised: [Equation]

Original: (13) where y ∈ [0 , 1] is the normalized property value and
Revised: where y ∈ [0 , 1] is the normalized property value and

Original: f ( y ) ∈ R determines the degree of the push down.
Revised: f ( y ) ∈ R determines the degree of the push down.

Original: We use f ( y )
Revised: We use f ( y )

Original: =
Revised: =

Original: 1 + e y in this work.
Revised: 1 + e y in this work.

Original: Thus, energies of molecules with higher property values are pushed down harder.
Revised: Thus, energies of molecules with higher property values are pushed down harder.

Original: After training, the generation process is the same as described in
Revised: After training, the generation process is the same as described in

Original: Section 2.4.
Revised: Section 2.4.

Original: Note that f ( y ) could also be a learnable function and we leave this as future work.
Revised: Note that f ( y ) could also be a learnable function and we leave this as future work.

Original: 2. C OMPOSITIONAL G ENERATION
Revised: 2. C OMPOSITIONAL G ENERATION

Original: In addition to single property constraints, it is commonly necessary to generate molecules with multiple property constraints in drug discovery (Jin et al., 2020c).
Revised: In addition to single property constraints, it is commonly necessary to generate molecules with multiple property constraints in drug discovery (Jin et al., 2020c).

Original: We observe that compositional generation with EBMs (Hinton, 2002), which has been shown to be effective in the image domain (Du et al., 2020a), can be naturally applied to generate molecules with multiple constraints based on our
Revised: We observe that compositional generation with EBMs (Hinton, 2002), which has been shown to be effective in the image domain (Du et al., 2020a), can be naturally applied to generate molecules with multiple constraints based on our

Original: GraphEBM.
Revised: GraphEBM.

Original: Thus, we investigate compositional generation in the graph domain.
Revised: Thus, we investigate compositional generation in the graph domain.

Original: Suppose we have two energy functions E θ (cid:63) 

Revised: Suppose we have two energy functions E θ (cid:63)

Original: ( X, A ) and E θ (cid:63) 2 ( X, A ) trained towards two property goals respectively, as described in Section 2.5.
Revised: ( X, A ) and E θ (cid:63) 2 ( X, A ) trained towards two property goals respectively, as described in Section 2.5.

Original: According to Hinton (2002) and Du et al.
Revised: According to Hinton (2002) and Du et al.

Original: (2020a), we can obtain a new energy function by summing the above two energy functions since the product of probabilities is equivalent to the sum of corresponding energies, according to Eq.
Revised: (2020a), we can obtain a new energy function by summing the above two energy functions since the product of probabilities is equivalent to the sum of corresponding energies, according to Eq.

Original: Formally,
Revised: Formally,

Original: [Equation]
Revised: [Equation]

Original:  Then we can apply the generation process described in Section 2.4 to E θ (cid:63) ( X, A ) to generate molecules towards multiple objectives in a compositional manner.
Revised: [Equation] Then we can apply the generation process described in Section 2.4 to E θ (cid:63) ( X, A ) to generate molecules towards multiple objectives in a compositional manner.

Original:  E XPERIMENTS
Revised:  E XPERIMENTS

Original: We evaluate our proposed method for molecule generation under three settings: random generation, goal-directed generation, and compositional generation.
Revised: We evaluate our proposed method for molecule generation under three settings: random generation, goal-directed generation, and compositional generation.

Original: We consider two widely used molecule datasets, QM9 (Ramakrishnan et al., 2014) and ZINC250k (Irwin et al., 2012).
Revised: We consider two widely used molecule datasets, QM9 (Ramakrishnan et al., 2014) and ZINC250k (Irwin et al., 2012).

Original: The details of the experimental setup for each setting are included in Appendix D
Revised: The details of the experimental setup for each setting are included in Appendix D.

Revised: Our implementation is included in

Revised: DIG 1 (Liu et al., 2021), a library for graph deep learning research.

Original: Random generation .
Revised: Random generation .

Original: The results on QM9 and ZINC250k are shown in Table 2 and Table 3 respectively in Appendix E.
Revised: The results on QM and ZINC250k are shown in Table 4 and Table 5 respectively in Appendix E.

Original: We can observe that GraphEBM performs competitively with existing methods, which is signiﬁcant considering that the study of EBMs is still in its early stage and GraphEBM is the ﬁrst EBM for molecule generation.
Revised: We can observe that GraphEBM performs competitively with existing methods, which is signiﬁcant considering that the study of EBMs is still in its early stage and GraphEBM is the ﬁrst EBM for molecule generation.

Original: Generated samples are visualized in Figure 4 in Appendix E, which further demonstrates that GraphEBM can generate non-trivial molecules.
Revised: Generated samples are visualized in Figure 6 in Appendix E, which further demonstrates that GraphEBM can generate non-trivial molecules.

Original: To better understand the implicit generation through Langevin dynamics, we visualize this process for an example in Figure 5 in AppendixE.
Revised: To better understand the implicit generation through Langevin dynamics, we visualize this process for an example in Figure 7 in Appendix

Original:  We can observe that Langevin dynamics effectively reﬁnes the random initialized sample to approach a data point that corresponds to a realistic molecule.
Revised: E. We can observe that Langevin dynamics effectively reﬁnes the random initialized sample to approach a data point that corresponds to a realistic molecule.

Original: Goal-directed generation .
Revised: Goal-directed generation .

Original: Figure 2 (a) and (b) compare the property value distribution between goal-directed generation and random generation.
Revised: Figure 3 (a) and (b) compare the property value distribution between goal-directed generation and random generation.

Original: It can be observed that goal-directed generation can generate more molecules with a high property value, indicating that our proposed strategy for goal-directed generation, which aims to assign lower energies to molecules with higher property values, is effective. 
Revised: It can be observed that goal-directed generation can generate more molecules with a high property value, indicating that our proposed strategy for goal-directed generation, which aims to assign lower energies to molecules with higher property values, is effective. [Table]

Original: Some examples on plogp optimization and QED optimization  are demonstrated in Figure 2 (d) and more results are available in 
Revised: The property optimization results are shown in Table 1. We observe that GraphEBM can ﬁnd more novel molecules with the best QED score ( 0 . 948 ) than baselines. This strongly demonstrates the effectiveness of our proposed goal-directed generation method. Examples of discovered novel molecules with high QED scores are illustrated in Figure

Revised: For constraint property optimization, as demonstrated in

Original: Appendix E. This indicates that the proposed approach for goal-directed generation based on GraphEBM can perform molecule optimization effectively and the improvements on the corresponding property can be signiﬁcant.
Revised: Table 2, GraphEBM can obtain higher property improvements over JT-VAE, GCPN, and MoFlow by signiﬁcant margins. In terms of success rate, our GraphEBM is not as strong as the methods using reinforcement learning ( i.e. , GCPN, GraphAF), and achieves results comparable to JT-VAE and MoFlow. Although GraphAF performs better than GraphEBM, it can be observed from Shi et al. (2019) that GraphAF learns to improve plogp by simply adding long carbon chains, while our GraphEBM learns more advanced chemical knowledge.

Revised: Several examples of constraint property optimization are shown in Figure 4.

Original: More importantly and interestingly, the modiﬁcations are interpretable to some degree.
Revised: It is interesting that the modiﬁcations are interpretable to some degree.

Original: Speciﬁcally, in the ﬁrst example of plogp optimization shown in Figure 2 (d), our model optimizes the plogp score with a remarkable margin of 18 .by replacing several carbon atoms with sulfur atoms, which could make the molecule more soluble in water, thus leading to a larger logP value.
Revised: Speciﬁcally, in the ﬁrst example, our model optimizes the plogp score with a remarkable margin of 18 . 03 by replacing several carbon atoms with sulfur atoms, which could make the molecule more soluble in water, thus leading to a larger logP value.

Original: Additionally, plogp is highly related to the number of long cycles and synthetic accessibility.
Revised: Additionally, plogp is highly related to the number of long cycles and synthetic accessibility.

Original: As shown in the second and third examples, our model improves the synthetic accessibility and reduces the number of long cycles by removing or breaking them.
Revised: As shown in the second and third examples, our model improves the synthetic accessibility and reduces the number of long cycles by removing or breaking them.

Original: These facts indicate that our goal-directed generation method can explore the underlying chemical knowledge related to the corresponding property.
Revised: These facts indicate that our goal-directed generation method can explore the underlying chemical knowledge related to the corresponding property [Table] [Table]

Original: Compositional generation .
Revised: Compositional generation .

Original: The comparison of the distributions on QED and plogp between compositional generation and random generation is illustrated in Figure 2 (c).
Revised: The comparison of the distributions on QED and plogp between compositional generation and random generation is illustrated in Figure 3 (c).

Original: We can observe that compositional generation tends to generate more molecules with high QED and plogp scores.
Revised: We can observe that compositional generation tends to generate more molecules with high QED and plogp scores.

Original: Additionally, the distribution of QED or plogp is similar to the corresponding distribution obtained by goal-directed generation towards a single objective (Figure 2 (a) and (b)).
Revised: Additionally, the distribution of QED or plogp is similar to the corresponding distribution obtained by goal-directed generation towards a single objective (Figure 3 (a) and (b)).

Original: These facts demonstrate that our GraphEBM is able to generate molecules towards multiple objectives in a composition manner, which provides a novel and promising way for multi-objective generation.
Revised: These facts demonstrate that our GraphEBM is able to generate molecules towards multiple objectives in a composition manner, which provides a novel and promising way for multi-objective generation.

Original:  C ONCLUSION AND O UTLOOK
Revised:  C ONCLUSION AND O UTLOOK

Original: In this paper, we propose GraphEBM, the ﬁrst energy-based model for generating molecular graphs that preserves the intrinsic property of permutation invariance.
Revised: In this paper, we propose GraphEBM, the ﬁrst energy-based model for generating molecular graphs that preserves the intrinsic property of permutation invariance.

Original: We propose to ﬂexibly push down energies for goal-directed generation and explore to generate molecules towards multiple objectives in a compositional manner, leading to a promising method for multi-objective generation.
Revised: We propose to ﬂexibly push down energies for goal-directed generation and explore to generate molecules towards multiple objectives in a compositional manner, leading to a promising method for multi-objective generation.

Original: Experimental results demonstrate that our GraphEBM can generate realistic molecules and the proposals of goal-directed generation and compositional generation are effective and promising.
Revised: Experimental results demonstrate that our GraphEBM can generate realistic molecules and the proposals of goal-directed generation and compositional generation are effective and promising.

Original: Since EBMs have unique advantages and have rarely been explored for graph generation, we hope our exploratory work would open the door for future research in this area.  
Revised: Since EBMs have unique advantages and have rarely been explored for graph generation, we hope our exploratory work would open the door for future research in this area.  A CKNOWLEDGMENTS

Revised: We thank Yilun Du and Chengxi Zang for their helpful discussions.

Revised: This work was supported in part by National Science Foundation grant DBI-1922969.

Original: A PPENDIX A R ELATED W ORK A. M OLECULAR G RAPH G ENERATION
Revised: A PPENDIX A R ELATED W ORK A. M OLECULAR G RAPH G ENERATION

Original: Since molecules can be represented as SMILES strings (Weininger, 1988), early studies generate molecules based on SMILES strings, such as CVAE (G ´ omez-Bombarelli et al., 2018), GVAE (Kusner et al., 2017), and SD-VAE (Dai et al., 2018).
Revised: Since molecules can be represented as SMILES strings (Weininger, 1988), early studies generate molecules based on SMILES strings, such as CVAE (G ´ omez-Bombarelli et al., 2018), GVAE (Kusner et al., 2017), and SD-VAE (Dai et al., 2018).

Original: Recent studies mostly represent and generate molecules as graphs (Simonovsky & Komodakis, 2018; De Cao & Kipf, 2018; Madhawa et al., 2019).
Revised: Recent studies mostly represent and generate molecules as graphs (Simonovsky & Komodakis, 2018; De Cao & Kipf, 2018; Madhawa et al., 2019).

Original: We can categorize existing molecular graph generation methods based on the underlying generative methods or the generation processes.
Revised: We can categorize existing molecular graph generation methods based on the underlying generative methods or the generation processes.

Original: Current molecular graph generation approaches can be grouped into four categories according to their underlying generative models, i.e. , VAEs,
Revised: Current molecular graph generation approaches can be grouped into four categories according to their underlying generative models, i.e. , VAEs,

Original: GANs, ﬂow models, and RNNs.
Revised: GANs, ﬂow models, and RNNs.

Original: They can also be classiﬁed into two primary types based on their generation processes; those are, sequential generation and one-shot generation.
Revised: They can also be classiﬁed into two primary types based on their generation processes; those are, sequential generation and one-shot generation.

Original: The sequential process generates nodes and edges in a sequential order by adding nodes and edges one by one.
Revised: The sequential process generates nodes and edges in a sequential order by adding nodes and edges one by one.

Original: The one-shot process generates all nodes and edges at one time.
Revised: The one-shot process generates all nodes and edges at one time.

Original: To facilitate comparison, we summarize existing methods in Table 1.
Revised: To facilitate comparison, we summarize existing methods in Table 3.

Original: We can observe that most of them fail to satisfy an intrinsic property of graphs; that is, permutation invariance.
Revised: We can observe that most of them fail to satisfy an intrinsic property of graphs; that is, permutation invariance.

Original: Speciﬁcally, a generative model should yield the same likelihood for different permutations of the same graph.
Revised: Speciﬁcally, a generative model should yield the same likelihood for different permutations of the same graph.

Original: Currently, permutation invariance remains to be a challenging goal to achieve.
Revised: Currently, permutation invariance remains to be a challenging goal to achieve.

Original: The sequential generation approaches have to choose a speciﬁc order of nodes, thus failing to preserve permutation invariance.
Revised: The sequential generation approaches have to choose a speciﬁc order of nodes, thus failing to preserve permutation invariance.

Original: Among the one-shot methods, Bresson & Laurent (2019) also use the speciﬁc node order given by the SMILES representation.
Revised: Among the one-shot methods, Bresson & Laurent (2019) also use the speciﬁc node order given by the SMILES representation.

Original: GraphVAE and RVAE perform an approximate and expensive graph matching to train the VAE model, and they cannot achieve exact permutation invariance.
Revised: GraphVAE and RVAE perform an approximate and expensive graph matching to train the VAE model, and they cannot achieve exact permutation invariance.

Original: MolGAN circumvents this issue by using a likelihood-free method.
Revised: MolGAN circumvents this issue by using a likelihood-free method.

Original: The recent one-shot ﬂow methods have the potential to satisfy this property.
Revised: The recent one-shot ﬂow methods have the potential to satisfy this property.

Original: However, GraphNVP, GRF, and MoFlow cannot preserve this property since the masking strategies in the coupling layers are sensitive to node order.
Revised: However, GraphNVP, GRF, and MoFlow cannot preserve this property since the masking strategies in the coupling layers are sensitive to node order.

Original: An exception is GraphCNF, which achieves permutation invariance by assigning likelihood independent of node ordering via categorical normalizing ﬂows.
Revised: An exception is GraphCNF, which achieves permutation invariance by assigning likelihood independent of node ordering via categorical normalizing ﬂows.

Original: In this work, we propose to develop energy-based models (EBMs) (LeCun et al., 2006) for molecular graph generation.
Revised: In this work, we propose to develop energy-based models (EBMs) (LeCun et al., 2006) for molecular graph generation.

Original: EBMs are a class of powerful methods for modeling richly structured data, but their use for graph generation has been under-explored.
Revised: EBMs are a class of powerful methods for modeling richly structured data, but their use for graph generation has been under-explored.

Original: We show that our method can achieve the desirable property of permutation invariance.
Revised: We show that our method can achieve the desirable property of permutation invariance.

Original: Additionally, our method has the potential to generate molecules in a compositional manner towards multiple objectives, which cannot be achieved by any existing methods. 
Revised: Additionally, our method has the potential to generate molecules in a compositional manner towards multiple objectives, which cannot be achieved by any existing methods. [Figure] NERGY [Figure]

Revised: ASED

Original: [Figure]
Revised: [Figure]

Original: Modeling variables by deﬁning an unnormalized probability density has been explored for decades Hopﬁeld (1982); Ackley et al.
Revised: Modeling variables by deﬁning an unnormalized probability density has been explored for decades Hopﬁeld (1982); Ackley et al.

Original: (1985); Cipra (1987);
Revised: (1985); Cipra (1987);

Original: Dayan et al.
Revised: Dayan et al.

Original: ; Cipra (1987);
Revised: ; Cipra (1987);

Original: Zhu et al.
Revised: Zhu et al.

Original: (1998); Hinton (2012).
Revised: (1998); Hinton (2012).

Original: Such methods are known and uniﬁed as energy-based models (EBMs) (LeCun et al., 2006) in machine learning.
Revised: Such methods are known and uniﬁed as energy-based models (EBMs) (LeCun et al., 2006) in machine learning.

Original: EBMs capture the dependencies of variables by assigning a scalar energy to each conﬁguration of the variables with a learnable energy function.
Revised: EBMs capture the dependencies of variables by assigning a scalar energy to each conﬁguration of the variables with a learnable energy function.

Original: Given a trained EBM, inference is to ﬁnd the conﬁgurations that yield low energies.
Revised: Given a trained EBM, inference is to ﬁnd the conﬁgurations that yield low energies.

Original: Training an
Revised: Training an

Original: EBM aims at obtaining an energy function where observed conﬁgurations are associated with lower energies than unobserved ones.
Revised: EBM aims at obtaining an energy function where observed conﬁgurations are associated with lower energies than unobserved ones.

Original: Currently, EBMs have been used as generative models in multiple domains, including images (Xie et al., 2015; 2016; Du & Mordatch, 2019; Du et al., 2020a;b), videos (Xie et al., 2017), 3D objects (Xie et al., 2018), and point sets (Xie et al., 2020).
Revised: Currently, EBMs have been used as generative models in multiple domains, including images (Xie et al., 2015; 2016; Du & Mordatch, 2019; Du et al., 2020a;b), videos (Xie et al., 2017), 3D objects (Xie et al., 2018), and point sets (Xie et al., 2020).

Original: To date, EBMs have rarely been studied in the graph domain.
Revised: To date, EBMs have rarely been studied in the graph domain.

Original: Liu et al.
Revised: Liu et al.

Original: (2020a) attempt to generate graphs by building EBMs based on graph neural networks.
Revised: (2020a) attempt to generate graphs by building EBMs based on graph neural networks.

Original: Niu et al.
Revised: Niu et al.

Original: (2020) model graphs using a score-based generative model (Song & Ermon, 2019), a method that is similar to EBMs.
Revised: (2020) model graphs using a score-based generative model (Song & Ermon, 2019), a method that is similar to EBMs.

Original: However, these two methods can only generate graph structures, and it is not straightforward to use them on attributed graphs.
Revised: However, these two methods can only generate graph structures, and it is not straightforward to use them on attributed graphs.

Original: In this work, we propose GraphEBM to generate attributed molecular graphs using
Revised: In this work, we propose GraphEBM to generate attributed molecular graphs using

Original: EBMs.
Revised: EBMs.

Original: Hence, we consider GraphEBM to be the ﬁrst energy-based model capable of generating attributed graphs.
Revised: Hence, we consider GraphEBM to be the ﬁrst energy-based model capable of generating attributed graphs.

Original: B G ENERATION P ROCESS OF G RAPH EBM
Revised: B G ENERATION P ROCESS OF G RAPH EBM

Original: The generation process of our GraphEBM is illustrated in Figure 3.
Revised: The generation process of our GraphEBM is illustrated in Figure 5.

Original: [Equation] E [Equation] G ENERATION S TRATEGIES
Revised: C E XISTING G OAL -D IRECTED G ENERATION S TRATEGIES

Original: There are mainly three approaches in the literature for goal-directed generation.
Revised: There are mainly three approaches in the literature for goal-directed generation.

Original: First, this task can be modeled as a conditional generation problem, where the property value can be utilized as the condition (Simonovsky & Komodakis, 2018).
Revised: First, this task can be modeled as a conditional generation problem, where the property value can be utilized as the condition (Simonovsky & Komodakis, 2018).

Original: Second, for methods using the latent space, a predictor can be applied to learn the property value from the latent representation (G ´ omez-Bombarelli et al., 2018).
Revised: Second, for methods using the latent space, a predictor can be applied to learn the property value from the latent representation (G ´ omez-Bombarelli et al., 2018).

Original: Third, reinforcement learning can be used to optimize the properties of generated molecules (You et al., 2018).
Revised: Third, reinforcement learning can be used to optimize the properties of generated molecules (You et al., 2018).

Original: However, it is not straightforward to apply these methods to our
Revised: However, it is not straightforward to apply these methods to our

Original: GraphEBM for goal-directed generation since GraphEBM generates molecules implicitly using
Revised: GraphEBM for goal-directed generation since GraphEBM generates molecules implicitly using

Original: Langevin dynamics and no latent space exists.
Revised: Langevin dynamics and no latent space exists.

Original: Also, using EBMs for generation that is conditional on continuous conditions is rarely studied by the community.
Revised: Also, using EBMs for generation that is conditional on continuous conditions is rarely studied by the community.

Original: Hence, it remains challenging to apply EBMs for goal-directed generation.
Revised: Hence, it remains challenging to apply EBMs for goal-directed generation.

Original: D E XPERIMENTAL S ETUP
Revised: D E XPERIMENTAL S ETUP

Original: Dataset .
Revised: Dataset .

Original: QM9 consists of 134 k organic molecules and the maximum number of atoms is 9 .
Revised: QM9 consists of 134 k organic molecules and the maximum number of atoms is 9 .

Original: It contains 4 atom types and 3 bond types.
Revised: It contains 4 atom types and 3 bond types.

Original: ZINC250k has 250 k drug-like molecules and the maximum number of atoms is 38 .
Revised: ZINC250k has 250 k drug-like molecules and the maximum number of atoms is 38 .

Original: It includes 9 atom types and 3 edge types.
Revised: It includes 9 atom types and 3 edge types.

Original: Implementation details .
Revised: Implementation details .

Original: We kekulize molecules and remove their hydrogen atoms using RDKit (Landrum et al., 2006).
Revised: We kekulize molecules and remove their hydrogen atoms using RDKit (Landrum et al., 2006).

Original: In our parameterized energy function, we adopt a network of L =layers with hidden dimension d = 64 .
Revised: In our parameterized energy function, we adopt a network of L =layers with hidden dimension d = 64 .

Original: We use Swish as the activation function.
Revised: We use Swish as the activation function.

Original: We set α = 1 in the loss function and the standard variance σ = 0 .
Revised: We set α = 1 in the loss function and the standard variance σ = 0 .

Original: 005 in the gaussian noise.
Revised: 005 in the gaussian noise.

Original: For training, we tune the following hyperparameters: the scale t of uniform noise ∈ [0, 1], the sample step K of Langevin dynamics ∈ [30, 300], and the step size λ 2 ∈ [10, 50].
Revised: For training, we tune the following hyperparameters: the scale t of uniform noise ∈ [0, 1], the sample step K of Langevin dynamics ∈ [30, 300], and the step size λ 2 ∈ [10, 50].

Original: All models are trained for up to 20 epochs with a learning rate of 0 . 0001 and a batch size of 128 .
Revised: All models are trained for up to 20 epochs with a learning rate of 0 . 0001 and a batch size of 128 .

Original: It is well known that it is difﬁcult to train
Revised: It is well known that it is difﬁcult to train

Original: EBMs.
Revised: EBMs.

Original: We follow the techniques adopted in Du & Mordatch (2019) to stabilize the training process.
Revised: We follow the techniques adopted in Du & Mordatch (2019) to stabilize the training process.

Original: Speciﬁcally, we add spectral normalization (Miyato et al., 2018) to all layers of the network.
Revised: Speciﬁcally, we add spectral normalization (Miyato et al., 2018) to all layers of the network.

Original: In addition, we clip the gradient used in Langevin dynamics so that its value magnitude can be less than 0 .
Revised: In addition, we clip the gradient used in Langevin dynamics so that its value magnitude can be less than 0 .

Original: GraphEBM is implemented with PyTorch (Paszke et al., 2017).
Revised: GraphEBM is implemented with PyTorch (Paszke et al., 2017).

Original: Random generation .
Revised: Random generation .

Original: We evaluate the ability of our proposed GraphEBM to model and generate molecules.
Revised: We evaluate the ability of our proposed GraphEBM to model and generate molecules.

Original: We consider most methods reviewed in Section A.1 as baselines.
Revised: We consider most methods reviewed in Section A.1 as baselines.

Original: The following widely used metrics are adopted.
Revised: The following widely used metrics are adopted.

Original: Validity is the percentage of chemically valid molecules among all generated molecules.
Revised: Validity is the percentage of chemically valid molecules among all generated molecules.

Original: Uniqueness denotes the percentage of unique molecules among all valid molecules.
Revised: Uniqueness denotes the percentage of unique molecules among all valid molecules.

Original: Novelty corresponds to the percentage of generated valid molecules that are not present in the training set.
Revised: Novelty corresponds to the percentage of generated valid molecules that are not present in the training set.

Original: The metrics are computed on 10 , 000 randomly generated molecules.
Revised: The metrics are computed on 10 , 000 randomly generated molecules.

Original: Results averaged over 5 runs are reported.
Revised: Results averaged over 5 runs are reported.

Original: Goal-directed generation .
Revised: Goal-directed generation .

Original: To evaluate the performance of goal-directed generation, the community usually compares the highest chemical property scores of generated molecules.

Original: We note that, due to the fact that EBMs have rarely been explored for graph generation, its performance in terms of this metric is not competitive with prior methods based on VAE, GAN, and Flow, which have been intensively studied and tuned for graph generation.

Original: We believe this could be improved with the development of this direction in the future.

Original: To empirically show the effectiveness of our goaldirected generation method proposed in Section 2.5, we train models on ZINC250k accordingly and compare the distribution of the property score between goal-directed generated molecules and random generated molecules.
Revised: To empirically show the effectiveness of our goal-directed generation method proposed in Section 2.5, we train models on ZINC250k accordingly and compare the distribution of the property score between goal-directed generated molecules and random generated molecules.

Original: We consider two chemical properties, including Quantitative Estimate of Druglikeness (QED) Bickerton et al. and penalized logP (plogp), which is the wateroctanol partition coefﬁcient penalized by the number of long cycles and synthetic accessibility.
Revised: We consider two chemical properties, including Quantitative Estimate of Druglikeness (QED) (Bickerton et al., 2012) and penalized logP (plogp), which is the water-octanol partition coefﬁcient penalized by the number of long cycles and synthetic accessibility.

Original: We further verify the effectiveness of our proposed goal-directed generation method by performing molecule  optimization.
Revised: We further verify the effectiveness of our proposed goal-directed generation method by performing molecule optimization, including property optimization and constrained property optimization.

Original: That is, given a molecule, we are expected to obtain a new molecule that has a better desirable chemical property.
Revised: Property optimization aims at generating novel molecules with high QED scores.

Original: Also, the obtained molecule should be similar to  the given molecule.
Revised: We directly use the model trained for goal-directed generation and leverage the molecules in the training set as initialization for Langevin dynamics, following prior works (Jin et al., 2018; Zang & Wang, 2020). We report the highest QED scores and the corresponding novel molecules discovered by our method. For constrained property optimization, given a molecule m , our task is to obtain a new molecule m (cid:48) that has a better desired chemical property with the molecular similarity sim ( m, m (cid:48) ) ≥ δ for some threshold δ .

Original: We adopt Tanimoto similarity of Morgan ﬁngerprint (Rogers & Hahn, 2010) to measure the similarity between molecules and we set the lower bound of the similarity constraint as 0 .
Revised: We adopt Tanimoto similarity of Morgan ﬁngerprint (Rogers & Hahn, 2010) to measure the similarity between molecules. We ﬁnd that there are two different settings in baselines.

Original: 2 inthis experiment. We directly use the model trained for goal-directed generation and leverage the given molecule as initialization for Langevin dynamics, instead of random initialization. Then we can obtain desirable molecules that satisfy the similarity constraint from samples given by Langevin dynamics.
Revised: JT-VAE and GCPN choose 800 molecules with the lowest plogp scores in the test set and use them as initialization, while GraphAF and MoFlow choose from the training set. We report our results on both of these two settings for extensive comparisons.

Original: Compositional generation .
Revised: Compositional generation .

Original: As investigated in Section 2.6, our GraphEBM has the potential to conduct compositional generation towards multiple objectives.
Revised: As investigated in Section 2.6, our GraphEBM has the potential to conduct compositional generation towards multiple objectives.

Original: To verify this, we combine the two energy functions obtained in goal-directed generation experiments, as formulated in Eq.
Revised: To verify this, we combine the two energy functions obtained in goal-directed generation experiments, as formulated in Eq.

Original: Then we apply the generation process described in Section 2.4 to the resulting energy function to generate molecules.
Revised: Then we apply the generation process described in Section 2.4 to the resulting energy function to generate molecules.

Original: E E XPERIMENTAL R ESULTS
Revised: E E XPERIMENTAL R ESULTS

Original: Random generation .

Original: The generation performance on QM9 and ZINC250k is given in Table 2 and Table 3.
Revised: The generation performance on QM9 and ZINC250k is given in Table 4 and Table 5.

Original: The generated molecule samples are visualized in Figure 4.
Revised: The generated molecule samples are visualized in Figure 6.

Original: The visualization of the implicit generation process of our GraphEBM is shown in Figure 5.
Revised: The visualization of the implicit generation process of our GraphEBM is shown in Figure 7.

Original: The results of CVAE and GVAE are obtained from Simonovsky & Komodakis (2018).
Revised: The results of CVAE and GVAE are obtained from Simonovsky & Komodakis (2018).

Original: The result of MoFlow is obtained by evaluating its public trained model.
Revised: The result of MoFlow is obtained by evaluating its public trained model.

Original: All other results are from their original papers.
Revised: All other results are from their original papers.

Original: [Table]
Revised: [Table]

Original: The results of GCPN and JTVAE are obtained from Shi et al.
Revised: The results of GCPN and JTVAE are obtained from Shi et al.

Original: The result of MoFlow is obtained by evaluating its public trained model.
Revised: The result of MoFlow is obtained by evaluating its public trained model.

Original: All other results are from their originalpapers
Revised: All other results are from their original

Original: [Table]
Revised: [Table]

Original: [Figure]
Revised: [Figure]

Original: Goal-directed generation .

Original: More examples of molecule optimization towards plogp and QED are available in Figure