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+# DISSECTING GRAPH MEASURES PERFORMANCE FOR NODE CLUSTERING IN LFR PARAMETER SPACE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Graph measures can be used for graph node clustering using metric clustering algorithms. There are multiple measures applicable to this task, and which one performs better is an open question. We study the performance of 25 graph measures on generated graphs with different parameters. While usually measure comparisons are limited to general measure ranking on a particular dataset, we aim to explore the performance of various measures depending on graph features. Using an LFR graph generator, we create a dataset of $\sim 7 5 0 0$ graphs covering the whole LFR parameter space. For each graph, we assess the quality of clustering with $k$ -means algorithm for every considered measure. We determine the best measure for every area of the parameter space. We find that the parameter space consists of distinct zones where one particular measure is the best. We analyze the geometry of the resulting zones and describe it with simple criteria. Given particular graph parameters, this allows us to choose the best measure to use for clustering.
+
+# 1 INTRODUCTION
+
+Graph node clustering is one of the central tasks in graph structure analysis. It provides a partition of nodes into disjoint clusters, which are groups of nodes that are characterized by strong mutual connections. It can be of practical use for graphs representing real-life systems, such as social networks or industrial processes. Clustering allows to infer some information about the system: the nodes of the same cluster are highly similar, while the nodes of different clusters are dissimilar. The technique can be applied without any labeled data to extract important insights about a network.
+
+There are different approaches to clustering, including ones based on modularity optimization (Newman & Girvan, 2004; Blondel et al., 2008), label propagation algorithm (Raghavan et al., 2007; Barber & Clark, 2009), Markov cluster process (Van Dongen, 2000; Enright et al., 2002), and spectral clustering (Von Luxburg, 2007). In this work, we use a different approach based on choosing a closeness measure on a graph, which allows one to use any metric clustering algorithm (e.g., Yen et al., 2009).
+
+The choice of the measure significantly affects the quality of clustering. Classical measures are the Shortest Path (Buckley & Harary, 1990) and the Commute Time (Gobel & Jagers, 1974) distances. ¨ The former is the minimum number of edges in a path between a given pair of nodes. The latter is the expected number of steps from one node to the other and back in a random walk on the graph. There is a number of other measures, including recent ones (e.g., Estrada & Silver, 2017; Jacobsen & Tien, 2018), many of them are parametric. Despite the fact that graph measures are compatible with any metric algorithm, in this paper we restrict ourselves to the kernel $k$ -means algorithm (e.g., Fouss et al., 2016).
+
+We base our research on a generated set of graphs. There are various algorithms to generate graphs with community structures. The well-known ones are the Stochastic Block Model (Holland et al., 1983) and Lancichinetti–Fortunato–Radicchi benchmark (Lancichinetti et al., 2008) (hereafter, LFR). The first one is an extension of the Erdos–R ˝ enyi model with different intra- and in- ´ tercluster probabilities of edge creation. The second one involves power law distributions of node degrees and community sizes. There are other generation models, e.g., Naive Scale-free Clustering (Pasta & Zaidi, 2017). We choose the LFR model: although it misses some key properties of real graphs, like diameter or the clustering coefficient, this model has been proven to be effective in meta-learning (Prokhorenkova, 2019).
+
+There are a lot of measure benchmarking studies considering node classification and clustering for both generated graphs and real-world datasets (Fouss et al., 2012; Sommer et al., 2016; 2017; Avrachenkov et al., 2017; Ivashkin & Chebotarev, 2016; Guex et al., 2018; 2019; Aynulin, 2019a;b; Courtain et al., 2020; Leleux et al., 2020), etc. Despite a large number of experimental results, theoretical results are still a matter of the future. One of the most interesting theoretical results on graph measures is the work by Luxburg et al. (2010), where some unattractive features of the Commute Time distance on large graphs were explained theoretically, and a reasonable amendment was proposed to fix the problem. Beyond the complexity of such proofs, there is still very little empirical understanding of what effects need to be proven. Our empirical work has two main differences from the previous ones. First, we consider a large number of graph measures, which for the first time gives a fairly complete picture. Second, unlike these studies concluding with a global leaderboard, we are looking for the leading measures for each set of the LFR parameters.
+
+We aim to explore the performance of of the 25 most popular measures in the graph clustering problem on a set of generated graphs with various parameters. We assess the quality of clustering with every considered measure and determine the best measure for every region of the graph parameter space.
+
+Our contributions are as follows:
+
+• We generate a dataset of $\sim 7 5 0 0$ graphs covering all parameter space of LFR generator;   
+• We consider a broad set of measures and rank measures by clustering performance on this dataset;   
+• We find the regions of certain measure leadership in the graph parameter space;   
+• We determine the graph features that are responsible for measure leadership;   
+• We check the applicability of the results on real-world graphs.
+
+Our framework for clustering with graph measures as well as a collected dataset are available on link_is_not_available_during_blind_review.
+
+# 2 DEFINITIONS
+
+# 2.1 KERNEL $k$ -MEANS
+
+The original $k$ -means algorithm (Lloyd, 1982; MacQueen et al., 1967) clusters objects in Euclidean space. It requires coordinates of the objects to determine the distances between them and centroids. The algorithm can be generalized to use the degree of closeness between the objects without defining a particular space. This technique is called the kernel trick, usually it is used to bring non-linearity to linear algorithms. The algorithm that uses the kernel trick is called kernel $k$ -means (see, e.g., Fouss et al., 2016). For graph node clustering scenario, we can use graph measures as kernels for the kernel $k$ -means.
+
+Initially, the number of clusters is known and we need to set initial state of centroids. The results of the clustering with $k$ -means are very sensitive to it. Usually, the algorithm runs several times with different initial states (trials) and chooses the best trial. There are different approaches to the initialization; we consider three of them: random data points, $k$ -means $^ { + + }$ (Arthur & Vassilvitskii, 2006), and random partition. We combine all these strategies to reduce the impact of the initialization strategy on the result.
+
+# 2.2 CLOSENESS MEASURES
+
+For a given graph $G$ , $V ( G )$ is the set of its vertices and $A$ is its adjacency matrix. A measure on $G$ is a function $\kappa \colon V ( G ) \times V ( G ) \to \mathbb { R }$ , which gets two nodes and returns closeness (bigger means closer) or distance (bigger means farther).
+
+A kernel on a graph is a graph nodes’ closeness measure that has an inner product representation. Any symmetric positive semidefinite matrix is an inner product matrix (also called Gram matrix). A kernel matrix $K$ is a square matrix that contains similarities for all pairs of nodes in a graph.
+
+To use kernel $k$ -means, we need kernels. Despite that not all closeness measures we consider are Gram matrices, we treat them as kernels. The applicability of this approach was confirmed in Fouss et al. (2016). For the list of measures bellow, we use the word “kernel” only for the measures that satisfy the strict definition of kernel.
+
+Classical measures are Shortest Path distance (Buckley & Harary, 1990) (SP) and Commute Time distance (Gobel & Jagers, 1974) (CT). SP is the minimum number of edges in a path between a ¨ given pair of nodes. CT is the expected lengths of random walks between two nodes. SP and CT are defined as distances, so we need to transform them into similarities to use as kernels. We apply the following distance to closeness transformation (Chebotarev & Shamis, 1998a; Borg & Groenen, 2005):
+
+$$
+K = - H { \mathcal { D } } H ; H = I - E / n ,
+$$
+
+where $\mathcal { D }$ is a distance matrix, $E$ is the matrix of ones, $I$ is the identity matrix, and $n$ is the number of nodes.
+
+In this paper, we examine 25 graph measures (or, more exactly, 25 parametric families of measures). We present these measures grouped by type similarly to (Avrachenkov et al., 2017):
+
+• Adjacency Matrix $A$ based kernels and measures. – Katz kernel: $K _ { \alpha } ^ { \mathrm { K a t z } } = ( I - \alpha A ) ^ { - 1 }$ , $0 < \alpha < \rho ^ { - 1 }$ , where $\rho$ is the spectral radius of $A$ . (Katz, 1953) (also known as Walk proximity (Chebotarev & Shamis, 1998b) or von Neumann diffusion kernel (Kandola et al., 2003; Shawe-Taylor & Cristianini et al., 2004)). – Communicability kernel $K _ { t } ^ { \mathrm { C o m m } } = \mathrm { e x p m } ( t A )$ , $t > 0$ , where expm means matrix exponential (Fouss et al., 2006; Estrada & Hatano, 2007; 2008). – Double Factorial closeness: $\begin{array} { r } { K _ { t } ^ { \mathrm { D F } } = \sum _ { k = 0 } ^ { \operatorname* { i n f } } \frac { t ^ { k } } { k ! ! } A ^ { k } } \end{array}$ , $t > 0$ (Estrada & Silver, 2017).
+
+• Laplacian Matrix $L = D - A$ based kernels and measures, where $D = \mathrm { D i a g } ( A \cdot { \bf 1 } )$ is the degree matrix of $G$ , $\operatorname { D i a g } ( \mathbf { x } )$ is the diagonal matrix with vector $\mathbf { X }$ on the main diagonal.
+
+– Forest kernel: $K _ { t } ^ { \mathrm { F o r } } ~ = ~ ( I + t L ) ^ { - 1 }$ , $t > 0$ (also known as Regularized Laplacian kernel) (Chebotarev & Shamis, 1995).   
+– Heat kernel: $K _ { t } ^ { \mathrm { H e a t } } = \mathrm { e x p m } ( - t L )$ , $t > 0$ (Chung & Yau, 1998).   
+– Normalized Heat kernel: $K _ { t } ^ { \mathrm { N H e a t } } = \mathrm { e x p m } ( - t \mathcal { L } )$ , $\mathcal { L } = D ^ { - \frac { 1 } { 2 } } L D ^ { - \frac { 1 } { 2 } }$ , $t > 0$ (Chung, 1997).   
+– Absorption kernel: $K _ { t } ^ { \mathrm { A b s } } = ( t A + L ) ^ { - 1 }$ , $t > 0$ (Jacobsen & Tien, 2018).
+
+• Markov Matrix $P = D ^ { - 1 } A$ based kernels and measures.
+
+– Personalized PageRank closeness: $K _ { \alpha } ^ { \mathrm { P P R } } = ( I - \alpha P ) ^ { - 1 }$ , $0 \textless \alpha \textless 1$ (Page et al., 1999).   
+– Modified Personalized PageRank: $K _ { \alpha } ^ { \mathrm { M P P R } } = ( I - \alpha P ) ^ { - 1 } D ^ { - 1 } = ( D - \alpha A ) ^ { - 1 } ,$ , $0 < \alpha < 1$ (Kirkland $\&$ Neumann, 2012).   
+– PageRank heat closeness: $K _ { t } ^ { \mathrm { H P R } } = \exp \mathrm { m } ( - t ( I - P ) )$ , $t > 0$ (Chung, 2007).   
+– Randomized Shortest Path distance. Using $P$ and the matrix of the SP distances $C$ first get $Z$ (Yen et al., 2008):
+
+$$
+\begin{array} { r } { W = P \circ \exp ( - \beta C ) ; Z = ( I - W ) ^ { - 1 } . } \end{array}
+$$
+
+Then $S = ( Z ( C \circ W ) Z ) \div Z$ ; $\bar { C } = S - \mathbf { e } \mathrm { d i a g } ( S ) ^ { T }$ , and finally, ${ \mathcal { D } } _ { \mathrm { R S P } } = ( { \bar { C } } + { \bar { C } } ^ { T } ) / 2$ . Here $\circ$ and $\div$ are element-wise multiplication and division. Kernel version $K ^ { \mathrm { R S P } } ( t )$ can be obtained with equation 1.
+
+– Free Energy distance. Using $Z$ from equation 2: $Z ^ { h } \ = \ Z { \mathrm { D i a g } } ( Z ) ^ { - 1 }$ ; $\Phi \_ { } = $ $- 1 / \beta \log Z ^ { h }$ ; $\mathcal { D } _ { \mathrm { F E } } = ( \Phi + \bar { \Phi ^ { T } } ) / 2$ (Kivimaki et al., 2014). Kernel version ¨ $K ^ { \mathrm { F E } } ( t )$ can be obtained with equation 1.
+
+• Sigmoid Commute Time kernels.
+
+– Sigmoid Commute Time kernel:
+
+$$
+K _ { t } ^ { \mathrm { S C T } } = \sigma ( - t K ^ { \mathrm { C T } } / \mathrm { s t d } ( K ^ { \mathrm { C T } } ) ) , t > 0 ,
+$$
+
+where $\sigma$ is an element-wise sigmoid function $\sigma ( x ) = 1 / ( 1 + e ^ { - x } )$ (Yen et al., 2007).
+
+Table 1: Short names of considered kernels and other measures.   
+
+<table><tr><td rowspan="3">Family Adjacency matrix</td><td colspan="2">Short name</td><td></td></tr><tr><td>Plain</td><td>Logarithmic</td><td></td></tr><tr><td>measure</td><td>measure</td><td>Full name Katz kernel</td></tr><tr><td rowspan="2">based kernels</td><td>Katz Comm</td><td>logKatz logComm</td><td>Communicability kernel</td></tr><tr><td>DF</td><td>logDF</td><td>Double Factorial closeness</td></tr><tr><td rowspan="4">Laplacian based kernels</td><td>For</td><td>logFor</td><td>Forest kernel</td></tr><tr><td>Heat</td><td>logHeat</td><td>Heat kernel</td></tr><tr><td>NHeat</td><td>logNHeat</td><td>Normalized Heat kernel</td></tr><tr><td>Abs</td><td>logAbs</td><td>Absorption kernel</td></tr><tr><td rowspan="4">Markovmatrix based kernels and measures</td><td>PPR MPPR</td><td>logPPR</td><td>Personalized PageRank closeness</td></tr><tr><td>HPR</td><td>logMPPR</td><td>Modified Personalized PageRank</td></tr><tr><td></td><td>logHPR</td><td>PageRank heat closeness</td></tr><tr><td>RSP FE</td><td>=</td><td>Randomized Shortest Path kernel</td></tr><tr><td rowspan="2">Sigmoid Commute Time</td><td>SCT</td><td>1</td><td>Free Energy kernel Sigmoid Commute Time kernel</td></tr><tr><td>SCCT</td><td>=</td><td>Sigmoid Corrected Commute Time kernel</td></tr><tr><td rowspan="2"></td><td>SP-CT</td><td></td><td></td></tr><tr><td></td><td>1</td><td>linear combination of SP and CT</td></tr></table>
+
+– Sigmoid Corrected Commute Time kernel. First of all, we need the Corrected Commute Time kernel (Luxburg et al., 2010):
+
+$$
+K ^ { \mathrm { C C T } } = H D ^ { - { \frac { 1 } { 2 } } } M ( I - M ) ^ { - 1 } M D ^ { - { \frac { 1 } { 2 } } } H ; M = D ^ { - { \frac { 1 } { 2 } } } \Big ( A - \frac { \vec { \bf d } \vec { \bf d } ^ { T } } { \mathrm { v o l } ( G ) } \Big ) D ^ { - { \frac { 1 } { 2 } } }
+$$
+
+where $H$ is the centering matrix $H = I - E / n$ , $\vec { \bf d }$ is the vector of diagonal elements of $D$ and $\mathrm { v o l } ( G )$ is the sum of all elements of $A$ . Then, apply equation 3 replacing $K ^ { \mathrm { C T } }$ with $K ^ { \mathrm { C C T } }$ to obtain $K ^ { \mathrm { S C C T } }$ .
+
+Occasionally, element-wise logarithm is applied to the resulting kernel matrix (Chebotarev, 2013; Ivashkin $\&$ Chebotarev, 2016). We apply it to almost all investigated measures and consider the resulting measures separately from their plain versions (see Table 1). For some measures, like Forest kernel, this is well-known practice (Chebotarev, 2013), while for others, like Double Factorial closeness, this transformation, to the best of our knowledge, is applied for the first time. The considered measures and their short names are summarized in Table 1.
+
+# 3 DATASET
+
+We collected a paired dataset of graphs and the corresponding results of clustering with each measure mentioned in Table 1. In this section, we describe the graph generator, the sampling strategy, the calculated graph features, and the pipeline for the measure score calculation.
+
+We use Lancichinetti–Fortunato–Radicchi (LFR) graph generator. It generates non-weighted graphs with ground truth non-overlapping communities. The model has mandatory parameters: the number of nodes $n$ ${ \mathrm { ~  ~ n ~ } } > 0 ,$ , the power law exponent for the degree distribution $\tau _ { 1 }$ $\mathit { \check { \tau } _ { 1 } } > 1 \mathit { \check { \tau } _ { \mathrm { , } } }$ , the power law exponent for the community size distribution $\tau _ { 2 }$ $( \tau _ { 2 } ~ > ~ 1 )$ , the fraction of intra-community edges incident to each node $\mu$ $0 \leq \mu \leq 1 )$ ), and either minimum degree (min degree) or average degree (avg degree). There are also extra parameters: maximum degree (max degree), minimum community size (min community), maximum community size (max community). Not the whole LFR parameter space corresponds to common real-world graphs; most of such graphs are described with $\tau _ { 1 } ~ \in ~ [ 1 , 4 ]$ and $\mu \ : < \ : 0 . 5$ (e.g., Fotouhi et al., 2019). However, there is also an interesting case of bipartite/multipartite-like graphs with $\mu > 0 . 5$ . Moreover, many of the datasets studied in Section 5 have $\tau _ { 1 } > 4$ . Our choice is to consider the entire parameter space to cover all theoretical and practical cases.
+
+For the generation, we consider $1 0 < n < 1 5 0 0$ . It is impossible to generate a dataset with a uniform distribution of all LFR parameters, because $\tau _ { 1 }$ and $\tau _ { 2 }$ parameters are located on rays. We transform $\tau _ { 1 }$ and $\tau _ { 2 }$ to $\tilde { \tau _ { i } } = 1 - ( 1 / \sqrt { \tau _ { i } } ) , i = 1 , 2$ to bring their scope to the $[ 0 , 1 ]$ interval. In this case, “realistic” settings with $\tau _ { 1 } \in [ 1 , 4 ]$ take up $50 \%$ of the variable range. Also, as avg degree feature is limited by the $n$ of a particular graph, we decided to replace it with density (avg degree/ $( n { - } 1 ) )$ . It is not dependent on $n$ and belongs to $[ 0 , 1 ]$ . Using all these considerations, we collected our dataset by uniformly sampling parameters for LFR generator from the set $[ n , \tilde { \tau } _ { 1 } , \tilde { \tau } _ { 2 } , \mu$ , density] and generating graphs with these parameters. Additionally, we filter out all disconnected graphs.
+
+![](images/94e9d4844b8285e314e071ffb3f2f6dd30ffde4310a9a475e836731ad9512533.jpg)  
+Figure 1: Distribution of graph features in the dataset
+
+In total, we generated 7396 graphs. It is worth noting that the generator fails for some sets of parameters, so the resulting dataset is not uniform (see Fig. 1). In our study, non-uniformity is not a very important issue, because we are interested in local effects, not global leadership. Moreover, true uniformity for LFR parameter space is impossible, due to the unlimited scope of parameters.
+
+For our research, we choose a minimum set of the features that describe particular properties of graphs and are not interchangeable.
+
+The LFR parameters can be divided in three groups by the graph properties they reflect:
+
+• The size of the graph and the communities: $n , \tau _ { 1 }$ , min community, max community;   
+• The density and uniformity of the node degrees distribution: $\tau _ { 2 }$ , min degree, avg degree, max degree. As avg degree depends on $n$ , it is distributed exponentially, so we use log(avg degree) instead;   
+• The cluster separability: $\mu$ . As $\mu$ parameter considers only the ratio between the number of inter-cluster edges and the number of nodes but ignores overall density, we use modularity (Newman & Girvan, 2004) as a more appropriate measure for cluster separability.
+
+![](images/f87f7ddfe7340ba93c18fe511c5012c5342ef0d84239b578efe2f3f391b6661d.jpg)  
+Figure 2: Measuring ARI clustering score for a particular graph, measure, and measure parameter
+
+Thus, the defined set of features $[ n , \tau _ { 1 } , \tau _ { 2 }$ , avg degree, modularity] is enough to consider all graph properties mentioned above. Although modularity is a widely used measure, it suffers from resolution limit problems (Fortunato & Barthelemy, 2007). We acknowledge that this may cause some limitations in our approach, which should be the topic of further research.
+
+Table 2: The overall leaderboard. The win percentage is calculated among all 7396 graphs in the dataset. The ARI column shows the mean ARI across the dataset.   
+
+<table><tr><td>#</td><td>Measure</td><td>Rank</td><td>Wins, %</td><td>ARI</td><td>#</td><td>Measure</td><td>Rank</td><td>Wins, %</td><td>ARI</td></tr><tr><td>1</td><td>SCCT</td><td>4.7</td><td>56.0</td><td>0.58</td><td>14</td><td>DF</td><td>11.0</td><td>8.1</td><td>0.28</td></tr><tr><td>2</td><td>NHeat</td><td>6.9</td><td>25.2</td><td>0.40</td><td>15</td><td>logAbs</td><td>12.1</td><td>15.9</td><td>0.33</td></tr><tr><td>3</td><td>RSP</td><td>7.7</td><td>22.4</td><td>0.42</td><td>16</td><td>logComm</td><td>12.6</td><td>15.5</td><td>0.23</td></tr><tr><td>4</td><td>SCT</td><td>8.4</td><td>19.5</td><td>0.40</td><td>17</td><td>logFor</td><td>13.0</td><td>12.7</td><td>0.22</td></tr><tr><td>5</td><td>Comm</td><td>8.4</td><td>15.8</td><td>0.36</td><td>18</td><td>HeatPR</td><td>13.4</td><td>12.5</td><td>0.22</td></tr><tr><td>6</td><td>logNHeat</td><td>8.8</td><td>18.0</td><td>0.37</td><td>19</td><td>logHeat</td><td>13.6</td><td>11.9</td><td>0.21</td></tr><tr><td>7</td><td>SP-CT</td><td>9.0</td><td>20.4</td><td>0.41</td><td>20</td><td>Heat</td><td>14.7</td><td>11.2</td><td>0.19</td></tr><tr><td>8</td><td>logHeatPR</td><td>9.2</td><td>18.0</td><td>0.37</td><td>21</td><td>logDF</td><td>14.8</td><td>11.1</td><td>0.18</td></tr><tr><td>9</td><td>FE</td><td>9.5</td><td>19.3</td><td>0.39</td><td>22</td><td>PPR</td><td>16.7</td><td>4.0</td><td>0.13</td></tr><tr><td>10</td><td>Katz</td><td>9.8</td><td>7.7</td><td>0.32</td><td>23</td><td>Abs</td><td>18.6</td><td>3.7</td><td>0.08</td></tr><tr><td>11</td><td>logKatz</td><td>9.9</td><td>18.1</td><td>0.35</td><td>24</td><td>For</td><td>19.3</td><td>2.8</td><td>0.07</td></tr><tr><td>12</td><td>logPPR</td><td>10.0</td><td>17.5</td><td>0.35</td><td>25</td><td>ModifPPR</td><td>20.5</td><td>1.7</td><td>0.05</td></tr><tr><td>13</td><td>logModifPPR</td><td>10.5</td><td>17.3</td><td>0.35</td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+For every generated graph, we calculate the top ARI score for every measure (Hubert & Arabie, 1985). We choose ARI as a clustering score which is both popular and unbiased (Gosgens et al., ¨ 2019). As soon as every measure has a parameter, we perform clustering for a range of parameter values (we transform the parameter to become in the [0, 1] interval and then choose 16 values linearly spaced from 0 to 1). For each value, we run $6 + 6 + 6$ trials of $k$ -means (6 trials for each of three initialization methods).
+
+Fig. 2 shows the pipeline we use to calculate ARI score for a given LFR parameter set, a measure, and a measure parameter. Measure parameters are not the subject of our experiments, so for every measure we just take the result of the measure with the value of the parameter that gives the best ARI score.
+
+Because of the need to iterate over graphs, measures, parameter values, and initializations, the task is computationally difficult. The total computation time was 20 days on 18 CPU cores and 6 GPUs.
+
+# 4 RESULTS
+
+# 4.1 GLOBAL LEADERSHIP IN LFR SPACE
+
+We rank the measures by their ARI score on every graph of the dataset. The rank is defined as the position of the measure in this list, averaged over the dataset (see Table 2). It is important to note that the global leadership does not give a comprehensive advice on which measure is better to use, because for a particular graph, the global leader can perform worse than the others. Here we consider the entire LFR space, not just its zone corresponding to common real-world graphs, so the ranking may differ from those obtained for restricted settings.
+
+As SCCT is the winner for both ranking and percentage of wins, we can say for sure that it is the global winner for the LFR space graphs. Other measures still can be leaders in some zones of the feature space.
+
+# 4.2 FEATURE IMPORTANCE STUDY
+
+First of all, we find out which graph features are important for the choice of the best measure and which are not. To do that, we use Linear Discriminant Analysis (Mika et al., 1999) (LDA). This method finds a new basis in the feature space to classify a dataset in the best way. It also shows how many components of basis are required to fit the majority of data.
+
+Fig. 3a shows that the first two components take about $90 \%$ of the explained variance. Fig. 3b shows that these components include only $\tau _ { 1 }$ , avg degree, and modularity. The fact that $n$ is not used means that the size of the graph as well as the density are not of primary importance for choosing the best measure. So is not $\tau _ { 2 }$ measuring the diversity of cluster sizes.
+
+![](images/0bd8fcae444182bb890f34dfe5fa2f534fccbc423006501df41aac01e87c3458.jpg)  
+Figure 3: The results of LDA components.
+
+Fig. 4 shows the point cloud projected on the space of the two main components of LDA. We see a confirmation that the measures are indeed zoned, but the areas are quite noisy. To detect the zones of measure leadership, we need to know the leadership on average in every area of space, rather than the wins in particular points. To define the local measure leadership in the whole space, we need to introduce a filtering algorithm that for every point of the space returns the measure leadership depending on the closest data points. As the choice of measure is actually dependent only on three features, we can limit our feature space to $[ \tau _ { 1 }$ , avg degree, modularity].
+
+![](images/987397c1772a927349e58b530d6fa6c770b22f058ba7a3de3c97fd0595b8fe5a.jpg)  
+Figure 4: The dataset projected on the two main components of LDA; the point color represents the winning measure for each graph
+
+# 4.3 GAUSSIAN FILTER IN FEATURE SPACE
+
+Using a filter in the feature space, we can suppress the noise and find actual zones of leadership for the measures. We use the Gaussian filter with a scale parameter $\sigma$ . For every given point of the space, it takes the data points that are closer than $3 \sigma$ and averages ARIs of the chosen points with a weight $e ^ { - \mathrm { d i s t } ^ { 2 } / 2 \sigma ^ { 2 } }$ . This allows to give larger weights to closer points. If there are less than three data points inside the sphere with a $3 \sigma$ radius, the filter returns nothing, allowing to ignore the points with insufficient data.
+
+Before applying the filter, we prepare the dataset. First, we only take the points with only one winning measure, because multiple winners can confuse the filter. Then we normalize the standard
+
+deviation of every feature distribution to one. Finally, we cut off the long tail of distant data points.   
+The resulting number of graphs is 5201.
+
+To choose $\sigma$ , we apply the filter with different $\sigma$ and look at the number of connected components in the feature space. The needed $\sigma$ should be large enough to suppress the noise, however, it should not suppress small zones. Guided by this heuristic, we choose $\sigma = 0 . 5$ .
+
+Table 3: The leaderboard of measure wins after filtering with $\sigma = 0 . 5$   
+
+<table><tr><td colspan="10">1oomoom</td><td rowspan="2"></td><td colspan="10">1H8o</td></tr><tr><td>Measure Wins</td><td>SCCS 4283</td><td>441</td><td>JHe 268</td><td></td><td>uom 78</td><td>64</td><td>3 56</td><td>E 3</td><td></td><td>A 3</td><td>5 2</td><td></td><td>1</td><td>J8o 1</td></tr></table>
+
+After filtering with $\sigma = 0 . 5$ , the leaderboard of measure wins is changed (see Table 3). Only six measures keep their positions: SCCT, NHeat, logComm, Comm, logDF, and RSP. This means that these measures do have zones of leadership, otherwise they would be filtered out. We can plot the entire feature space colored by the leadership zones of the measures (Fig. 5). As the resulting space is 3D, we show slices of it by each of the three coordinates.
+
+![](images/6753805399d1d24f663771f2e1b233600464691b7c41335d14ebeabe0ec5aa53.jpg)  
+Figure 5: The feature space $[ \tau _ { 1 }$ , avg degree, modularity] divided into the leadership zones of six measures. The yellow points represent the positions of real-world datasets under study in the space.
+
+# 5 REAL-WORLD DATASETS
+
+Even though LFR lacks some characteristics of real-world graphs, there is evidence that the optimal parameter of the Louvain clustering for a real graph is close to the parameter for LFR graphs generated from the features of a real one (Prokhorenkova, 2019). So, there is a chance that the learned space might be helpful for choosing measures in the wild.
+
+For evaluation, we use 29 graphs of standard datasets: Dolphins (Lusseau et al., 2003), Football (Newman & Girvan, 2004), Karate club (Zachary, 1977), Newsgroups (9 subsets, weights are binarized with threshold 0.1) (Yen et al., 2007), Political blogs (Adamic & Glance, 2005), Political books (Newman, 2006), SocioPatterns Primary school day (2 graphs) (Stehle et al., 2011), Cora (11 ´ subsets) (McCallum et al., 2000), Eu-core (Leskovec et al., 2007), EuroSIS (WebAtlas, 2009). The parameters of these graphs are marked in Fig. 5. For each graph, we found the best ARI for every measure (iterating over the measure parameter value). Now we can check the quality of measure choice, based on the found LFR data. The result of LFR recommendation is the measure that is chosen for the set of parameters corresponding to the dataset in hand.
+
+Table 4: Mean ARI of the LFR recommended strategies for datasets. Top6 stands for the set of measures that have their zones in the LFR parameter space.
+
+<table><tr><td>Strategy</td><td>Mean ARI</td></tr><tr><td>AlwaystakeSCCT Based on LFR space, top6 measures</td><td>0.61 0.62</td></tr><tr><td>Based on LFR space,all measures</td><td>0.62</td></tr><tr><td>Upper bound</td><td>0.64</td></tr></table>
+
+The best measures on the considered datasets are SCCT (by the mean ARI) and SCT (by the rank). This is pretty similar to the results obtained for LFR. Moreover, Spearman correlation between the ranks of measures for the datasets and for the corresponding LFR recommendations is 0.90.
+
+Let us use “always take SCCT” as our baseline strategy. In Table 4 we compare it with strategies based on the LFR space. We obtain LFR recommendation using $k \mathrm { n n }$ as a well-proven method for meta-learning. Since each graph is unique, the result of 1nn can be very noisy, thus we use 5nn.
+
+Table 4 shows that the recommendation approach slightly beats the baseline. Reducing the number of measures from 25 to 6 do not drop the quality. However, this quality increase is not enough to draw confident conclusions about the advantages of the method. Using this fact and the fact that the ranks on datasets and recommendations are highly correlated, we conclude that the metalearning procedure is adequate to give a robust recommendation, but not precise enough to beat the baseline confidently. This may be due to the fact that that the nodes of real graphs were not labeled systematically since they were created in the wild. A larger dataset could help separate the signal from the noise and pinpoint where the limits of the method are. At least, the good news is that the conclusions made on the LFR basis do not contradict the results obtained on the datasets.
+
+# 6 CONCLUSIONS
+
+In this work, we have shown that the global leadership of measures does not provide comprehensive knowledge about graph measures. We demonstrated that among 25 measures, SCCT is the best measure for the LFR graphs both by winning rate and ranking. However, there are also smaller confident zones of leadership for NHeat, Comm, logComm, logDF, and RSP.
+
+Our results do not contradict those of other experimental works and rather expand them by providing new findings. LogComm was first introduced in Ivashkin & Chebotarev (2016) and won in the competitions on graphs generated with a fixed set of SBM parameters. This study confirms its leadership, but only for a certain type of graphs. Another interesting finding is logDF, which unexpectedly shows good performance for the graphs with low modularity and low average degree.
+
+This study is based on the LFR benchmark data. An attempt to apply the results to real data gives the quality slightly above the baseline. However, there is a strong correlation between the ranking of measures for datasets and the ranking of LFR recommendation, which indicates that the leading measures are the same, while the recommendations are not precise enough.
+
+It can be noted that our study is insensitive to the non-uniformity of the generated dataset. While manipulations with this dataset may affect the global leaderboard, they cannot change the local leadership studied in this work.
+
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parse/train/HkUfnZFt1Rw/HkUfnZFt1Rw_content_list.json

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+    {
+        "type": "text",
+        "text": "DISSECTING GRAPH MEASURES PERFORMANCE FOR NODE CLUSTERING IN LFR PARAMETER SPACE ",
+        "text_level": 1,
+        "bbox": [
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+            101,
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+        "page_idx": 0
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+    {
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+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph measures can be used for graph node clustering using metric clustering algorithms. There are multiple measures applicable to this task, and which one performs better is an open question. We study the performance of 25 graph measures on generated graphs with different parameters. While usually measure comparisons are limited to general measure ranking on a particular dataset, we aim to explore the performance of various measures depending on graph features. Using an LFR graph generator, we create a dataset of $\\sim 7 5 0 0$ graphs covering the whole LFR parameter space. For each graph, we assess the quality of clustering with $k$ -means algorithm for every considered measure. We determine the best measure for every area of the parameter space. We find that the parameter space consists of distinct zones where one particular measure is the best. We analyze the geometry of the resulting zones and describe it with simple criteria. Given particular graph parameters, this allows us to choose the best measure to use for clustering. ",
+        "bbox": [
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+            263,
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            468,
+            336,
+            484
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph node clustering is one of the central tasks in graph structure analysis. It provides a partition of nodes into disjoint clusters, which are groups of nodes that are characterized by strong mutual connections. It can be of practical use for graphs representing real-life systems, such as social networks or industrial processes. Clustering allows to infer some information about the system: the nodes of the same cluster are highly similar, while the nodes of different clusters are dissimilar. The technique can be applied without any labeled data to extract important insights about a network. ",
+        "bbox": [
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+            500,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There are different approaches to clustering, including ones based on modularity optimization (Newman & Girvan, 2004; Blondel et al., 2008), label propagation algorithm (Raghavan et al., 2007; Barber & Clark, 2009), Markov cluster process (Van Dongen, 2000; Enright et al., 2002), and spectral clustering (Von Luxburg, 2007). In this work, we use a different approach based on choosing a closeness measure on a graph, which allows one to use any metric clustering algorithm (e.g., Yen et al., 2009). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The choice of the measure significantly affects the quality of clustering. Classical measures are the Shortest Path (Buckley & Harary, 1990) and the Commute Time (Gobel & Jagers, 1974) distances. ¨ The former is the minimum number of edges in a path between a given pair of nodes. The latter is the expected number of steps from one node to the other and back in a random walk on the graph. There is a number of other measures, including recent ones (e.g., Estrada & Silver, 2017; Jacobsen & Tien, 2018), many of them are parametric. Despite the fact that graph measures are compatible with any metric algorithm, in this paper we restrict ourselves to the kernel $k$ -means algorithm (e.g., Fouss et al., 2016). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We base our research on a generated set of graphs. There are various algorithms to generate graphs with community structures. The well-known ones are the Stochastic Block Model (Holland et al., 1983) and Lancichinetti–Fortunato–Radicchi benchmark (Lancichinetti et al., 2008) (hereafter, LFR). The first one is an extension of the Erdos–R ˝ enyi model with different intra- and in- ´ tercluster probabilities of edge creation. The second one involves power law distributions of node degrees and community sizes. There are other generation models, e.g., Naive Scale-free Clustering (Pasta & Zaidi, 2017). We choose the LFR model: although it misses some key properties of real graphs, like diameter or the clustering coefficient, this model has been proven to be effective in meta-learning (Prokhorenkova, 2019). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "There are a lot of measure benchmarking studies considering node classification and clustering for both generated graphs and real-world datasets (Fouss et al., 2012; Sommer et al., 2016; 2017; Avrachenkov et al., 2017; Ivashkin & Chebotarev, 2016; Guex et al., 2018; 2019; Aynulin, 2019a;b; Courtain et al., 2020; Leleux et al., 2020), etc. Despite a large number of experimental results, theoretical results are still a matter of the future. One of the most interesting theoretical results on graph measures is the work by Luxburg et al. (2010), where some unattractive features of the Commute Time distance on large graphs were explained theoretically, and a reasonable amendment was proposed to fix the problem. Beyond the complexity of such proofs, there is still very little empirical understanding of what effects need to be proven. Our empirical work has two main differences from the previous ones. First, we consider a large number of graph measures, which for the first time gives a fairly complete picture. Second, unlike these studies concluding with a global leaderboard, we are looking for the leading measures for each set of the LFR parameters. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We aim to explore the performance of of the 25 most popular measures in the graph clustering problem on a set of generated graphs with various parameters. We assess the quality of clustering with every considered measure and determine the best measure for every region of the graph parameter space. ",
+        "bbox": [
+            174,
+            277,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our contributions are as follows: ",
+        "bbox": [
+            174,
+            340,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We generate a dataset of $\\sim 7 5 0 0$ graphs covering all parameter space of LFR generator;   \n• We consider a broad set of measures and rank measures by clustering performance on this dataset;   \n• We find the regions of certain measure leadership in the graph parameter space;   \n• We determine the graph features that are responsible for measure leadership;   \n• We check the applicability of the results on real-world graphs. ",
+        "bbox": [
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+            367,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our framework for clustering with graph measures as well as a collected dataset are available on link_is_not_available_during_blind_review. ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 DEFINITIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            540,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 KERNEL $k$ -MEANS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            571,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The original $k$ -means algorithm (Lloyd, 1982; MacQueen et al., 1967) clusters objects in Euclidean space. It requires coordinates of the objects to determine the distances between them and centroids. The algorithm can be generalized to use the degree of closeness between the objects without defining a particular space. This technique is called the kernel trick, usually it is used to bring non-linearity to linear algorithms. The algorithm that uses the kernel trick is called kernel $k$ -means (see, e.g., Fouss et al., 2016). For graph node clustering scenario, we can use graph measures as kernels for the kernel $k$ -means. ",
+        "bbox": [
+            174,
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+    },
+    {
+        "type": "text",
+        "text": "Initially, the number of clusters is known and we need to set initial state of centroids. The results of the clustering with $k$ -means are very sensitive to it. Usually, the algorithm runs several times with different initial states (trials) and chooses the best trial. There are different approaches to the initialization; we consider three of them: random data points, $k$ -means $^ { + + }$ (Arthur & Vassilvitskii, 2006), and random partition. We combine all these strategies to reduce the impact of the initialization strategy on the result. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2.2 CLOSENESS MEASURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "For a given graph $G$ , $V ( G )$ is the set of its vertices and $A$ is its adjacency matrix. A measure on $G$ is a function $\\kappa \\colon V ( G ) \\times V ( G ) \\to \\mathbb { R }$ , which gets two nodes and returns closeness (bigger means closer) or distance (bigger means farther). ",
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+    {
+        "type": "text",
+        "text": "A kernel on a graph is a graph nodes’ closeness measure that has an inner product representation. Any symmetric positive semidefinite matrix is an inner product matrix (also called Gram matrix). A kernel matrix $K$ is a square matrix that contains similarities for all pairs of nodes in a graph. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To use kernel $k$ -means, we need kernels. Despite that not all closeness measures we consider are Gram matrices, we treat them as kernels. The applicability of this approach was confirmed in Fouss et al. (2016). For the list of measures bellow, we use the word “kernel” only for the measures that satisfy the strict definition of kernel. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Classical measures are Shortest Path distance (Buckley & Harary, 1990) (SP) and Commute Time distance (Gobel & Jagers, 1974) (CT). SP is the minimum number of edges in a path between a ¨ given pair of nodes. CT is the expected lengths of random walks between two nodes. SP and CT are defined as distances, so we need to transform them into similarities to use as kernels. We apply the following distance to closeness transformation (Chebotarev & Shamis, 1998a; Borg & Groenen, 2005): ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8c2756638a171aebf9fcc9c2fdcf603ad54a32f10c426235d8b6bedd46066d6b.jpg",
+        "text": "$$\nK = - H { \\mathcal { D } } H ; H = I - E / n ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            248,
+            602,
+            265
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { D }$ is a distance matrix, $E$ is the matrix of ones, $I$ is the identity matrix, and $n$ is the number of nodes. ",
+        "bbox": [
+            174,
+            268,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we examine 25 graph measures (or, more exactly, 25 parametric families of measures). We present these measures grouped by type similarly to (Avrachenkov et al., 2017): ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Adjacency Matrix $A$ based kernels and measures. – Katz kernel: $K _ { \\alpha } ^ { \\mathrm { K a t z } } = ( I - \\alpha A ) ^ { - 1 }$ , $0 < \\alpha < \\rho ^ { - 1 }$ , where $\\rho$ is the spectral radius of $A$ . (Katz, 1953) (also known as Walk proximity (Chebotarev & Shamis, 1998b) or von Neumann diffusion kernel (Kandola et al., 2003; Shawe-Taylor & Cristianini et al., 2004)). – Communicability kernel $K _ { t } ^ { \\mathrm { C o m m } } = \\mathrm { e x p m } ( t A )$ , $t > 0$ , where expm means matrix exponential (Fouss et al., 2006; Estrada & Hatano, 2007; 2008). – Double Factorial closeness: $\\begin{array} { r } { K _ { t } ^ { \\mathrm { D F } } = \\sum _ { k = 0 } ^ { \\operatorname* { i n f } } \\frac { t ^ { k } } { k ! ! } A ^ { k } } \\end{array}$ , $t > 0$ (Estrada & Silver, 2017). ",
+        "bbox": [
+            215,
+            343,
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+            470
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Laplacian Matrix $L = D - A$ based kernels and measures, where $D = \\mathrm { D i a g } ( A \\cdot { \\bf 1 } )$ is the degree matrix of $G$ , $\\operatorname { D i a g } ( \\mathbf { x } )$ is the diagonal matrix with vector $\\mathbf { X }$ on the main diagonal. ",
+        "bbox": [
+            215,
+            473,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "– Forest kernel: $K _ { t } ^ { \\mathrm { F o r } } ~ = ~ ( I + t L ) ^ { - 1 }$ , $t > 0$ (also known as Regularized Laplacian kernel) (Chebotarev & Shamis, 1995).   \n– Heat kernel: $K _ { t } ^ { \\mathrm { H e a t } } = \\mathrm { e x p m } ( - t L )$ , $t > 0$ (Chung & Yau, 1998).   \n– Normalized Heat kernel: $K _ { t } ^ { \\mathrm { N H e a t } } = \\mathrm { e x p m } ( - t \\mathcal { L } )$ , $\\mathcal { L } = D ^ { - \\frac { 1 } { 2 } } L D ^ { - \\frac { 1 } { 2 } }$ , $t > 0$ (Chung, 1997).   \n– Absorption kernel: $K _ { t } ^ { \\mathrm { A b s } } = ( t A + L ) ^ { - 1 }$ , $t > 0$ (Jacobsen & Tien, 2018). ",
+        "bbox": [
+            246,
+            503,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Markov Matrix $P = D ^ { - 1 } A$ based kernels and measures. ",
+        "bbox": [
+            215,
+            602,
+            604,
+            617
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "– Personalized PageRank closeness: $K _ { \\alpha } ^ { \\mathrm { P P R } } = ( I - \\alpha P ) ^ { - 1 }$ , $0 \\textless \\alpha \\textless 1$ (Page et al., 1999).   \n– Modified Personalized PageRank: $K _ { \\alpha } ^ { \\mathrm { M P P R } } = ( I - \\alpha P ) ^ { - 1 } D ^ { - 1 } = ( D - \\alpha A ) ^ { - 1 } ,$ , $0 < \\alpha < 1$ (Kirkland $\\&$ Neumann, 2012).   \n– PageRank heat closeness: $K _ { t } ^ { \\mathrm { H P R } } = \\exp \\mathrm { m } ( - t ( I - P ) )$ , $t > 0$ (Chung, 2007).   \n– Randomized Shortest Path distance. Using $P$ and the matrix of the SP distances $C$ first get $Z$ (Yen et al., 2008): ",
+        "bbox": [
+            246,
+            619,
+            825,
+            726
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/64f92fabe391830bdc8f20af00ccb901771a2162ae13398b46e363acc1d18aea.jpg",
+        "text": "$$\n\\begin{array} { r } { W = P \\circ \\exp ( - \\beta C ) ; Z = ( I - W ) ^ { - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            408,
+            731,
+            679,
+            750
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Then $S = ( Z ( C \\circ W ) Z ) \\div Z$ ; $\\bar { C } = S - \\mathbf { e } \\mathrm { d i a g } ( S ) ^ { T }$ , and finally, ${ \\mathcal { D } } _ { \\mathrm { R S P } } = ( { \\bar { C } } + { \\bar { C } } ^ { T } ) / 2$ . Here $\\circ$ and $\\div$ are element-wise multiplication and division. Kernel version $K ^ { \\mathrm { R S P } } ( t )$ can be obtained with equation 1. ",
+        "bbox": [
+            261,
+            755,
+            823,
+            797
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "– Free Energy distance. Using $Z$ from equation 2: $Z ^ { h } \\ = \\ Z { \\mathrm { D i a g } } ( Z ) ^ { - 1 }$ ; $\\Phi \\_ { } = $ $- 1 / \\beta \\log Z ^ { h }$ ; $\\mathcal { D } _ { \\mathrm { F E } } = ( \\Phi + \\bar { \\Phi ^ { T } } ) / 2$ (Kivimaki et al., 2014). Kernel version ¨ $K ^ { \\mathrm { F E } } ( t )$ can be obtained with equation 1. ",
+        "bbox": [
+            248,
+            799,
+            823,
+            842
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "• Sigmoid Commute Time kernels. ",
+        "bbox": [
+            217,
+            847,
+            450,
+            861
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "– Sigmoid Commute Time kernel: ",
+        "bbox": [
+            248,
+            864,
+            473,
+            880
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a379bfc47a9404e964889c22c54290a1f8c8e0c667f3ba16e53ab50b5628eedb.jpg",
+        "text": "$$\nK _ { t } ^ { \\mathrm { S C T } } = \\sigma ( - t K ^ { \\mathrm { C T } } / \\mathrm { s t d } ( K ^ { \\mathrm { C T } } ) ) , t > 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            883,
+            671,
+            904
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is an element-wise sigmoid function $\\sigma ( x ) = 1 / ( 1 + e ^ { - x } )$ (Yen et al., 2007). ",
+        "bbox": [
+            266,
+            909,
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+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/f5059267794eaebe9a354e398abc6c919bd2b2f09f82c7f74672762e8fd280ab.jpg",
+        "table_caption": [
+            "Table 1: Short names of considered kernels and other measures. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">Family Adjacency matrix</td><td colspan=\"2\">Short name</td><td></td></tr><tr><td>Plain</td><td>Logarithmic</td><td></td></tr><tr><td>measure</td><td>measure</td><td>Full name Katz kernel</td></tr><tr><td rowspan=\"2\">based kernels</td><td>Katz Comm</td><td>logKatz logComm</td><td>Communicability kernel</td></tr><tr><td>DF</td><td>logDF</td><td>Double Factorial closeness</td></tr><tr><td rowspan=\"4\">Laplacian based kernels</td><td>For</td><td>logFor</td><td>Forest kernel</td></tr><tr><td>Heat</td><td>logHeat</td><td>Heat kernel</td></tr><tr><td>NHeat</td><td>logNHeat</td><td>Normalized Heat kernel</td></tr><tr><td>Abs</td><td>logAbs</td><td>Absorption kernel</td></tr><tr><td rowspan=\"4\">Markovmatrix based kernels and measures</td><td>PPR MPPR</td><td>logPPR</td><td>Personalized PageRank closeness</td></tr><tr><td>HPR</td><td>logMPPR</td><td>Modified Personalized PageRank</td></tr><tr><td></td><td>logHPR</td><td>PageRank heat closeness</td></tr><tr><td>RSP FE</td><td>=</td><td>Randomized Shortest Path kernel</td></tr><tr><td rowspan=\"2\">Sigmoid Commute Time</td><td>SCT</td><td>1</td><td>Free Energy kernel Sigmoid Commute Time kernel</td></tr><tr><td>SCCT</td><td>=</td><td>Sigmoid Corrected Commute Time kernel</td></tr><tr><td rowspan=\"2\"></td><td>SP-CT</td><td></td><td></td></tr><tr><td></td><td>1</td><td>linear combination of SP and CT</td></tr></table>",
+        "bbox": [
+            173,
+            127,
+            839,
+            381
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "– Sigmoid Corrected Commute Time kernel. First of all, we need the Corrected Commute Time kernel (Luxburg et al., 2010): ",
+        "bbox": [
+            246,
+            405,
+            823,
+            434
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e2f6e962349a77cfa0bc8ac1e93a0fe49b7d74de57c7d1c9ad855cd6c1d59748.jpg",
+        "text": "$$\nK ^ { \\mathrm { C C T } } = H D ^ { - { \\frac { 1 } { 2 } } } M ( I - M ) ^ { - 1 } M D ^ { - { \\frac { 1 } { 2 } } } H ; M = D ^ { - { \\frac { 1 } { 2 } } } \\Big ( A - \\frac { \\vec { \\bf d } \\vec { \\bf d } ^ { T } } { \\mathrm { v o l } ( G ) } \\Big ) D ^ { - { \\frac { 1 } { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            438,
+            790,
+            478
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $H$ is the centering matrix $H = I - E / n$ , $\\vec { \\bf d }$ is the vector of diagonal elements of $D$ and $\\mathrm { v o l } ( G )$ is the sum of all elements of $A$ . Then, apply equation 3 replacing $K ^ { \\mathrm { C T } }$ with $K ^ { \\mathrm { C C T } }$ to obtain $K ^ { \\mathrm { S C C T } }$ . ",
+        "bbox": [
+            264,
+            483,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Occasionally, element-wise logarithm is applied to the resulting kernel matrix (Chebotarev, 2013; Ivashkin $\\&$ Chebotarev, 2016). We apply it to almost all investigated measures and consider the resulting measures separately from their plain versions (see Table 1). For some measures, like Forest kernel, this is well-known practice (Chebotarev, 2013), while for others, like Double Factorial closeness, this transformation, to the best of our knowledge, is applied for the first time. The considered measures and their short names are summarized in Table 1. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            641,
+            282,
+            657
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We collected a paired dataset of graphs and the corresponding results of clustering with each measure mentioned in Table 1. In this section, we describe the graph generator, the sampling strategy, the calculated graph features, and the pipeline for the measure score calculation. ",
+        "bbox": [
+            176,
+            672,
+            825,
+            714
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We use Lancichinetti–Fortunato–Radicchi (LFR) graph generator. It generates non-weighted graphs with ground truth non-overlapping communities. The model has mandatory parameters: the number of nodes $n$ ${ \\mathrm { ~  ~ n ~ } } > 0 ,$ , the power law exponent for the degree distribution $\\tau _ { 1 }$ $\\mathit { \\check { \\tau } _ { 1 } } > 1 \\mathit { \\check { \\tau } _ { \\mathrm { , } } }$ , the power law exponent for the community size distribution $\\tau _ { 2 }$ $( \\tau _ { 2 } ~ > ~ 1 )$ , the fraction of intra-community edges incident to each node $\\mu$ $0 \\leq \\mu \\leq 1 )$ ), and either minimum degree (min degree) or average degree (avg degree). There are also extra parameters: maximum degree (max degree), minimum community size (min community), maximum community size (max community). Not the whole LFR parameter space corresponds to common real-world graphs; most of such graphs are described with $\\tau _ { 1 } ~ \\in ~ [ 1 , 4 ]$ and $\\mu \\ : < \\ : 0 . 5$ (e.g., Fotouhi et al., 2019). However, there is also an interesting case of bipartite/multipartite-like graphs with $\\mu > 0 . 5$ . Moreover, many of the datasets studied in Section 5 have $\\tau _ { 1 } > 4$ . Our choice is to consider the entire parameter space to cover all theoretical and practical cases. ",
+        "bbox": [
+            174,
+            722,
+            825,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "For the generation, we consider $1 0 < n < 1 5 0 0$ . It is impossible to generate a dataset with a uniform distribution of all LFR parameters, because $\\tau _ { 1 }$ and $\\tau _ { 2 }$ parameters are located on rays. We transform $\\tau _ { 1 }$ and $\\tau _ { 2 }$ to $\\tilde { \\tau _ { i } } = 1 - ( 1 / \\sqrt { \\tau _ { i } } ) , i = 1 , 2$ to bring their scope to the $[ 0 , 1 ]$ interval. In this case, “realistic” settings with $\\tau _ { 1 } \\in [ 1 , 4 ]$ take up $50 \\%$ of the variable range. Also, as avg degree feature is limited by the $n$ of a particular graph, we decided to replace it with density (avg degree/ $( n { - } 1 ) )$ . It is not dependent on $n$ and belongs to $[ 0 , 1 ]$ . Using all these considerations, we collected our dataset by uniformly sampling parameters for LFR generator from the set $[ n , \\tilde { \\tau } _ { 1 } , \\tilde { \\tau } _ { 2 } , \\mu$ , density] and generating graphs with these parameters. Additionally, we filter out all disconnected graphs. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/94e9d4844b8285e314e071ffb3f2f6dd30ffde4310a9a475e836731ad9512533.jpg",
+        "image_caption": [
+            "Figure 1: Distribution of graph features in the dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            107,
+            784,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In total, we generated 7396 graphs. It is worth noting that the generator fails for some sets of parameters, so the resulting dataset is not uniform (see Fig. 1). In our study, non-uniformity is not a very important issue, because we are interested in local effects, not global leadership. Moreover, true uniformity for LFR parameter space is impossible, due to the unlimited scope of parameters. ",
+        "bbox": [
+            173,
+            470,
+            825,
+            526
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For our research, we choose a minimum set of the features that describe particular properties of graphs and are not interchangeable. ",
+        "bbox": [
+            174,
+            534,
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+            563
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The LFR parameters can be divided in three groups by the graph properties they reflect: ",
+        "bbox": [
+            173,
+            569,
+            485,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "• The size of the graph and the communities: $n , \\tau _ { 1 }$ , min community, max community;   \n• The density and uniformity of the node degrees distribution: $\\tau _ { 2 }$ , min degree, avg degree, max degree. As avg degree depends on $n$ , it is distributed exponentially, so we use log(avg degree) instead;   \n• The cluster separability: $\\mu$ . As $\\mu$ parameter considers only the ratio between the number of inter-cluster edges and the number of nodes but ignores overall density, we use modularity (Newman & Girvan, 2004) as a more appropriate measure for cluster separability. ",
+        "bbox": [
+            215,
+            609,
+            485,
+            856
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/f87f7ddfe7340ba93c18fe511c5012c5342ef0d84239b578efe2f3f391b6661d.jpg",
+        "image_caption": [
+            "Figure 2: Measuring ARI clustering score for a particular graph, measure, and measure parameter "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            500,
+            568,
+            820,
+            804
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Thus, the defined set of features $[ n , \\tau _ { 1 } , \\tau _ { 2 }$ , avg degree, modularity] is enough to consider all graph properties mentioned above. Although modularity is a widely used measure, it suffers from resolution limit problems (Fortunato & Barthelemy, 2007). We acknowledge that this may cause some limitations in our approach, which should be the topic of further research. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/bf31236aac113b5c29a823a5c94adde266d40c14b068089f26234eb856bc8316.jpg",
+        "table_caption": [
+            "Table 2: The overall leaderboard. The win percentage is calculated among all 7396 graphs in the dataset. The ARI column shows the mean ARI across the dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>#</td><td>Measure</td><td>Rank</td><td>Wins, %</td><td>ARI</td><td>#</td><td>Measure</td><td>Rank</td><td>Wins, %</td><td>ARI</td></tr><tr><td>1</td><td>SCCT</td><td>4.7</td><td>56.0</td><td>0.58</td><td>14</td><td>DF</td><td>11.0</td><td>8.1</td><td>0.28</td></tr><tr><td>2</td><td>NHeat</td><td>6.9</td><td>25.2</td><td>0.40</td><td>15</td><td>logAbs</td><td>12.1</td><td>15.9</td><td>0.33</td></tr><tr><td>3</td><td>RSP</td><td>7.7</td><td>22.4</td><td>0.42</td><td>16</td><td>logComm</td><td>12.6</td><td>15.5</td><td>0.23</td></tr><tr><td>4</td><td>SCT</td><td>8.4</td><td>19.5</td><td>0.40</td><td>17</td><td>logFor</td><td>13.0</td><td>12.7</td><td>0.22</td></tr><tr><td>5</td><td>Comm</td><td>8.4</td><td>15.8</td><td>0.36</td><td>18</td><td>HeatPR</td><td>13.4</td><td>12.5</td><td>0.22</td></tr><tr><td>6</td><td>logNHeat</td><td>8.8</td><td>18.0</td><td>0.37</td><td>19</td><td>logHeat</td><td>13.6</td><td>11.9</td><td>0.21</td></tr><tr><td>7</td><td>SP-CT</td><td>9.0</td><td>20.4</td><td>0.41</td><td>20</td><td>Heat</td><td>14.7</td><td>11.2</td><td>0.19</td></tr><tr><td>8</td><td>logHeatPR</td><td>9.2</td><td>18.0</td><td>0.37</td><td>21</td><td>logDF</td><td>14.8</td><td>11.1</td><td>0.18</td></tr><tr><td>9</td><td>FE</td><td>9.5</td><td>19.3</td><td>0.39</td><td>22</td><td>PPR</td><td>16.7</td><td>4.0</td><td>0.13</td></tr><tr><td>10</td><td>Katz</td><td>9.8</td><td>7.7</td><td>0.32</td><td>23</td><td>Abs</td><td>18.6</td><td>3.7</td><td>0.08</td></tr><tr><td>11</td><td>logKatz</td><td>9.9</td><td>18.1</td><td>0.35</td><td>24</td><td>For</td><td>19.3</td><td>2.8</td><td>0.07</td></tr><tr><td>12</td><td>logPPR</td><td>10.0</td><td>17.5</td><td>0.35</td><td>25</td><td>ModifPPR</td><td>20.5</td><td>1.7</td><td>0.05</td></tr><tr><td>13</td><td>logModifPPR</td><td>10.5</td><td>17.3</td><td>0.35</td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For every generated graph, we calculate the top ARI score for every measure (Hubert & Arabie, 1985). We choose ARI as a clustering score which is both popular and unbiased (Gosgens et al., ¨ 2019). As soon as every measure has a parameter, we perform clustering for a range of parameter values (we transform the parameter to become in the [0, 1] interval and then choose 16 values linearly spaced from 0 to 1). For each value, we run $6 + 6 + 6$ trials of $k$ -means (6 trials for each of three initialization methods). ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 shows the pipeline we use to calculate ARI score for a given LFR parameter set, a measure, and a measure parameter. Measure parameters are not the subject of our experiments, so for every measure we just take the result of the measure with the value of the parameter that gives the best ARI score. ",
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+    {
+        "type": "text",
+        "text": "Because of the need to iterate over graphs, measures, parameter values, and initializations, the task is computationally difficult. The total computation time was 20 days on 18 CPU cores and 6 GPUs. ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 GLOBAL LEADERSHIP IN LFR SPACE ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We rank the measures by their ARI score on every graph of the dataset. The rank is defined as the position of the measure in this list, averaged over the dataset (see Table 2). It is important to note that the global leadership does not give a comprehensive advice on which measure is better to use, because for a particular graph, the global leader can perform worse than the others. Here we consider the entire LFR space, not just its zone corresponding to common real-world graphs, so the ranking may differ from those obtained for restricted settings. ",
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+    },
+    {
+        "type": "text",
+        "text": "As SCCT is the winner for both ranking and percentage of wins, we can say for sure that it is the global winner for the LFR space graphs. Other measures still can be leaders in some zones of the feature space. ",
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+    {
+        "type": "text",
+        "text": "4.2 FEATURE IMPORTANCE STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "First of all, we find out which graph features are important for the choice of the best measure and which are not. To do that, we use Linear Discriminant Analysis (Mika et al., 1999) (LDA). This method finds a new basis in the feature space to classify a dataset in the best way. It also shows how many components of basis are required to fit the majority of data. ",
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+    },
+    {
+        "type": "text",
+        "text": "Fig. 3a shows that the first two components take about $90 \\%$ of the explained variance. Fig. 3b shows that these components include only $\\tau _ { 1 }$ , avg degree, and modularity. The fact that $n$ is not used means that the size of the graph as well as the density are not of primary importance for choosing the best measure. So is not $\\tau _ { 2 }$ measuring the diversity of cluster sizes. ",
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+    {
+        "type": "image",
+        "img_path": "images/0bd8fcae444182bb890f34dfe5fa2f534fccbc423006501df41aac01e87c3458.jpg",
+        "image_caption": [
+            "Figure 3: The results of LDA components. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fig. 4 shows the point cloud projected on the space of the two main components of LDA. We see a confirmation that the measures are indeed zoned, but the areas are quite noisy. To detect the zones of measure leadership, we need to know the leadership on average in every area of space, rather than the wins in particular points. To define the local measure leadership in the whole space, we need to introduce a filtering algorithm that for every point of the space returns the measure leadership depending on the closest data points. As the choice of measure is actually dependent only on three features, we can limit our feature space to $[ \\tau _ { 1 }$ , avg degree, modularity]. ",
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+    {
+        "type": "image",
+        "img_path": "images/987397c1772a927349e58b530d6fa6c770b22f058ba7a3de3c97fd0595b8fe5a.jpg",
+        "image_caption": [
+            "Figure 4: The dataset projected on the two main components of LDA; the point color represents the winning measure for each graph "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.3 GAUSSIAN FILTER IN FEATURE SPACE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Using a filter in the feature space, we can suppress the noise and find actual zones of leadership for the measures. We use the Gaussian filter with a scale parameter $\\sigma$ . For every given point of the space, it takes the data points that are closer than $3 \\sigma$ and averages ARIs of the chosen points with a weight $e ^ { - \\mathrm { d i s t } ^ { 2 } / 2 \\sigma ^ { 2 } }$ . This allows to give larger weights to closer points. If there are less than three data points inside the sphere with a $3 \\sigma$ radius, the filter returns nothing, allowing to ignore the points with insufficient data. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Before applying the filter, we prepare the dataset. First, we only take the points with only one winning measure, because multiple winners can confuse the filter. Then we normalize the standard ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "deviation of every feature distribution to one. Finally, we cut off the long tail of distant data points.   \nThe resulting number of graphs is 5201. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To choose $\\sigma$ , we apply the filter with different $\\sigma$ and look at the number of connected components in the feature space. The needed $\\sigma$ should be large enough to suppress the noise, however, it should not suppress small zones. Guided by this heuristic, we choose $\\sigma = 0 . 5$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/fb155bf8a3e80d0274186eb7406b698a99978ac1180ce3713844f2823a2cbf88.jpg",
+        "table_caption": [
+            "Table 3: The leaderboard of measure wins after filtering with $\\sigma = 0 . 5$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"10\">1oomoom</td><td rowspan=\"2\"></td><td colspan=\"10\">1H8o</td></tr><tr><td>Measure Wins</td><td>SCCS 4283</td><td>441</td><td>JHe 268</td><td></td><td>uom 78</td><td>64</td><td>3 56</td><td>E 3</td><td></td><td>A 3</td><td>5 2</td><td></td><td>1</td><td>J8o 1</td></tr></table>",
+        "bbox": [
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+            220,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "After filtering with $\\sigma = 0 . 5$ , the leaderboard of measure wins is changed (see Table 3). Only six measures keep their positions: SCCT, NHeat, logComm, Comm, logDF, and RSP. This means that these measures do have zones of leadership, otherwise they would be filtered out. We can plot the entire feature space colored by the leadership zones of the measures (Fig. 5). As the resulting space is 3D, we show slices of it by each of the three coordinates. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/6753805399d1d24f663771f2e1b233600464691b7c41335d14ebeabe0ec5aa53.jpg",
+        "image_caption": [
+            "Figure 5: The feature space $[ \\tau _ { 1 }$ , avg degree, modularity] divided into the leadership zones of six measures. The yellow points represent the positions of real-world datasets under study in the space. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 REAL-WORLD DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Even though LFR lacks some characteristics of real-world graphs, there is evidence that the optimal parameter of the Louvain clustering for a real graph is close to the parameter for LFR graphs generated from the features of a real one (Prokhorenkova, 2019). So, there is a chance that the learned space might be helpful for choosing measures in the wild. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "For evaluation, we use 29 graphs of standard datasets: Dolphins (Lusseau et al., 2003), Football (Newman & Girvan, 2004), Karate club (Zachary, 1977), Newsgroups (9 subsets, weights are binarized with threshold 0.1) (Yen et al., 2007), Political blogs (Adamic & Glance, 2005), Political books (Newman, 2006), SocioPatterns Primary school day (2 graphs) (Stehle et al., 2011), Cora (11 ´ subsets) (McCallum et al., 2000), Eu-core (Leskovec et al., 2007), EuroSIS (WebAtlas, 2009). The parameters of these graphs are marked in Fig. 5. For each graph, we found the best ARI for every measure (iterating over the measure parameter value). Now we can check the quality of measure choice, based on the found LFR data. The result of LFR recommendation is the measure that is chosen for the set of parameters corresponding to the dataset in hand. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Mean ARI of the LFR recommended strategies for datasets. Top6 stands for the set of measures that have their zones in the LFR parameter space. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/31b6cc5f7c4678b49ab89da5c7af2106c38dc0dc46c5c93a1640ad9c404528b4.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Strategy</td><td>Mean ARI</td></tr><tr><td>AlwaystakeSCCT Based on LFR space, top6 measures</td><td>0.61 0.62</td></tr><tr><td>Based on LFR space,all measures</td><td>0.62</td></tr><tr><td>Upper bound</td><td>0.64</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The best measures on the considered datasets are SCCT (by the mean ARI) and SCT (by the rank). This is pretty similar to the results obtained for LFR. Moreover, Spearman correlation between the ranks of measures for the datasets and for the corresponding LFR recommendations is 0.90. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Let us use “always take SCCT” as our baseline strategy. In Table 4 we compare it with strategies based on the LFR space. We obtain LFR recommendation using $k \\mathrm { n n }$ as a well-proven method for meta-learning. Since each graph is unique, the result of 1nn can be very noisy, thus we use 5nn. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4 shows that the recommendation approach slightly beats the baseline. Reducing the number of measures from 25 to 6 do not drop the quality. However, this quality increase is not enough to draw confident conclusions about the advantages of the method. Using this fact and the fact that the ranks on datasets and recommendations are highly correlated, we conclude that the metalearning procedure is adequate to give a robust recommendation, but not precise enough to beat the baseline confidently. This may be due to the fact that that the nodes of real graphs were not labeled systematically since they were created in the wild. A larger dataset could help separate the signal from the noise and pinpoint where the limits of the method are. At least, the good news is that the conclusions made on the LFR basis do not contradict the results obtained on the datasets. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we have shown that the global leadership of measures does not provide comprehensive knowledge about graph measures. We demonstrated that among 25 measures, SCCT is the best measure for the LFR graphs both by winning rate and ranking. However, there are also smaller confident zones of leadership for NHeat, Comm, logComm, logDF, and RSP. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our results do not contradict those of other experimental works and rather expand them by providing new findings. LogComm was first introduced in Ivashkin & Chebotarev (2016) and won in the competitions on graphs generated with a fixed set of SBM parameters. This study confirms its leadership, but only for a certain type of graphs. Another interesting finding is logDF, which unexpectedly shows good performance for the graphs with low modularity and low average degree. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This study is based on the LFR benchmark data. An attempt to apply the results to real data gives the quality slightly above the baseline. However, there is a strong correlation between the ranking of measures for datasets and the ranking of LFR recommendation, which indicates that the leading measures are the same, while the recommendations are not precise enough. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "It can be noted that our study is insensitive to the non-uniformity of the generated dataset. While manipulations with this dataset may affect the global leaderboard, they cannot change the local leadership studied in this work. ",
+        "bbox": [
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+    {
+        "type": "text",
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parse/train/rkZvSe-RZ/rkZvSe-RZ.md

@@ -0,0 +1,504 @@
+# ENSEMBLE ADVERSARIAL TRAINING: ATTACKS AND DEFENSES
+
+Florian Tramer\`   
+Stanford University   
+tramer@cs.stanford.edu   
+Alexey Kurakin   
+Google Brain   
+kurakin@google.com
+
+Nicolas Papernot∗ Pennsylvania State University ngp5056@cse.psu.edu
+
+Ian Goodfellow   
+Google Brain   
+goodfellow@google.com
+
+Dan Boneh Stanford University dabo@cs.stanford.edu
+
+Patrick McDaniel Pennsylvania State University mcdaniel@cse.psu.edu
+
+# ABSTRACT
+
+Adversarial examples are perturbed inputs designed to fool machine learning models. Adversarial training injects such examples into training data to increase robustness. To scale this technique to large datasets, perturbations are crafted using fast single-step methods that maximize a linear approximation of the model’s loss.
+
+We show that this form of adversarial training converges to a degenerate global minimum, wherein small curvature artifacts near the data points obfuscate a linear approximation of the loss. The model thus learns to generate weak perturbations, rather than defend against strong ones. As a result, we find that adversarial training remains vulnerable to black-box attacks, where we transfer perturbations computed on undefended models, as well as to a powerful novel single-step attack that escapes the non-smooth vicinity of the input data via a small random step.
+
+We further introduce Ensemble Adversarial Training, a technique that augments training data with perturbations transferred from other models. On ImageNet, Ensemble Adversarial Training yields models with strong robustness to black-box attacks. In particular, our most robust model won the first round of the NIPS 2017 competition on Defenses against Adversarial Attacks (Kurakin et al., 2017c).
+
+# 1 INTRODUCTION
+
+Machine learning (ML) models are often vulnerable to adversarial examples, maliciously perturbed inputs designed to mislead a model at test time (Biggio et al., 2013; Szegedy et al., 2013; Goodfellow et al., 2014b; Papernot et al., 2016a). Furthermore, Szegedy et al. (2013) showed that these inputs transfer across models: the same adversarial example is often misclassified by different models, thus enabling simple black-box attacks on deployed models (Papernot et al., 2017; Liu et al., 2017).
+
+Adversarial training (Szegedy et al., 2013) increases robustness by augmenting training data with adversarial examples. Madry et al. (2017) showed that adversarially trained models can be made robust to white-box attacks (i.e., with knowledge of the model parameters) if the perturbations computed during training closely maximize the model’s loss. However, prior attempts at scaling this approach to ImageNet-scale tasks (Deng et al., 2009) have proven unsuccessful (Kurakin et al., 2017b).
+
+It is thus natural to ask whether it is possible, at scale, to achieve robustness against the class of black-box adversaries Towards this goal, Kurakin et al. (2017b) adversarially trained an Inception v3 model (Szegedy et al., 2016b) on ImageNet using a “single-step” attack based on a linearization of the model’s loss (Goodfellow et al., 2014b). Their trained model is robust to single-step perturbations but remains vulnerable to more costly “multi-step” attacks. Yet, Kurakin et al. (2017b) found that these attacks fail to reliably transfer between models, and thus concluded that the robustness of their model should extend to black-box adversaries. Surprisingly, we show that this is not the case.
+
+We demonstrate, formally and empirically, that adversarial training with single-step methods admits a degenerate global minimum, wherein the model’s loss can not be reliably approximated by a linear function. Specifically, we find that the model’s decision surface exhibits sharp curvature near the data points, thus degrading attacks based on a single gradient computation. In addition to the model of Kurakin et al. (2017b), we reveal similar overfitting in an adversarially trained Inception ResNet v2 model (Szegedy et al., 2016a), and a variety of models trained on MNIST (LeCun et al., 1998).
+
+We harness this result in two ways. First, we show that adversarially trained models using single-step methods remain vulnerable to simple attacks. For black-box adversaries, we find that perturbations crafted on an undefended model often transfer to an adversarially trained one. We also introduce a simple yet powerful single-step attack that applies a small random perturbation—to escape the nonsmooth vicinity of the data point—before linearizing the model’s loss. While seemingly weaker than the Fast Gradient Sign Method of Goodfellow et al. (2014b), our attack significantly outperforms it for a same perturbation norm, for models trained with or without adversarial training.
+
+Second, we propose Ensemble Adversarial Training, a training methodology that incorporates perturbed inputs transferred from other pre-trained models. Our approach decouples adversarial example generation from the parameters of the trained model, and increases the diversity of perturbations seen during training. We train Inception v3 and Inception ResNet v2 models on ImageNet that exhibit increased robustness to adversarial examples transferred from other holdout models, using various single-step and multi-step attacks (Goodfellow et al., 2014b; Carlini & Wagner, 2017a; Kurakin et al., 2017a; Madry et al., 2017). We also show that our methods globally reduce the dimensionality of the space of adversarial examples (Tramer et al., 2017). Our Inception ResNet v2 model won the \` first round of the NIPS 2017 competition on Defenses Against Adversarial Attacks (Kurakin et al., 2017c), where it was evaluated on other competitors’ attacks in a black-box setting.
+
+# 2 RELATED WORK
+
+Various defensive techniques against adversarial examples in deep neural networks have been proposed (Gu & Rigazio, 2014; Luo et al., 2015; Papernot et al., 2016c; Nayebi & Ganguli, 2017; Cisse et al., 2017) and many remain vulnerable to adaptive attackers (Carlini & Wagner, 2017a;b; Baluja & Fischer, 2017). Adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b; Kurakin et al., 2017b; Madry et al., 2017) appears to hold the greatest promise for learning robust models.
+
+Madry et al. (2017) show that adversarial training on MNIST yields models that are robust to whitebox attacks, if the adversarial examples used in training closely maximize the model’s loss. Moreover, recent works by Sinha et al. (2018), Raghunathan et al. (2018) and Kolter & Wong (2017) even succeed in providing certifiable robustness for small perturbations on MNIST. As we argue in Appendix C, the MNIST dataset is peculiar in that there exists a simple “closed-form” denoising procedure (namely feature binarization) which leads to similarly robust models without adversarial training. This may explain why robustness to white-box attacks is hard to scale to tasks such as ImageNet (Kurakin et al., 2017b). We believe that the existence of a simple robust baseline for MNIST can be useful for understanding some limitations of adversarial training techniques.
+
+Szegedy et al. (2013) found that adversarial examples transfer between models, thus enabling blackbox attacks on deployed models. Papernot et al. (2017) showed that black-box attacks could succeed with no access to training data, by exploiting the target model’s predictions to extract (Tramer et al., \` 2016) a surrogate model. Some prior works have hinted that adversarially trained models may remain vulnerable to black-box attacks: Goodfellow et al. (2014b) found that an adversarial maxout network on MNIST has slightly higher error on transferred examples than on white-box examples. Papernot et al. (2017) further showed that a model trained on small perturbations can be evaded by transferring perturbations of larger magnitude. Our finding that adversarial training degrades the accuracy of linear approximations of the model’s loss is as an instance of a gradient-masking phenomenon (Papernot et al., 2016b), which affects other defensive techniques (Papernot et al., 2016c; Carlini & Wagner, 2017a; Nayebi & Ganguli, 2017; Brendel & Bethge, 2017; Athalye et al., 2018).
+
+# 3 THE ADVERSARIAL TRAINING FRAMEWORK
+
+We consider a classification task with data $x \in [ 0 , 1 ] ^ { d }$ and labels $y _ { \mathrm { t r u e } } \in \mathbb { Z } _ { k }$ sampled from a distribution $\mathcal { D }$ . We identify a model with an hypothesis $h$ from a space $\mathcal { H }$ . On input $x$ , the model outputs class scores $h ( x ) \in \mathbb { R } ^ { k }$ . The loss function used to train the model, e.g., cross-entropy, is $L ( h ( x ) , y )$ .
+
+# 3.1 THREAT MODEL
+
+For some target model $h \in \mathcal H$ and inputs $( x , y _ { \mathrm { t r u e } } )$ the adversary’s goal is to find an adversarial example $x ^ { \mathrm { a d v } }$ such that $x ^ { \mathrm { a d v } }$ and $x$ are “close” yet the model misclassifies $x ^ { \mathrm { a d v } }$ . We consider the wellstudied class of $\ell _ { \infty }$ bounded adversaries (Goodfellow et al., 2014b; Madry et al., 2017) that, given some budget $\epsilon$ , output examples $x ^ { \mathrm { a d v } }$ where $\| x ^ { \mathrm { a d v } } - x \| _ { \infty } \leq \epsilon .$ . As we comment in Appendix C.1, $\ell _ { \infty }$ robustness is of course not an end-goal for secure ML. We use this standard model to showcase limitations of prior adversarial training methods, and evaluate our proposed improvements.
+
+We distinguish between white-box adversaries that have access to the target model’s parameters (i.e., $h _ { . }$ ), and black-box adversaries with only partial information about the model’s inner workings. Formal definitions for these adversaries are in Appendix A. Although security against white-box attacks is the stronger notion (and the one we ideally want ML models to achieve), black-box security is a reasonable and more tractable goal for deployed ML models.
+
+# 3.2 ADVERSARIAL TRAINING
+
+Following Madry et al. (2017), we consider an adversarial variant of standard Empirical Risk Minimization (ERM), where our aim is to minimize the risk over adversarial examples:
+
+$$
+\begin{array} { r l } { h ^ { * } = \underset { h \in \mathcal { H } } { \arg \operatorname* { m i n } } } & { \underset { ( x , y _ { \mathrm { t u e } } ) \sim \mathcal { D } } { \mathbb { E } } \left[ \underset { \| x ^ { \mathrm { a d v } } - x \| _ { \infty } \leq \epsilon } { \operatorname* { m a x } } L ( h ( x ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) \right] . } \end{array}
+$$
+
+Madry et al. (2017) argue that adversarial training has a natural interpretation in this context, where a given attack (see below) is used to approximate solutions to the inner maximization problem, and the outer minimization problem corresponds to training over these examples. Note that the original formulation of adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b), which we use in our experiments, trains on both the “clean” examples x and adversarial examples xadv.
+
+We consider three algorithms to generate adversarial examples with bounded $\ell _ { \infty }$ norm. The first two are single-step (i.e., they require a single gradient computation); the third is iterative—it computes multiple gradient updates. We enforce $x ^ { \mathrm { a d v } } \in [ 0 , 1 ] ^ { d }$ by clipping all components of $x ^ { \mathrm { a d v } }$ .
+
+Fast Gradient Sign Method (FGSM). This method (Goodfellow et al., 2014b) linearizes the inner maximization problem in (1):
+
+$$
+x _ { \mathrm { F G S M } } ^ { \mathrm { a d v } } : = x + \varepsilon \cdot \mathrm { s i g n } \left( \nabla _ { x } L ( h ( x ) , y _ { \mathrm { t r u e } } ) \right) \ .
+$$
+
+Single-Step Least-Likely Class Method (Step-LL). This variant of FGSM introduced by Kurakin et al. (2017a;b) targets the least-likely class, $y _ { \mathrm { L L } } = \arg \operatorname* { m i n } \{ h ( x ) \}$ :
+
+$$
+x _ { \mathrm { L L } } ^ { \mathrm { a d v } } : = x - \varepsilon \cdot \mathrm { s i g n } \left( \nabla _ { x } L ( h ( x ) , y _ { \mathrm { L L } } ) \right) .
+$$
+
+Although this attack only indirectly tackles the inner maximization in (1), Kurakin et al. (2017b) find it to be the most effective for adversarial training on ImageNet.
+
+Iterative Attack (I-FGSM or Iter-LL). This method iteratively applies the FGSM or Step-LL $k$ times with step-size $\alpha \geq \epsilon / k$ and projects each step onto the $\ell _ { \infty }$ ball of norm $\epsilon$ around $x$ . It uses projected gradient descent to solve the maximization in (1). For fixed $\epsilon$ , iterative attacks induce higher error rates than single-step attacks, but transfer at lower rates (Kurakin et al., 2017a;b).
+
+# 3.3 A DEGENERATE GLOBAL MINIMUM FOR SINGLE-STEP ADVERSARIAL TRAINING
+
+When performing adversarial training with a single-step attack (e.g., the FGSM or Step-LL methods above), we approximate Equation (1) by replacing the solution to the inner maximization problem
+
+in with the output of the single-step attack (e.g., $x _ { \mathrm { F G S M } } ^ { \mathrm { a d v } }$ in (2)). That is, we solve
+
+$$
+h ^ { * } = \underset { h \in \mathcal { H } } { \arg \operatorname* { m i n } } \quad \underset { ( x , y _ { \mathrm { t u c } } ) \sim \mathcal { D } } { \mathbb { E } } \left[ L \big ( h ( x _ { \mathrm { F G S M } } ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } \big ) \right] .
+$$
+
+For model families $\mathcal { H }$ with high expressive power, this alternative optimization problem admits at least two substantially different global minima $h ^ { * }$ :
+
+• For an input $x$ from $\mathcal { D }$ , there is no $x ^ { \mathrm { a d v } }$ close to $x$ (in $\ell _ { \infty }$ norm) that induces a high loss. That is,
+
+$$
+L ( h ^ { * } ( x _ { \mathrm { F G S M } } ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) \approx \operatorname* { m a x } _ { \| x ^ { \mathrm { a d v } } - x \| _ { \infty } \leq \epsilon } L ( h ^ { * } ( x ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) ] \approx 0 .
+$$
+
+In other words, $h ^ { * }$ is robust to all $\ell _ { \infty }$ bounded perturbations.
+
+• The minimizer $h ^ { * }$ is a model for which the approximation method underlying the attack (i.e., linearization in our case) poorly fits the model’s loss function. That is,
+
+$$
+L ( h ^ { * } ( x _ { \mathrm { F G S M } } ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) \ll \operatorname* { m a x } _ { \| x ^ { \mathrm { a d v } } - x \| _ { \infty } \le \epsilon } L ( h ^ { * } ( x ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) ] .
+$$
+
+Thus the attack when applied to $h ^ { * }$ produces samples $x ^ { \mathrm { a d v } }$ that are far from optimal.
+
+Note that this second “degenerate” minimum can be more subtle than a simple case of overfitting to samples produced from single-step attacks. Indeed, we show in Section 4.1 that single-step attacks applied to adversarially trained models create “adversarial” examples that are easy to classify even for undefended models. Thus, adversarial training does not simply learn to resist the particular attack used during training, but actually to make that attack perform worse overall. This phenomenon relates to the notion of Reward Hacking (Amodei et al., 2016) wherein an agent maximizes its formal objective function via unintended behavior that fails to captures the designer’s true intent.
+
+# 3.4 ENSEMBLE ADVERSARIAL TRAINING
+
+The degenerate minimum described in Section 3.3 is attainable because the learned model’s parameters influence the quality of both the minimization and maximization in (1). One solution is to use a stronger adversarial example generation process, at a high performance cost (Madry et al., 2017). Alternatively, Baluja & Fischer (2017) suggest training an adversarial generator model as in the GAN framework (Goodfellow et al., 2014a). The power of this generator is likely to require careful tuning, to avoid similar degenerate minima (where the generator or classifier overpowers the other).
+
+We propose a conceptually simpler approach to decouple the generation of adversarial examples from the model being trained, while simultaneously drawing an explicit connection with robustness to black-box adversaries. Our method, which we call Ensemble Adversarial Training, augments a model’s training data with adversarial examples crafted on other static pre-trained models. Intuitively, as adversarial examples transfer between models, perturbations crafted on an external model are good approximations for the maximization problem in (1). Moreover, the learned model can not influence the “strength” of these adversarial examples. As a result, minimizing the training loss implies increased robustness to black-box attacks from some set of models.
+
+Domain Adaptation with multiple sources. We can draw a connection between Ensemble Adversarial Training and multiple-source Domain Adaptation (Mansour et al., 2009; Zhang et al., 2012). In Domain Adaptation, a model trained on data sampled from one or more source distributions $\boldsymbol { S } _ { 1 } , \ldots , \boldsymbol { S } _ { k }$ is evaluated on samples $x$ from a different target distribution $\tau$ .
+
+Let $A _ { i }$ be an adversarial distribution obtained by sampling $( x , y _ { \mathrm { t r u e } } )$ from $\mathcal { D }$ , computing an adversarial example $x ^ { \mathrm { a d v } }$ for some model such that $\| x ^ { \mathrm { a d v } } - x \| _ { \infty } \leq \epsilon$ , and outputting $( x ^ { \mathrm { a d v } } , y _ { \mathrm { t r u e } } )$ . In Ensemble Adversarial Training, the source distributions are $\mathcal { D }$ (the clean data) and $\mathcal { A } _ { 1 } , \ldots , \mathcal { A } _ { k }$ (the attacks overs the currently trained model and the static pre-trained models). The target distribution takes the form of an unseen black-box adversary $\mathcal { A } ^ { * }$ . Standard generalization bounds for Domain Adaptation (Mansour et al., 2009; Zhang et al., 2012) yield the following result.
+
+Theorem 1 (informal). Let $h ^ { \ast } \in \mathcal { H }$ be a model learned with Ensemble Adversarial Training and static black-box adversaries $\mathcal { A } _ { 1 } , \ldots , \mathcal { A } _ { k }$ . Then, if $h ^ { * }$ is robust against the black-box adversaries $\mathcal { A } _ { 1 } , \ldots \mathcal { A } _ { k }$ used at training time, then $h ^ { * }$ has bounded error on attacks from a future black-box adversary $\mathcal { A } ^ { * }$ , if $\mathcal { A } ^ { * }$ is not “much stronger”, on average, than the static adversaries $\mathcal { A } _ { 1 } , \ldots , \mathcal { A } _ { k }$ .
+
+![](images/364615ab0fa795bce9de81e64d675fc54bea01c98991140721a6039e35625055.jpg)  
+Figure 1: Gradient masking in single-step adversarial training. We plot the loss of model ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ on points $x ^ { * } = x + \epsilon _ { 1 } \cdot g + \epsilon _ { 2 } \cdot g ^ { \perp }$ , where $g$ is the signed gradient and $g ^ { \perp }$ is an orthogonal adversarial direction. Plot (b) is a zoom of (a) near $x$ . The gradient poorly approximates the global loss.
+
+We give a formal statement of this result and of the assumptions on $\mathcal { A } ^ { * }$ in Appendix B. Of course, ideally we would like guarantees against arbitrary future adversaries. For very low-dimensional tasks (e.g., MNIST), stronger guarantees are within reach for specific classes of adversaries (e.g., $\ell _ { \infty }$ bounded perturbations (Madry et al., 2017; Sinha et al., 2018; Raghunathan et al., 2018; Kolter & Wong, 2017)), yet they also fail to extend to other adversaries not considered at training time (see Appendix C.1 for a discussion). For ImageNet-scale tasks, stronger formal guarantees appear out of reach, and we thus resort to an experimental assessment of the robustness of Ensemble Adversarially Trained models to various non-interactive black-box adversaries in Section 4.2.
+
+# 4 EXPERIMENTS
+
+We show the existence of a degenerate minimum, as described in Section 3.3, for the adversarially trained Inception v3 model of Kurakin et al. (2017b). Their model (denoted ${ \mathrm { v } } 3 _ { \mathrm { a d v } , }$ ) was trained on a Step-LL attack with $\epsilon \leq 1 6 / 2 5 6$ . We also adversarially train an Inception ResNet v2 model (Szegedy et al., 2016a) using the same setup. We denote this model by $\mathrm { I R v } 2 _ { \mathrm { a d v } }$ . We refer the reader to (Kurakin et al., 2017b) for details on the adversarial training procedure.
+
+We first measure the approximation-ratio of the Step-LL attack for the inner maximization in (1). As we do not know the true maximum, we lower-bound it using an iterative attack. For 1,000 random test points, we find that for a standard Inception v3 model, step-LL gets within $1 9 \%$ of the optimum loss on average. This attack is thus a good candidate for adversarial training. Yet, for the ${ \bf V } { \boldsymbol 3 } _ { \mathrm { a d v } }$ model, the approximation ratio drops to $7 \%$ , confirming that the learned model is less amenable to linearization. We obtain similar results for Inception ResNet v2 models. The ratio is $1 7 \%$ for a standard model, and $8 \%$ for $\mathrm { I R v } 2 _ { \mathrm { a d v } }$ . Similarly, we look at the cosine similarity between the perturbations given by a single-step and multi-step attack. The more linear the model, the more similar we expect both perturbations to be. The average similarity drops from 0.13 for Inception v3 to 0.02 for ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ . This effect is not due to the decision surface of ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ being “too flat” near the data points: the average gradient norm is larger for ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ (0.17) than for the standard v3 model (0.10).
+
+We visualize this “gradient-masking” effect (Papernot et al., 2016b) by plotting the loss of ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ on examples $x ^ { * } = \bar { x } + \epsilon _ { 1 } \cdot g + \epsilon _ { 2 } \cdot \bar { g } ^ { ^ { \perp } }$ , where $g$ is the signed gradient of model ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ and $g ^ { \bot }$ is a signed vector orthogonal to $g$ . Looking forward to Section 4.1, we actually chose $g ^ { \bot }$ to be the signed gradient of another Inception model, from which adversarial examples transfer to ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ . Figure 1 shows that the loss is highly curved in the vicinity of the data point $x$ , and that the gradient poorly reflects the global loss landscape. Similar plots for additional data points are in Figure 4.
+
+We show similar results for adversarially trained MNIST models in Appendix C.2. On this task, input dropout (Srivastava et al., 2014) mitigates adversarial training’s overfitting problem, in some cases. Presumably, the random input mask diversifies the perturbations seen during training (dropout at intermediate layers does not mitigate the overfitting effect). Mishkin et al. (2017) find that input dropout significantly degrades accuracy on ImageNet, so we did not include it in our experiments.
+
+Table 1: Error rates $( \mathbf { i n \% } )$ of adversarial examples transferred between models. We use StepLL with $\epsilon = { ^ { 1 6 } } / { 2 5 6 }$ for 10,000 random test inputs. Diagonal elements represent a white-box attack. The best attack for each target appears in bold. Similar results for MNIST models appear in Table 7.   
+
+<table><tr><td colspan="6">Source</td></tr><tr><td>Target</td><td>V4</td><td>v3</td><td>v3adv</td><td>IRv2</td><td>IRv2adv</td></tr><tr><td>v4</td><td>60.2</td><td>39.2</td><td>31.1</td><td>36.6</td><td>30.9</td></tr><tr><td>v3</td><td>43.8</td><td>69.6</td><td>36.4</td><td>42.1</td><td>35.1</td></tr><tr><td>v3ady</td><td>36.3</td><td>35.6</td><td>26.6</td><td>35.2</td><td>35.9</td></tr><tr><td>IRv2</td><td>38.0</td><td>38.0</td><td>30.8</td><td>50.7</td><td>31.9</td></tr><tr><td>IRv2adv</td><td>31.0</td><td>30.3</td><td>25.7</td><td>30.6</td><td>21.4</td></tr></table>
+
+<table><tr><td colspan="6">Source</td></tr><tr><td>Target</td><td>v4</td><td>v3</td><td>v3adv</td><td>IRv2</td><td>IRv2adv</td></tr><tr><td>v4</td><td>31.0</td><td>14.9</td><td>10.2</td><td>13.6</td><td>9.9</td></tr><tr><td>v3</td><td>18.7</td><td>42.7</td><td>13.0</td><td>17.8</td><td>12.8</td></tr><tr><td>v3ady</td><td>13.6</td><td>13.5</td><td>9.0</td><td>13.0</td><td>14.5</td></tr><tr><td>IRv2</td><td>14.1</td><td>14.8</td><td>9.9</td><td>24.0</td><td>10.6</td></tr><tr><td>IRv2adv</td><td>10.3</td><td>10.5</td><td>7.7</td><td>10.4</td><td>58</td></tr></table>
+
+# 4.1 ATTACKS AGAINST ADVERSARIALLY TRAINED NETWORKS
+
+Kurakin et al. (2017b) found their adversarially trained model to be robust to various single-step attacks. They conclude that this robustness should translate to attacks transferred from other models. As we have shown, the robustness to single-step attacks is actually misleading, as the model has learned to degrade the information contained in the model’s gradient. As a consequence, we find that the ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ model is substantially more vulnerable to single-step attacks than Kurakin et al. (2017b) predicted, both in a white-box and black-box setting. The same holds for the $\mathrm { I R v } 2 _ { \mathrm { a d v } }$ model.
+
+In addition to the ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ and $\mathrm { I R v } 2 _ { \mathrm { a d v } }$ models, we consider standard Inception v3, Inception v4 and Inception ResNet v2 models. These models are available in the TensorFlow-Slim library (Abadi et al., 2015). We describe similar results for a variety of models trained on MNIST in Appendix C.2.
+
+Black-box attacks. Table 1 shows error rates for single-step attacks transferred between models. We compute perturbations on one model (the source) and transfer them to all others (the targets). When the source and target are the same, the attack is white-box. Adversarial training greatly increases robustness to white-box single-step attacks, but incurs a higher error rate in a black-box setting. Thus, the robustness gain observed when evaluating defended models in isolation is misleading. Given the ubiquity of this pitfall among proposed defenses against adversarial examples (Carlini & Wagner, $2 0 1 7 \mathrm { a }$ ; Brendel & Bethge, 2017; Papernot et al., 2016b), we advise researchers to always consider both white-box and black-box adversaries when evaluating defensive strategies. Notably, a similar discrepancy between white-box and black-box attacks was recently observed in Buckman et al. (2018).
+
+Attacks crafted on adversarial models are found to be weaker even against undefended models (i.e., when using ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ or $\mathrm { I R v } 2 _ { \mathrm { a d v } }$ as source, the attack transfers with lower probability). This confirms our intuition from Section 3.3: adversarial training does not just overfit to perturbations that affect standard models, but actively degrades the linear approximation underlying the single-step attack.
+
+A new randomized single-step attack. The loss function visualization in Figure 1 shows that sharp curvature artifacts localized near the data points can mask the true direction of steepest ascent. We thus suggest to prepend single-step attacks by a small random step, in order to “escape” the non-smooth vicinity of the data point before linearizing the model’s loss. Our new attack, called $\mathrm { R + F G S M }$ (alternatively, $\mathrm { R } { + } ,$ Step-LL), is defined as follows, for parameters $\epsilon$ and $\alpha$ (where $\alpha < \epsilon$ ):
+
+$$
+x ^ { \mathrm { a d v } } = x ^ { \prime } + ( \varepsilon - \alpha ) \cdot \mathsf { s i g n } \left( \nabla _ { x ^ { \prime } } J ( x ^ { \prime } , y _ { \mathrm { t r u e } } ) \right) , \quad \mathrm { w h e r e } \quad x ^ { \prime } = x + \alpha \cdot \mathsf { s i g n } ( \mathcal { N } ( \mathbf { 0 } ^ { d } , \mathbf { I } ^ { d } ) ) ~ .
+$$
+
+Note that the attack requires a single gradient computation. The $\mathrm { R + F G S M }$ is a computationally efficient alternative to iterative methods that have high success rates in a white-box setting. Our attack can be seen as a single-step variant of the general PGD method from (Madry et al., 2017).
+
+Table 2 compares error rates for the Step-LL and $\mathsf { R } { + } \mathsf { S }$ tep-LL methods (with $\epsilon \ : = \ : 1 6 / 2 5 6$ and $\alpha = \epsilon / 2 )$ ). The extra random step yields a stronger attack for all models, even those without adversarial training. This suggests that a model’s loss function is generally less smooth near the data points. We further compared the $\mathsf { R } { + } \iota$ Step-LL attack to a two-step Iter-LL attack, which computes two gradient steps. Surprisingly, we find that for the adversarially trained Inception v3 model, the $\mathbf { R } { + } \mathbf { S }$ tep-LL attack is stronger than the two-step Iter-LL attack. That is, the local gradients learned by the adversarially trained model are worse than random directions for finding adversarial examples!
+
+<table><tr><td>v4</td><td>v3</td><td>v3adv</td><td>IRv2</td><td>IRv2adv</td></tr><tr><td>60.2</td><td>69.6</td><td>26.6</td><td>50.7</td><td>21.4</td></tr><tr><td>70.5</td><td>80.0</td><td>64.8</td><td>56.3</td><td>37.5</td></tr><tr><td>78.5</td><td>86.3</td><td>58.3</td><td>69.9</td><td>41.6</td></tr></table>
+
+Table 2: Error rates $( \mathbf { i n \% } )$ for Step-LL, $\mathbf { R } +$ Step-LL and a two-step Iter-LL on ImageNet. We use $\epsilon = { ^ { 1 6 } } / { 2 5 6 }$ , $\alpha = \epsilon / 2$ on 10,000 random test inputs. $\mathrm { R + F G S M }$ results on MNIST are in Table 7.   
+  
+
+<table><tr><td>v4</td><td>v3</td><td>v3adv</td><td>IRv2</td><td>IRv2adv</td></tr><tr><td>31.0</td><td>42.7</td><td>9.0</td><td>24.0</td><td>5.8</td></tr><tr><td>42.8</td><td>57.1</td><td>37.1</td><td>29.3</td><td>15.0</td></tr><tr><td>56.2</td><td>70.2</td><td>29.6</td><td>45.4</td><td>16.5</td></tr></table>
+
+Table 3: Models used for Ensemble Adversarial Training on ImageNet. The ResNets (He et al., 2016) use either 50 or 101 layers. IncRes stands for Inception ResNet (Szegedy et al., 2016a).   
+
+<table><tr><td>Trained Model</td><td>Pre-trained Models</td><td>Holdout Models</td></tr><tr><td>Inception v3 (v3adv-ens3)</td><td>Inception v3,ResNet v2 (50)</td><td>Inception v4</td></tr><tr><td>Inception v3 (v3adv-ens4)</td><td>Inception v3,ResNet v2 (5O), IncRes v2</td><td>ResNet v1 (50)</td></tr><tr><td>IncRes v2 (IRv2adv-ens)</td><td>Inception v3, IncRes v2</td><td>ResNetv2(101))</td></tr></table>
+
+We find that the addition of this random step hinders transferability (see Table 9). We also tried adversarial training using $\mathrm { R + F G S M }$ on MNIST, using a similar approach as (Madry et al., 2017). We adversarially train a CNN (model A in Table 5) for 100 epochs, and attain $> 9 0 . 0 \%$ accuracy on $\mathrm { R + F G S M }$ samples. However, training on $\mathrm { R + F G S M }$ provides only little robustness to iterative attacks. For the PGD attack of (Madry et al., 2017) with 20 steps, the model attains $1 8 . 0 \%$ accuracy.
+
+# 4.2 ENSEMBLE ADVERSARIAL TRAINING
+
+We now evaluate our Ensemble Adversarial Training strategy described in Section 3.4. We recall our intuition: by augmenting training data with adversarial examples crafted from static pre-trained models, we decouple the generation of adversarial examples from the model being trained, so as to avoid the degenerate minimum described in Section 3.3. Moreover, our hope is that robustness to attacks transferred from some fixed set of models will generalize to other black-box adversaries.
+
+We train Inception v3 and Inception ResNet v2 models (Szegedy et al., 2016a) on ImageNet, using the pre-trained models shown in Table 3. In each training batch, we rotate the source of adversarial examples between the currently trained model and one of the pre-trained models. We select the source model at random in each batch, to diversify examples across epochs. The pre-trained models’ gradients can be precomputed for the full training set. The per-batch cost of Ensemble Adversarial Training is thus lower than that of standard adversarial training: using our method with $n - 1$ pre-trained models, only every $n ^ { \mathrm { t h } }$ batch requires a forward-backward pass to compute adversarial gradients. We use synchronous distributed training on 50 machines, with minibatches of size 16 (we did not pre-compute gradients, and thus lower the batch size to fit all models in memory). Half of the examples in a minibatch are replaced by Step-LL examples. As in Kurakin et al. (2017b), we use RMSProp with a learning rate of 0.045, decayed by a factor of 0.94 every two epochs.
+
+To evaluate how robustness to black-box attacks generalizes across models, we transfer various attacks crafted on three different holdout models (see Table 3), as well as on an ensemble of these models (as in Liu et al. (2017)). We use the Step-LL, $\mathbf { R } +$ Step-LL, FGSM, I-FGSM and the PGD attack from Madry et al. (2017) using the hinge-loss function from Carlini & Wagner (2017a). Our results are in Table 4. For each model, we report the worst-case error rate over all black-box attacks transfered from each of the holdout models (20 attacks in total). Results for MNIST are in Table 8.
+
+Convergence speed. Convergence of Ensemble Adversarial Training is slower than for standard adversarial training, a result of training on “hard” adversarial examples and lowering the batch size. Kurakin et al. (2017b) report that after 187 epochs ( $1 5 0 k$ iterations with minibatches of size 32), the ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ model achieves $7 8 \%$ accuracy. Ensemble Adversarial Training for models ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 3 }$ and ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 4 }$ converges after 280 epochs ( $4 5 0 k$ iterations with minibatches of size 16). The Inception ResNet v2 model is trained for 175 epochs, where a baseline model converges at around 160 epochs.
+
+Table 4: Error rates $( \mathbf { i n \ \% } )$ ) for Ensemble Adversarial Training on ImageNet. Error rates on clean data are computed over the full test set. For 10,000 random test set inputs, and $\epsilon = { ^ { 1 6 } } / { 2 5 6 }$ , we report error rates on white-box Step-LL and the worst-case error over a series of black-box attacks (Step-LL, $R +$ Step-LL, FGSM, I-FGSM, PGD) transferred from the holdout models in Table 3. For both architectures, we mark methods tied for best in bold (based on $9 5 \%$ confidence).   
+
+<table><tr><td rowspan="2">Model</td><td colspan="3">Top 1</td><td colspan="3">Top 5</td></tr><tr><td>Clean</td><td>Step-LL</td><td>Max.Black-Box</td><td>Clean</td><td>Step-LL</td><td>Max.Black-Box</td></tr><tr><td>v3</td><td>22.0</td><td>69.6</td><td>51.2</td><td>6.1</td><td>42.7</td><td>24.5</td></tr><tr><td>v3adv</td><td>22.0</td><td>26.6</td><td>40.8</td><td>6.1</td><td>9.0</td><td>17.4</td></tr><tr><td>v3adv-ens3</td><td>23.6</td><td>30.0</td><td>34.0</td><td>7.6</td><td>10.1</td><td>11.2</td></tr><tr><td>v3adv-ens4</td><td>24.2</td><td>43.3</td><td>33.4</td><td>7.8</td><td>19.4</td><td>10.7</td></tr><tr><td>IRv2</td><td>19.6</td><td>50.7</td><td>44.4</td><td>4.8</td><td>24.0</td><td>17.8</td></tr><tr><td>IRv2adv</td><td>19.8</td><td>21.4</td><td>34.5</td><td>4.9</td><td>5.8</td><td>11.7</td></tr><tr><td>IRv2adv-ens</td><td>20.2</td><td>26.0</td><td>27.0</td><td>5.1</td><td>7.6</td><td>7.9</td></tr></table>
+
+White-box attacks. For both architectures, the models trained with Ensemble Adversarial Training are slightly less accurate on clean data, compared to standard adversarial training. Our models are also more vulnerable to white-box single-step attacks, as they were only partially trained on such perturbations. Note that for ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 4 }$ , the proportion of white-box Step-LL samples seen during training is $^ 1 / 4$ (instead of $^ 1 / 3$ for model ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 3 }$ ). The negative impact on the robustness to white-box attacks is large, for only a minor gain in robustness to transferred samples. Thus it appears that while increasing the diversity of adversarial examples seen during training can provide some marginal improvement, the main benefit of Ensemble Adversarial Training is in decoupling the attacks from the model being trained, which was the goal we stated in Section 3.4.
+
+Ensemble Adversarial Training is not robust to white-box Iter-LL and $\mathrm { R } { + } \mathrm { : }$ Step-LL samples: the error rates are similar to those for the ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ model, and omitted for brevity (see Kurakin et al. (2017b) for Iter-LL attacks and Table 2 for $\mathrm { R } { + } \mathrm { i }$ Step-LL attacks). Kurakin et al. (2017b) conjecture that larger models are needed to attain robustness to such attacks. Yet, against black-box adversaries, these attacks are only a concern insofar as they reliably transfer between models.
+
+Black-box attacks. Ensemble Adversarial Training significantly boosts robustness to all attacks transferred from the holdout models. For the $\mathrm { I R v } 2 _ { \mathrm { a d v - e n s } }$ model, the accuracy loss (compared to IRv2’s accuracy on clean data) is $7 . 4 \%$ (top 1) and $\mathbf { 3 . 1 \% }$ (top 5). We find that the strongest attacks in our test suite (i.e., with highest transfer rates) are the FGSM attacks. Black-box $\mathbf { R } { + } \mathbf { S }$ tep-LL or iterative attacks are less effective, as they do not transfer with high probability (see Kurakin et al. (2017b) and Table 9). Attacking an ensemble of all three holdout models, as in Liu et al. (2017), did not lead to stronger black-box attacks than when attacking the holdout models individually.
+
+Our results have little variance with respect to the attack parameters (e.g., smaller $\epsilon$ ) or to the use of other holdout models for black-box attacks (e.g., we obtain similar results by attacking the v3adv-ens3 and ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 4 }$ models with the IRv2 model). We also find that ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 3 }$ is not vulnerable to perturbations transferred from ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 4 }$ . We obtain similar results on MNIST (see Appendix C.2), thus demonstrating the applicability of our approach to different datasets and model architectures.
+
+The NIPS 2017 competition on adversarial examples. Our Inception ResNet v2 model was included as a baseline defense in the NIPS 2017 competition on Adversarial Examples (Kurakin et al., 2017c). Participants of the attack track submitted non-interactive black-box attacks that produce adversarial examples with bounded $\ell _ { \infty }$ norm. Models submitted to the defense track were evaluated on all attacks over a subset of the ImageNet test set. The score of a defense was defined as the average accuracy of the model over all adversarial examples produced by all attacks.
+
+Our $\mathrm { I R v } 2 _ { \mathrm { a d v - e n s } }$ model finished $\boldsymbol { l } ^ { s t }$ among 70 submissions in the first development round, with a score of $9 5 . 3 \%$ (the second placed defense scored $8 9 . 9 \%$ ). The test data was intentionally chosen as an “easy” subset of ImageNet. Our model achieved $9 7 . 9 \%$ accuracy on the clean test data.
+
+After the first round, we released our model publicly, which enabled other users to launch white-box attacks against it. Nevertheless, a majority of the final submissions built upon our released model. The winning submission (team “liaofz” with a score of $9 5 . 3 \%$ ) made use of a novel adversarial denoising technique. The second placed defense (team “cihangxie” with a score of $9 2 . 4 \%$ ) prepends our $\mathrm { I R v } 2 _ { \mathrm { a d v - e n s } }$ model with random padding and resizing of the input image (Xie et al., 2018).
+
+![](images/f449b38b69b45a3191580e87ac4bd69f0109119052e927f54a66d5fb57558e43.jpg)  
+Figure 2: The dimensionality of the adversarial cone. For 500 correctly classified points $x$ , and for $\epsilon \in \{ 4 , 1 0 , 1 6 \}$ , we plot the probability that we find at least $k$ orthogonal vectors $r _ { i }$ such that $\| r _ { i } \| _ { \infty } = \epsilon$ and $\boldsymbol { x } + \boldsymbol { r } _ { i }$ is misclassified. For $\epsilon \geq 1 0$ , model ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ shows a bimodal phenomenon: most points $x$ either have 0 adversarial directions or more than 90.
+
+It is noteworthy that the defenses that incorporated Ensemble Adversarial Training faired better against the worst-case black-box adversary. Indeed, although very robust on average, the winning defense achieved as low as $1 1 . 8 \%$ accuracy on some attacks. The best defense under this metric (team “rafaelmm” which randomly perturbed images before feeding them to our $\mathrm { I R v } 2 _ { \mathrm { a d v - e n s } }$ model) achieved at least $5 3 . 6 \%$ accuracy against all submitted attacks, including the attacks that explicitly targeted our released model in a white-box setting.
+
+Decreasing gradient masking. Ensemble Adversarial Training decreases the magnitude of the gradient masking effect described previously. For the ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 3 }$ and ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 4 }$ models, we find that the loss incurred on a Step-LL attack gets within respectively $1 3 \%$ and $1 8 \%$ of the optimum loss (we recall that for models v3 and ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ , the approximation ratio was respectively $1 9 \%$ and $7 \%$ ). Similarly, for the $\mathrm { I R v } 2 _ { \mathrm { a d v - e n s } }$ model, the ratio improves from $8 \%$ (for $\mathrm { I R } \mathrm { v } 2 _ { \mathrm { a d v } } ,$ ) to $1 4 \%$ . As expected, not solely training on a white-box single-step attack reduces gradient masking. We also verify that after Ensemble Adversarial Training, a two-step iterative attack outperforms the $\mathrm { R } +$ Step-LL attack from Section 4.1, thus providing further evidence that these models have meaningful gradients.
+
+Finally, we revisit the “Gradient-Aligned Adversarial Subspace” (GAAS) method of Tramer et al. \` (2017). Their method estimates the size of the space of adversarial examples in the vicinity of a point, by finding a set of orthogonal perturbations of norm $\epsilon$ that are all adversarial. We note that adversarial perturbations do not technically form a “subspace” (e.g., the 0 vector is not adversarial). Rather, they may form a “cone”, the dimension of which varies as we increase $\epsilon$ . By linearizing the loss function, estimating the dimensionality of this cone reduces to finding vectors $r _ { i }$ that are strongly aligned with the model’s gradient $g = \nabla _ { x } L ( h ( x ) , y _ { \mathrm { t r u e } } )$ . Tramer et al. (2017) give a method \` that finds $k$ orthogonal vectors $r _ { i }$ that satisfy $\begin{array} { r } { \boldsymbol { g } ^ { \intercal } \boldsymbol { r } _ { i } \geq \dot { \epsilon } \cdot \| \boldsymbol { g } \| _ { 2 } \cdot \frac { 1 } { \sqrt { k } } } \end{array}$ (this bound is tight). We extend this result to the $\ell _ { \infty }$ norm, an open question in Tramer et al. (2017). In Section E, we give a randomized \` combinatorial construction (Colbourn, 2010), that finds $k$ orthogonal vectors $r _ { i }$ satisfying $\| r _ { i } \| _ { \infty } =$ $\epsilon$ and $\begin{array} { r } { \mathbb { E } \left[ \boldsymbol { g } ^ { \top } \boldsymbol { r } _ { i } \right] \ge \epsilon \cdot \| \boldsymbol { g } \| _ { 1 } \cdot \frac { 1 } { \sqrt { k } } } \end{array}$ . We show that this result is tight as well.
+
+For models v3, ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ and ${ \mathbf { v } } 3 _ { \mathrm { a d v - e n s } 3 }$ , we select 500 correctly classified test points. For each $x$ , we search for a maximal number of orthogonal adversarial perturbations $r _ { i }$ with $\| r _ { i } \| _ { \infty } = \epsilon$ . We limit our search to $k \leq 1 0 0$ directions per point. The results are in Figure 2. For $\epsilon \in \{ 4 , 1 0 , 1 6 \}$ , we plot the proportion of points that have at least $k$ orthogonal adversarial perturbations. For a fixed $\epsilon$ , the value of $k$ can be interpreted as the dimension of a “slice” of the cone of adversarial examples near a data point. For the standard Inception v3 model, we find over 50 orthogonal adversarial directions for $3 0 \%$ of the points. The ${ \bf V } { \boldsymbol 3 } _ { \mathrm { a d v } }$ model shows a curious bimodal phenomenon for $\epsilon \geq 1 0$ : for most points $( \approx 8 0 \%$ ), we find no adversarial direction aligned with the gradient, which is consistent with the gradient masking effect. Yet, for most of the remaining points, the adversarial space is very high-dimensional $k \geq 9 0$ ). Ensemble Adversarial Training yields a more robust model, with only a small fraction of points near a large adversarial space.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+Previous work on adversarial training at scale has produced encouraging results, showing strong robustness to (single-step) adversarial examples (Goodfellow et al., 2014b; Kurakin et al., 2017b). Yet, these results are misleading, as the adversarially trained models remain vulnerable to simple black-box and white-box attacks. Our results, generic with respect to the application domain, suggest that adversarial training can be improved by decoupling the generation of adversarial examples from the model being trained. Our experiments with Ensemble Adversarial Training show that the robustness attained to attacks from some models transfers to attacks from other models.
+
+We did not consider black-box adversaries that attack a model via other means than by transferring examples from a local model. For instance, generative techniques (Baluja & Fischer, 2017) might provide an avenue for stronger attacks. Yet, a recent work by Xiao et al. (2018) found Ensemble Adversarial Training to be resilient to such attacks on MNIST and CIFAR10, and often attaining higher robustness than models that were adversarially trained on iterative attacks.
+
+Moreover, interactive adversaries (see Appendix A) could try to exploit queries to the target model’s prediction function in their attack, as demonstrated in Papernot et al. (2017). If queries to the target model yield prediction confidences, an adversary can estimate the target’s gradient at a given point (e.g., using finite-differences as in Chen et al. (2017)) and fool the target with our $\mathrm { R + F G S M }$ attack. Note that if queries only return the predicted label, the attack does not apply. Exploring the impact of these classes of black-box attacks and evaluating their scalability to complex tasks is an interesting avenue for future work.
+
+# ACKNOWLEDGMENTS
+
+We thank Ben Poole and Jacob Steinhardt for feedback on early versions of this work. Nicolas Papernot is supported by a Google PhD Fellowship in Security. Research was supported in part by the Army Research Laboratory, under Cooperative Agreement Number W911NF-13-2-0045 (ARL Cyber Security CRA), and the Army Research Office under grant W911NF-13-1-0421. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for government purposes notwithstanding any copyright notation hereon.
+
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+# A THREAT MODEL: FORMAL DEFINITIONS
+
+We provide formal definitions for the threat model introduced in Section 3.1. In the following, we explicitly identify the hypothesis space $\mathcal { H }$ that a model belongs to as describing the model’s architecture. We consider a target model $h \in \mathcal H$ trained over inputs $( x , y _ { \mathrm { t r u e } } )$ sampled from a data distribution $\mathcal { D }$ . More precisely, we write
+
+$$
+h \gets \mathsf { t r a i n } ( \mathcal { H } , X _ { \mathrm { t r a i n } } , Y _ { \mathrm { t r a i n } } , r ) \ ,
+$$
+
+where train is a randomized training procedure that takes in a description of the model architecture $\mathcal { H }$ , a training set $X _ { \mathrm { t r a i n } } , Y _ { \mathrm { t r a i n } }$ sampled from $\mathcal { D }$ , and randomness $r$ .
+
+iven a set of test inputs produces adversarial $X , Y = \{ ( x _ { 1 } , y _ { 1 } ) , \dots , ( x _ { m } , y _ { m } ) \}$ om, su $\mathcal { D }$ and a that $\epsilon > 0$ versaryfor all $\mathcal { A }$ $X ^ { \mathrm { a d v } } = \{ x _ { 1 } ^ { \mathrm { a d v } } , \ldots , x _ { m } ^ { \mathrm { a d v } } \}$ $\| x _ { i } - x _ { i } ^ { \mathrm { a d v } } \| _ { \infty } ~ \leq ~ \epsilon$ $i \in [ 1 , m ]$ . We evaluate success of the attack as the error rate of the target model over $X ^ { \mathrm { a d v } }$ :
+
+$$
+\frac { 1 } { m } \sum _ { i = 1 } ^ { m } \mathbb { 1 } ( \arg \operatorname* { m a x } h ( x _ { i } ^ { \mathrm { a d v } } ) \neq y _ { i } ) ~ .
+$$
+
+We assume $\mathcal { A }$ can sample inputs according to the data distribution $\mathcal { D }$ . We define three adversaries.
+
+Definition 2 (White-Box Adversary). For a target model $h \in \mathcal H$ , a white-box adversary is given access to all elements of the training procedure, that is train (the training algorithm), $\mathcal { H }$ (the model architecture), the training data $X _ { t r a i n } , Y _ { t r a i n } ,$ , the randomness $r$ and the parameters $h$ . The adversary can use any attack (e.g., those in Section 3.2) to find adversarial inputs.
+
+White-box access to the internal model weights corresponds to a very strong adversarial model. We thus also consider the following relaxed and arguably more realistic notion of a black-box adversary. Definition 3 (Non-Interactive Black-Box Adversary). For a target model $h \in \mathcal H$ , a non-interactive black-box adversary only gets access to train (the target model’s training procedure) and $\mathcal { H }$ (the model architecture). The adversary can sample from the data distribution $\mathcal { D }$ , and uses a local algorithm to craft adversarial examples Xadv.
+
+Attacks based on transferability (Szegedy et al., 2013) fall in this category, wherein the adversary selects a procedure train0 and model architecture $\mathcal { H } ^ { \prime }$ , trains a local model $\mathit { \Pi } _ { h ^ { ' } }$ over $\mathcal { D }$ , and computes adversarial examples on its local model $h ^ { \prime }$ using white-box attack strategies.
+
+Most importantly, a black-box adversary does not learn the randomness $r$ used to train the target, nor the target’s parameters $h$ . The black-box adversaries in our paper are actually slightly stronger than the ones defined above, in that they use the same training data $X _ { \mathrm { t r a i n } } , Y _ { \mathrm { t r a i n } }$ as the target model.
+
+We provide $\mathcal { A }$ with the target’s training procedure train to capture knowledge of defensive strategies applied at training time, e.g., adversarial training (Szegedy et al., 2013; Goodfellow et al., 2014b) or ensemble adversarial training (see Section 4.2). For ensemble adversarial training, $\mathcal { A }$ also knows the architectures of all pre-trained models. In this work, we always mount black-box attacks that train a local model with a different architecture than the target model. We actually find that black-box attacks on adversarially trained models are stronger in this case (see Table 1).
+
+The main focus of our paper is on non-interactive black-box adversaries as defined above. For completeness, we also formalize a stronger notion of interactive black-box adversaries that additionally issue prediction queries to the target model (Papernot et al., 2017). We note that in cases where ML models are deployed as part of a larger system (e.g., a self driving car), an adversary may not have direct access to the model’s query interface.
+
+Definition 4 (Interactive Black-Box Adversary). For a target model $h \in \mathcal H$ , an interactive blackbox adversary only gets access to train (the target model’s training procedure) and $\mathcal { H }$ (the model architecture). The adversary issues (adaptive) oracle queries to the target model. That is, for arbitrary inputs $x \in [ 0 , 1 ] ^ { d }$ , the adversary obtains $y = \arg \operatorname* { m a x } h ( x )$ and uses a local algorithm to craft adversarial examples (given knowledge of $\mathcal { H }$ , train, and tuples $( x , y )$ ).
+
+Papernot et al. (2017) show that such attacks are possible even if the adversary only gets access to a small number of samples from $\mathcal { D }$ . Note that if the target model’s prediction interface additionally returns class scores $h ( x )$ , interactive black-box adversaries could use queries to the target model to estimate the model’s gradient (e.g., using finite differences) (Chen et al., 2017), and then apply the attacks in Section 3.2. We further discuss interactive black-box attack strategies in Section 5.
+
+# B GENERALIZATION BOUND FOR ENSEMBLE ADVERSARIAL TRAINING
+
+We provide a formal statement of Theorem 1 in Section 3.4, regarding the generalization guarantees of Ensemble Adversarial Training. For simplicity, we assume that the model is trained solely on adversarial examples computed on the pre-trained models (i.e., we ignore the clean training data and the adversarial examples computed on the model being trained). Our results are easily extended to also consider these data points.
+
+Let $\mathcal { D }$ be the data distribution and $\mathcal { A } _ { 1 } , \ldots , \mathcal { A } _ { k } , \mathcal { A } ^ { * }$ be adversarial distributions where a sample $( x , y )$ is obtained by sampling $( x , y _ { \mathrm { t r u e } } )$ from $\mathcal { D }$ , computing an $x ^ { \mathrm { a d v } }$ such that $\| x ^ { \mathrm { a d v } } - x \| _ { \infty } \leq \epsilon$ and returning $( x ^ { \mathrm { a d v } } , y _ { \mathrm { t r u e } } )$ . We assume the model is trained on $N$ data points $Z _ { \mathrm { t r a i n } }$ , where $\textstyle { \frac { N } { k } }$ data points are sampled from each distribution $A _ { i }$ , for $1 \leq i \leq k$ . We denote $\mathcal { A } _ { \mathrm { t r a i n } } = \{ \mathcal { A } _ { 1 } , . . . , \overset { \because } { \mathcal { A } } _ { k } \}$ . At test time, the model is evaluated on adversarial examples from $\mathcal { A } ^ { * }$ .
+
+For a model $h \in \mathcal H$ we define the empirical risk
+
+$$
+\hat { R } ( h , { \mathcal A } _ { \mathrm { t r a i n } } ) : = \frac { 1 } { N } \sum _ { ( x ^ { \mathrm { a d v } } , y _ { \mathrm { t r u e } } ) \in Z _ { \mathrm { t r a i n } } } L ( h ( x ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) \ ,
+$$
+
+and the risk over the target distribution (or future adversary)
+
+$$
+R ( h , \mathcal { A } ^ { \ast } ) : = \underset { ( x ^ { \mathrm { a d v } } , y _ { \mathrm { t r u e } } ) \sim \mathcal { A } ^ { \ast } } { \mathbb { E } } [ L ( h ( x ^ { \mathrm { a d v } } ) , y _ { \mathrm { t r u e } } ) ] .
+$$
+
+We further define the average discrepancy distance (Mansour et al., 2009) between distributions $A _ { i }$ and $\mathcal { A } ^ { * }$ with respect to a hypothesis space $\mathcal { H }$ as
+
+$$
+\operatorname { d i s c } _ { { \mathcal { H } } } ( { \mathcal { A } } _ { \mathrm { t r a i n } } , { \mathcal { A } } ^ { * } ) : = \frac { 1 } { k } \sum _ { i = 1 } ^ { k } \operatorname* { s u p } _ { h _ { 1 } , h _ { 2 } \in { \mathcal { H } } } \left| \frac { \mathbb { E } } { A _ { i } } \big [ \mathbb { 1 } _ { \{ h _ { 1 } ( x ^ { \mathrm { a b } } ) = h _ { 2 } ( x ^ { \mathrm { a b } } ) \} } \big ] - \underline { { \mathbb { E } } } _ { * } \big [ \mathbb { 1 } _ { \{ h _ { 1 } ( x ^ { \mathrm { a b } } ) = h _ { 2 } ( x ^ { \mathrm { a b } } ) \} } \big ] \right| .
+$$
+
+This quantity characterizes how “different” the future adversary is from the train-time adversaries. Intuitively, the distance disc $( \mathcal { A } _ { \mathrm { t r a i n } } , \mathcal { A } ^ { * } )$ is small if the difference in robustness between two models to the target attack $\mathcal { A } ^ { * }$ is somewhat similar to the difference in robustness between these two models to the attacks used for training (e.g., if the static black-box attacks $\mathbf { \mathcal { A } } _ { i }$ induce much higher error on some model $h _ { 1 }$ than on another model $h _ { 2 }$ , then the same should hold for the target attack $\mathcal { A } ^ { * }$ ). In other words, the ranking of the robustness of models $h \in \mathcal H$ should be similar for the attacks in $\mathcal { A } _ { \mathrm { t r a i n } }$ as for $\mathcal { A } ^ { * }$ .
+
+Finally, let $R _ { N } ( \mathscr { H } )$ be the average Rademacher complexity of the distributions $\mathcal { A } _ { 1 } , \ldots , \mathcal { A } _ { k }$ (Zhang et al., 2012). Note that $R _ { N } ( \mathscr { H } ) \dot { \mathbf { \Omega } } \to 0$ as $N  \infty$ . The following theorem is a corollary of Zhang et al. (2012, Theorem 5.2):
+
+Theorem 5. Assume that $\mathcal { H }$ is a function class consisting of bounded functions. Then, with probability at least $1 - \epsilon ,$ ,
+
+$$
+\operatorname* { s u p } _ { h \in \mathcal { H } } \vert \hat { R } ( h , { \mathcal A } _ { t r a i n } ) - R ( h , { \mathcal A } ^ { * } ) \vert \leq d i s c _ { \mathcal { H } } ( { \mathcal A } _ { t r a i n } , { \mathcal A } ^ { * } ) + 2 R _ { N } ( \mathcal { H } ) + O \left( \sqrt { \frac { \ln ( 1 / \epsilon ) } { N } } \right) .
+$$
+
+Compared to the standard generalization bound for supervised learning, the generalization bound for Domain Adaptation incorporates the extra term $\mathrm { d i s c } _ { \mathcal { H } } ( \mathcal { A } _ { \mathrm { t r a i n } } , \mathcal { A } ^ { * } )$ to capture the divergence between the target and source distributions. In our context, this means that the model $h ^ { * }$ learned by Ensemble Adversarial Training has guaranteed generalization bounds with respect to future adversaries that are not “too different” from the ones used during training. Note that $\mathcal { A } ^ { * }$ need not restrict itself to perturbation with bounded $\ell _ { \infty }$ norm for this result to hold.
+
+# C EXPERIMENTS ON MNIST
+
+We re-iterate our ImageNet experiments on MNIST. For this simpler task, Madry et al. (2017) show that training on iterative attacks conveys robustness to white-box attacks with bounded $\ell _ { \infty }$ norm. Our goal is not to attain similarly strong white-box robustness on MNIST, but to show that our observations on limitations of single-step adversarial training, extend to other datasets than ImageNet.
+
+![](images/2b6af3b5b8bdd08fb1a44e32ac65797040e19643a1e40965efbd54da313b7abf.jpg)  
+Figure 3: Adversarial Examples on MNIST. (top) clean examples. (middle) inputs are rotated by $2 0 ^ { \circ }$ and 5 random pixels are flipped. (bottom) The I-FGSM with $\epsilon = 0 . 3$ is applied.
+
+C.1 A NOTE ON $\ell _ { \infty }$ ROBUSTNESS ON MNIST
+
+The MNIST dataset is a simple baseline for assessing the potential of a defense, but the obtained results do not always generalize to harder tasks. We suggest that this is because achieving robustness to $\ell _ { \infty }$ perturbations admits a simple “closed-form” solution, given the near-binary nature of the data. Indeed, for an average MNIST image, over $8 0 \%$ of the pixels are in $\{ 0 , 1 \}$ and only $6 \%$ are in the range [0.2, 0.8]. Thus, for a perturbation with $\epsilon \leq 0 . 3$ , binarized versions of $x$ and $x ^ { \mathrm { a d v } }$ can differ in at most $6 \%$ of the input dimensions. By binarizing the inputs of a standard CNN trained without adversarial training, we obtain a model that enjoys robustness similar to the model trained by Madry et al. (2017). Concretely, for a white-box I-FGSM attack, we get at most $1 1 . 4 \%$ error.
+
+The existence of such a simple robust representation begs the question of why learning a robust model with adversarial training takes so much effort. Finding techniques to improve the performance of adversarial training, even on simple tasks, could provide useful insights for more complex tasks such as ImageNet, where we do not know of a similarly simple “denoising” procedure.
+
+These positive results on MNIST for the $\ell _ { \infty }$ norm also leave open the question of defining a general norm for adversarial examples. Let us motivate the need for such a definition: we find that if we first rotate an MNIST digit by $2 0 ^ { \circ }$ , and then use the I-FGSM, our rounding model and the model from Madry et al. (2017) achieve only $6 5 \%$ accuracy (on “clean” rotated inputs, the error is $< 5 \%$ ). If we further randomly “flip” 5 pixels per image, the accuracy of both models drops to under $5 0 \%$ . Thus, we successfully evade the model by slightly extending the threat model (see Figure 3).
+
+Of course, we could augment the training set with such perturbations (see Engstrom et al. (2017)). An open question is whether we can enumerate all types of “adversarial” perturbations. In this work, we focus on the $\ell _ { \infty }$ norm to illustrate our findings on the limitations of single-step adversarial training on ImageNet and MNIST, and to showcase the benefits of our Ensemble Adversarial Training variant. Our approach can easily be extended to consider multiple perturbation metrics. We leave such an evaluation to future work.
+
+# C.2 RESULTS
+
+We repeat experiments from Section 4 on MNIST. We use the architectures in Table 5. We train a standard model for 6 epochs, and an adversarial model with the FGSM $\epsilon = 0 . 3$ ) for 12 epochs.
+
+During adversarial training, we avoid the label leaking effect described by Kurakin et al. (2017b) by using the model’s predicted class arg max $h ( x )$ instead of the true label $y _ { \mathrm { t r u e } }$ in the FGSM,
+
+We first analyze the “degenerate” minimum of adversarial training, described in Section 3.3. For each trained model, we compute the approximation-ratio of the FGSM for the inner maximization problem in equation (1). That is, we compare the loss produced by the FGSM with the loss of a strong iterative attack. The results appear in Table 6. As we can see, for all model architectures, adversarial training degraded the quality of a linear approximation to the model’s loss.
+
+Table 5: Neural network architectures used in this work for the MNIST dataset. Conv: convolutional layer, FC: fully connected layer.   
+
+<table><tr><td>A</td><td>B</td><td>C</td><td>D</td></tr><tr><td>Conv(64,5,5) +Relu</td><td>Dropout(0.2)</td><td>Conv(128,3,3) + Tanh</td><td>[FC(300) + Relu] ×4</td></tr><tr><td>Conv(64,5,5) + Relu</td><td>Conv(64,8,8) + Relu</td><td>MaxPool(2,2)</td><td>Dropout(0.5）</td></tr><tr><td>Dropout(0.25)</td><td>Conv(128,6,6) +Relu</td><td>Conv(64,3,3) + Tanh</td><td>FC +Softmax</td></tr><tr><td>FC(128) + Relu</td><td>Conv(128,5,5) +Relu</td><td>MaxPool(2,2)</td><td></td></tr><tr><td>Dropout(0.5)</td><td>Dropout(0.5)</td><td>FC(128) + Relu</td><td></td></tr><tr><td>FC + Softmax</td><td>FC + Softmax</td><td>FC + Softmax</td><td></td></tr></table>
+
+Table 6: Approximation ratio between optimal loss and loss induced by single-step attack on MNIST. Architecture $\mathbf { B } ^ { \prime }$ is the same as B without the input dropout layer.   
+
+<table><tr><td>A</td><td>Aadv</td><td>B</td><td>Badv</td><td>B*</td><td>Bdv</td><td>C</td><td>Cadv</td><td>D</td><td>Dady</td></tr><tr><td>17%</td><td>0%</td><td>25%</td><td>8%</td><td>23%</td><td>1%</td><td>25%</td><td>0%</td><td>49%</td><td>16%</td></tr></table>
+
+We find that input dropout (Srivastava et al., 2014) (i.e., randomly dropping a fraction of input features during training) as used in architecture B limits this unwarranted effect of adversarial training.2 If we omit the input dropout (we call this architecture $\boldsymbol { \mathrm { B } } ^ { * }$ ) the single-step attack degrades significantly. We discuss this effect in more detail below. For the fully connected architecture D, we find that the learned model is very close to linear and thus also less prone to the degenerate solution to the min-max problem, as we postulated in Section 3.3.
+
+Attacks. Table 7 compares error rates of undefended and adversarially trained models on whitebox and black-box attacks, as in Section 4.1. Again, model B presents an anomaly. For all other models, we corroborate our findings on ImageNet for adversarial training: (1) black-box attacks trump white-box single-step attacks; (2) white-box single-step attacks are significantly stronger if prepended by a random step. For model $\mathrm { \Delta B _ { a d v } }$ , the opposite holds true. We believe this is because input dropout increases diversity of attack samples similarly to Ensemble Adversarial Training.
+
+Table 7: White-box and black-box attacks against standard and adversarially trained models. For each model, the strongest single-step white-box and black box attacks are marked in bold.   
+
+<table><tr><td rowspan="2"></td><td colspan="2">white-box</td><td colspan="5">black-box</td></tr><tr><td>FGSM</td><td>R+FGSM</td><td>FGSMA</td><td>FGSMB</td><td>FGSMB*</td><td>FGSMc</td><td>FGSMD</td></tr><tr><td>A</td><td>64.7</td><td>69.7</td><td>1</td><td>61.5</td><td>53.2</td><td>46.8</td><td>41.5</td></tr><tr><td>Aadv</td><td>2.2</td><td>14.8</td><td>6.6</td><td>10.7</td><td>8.8</td><td>6.5</td><td>8.3</td></tr><tr><td>B</td><td>85.0</td><td>86.0</td><td>45.7</td><td>1</td><td>69.9</td><td>59.9</td><td>85.9</td></tr><tr><td>Badv</td><td>11.6</td><td>11.1</td><td>6.4</td><td>8.9</td><td>8.5</td><td>4.9</td><td>6.1</td></tr><tr><td>B*</td><td>75.7</td><td>74.1</td><td>44.3</td><td>72.8</td><td>-</td><td>46.0</td><td>62.6</td></tr><tr><td>Badv</td><td>4.3</td><td>40.6</td><td>16.1</td><td>14.7</td><td>15.0</td><td>17.9</td><td>9.1</td></tr><tr><td>C</td><td>81.8</td><td>81.8</td><td>40.2</td><td>55.8</td><td>49.5</td><td>1</td><td>59.4</td></tr><tr><td>Cadv</td><td>3.7</td><td>17.1</td><td>9.8</td><td>29.3</td><td>21.5</td><td>11.9</td><td>21.9</td></tr><tr><td>D</td><td>92.4</td><td>95.4</td><td>61.3</td><td>74.1</td><td>68.9</td><td>65.1</td><td>-</td></tr><tr><td>Dadv</td><td>25.5</td><td>47.5</td><td>32.1</td><td>30.5</td><td>29.3</td><td>28.2</td><td>21.8</td></tr></table>
+
+While training with input dropout helps avoid the degradation of the single-step attack, it also significantly delays convergence of the model. Indeed, model $\mathrm { \Delta B _ { a d v } }$ retains relatively high error on white-box FGSM examples. Adversarial training with input dropout can be seen as comparable to training with a randomized single-step attack, as discussed in Section 4.1.
+
+The positive effect of input dropout is architecture and dataset specific: Adding an input dropout layer to models A, C and D confers only marginal benefit, and is outperformed by Ensemble Adversarial Training, discussed below. Moreover, Mishkin et al. (2017) find that input dropout significantly degrades accuracy on ImageNet. We thus did not incorporate it into our models on ImageNet.
+
+Ensemble Adversarial Training. To evaluate Ensemble Adversarial Training 3.4, we train two models per architecture. The first, denoted $[ \mathrm { A } \mathrm { - } \mathrm { D } ] _ { \mathrm { a d v - e n s } }$ , uses a single pre-trained model of the same type (i.e., $\mathbf { A } _ { \mathrm { a d v - e n s } }$ is trained on perturbations from another model A). The second model, denoted [A$\mathrm { D l a d v } { \mathrm { - e n s } } 3$ , uses 3 pre-trained models $( \{ A , C , D \}$ or $\{ B , C , D \} )$ . We train all models for 12 epochs.
+
+We evaluate our models on black-box attacks crafted on models A,B,C,D (for a fair comparison, we do not use the same pre-trained models for evaluation, but retrain them with different random seeds).
+
+Table 8: Ensemble Adversarial Training on MNIST. For black-box robustness, we report the maximum and average error rate over a suite of 12 attacks, comprised of the FGSM, I-FGSM and PGD (Madry et al., 2017) attacks applied to models A,B,C and D. We use $\epsilon = 1 6$ in all cases. For each model architecture, we mark the models tied for best (at a $9 5 \%$ confidence level) in bold.   
+
+<table><tr><td></td><td>Clean</td><td>FGSM</td><td>Max.Black Box</td><td>Avg. Black Box</td></tr><tr><td>Aadv</td><td>0.8</td><td>2.2</td><td>10.8</td><td>7.7</td></tr><tr><td>Aadv-ens</td><td>0.8</td><td>7.0</td><td>6.6</td><td>5.2</td></tr><tr><td>Aadv-ens3</td><td>0.7</td><td>5.4</td><td>6.5</td><td>4.3</td></tr><tr><td>Badv</td><td>0.8</td><td>11.6</td><td>8.9</td><td>5.5</td></tr><tr><td>Badv-ens</td><td>0.7</td><td>10.5</td><td>6.8</td><td>5.3</td></tr><tr><td>Badv-ens3</td><td>0.8</td><td>14.0</td><td>8.8</td><td>5.1</td></tr><tr><td>Cadv</td><td>1.0</td><td>3.7</td><td>29.3</td><td>18.7</td></tr><tr><td>Cadv-ens</td><td>1.3</td><td>1.9</td><td>17.2</td><td>10.7</td></tr><tr><td>Cadv-ens3</td><td>1.4</td><td>3.6</td><td>14.5</td><td>8.4</td></tr><tr><td>Dadv</td><td>2.6</td><td>25.5</td><td>32.5</td><td>23.5</td></tr><tr><td>Dadv-ens</td><td>2.6</td><td>21.5</td><td>38.6</td><td>28.0</td></tr><tr><td>Dadv-ens3</td><td>2.6</td><td>29.4</td><td>29.8</td><td>15.6</td></tr></table>
+
+The attacks we consider are the FGSM, I-FGSM and the PGD attack from Madry et al. (2017) with the loss function from Carlini & Wagner (2017a)), all with $\epsilon = 0 . 3$ . The results appear in Table 8. For each model, we report the worst-case and average-case error rate over all black-box attacks.
+
+Ensemble Adversarial Training significantly increases robustness to black-box attacks, except for architecture B, which we previously found to not suffer from the same overfitting phenomenon that affects the other adversarially trained networks. Nevertheless, model $\mathrm { B _ { a d v - e n s } }$ achieves slightly better robustness to white-box and black-box attacks than $\mathrm { \Delta B _ { a d v } }$ . In the majority of cases, we find that using a single pre-trained model produces good results, but that the extra diversity of including three pre-trained models can sometimes increase robustness even further. Our experiments confirm our conjecture that robustness to black-box attacks generalizes across models. Indeed, we find that when training with three external models, we attain very good robustness against attacks initiated from models with the same architecture (as evidenced by the average error on our attack suite), but also increased robustness to attacks initiated from the fourth holdout model
+
+# D TRANSFERABILITY OF RANDOMIZED SINGLE-STEP PERTURBATIONS.
+
+In Section 4.1, we introduced the $\mathbb { R } { + } \mathbb { S }$ tep-LL attack, an extension of the Step-LL method that prepends the attack with a small random perturbation. In Table 9, we evaluate the transferability of $\mathrm { R } +$ Step-LL adversarial examples on ImageNet. We find that the randomized variant produces perturbations that transfer at a much lower rate (see Table 1 for the deterministic variant).
+
+Table 9: Error rates $( \mathbf { i n \ } \% )$ of randomized single-step attacks transferred between models on ImageNet. We use $\mathrm { R } +$ Step-LL with $\epsilon = 1 6 / 2 5 6 , \alpha = \epsilon / 2$ for 10,000 random test set samples. The white-box attack always outperforms black-box attacks.   
+
+<table><tr><td colspan="6">Source</td></tr><tr><td>Target</td><td>v4</td><td>v3</td><td>v3adv</td><td>IRv2</td><td>IRv2ady</td></tr><tr><td>v4</td><td>70.5</td><td>37.2</td><td>23.2</td><td>34.0</td><td>24.6</td></tr><tr><td>v3</td><td>42.6</td><td>80.0</td><td>26.7</td><td>38.5</td><td>27.6</td></tr><tr><td>v3ady</td><td>31.4</td><td>30.7</td><td>64.8</td><td>30.4</td><td>34.0</td></tr><tr><td>IRv2</td><td>36.2</td><td>35.7</td><td>23.0</td><td>56.3</td><td>24.6</td></tr><tr><td>IRv2adv</td><td>26.8</td><td>26.3</td><td>25.2</td><td>26.9</td><td>37.5</td></tr></table>
+
+<table><tr><td colspan="6">Source</td></tr><tr><td>Target</td><td>v4</td><td>v3</td><td>v3adv</td><td>IRv2</td><td>IRv2adv</td></tr><tr><td>v4</td><td>42.8</td><td>14.3</td><td>6.3</td><td>11.9</td><td>6.9</td></tr><tr><td>v3</td><td>18.0</td><td>57.1</td><td>8.0</td><td>15.6</td><td>8.6</td></tr><tr><td>v3ady</td><td>10.7</td><td>10.4</td><td>37.1</td><td>10.1</td><td>12.9</td></tr><tr><td>IRv2</td><td>12.8</td><td>13.6</td><td>6.1</td><td>29.3</td><td>7.0</td></tr><tr><td>IRv2adv</td><td>8.0</td><td>8.0</td><td>7.7</td><td>8.3</td><td>15.0</td></tr></table>
+
+# E GRADIENT ALIGNED ADVERSARIAL SUBSPACES FOR THE $\ell _ { \infty }$ NORM
+
+Tramer et al. (2017) consider the following task for a given model \` $h$ : for a (correctly classified) point $x$ , find $k$ orthogonal vectors $\{ r _ { 1 } , \ldots , r _ { k } \}$ such that $\lVert \boldsymbol { r } _ { i } \rVert _ { 2 } \leq \epsilon$ and all the $\boldsymbol { x } + \boldsymbol { r } _ { i }$ are adversarial (i.e., arg max $h ( x + r _ { i } ) \neq y _ { \mathrm { t r u e } } )$ . By linearizing the model’s loss function, this reduces to finding $k$ orthogonal vectors $r _ { i }$ that are maximally aligned with the model’s gradient $g = \nabla _ { x } L ( h ( x ) , \hat { y _ { \mathrm { t r u e } } } )$ . Tramer et al. (2017) left a construction for the\` $\ell _ { \infty }$ norm as an open problem.
+
+We provide an optimal construction for the $\ell _ { \infty }$ norm, based on Regular Hadamard Matrices (Colbourn, 2010). Given the $\ell _ { \infty }$ constraint, we find orthogonal vectors $r _ { i }$ that are maximally aligned with the signed gradient, $\mathtt { s i g n } ( g )$ . We first prove an analog of (Tramer et al., 2017, Lemma 1). \`
+
+Lemma 6. Let $v \in \{ - 1 , 1 \} ^ { d }$ and $\alpha \in ( 0 , 1 )$ . Suppose there are $k$ orthogonal vectors $r _ { 1 } , \ldots r _ { n } \in$ $\{ - 1 , 1 \} ^ { d }$ satisfying $v ^ { \top } r _ { i } \geq \alpha \cdot d .$ . Then $\alpha \leq { k ^ { - \frac { 1 } { 2 } } }$ .
+
+Proof. Let $\begin{array} { r } { \hat { r _ { i } } = \frac { r _ { i } } { \Vert r _ { i } \Vert _ { 2 } } = \frac { r _ { i } } { \sqrt { d } } } \end{array}$ . Then, we have
+
+$$
+d = \| \boldsymbol { v } \| _ { 2 } ^ { 2 } \geq \sum _ { i = 1 } ^ { k } | \boldsymbol { v } ^ { \top } \hat { \boldsymbol { r _ { i } } } | ^ { 2 } = d ^ { - 1 } \sum _ { i = 1 } ^ { k } | \boldsymbol { v } ^ { \top } \boldsymbol { r _ { i } } | ^ { 2 } \geq d ^ { - 1 } \cdot k \cdot \left( { \boldsymbol { \alpha } } \cdot d \right) ^ { 2 } = k \cdot \alpha ^ { 2 } \cdot d \ ,
+$$
+
+from which we obtain $\alpha \leq k ^ { - { \frac { 1 } { 2 } } }$ .
+
+This result bounds the number of orthogonal perturbations we can expect to find, for a given alignment with the signed gradient. As a warm-up consider the following trivial construction of $k$ orthogonal vectors in $\{ - 1 , 1 \} ^ { d }$ that are “somewhat” aligned with $\mathtt { s i g n } ( g )$ . We split $\mathtt { s i g n } ( g )$ into $k$ “chunks” of size $\frac { d } { k }$ and define $r _ { i }$ to be the vector that is equal to $\mathtt { s i g n } ( g )$ in the $i ^ { \mathrm { t h } }$ chunk and zero otherwise. We obtain $\begin{array} { r } { \mathrm { s i g n } ( g ) ^ { \top } r _ { i } = \frac { d } { k } } \end{array}$ , a factor $\sqrt { k }$ worse than the the bound in Lemma 6.
+
+We now provide a construction that meets this upper bound. We make use of Regular Hadamard Matrices of order $k$ (Colbourn, 2010). These are square matrices $H _ { k }$ such that: (1) all entries of √ $H _ { k }$ are in $\{ - 1 , 1 \} ^ { k }$ ; (2) the rows of $H _ { k }$ are mutually orthogonal; (3) All row sums are equal to $\sqrt { k }$ .
+
+The order of a Regular Hadamard Matrix is of the form $4 u ^ { 2 }$ for an integer $u$ . We use known constructions for $k \overset { \cdot } { \in } \{ 4 , 1 6 , 3 6 , 6 4 , 1 0 0 \}$ .
+
+Lemma 7. Let $\boldsymbol { g } \in \mathbb { R } ^ { d }$ and $k$ be an integer for which a Regular Hadamard Matrix of order $k$ exists. Then, there is a randomized construction of $k$ orthogonal vectors $r _ { 1 } , . . . r _ { n } \in \{ - 1 , 1 \} ^ { d }$ , such that $s i g n ( g ) ^ { \top } r _ { i } = d \cdot k ^ { - 1 / 2 }$ . Moreover, $\mathbb { E } [ g ^ { \top } r _ { i } ] = k ^ { - 1 / 2 } \cdot \lVert g \rVert _ { 1 }$ .
+
+Proof. We construct $k$ orthogonal vectors $r _ { 1 } , \ldots , r _ { k } \in \{ - 1 , 1 \} ^ { d }$ , where $r _ { i }$ is obtained by repeating the $\mathrm { i } ^ { \mathrm { t h } }$ row of $H _ { k } \ d / k$ times (for simplicity, we assume that $k$ divides $d$ . Otherwise we pad $r _ { i }$ with zeros). We then multiply each $r _ { i }$ component-wise with $\mathtt { s i g n } ( g )$ . By construction, the $k$ vectors $r _ { i } \in \{ - 1 , 1 \} ^ { d }$ are mutually orthogonal, and we have $\begin{array} { r } { \mathrm { s i g n } ( g ) ^ { \top } r _ { i } = \frac { d } { k } \cdot \sqrt { k } = d \cdot k ^ { - 1 / 2 } } \end{array}$ , which is tight according to Lemma 6.
+
+As the weight of the gradient $g$ may not be uniformly distributed among its $d$ components, we apply our construction to a random permutation of the signed gradient. We then obtain
+
+$$
+\begin{array} { r l } & { \mathbb { E } [ g ^ { \top } \boldsymbol { r } _ { i } ] = \mathbb { E } \left[ \displaystyle \sum _ { j = 1 } ^ { d } \vert g ^ { ( j ) } \vert \cdot \mathsf { s i g n } ( g ^ { ( j ) } ) \cdot r _ { i } ^ { ( j ) } \right] } \\ & { \quad \quad = \displaystyle \sum _ { j = 1 } ^ { d } \vert g ^ { ( j ) } \vert \cdot \mathbb { E } \left[ \mathsf { s i g n } ( g ^ { ( j ) } ) \cdot r _ { i } ^ { ( j ) } \right] = k ^ { - 1 / 2 } \cdot \| g \| _ { 1 } . } \end{array}
+$$
+
+It can be shown that the bound in Lemma 7 can be attained if and only if the $r _ { i }$ are constructed from the rows of a Regular Hadamard Matrix (Colbourn, 2010). For general integers $k$ for which no such matrix exists, other combinatorial designs may be useful for achieving looser bounds.
+
+# F ILLUSTRATIONS OF GRADIENT MASKING IN ADVERSARIAL TRAINING
+
+In Section 3.3, we show that adversarial training introduces spurious curvature artifacts in the model’s loss function around data points. As a result, one-shot attack strategies based on first-order approximations of the model loss produce perturbations that are non-adversarial. In Figures 4 and 5 we show further illustrations of this phenomenon for the Inception ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ model trained on ImageNet by Kurakin et al. (2017b) as well as for the model $\mathbf { A } _ { \mathrm { a d v } }$ we trained on MNIST.
+
+![](images/923595c61cad34fb814eb73c6c063c03833140695ae2f1302f3f857f5a158110.jpg)  
+Figure 4: Additional illustrations of the local curvature artifacts introduced by adversarial training on ImageNet. We plot the loss of model ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ on samples of the form $x ^ { * } = \stackrel { \cdot } { x } + \epsilon _ { 1 } \cdot g + \epsilon _ { 2 } .$ · $g ^ { \perp }$ , where $g$ is the signed gradient of ${ \mathrm { v } } 3 _ { \mathrm { a d v } }$ and $g ^ { \perp }$ is an orthogonal adversarial direction, obtained from an Inception v4 model. The right-side plots are zoomed in versions of the left-side plots.
+
+![](images/80a137c74b4999940442ab91d53cd66f2fe0e469a712fe75d67af8e599265474.jpg)  
+Figure 5: Illustrations of the local curvature artifacts introduced by adversarial training on MNIST. We plot the loss of model $\mathbf { A _ { \mathrm { a d v } } }$ on samples of the form $x ^ { * } = x + \epsilon _ { 1 } \cdot g + \epsilon _ { 2 } \cdot g ^ { \perp }$ , where $g$ is the signed gradient of model $\mathbf { A } _ { \mathrm { a d v } }$ and $g ^ { \perp }$ is an orthogonal adversarial direction, obtained from model B. The right-side plots are zoomed in versions of the left-side plots.

parse/train/uXc42E9ZPFs/uXc42E9ZPFs.md

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+# Language models enable zero-shot prediction of the effects of mutations on protein function
+
+Joshua Meier 1 2 Roshan Rao 3 Robert Verkuil 1 Jason Liu 1 Tom Sercu 1 Alexander Rives 1 2
+
+# Abstract
+
+Modeling the effect of sequence variation on function is a fundamental problem for understanding and designing proteins. Since evolution encodes information about function into patterns in protein sequences, unsupervised models of variant effects can be learned from sequence data. The approach to date has been to fit a model to a family of related sequences. The conventional setting is limited, since a new model must be trained for each prediction task. We show that using only zero-shot inference, without any supervision from experimental data or additional training, protein language models capture the functional effects of sequence variation, performing at state-of-the-art.
+
+# 1 Introduction
+
+Proteins have a myriad of diverse functions that underlie the complexity of life. Protein sequences encode function via structure through the spontaneous folding of the sequence into the three dimensional structure of the protein [1]. The effects of sequence mutations on function form a landscape that reveals how function constrains sequence. Alterations at some sites in a protein sequence cannot be tolerated because they are essential to the protein’s function. Other sites evolve together because the structure and function is determined by them collectively. Mutations can enhance the activity of a protein, attenuate it, or leave it unchanged.
+
+The functional effect of sequence variations can be measured through deep mutational scanning experiments [2]. Consisting of thousands to hundreds of thousands of measurements of protein function, deep mutational scans give insight into the intrinsic constraints on a protein’s structure and function. Due to the cost and difficulty of implementing such experiments, compilations of deep mutational scanning data include experiments on a few dozens of proteins at most, relative to the tens of thousands of proteins encoded in the human genome, and the millions more across the tree of life that we would like to understand.
+
+A model that learns the landscape linking sequence to function can provide insight into function without having to do experiments. Unsupervised models of mutational effects can be learned from sequences [3, 4]. Statistical patterns in a family of evolutionarily related protein sequences contain information about structure and function [5–7]. This is because the properties of a protein act as constraints on the selection of sequences through evolution [8].
+
+In the natural language modeling community, there has been interest in zero-shot transfer of models to new tasks. Massive language models can solve tasks they haven’t been directly trained on [9–11]. Recently protein language models have achieved state-of-the-art in various structure prediction tasks [12–14]. Work to date has mainly focused on transfer in the classical representation learning setting, using pre-trained features with supervision on the downstream task.
+
+![](images/1652ff3fe8fd19a92d3d1daa55b292f281bfda542c9b42a504809d8893cff617.jpg)  
+Figure 1: Depiction of a mutational effect prediction task. The objective is to score the effect of sequence mutations on the function of a protein. Deep mutational scanning experiments provide ground truth experimental measurements of the protein’s function (fluorescence activity in the example here) for a large set of single mutations or combinations of mutations. For each protein, the prediction task is to score each possible mutation and rank its relative activity. Predictions for single substitutions can be described in a score matrix. The columns are the positions in the sequence. The rows are the possible variations at each position.
+
+In this work we show that language models trained on large and diverse protein sequence databases can predict experimental measurements of protein function without further supervision. Prior work has focused on transferring the representations using supervision from experimental data [15, 16]. We find that language models can transfer to predict functional measurements without supervision. Language models perform zero-shot and few-shot prediction of mutational effects across a variety of proteins with widely differing functions. We perform experiments with state-of-the-art protein language models ESM-1b [12] and MSA Transformer [13]. We introduce a new protein language model, ESM-1v, with zero-shot performance comparable to state-of-the-art mutational effect predictors. Performance can be further improved by fine-tuning the model with sequences from the protein family. Predictions capture the functional landscape of the protein, correlate with amino acid conservation patterns in the core and surface, and identify residues responsible for binding and activity.
+
+# 2 Zero-shot transfer
+
+Zero-shot learning has classically described the extension of a classifier to a new set of classes that have not been seen in training [17]. In natural language processing this idea has been extended to describe the transfer of models to entirely new tasks without further training. Proposed as zero-data learning by Larochelle et al. [18], this perspective on transfer has been at the center of recent work understanding the generalization capabilities of large language models [9–11, 19]. The distinction from representation learning is that the models are used directly without additional supervision for the task. This means that the tasks must be learned purely from pre-training.
+
+In this work we take a similar perspective on zero-shot transfer to that of GPT-3, described in Brown et al. [10]. We define zero-shot transfer to be transfer of a model to a new task without any further supervision to specialize the model to the task. We also consider the closely related idea of few-shot transfer. Here as in Brown et al. [10] we define the few-shot setting to be one in which a few positive examples are given to the model as inputs at inference time. As in the zero-shot setting, no gradient updates are performed to specialize the model. Similar to Brown et al. [10], the claim is not one of out-of-distribution generalization. The assumption is that in the pre-training stage, the model learns information relevant to the tasks to which it will later be transferred. In the case of protein language models, the pre-training dataset includes sequences from across evolution, which implies the model may see examples of sequences from protein families on which it will be evaluated. The essential departure from the standard approach in computational biology is that the model is general purpose and can be applied across a variety of tasks without specialization.
+
+![](images/bbf328ffdfa3e742bec3e891fde1e83564ab08c3637186440dcae4002cccba89.jpg)  
+Figure 2: Steps involved in variant effect prediction methods. Compared with EVMutation [4] and DeepSequence [20], MSA Transformer and ESM-1v require no task-specific model training for inference. Moreover, ESM-1v does not require MSA generation.
+
+Measurements of function, a property of central importance to the understanding and design of proteins, are a practical ground for studying the generalization capability of protein language models. Deep mutational scanning experiments measure the effects of thousands to hundreds of thousands of mutations on a single protein, and have been performed on a variety of proteins having different functions and using various forms of experimental measurement. We study zero-shot and few-shot transfer of protein language models to function prediction using this data.
+
+Supervised methods trained with data from experimental measurements [15, 16], and unsupervised methods trained only on sequences [3, 4] have been developed for prediction of mutational effects. Unsupervised mutational effect predictors are trained as task specific models on sequences from an individual protein family. In this view every protein is an independent prediction task where the objective is to score the effect of mutations on the protein’s function. While mutational effect predictors trained on multiple sequence alignments (MSAs) are typically described as unsupervised, they can also be seen as weakly supervised. Hsu et al. [15] observe that such models have weak supervision on the task through the MSA, which describes the fitness landscape of the protein through positive examples.
+
+If protein language models can learn the information necessary to solve a task from pre-training, then they can be applied directly to new instances of the task, without specialization. This would mean that in practice a single general purpose model can be trained once and then applied to a variety of possible tasks. Thus zero-shot and few-shot transfer represent fundamentally new unsupervised learning capabilities that protein language models can bring to the computational biology toolkit.
+
+# 3 Method
+
+Protein language models trained with the masked language modeling objective are supervised to output the probability that an amino acid occurs at a position in a protein given the surrounding context. We use this capability to score sequence variations. For a given mutation we can consider the amino acid in the wildtype protein as a reference state, comparing the probability assigned to the mutated amino acid with the probability assigned to the wildtype.
+
+<table><tr><td>Models</td><td>Full</td><td>Test</td></tr><tr><td>PSSM EVMutation (published)</td><td>0.460 0.508</td><td>0.460 0.495</td></tr><tr><td>EVMutation (replicated)</td><td>0.511</td><td>0.498</td></tr><tr><td>DeepSequence (published)</td><td>0.514</td><td>0.499</td></tr><tr><td>DeepSequence (replicated)</td><td>0.520</td><td>0.506</td></tr><tr><td>MSA Transformer</td><td>0.542</td><td>0.524</td></tr><tr><td>ESM-1v (zero shot) ESM-1v (+further training)</td><td>0.509</td><td>0.482</td></tr></table>
+
+Table 1: Comparison of protein language models to state-of-the-art methods. Average |Spearman $\rho |$ on full and test sets. DeepSequence and ESM-1v models are each ensembles of 5 models. MSA Transformer is a single model, but is ensembled across 5 random samples of the MSA.
+
+We score mutations using the log odds ratio at the mutated position, assuming an additive model when multiple mutations $T$ exist in the same sequence:
+
+$$
+\sum _ { t \in T } \log p ( x _ { t } = x _ { t } ^ { m t } | x _ { \backslash T } ) - \log p ( x _ { t } = x _ { t } ^ { w t } | x _ { \backslash T } )
+$$
+
+Here the sum is over the mutated positions, and the sequence input to the model is masked at every mutated position.
+
+# 3.1 Zero-shot and few-shot transfer
+
+In the zero-shot setting, inference is performed directly on the sequence to be evaluated. Since the MSA Transformer can take multiple sequences as input at inference time, we use this model in the few-shot setting, where additional sequences from the protein family are provided along with the sequence to be evaluated. In both the zero-shot and few-shot settings, only forward passes of the models are performed during inference; no gradient updates are taken. Fig. 2 illustrates the approach in comparison to the current practice of fitting a new model for each task.
+
+# 3.2 Inference efficiency
+
+Inference with ESM-1v is more efficient than current state-of-the-art methods. This is a result of two important differences: (i) the effect of mutations can be inferred directly without training a task-specific model; (ii) fitness landscapes can be predicted with a single forward pass. Time requirements are summarized in Fig. 7.
+
+# 3.3 Scoring with MSA Transformer
+
+We score mutations with MSA Transformer using the log odds ratio and additive model in Eq. (1). However, since MSA Transformer uses a set of sequences for inference, we input the sequence to be evaluated as the first sequence, and provide additional sequences from the MSA as context. Masking and scoring are performed on the first sequence only.
+
+# 4 Results
+
+# 4.1 Experimental setup
+
+Prediction Models We compare to state-of-the-art unsupervised variant prediction methods, EVMutation [4] and DeepSequence [20]. We also examine performance of a variety of protein language models that have been recently introduced in the literature.
+
+The position specific scoring matrix (PSSM), EVmutation [4], and DeepSequence [20] methods are all MSA based. The PSSM treats each position in the sequence independently, factorizing the likelihood into one term per sequence position. EVmutation is a Potts model, which adds pairwise terms modeling the interactions between positions. DeepSequence introduces a latent code, allowing potential higher-order interactions between positions.
+
+![](images/43c45825cfe5cd5eb060a1a70bb3776601e4c547fd35142ac8a655968e6b084e.jpg)  
+Figure 3: Per task performance. Comparison across 41 deep mutational scanning datasets. Points are |Spearman $\rho |$ on each dataset, error bars show standard deviation of 20 bootstrapped samples. Validation proteins are shown to the left of the dividing line and test proteins to the right. In 17 out of the 41 tasks, ESM-1v zero-shot has a higher |Spearman $\rho |$ than DeepSequence.
+
+UniRep [21], TAPE [22], ProtBERT-BFD [14], ESM-1b [12], and ESM-1v (introduced here), are all single-sequence language models trained on large databases of unaligned and unrelated protein sequences (e.g. Pfam [23] or UniRef [24]). With the exception of UniRep, which is trained using next token prediction, all models are trained with masked language modeling [25].
+
+Finally, the MSA Transformer [13] is a combination of both approaches; it is trained on a large database of MSAs using masked language modeling and takes an MSA as input during inference.
+
+ESM-1v We train ESM-1v, a 650M parameter transformer language model for prediction of variant effects, on 98 million diverse protein sequences across evolution. The model is trained only on sequences, without any supervision from experimental measurements of function. We use Uniref90 2020-03 [24], employing the ESM-1b architecture and masked language modeling approach of Rives et al. [12]. The model attains a perplexity of 7.29 on a set of held-out Uniref90 sequences (Table 10). We train five models with different seeds to produce an ensemble.
+
+Evaluation Models are evaluated on a set of 41 deep mutational scans collected by Riesselman et al. [20], which comprise a variety of tasks assessing a diverse set of proteins. Across tasks, the experiments differ in the functions tested and in the measurements performed. We treat each deep mutational scanning dataset as a separate prediction task, scoring each of the variants in the dataset with the model. The tasks are split into a validation set of ten mutational scanning datasets and a test set consisting of the remaining datasets. We evaluate performance by comparing the scores with the experimental measurements using Spearman rank correlation.
+
+Comparisons Since the published versions of EVMutation and DeepSequence use MSAs generated from an earlier version of Uniref100, we generate new MSAs using EVMutation methodology and the version of Uniref100 concurrent with our pretraining dataset. We train replications of EVMutation and DeepSequence using their open source code. The same MSAs are also used in few-shot experiments with MSA Transformer and unsupervised fine-tuning experiments with ESM-1v.
+
+# 4.2 Language models enable zero-shot and few-shot prediction of the effects of mutations
+
+ESM-1v and MSA Transformer models make state-of-the-art predictions. Table 1 compares overall performance of the models across the 41 mutational scanning datasets. Fig. 3 presents a comparison between ESM-1v and DeepSequence on each of the tasks. Zero-shot inference with ESM-1v has a better correlation with experimental measurements than DeepSequence on 17 of the 41 datasets. The two methods are not statistically distinguishable via a paired $t$ -test.
+
+<table><tr><td>Models</td><td>Full</td><td>Test</td></tr><tr><td>UniRep</td><td>0.156</td><td>0.151</td></tr><tr><td>TAPE</td><td>0.171</td><td>0.175</td></tr><tr><td>ProtBERT-BFD</td><td>0.428</td><td>0.399</td></tr><tr><td>ESM-1b</td><td>0.459</td><td>0.424</td></tr><tr><td>ESM-1vt</td><td>0.484</td><td>0.457</td></tr><tr><td>ESM-1v*</td><td>0.509</td><td>0.482</td></tr></table>
+
+Table 2: Zero-shot performance. Average |Spearman $\rho |$ on full and test sets. †Average performance of five ESM-1v models. ?Ensemble of the five ESM-1v models.
+
+Table 2 compares protein language models in the zero-shot setting. ESM-1v outperforms existing protein language models TAPE [22], UniRep [21], ProtBERT-BFD [14], and ESM-1b [12]. Fig. 8 breaks down performance across each of the tasks.
+
+Pre-training data We examine the effect of the clustering level of pre-training data. Fig. 4 compares models pre-trained on datasets clustered at increasing sequence identity thresholds. ESM1b is trained on sequences clustered at a $50 \%$ identity threshold. Improvements are seen using a $70 \%$ threshold with greatest improvement at $90 \%$ . Uniref100 performance appears to deteriorate early in training despite being the largest of the datasets. These results establish a link between model performance and the data distribution, highlighting the importance of training data in the design of protein language models.
+
+Scoring methods We compare four scoring methods on the validation set - masked marginals, wildtype marginals, mutant marginals, and psuedolikelihood. Table 5 shows that the masked marginal approach described in Eq. (1) outperforms other scoring methods, including ones in which the likelihood changes at non-mutated positions are considered. The scoring methods are described in detail in Appendix A.
+
+Parameter count Previous work with protein language models has established a link between model scale and learning of protein structure [12, 26]. We examine zero-shot transfer performance as a function of parameter count. We train models using the same width, depth, and learning rate as described in Henighan et al. [27], observing improvements with scale (Fig. 9). These findings suggest that continued scaling of the models will further improve results.
+
+# 4.3 MSA Transformer
+
+We examine how the sequences provided to MSA Transformer affect few-shot transfer. Table 8 compares sequence selection methods that vary the diversity of the sequences. Providing a more diverse set of sequences improves few-shot performance. Selecting a set of sequences to maximize diversity outperforms selecting a diversity minimizing set of sequences. Random sampling performs even better, and sampling sequences according to sequence weights [28] performs best.
+
+We also vary the number of sequences used for inference. Fig. 11 shows few-shot performance as a function of the number of sequences given as input. The model performs well using only a few sequences, but performs best with 384 total sequences. In the main tables we report results sampling 384 sequences using sequence reweighting and ensembling predictions over five different subsamples from the MSA.
+
+# 4.4 Unsupervised fine-tuning on MSAs
+
+While ESM-1v performs well when evaluated in the zero-shot setting, we explore whether results can be improved by fine-tuning on the MSA. Fine-tuning on MSAs has been used in previous work [21, 16] as a stage in transfer learning to specialize a pre-trained model to a protein family, before applying supervision with labeled data. Here we consider using the fine-tuned model to make unsupervised predictions directly, without adding supervision from experimental data.
+
+![](images/8091e2784327233652acd1ed4fe80d706fc49d6d0f362f8f9713999d7747cccc.jpg)  
+Figure 4: Comparison of pre-training datasets. Average |Spearman $\rho |$ on the single-mutation validation set. While a $50 \%$ clustering threshold was used for ESM-1b, training with $90 \%$ clustering results in a significant improvement on variant prediction tasks. Notably, models trained on Uniref100, the largest dataset in this figure, appear to deteriorate early in training. These results establish a link between model performance and the data distribution, and highlight the importance of training data in the design of protein language models.
+
+We observe that naively fine-tuning the model on the MSA results in rapid overfitting and poor performance on the prediction tasks (Fig. 12). While we experiment with a variety of approaches to freezing parameters during fine-tuning, detailed in Appendix B, none produce significant improvements. We find that an approach using pre-training sequences to regularize the fine-tuning performs well and enables training of all parameters without overfitting (Fig. 13). Spiked fine-tuning improves average absolute Spearman rho on the full dataset from 0.510 for zero-shot evaluation to 0.537 with fine-tuning.
+
+# 5 Analysis of models
+
+Protein structure and function ESM-1v probabilities reflect the functional properties of sites within the protein. We use the entropy of the model’s predictions for a position as a measure of its estimation of conservation. The lowest entropy predictions cluster at binding sites. Fig. 14 compares the distribution of the model’s entropy between binding sites and non-binding sites. A significant difference is observed between the entropy assignment to binding and non-binding site residues. Fig. 5 visualizes the side chains of the 10 lowest entropy residues as predicted by the model on the crystal structure of DNA methyltransferase M.HaeIII interacting with its DNA substrate. In the crystal structure a cytosine of the substrate is inserted into the active site of the enzyme. The low entropy residues cluster in the active site and interact with the cytosine. Additional examples are visualized in Fig. 18.
+
+The model probabilities also correspond to structure. Fig. 15 compares the entropy assigned to sites that are buried in the core of the protein vs. exposed on the surface. The model assigns significantly lower entropy to sites that are in the core of the protein, consistent with the idea that tight packing in the core places greater constraints on the selection of residues. Fig. 5B visualizes the entropy assigned by the model to each position overlayed on the structure of Indole-3-glycerolphosphate Synthase, a TIM barrel protein. Higher entropy is assigned to residues having outward facing side chains on the alpha helices, while lower entropy is assigned to the inward facing positions. Fig. 17 compares the probability assigned to hydrophobic, polar, and charged amino acids for buried sites vs. non-buried sites. The model prefers hydrophobic residues in the core and hydrophilic residues on the surface. The model probabilities closely match the empirical probabilities and those from the PSSM. Fig. 5C visualizes probability assigned to hydrophobic amino acids on the structure of
+
+![](images/2ea85c03a08acec02f38e3c0f7439ab7a8794f0fc9dd070ba947f7ef79ed915f.jpg)  
+Figure 5: ESM-1v reflects the molecular basis of function in proteins. (A) DNA methylase HaeIII (pdbid: 1DCT [29]). Side chains for the top 10 positions with lowest prediction entropy shown in blue. Low-entropy positions cluster in the active site. (B) TIM Barrel (pdbid: 1IGS [30]) with residues colored by entropy. The model’s predictions for residues on the surface have highest entropy (red) while those in the core have lower entropy (blue). Notably, residues on the alpha helices show a clear gradient from high to low entropy as residues transition from surface-facing to core-facing. (C) Sucrose-specific Porin (pdbid: 1A0T [31]), a transmembrane protein. The model predicts a hydrophobic band where the protein is embedded in the membrane.
+
+Sucrose-specific Porin, a transmembrane protein. The model predicts a hydrophobic band in the center where the protein embeds in the membrane.
+
+Calibration We evaluate model calibration using 15008 sequences with length $< 1 0 2 4$ from the trRosetta [32] dataset. ESM-1v probabilities for each amino acid at each position are calculated with the masked marginal probability in Eq. (1). Fig. 6 shows that the model is generally well calibrated for all amino acids except Methionine. ESM-1v always predicts Methionine as the first position in the sequence since full protein sequences always start with it, so care must be used when applying the model to subsequences. When excepting the first residue, the model achieves an average calibration error (defined for the multi-class setting in Appendix D.4) of 0.006.
+
+We also explore the relationship between conservation (entropy of the PSSM) and the model’s predicted entropy. Fig. 16 shows that these are well correlated (Pearson’s $r = 0 . 4 4$ ), suggesting the model is able to identify conserved positions.
+
+# 6 Related Work
+
+# 6.1 Protein language models
+
+In the past few years, a number of groups have developed language models for protein sequences [21, 33, 14, 34, 35, 22, 13, 12]. These models have been used for many tasks, including supervised low-N function prediction [16, 12], remote homology detection [22, 12], and protein generation [35]. The approach to the tasks typically involves transfer learning, where a pretrained language model is fine-tuned for a particular problem. Vig et al. [36] and Rao et al. [26] found that transformer attention corresponds to known biological properties such as structure and binding sites and can be used to predict contacts.
+
+![](images/22719229610aa8a6b0b3b3a8d8d5b6b256a8c4636893a0c752af2b26b6e02432.jpg)  
+Figure 6: Calibration plot for ESM-1v predictions on each of the 20 naturally occurring amino acids on the trRosetta dataset. The multi-class classification is converted into a set of 20 oneversus-all classifications for the purpose of this analysis. Left and right plots show calibration of all positions and positions excluding the first residue, respectively. Since full sequences always start with Methionine, the model overwhelmingly predicts it in the first position. When evaluating the model on subsequences, such as those in the trRosetta dataset, this causes a miscalibration at the first residue. Including the first residue, the model has an average calibration error (ACE) of 0.011 in the first case and 0.006 in the second.
+
+# 6.2 Mutation effect prediction
+
+Supervised and unsupervised methods have been developed for prediction of mutational effects. Supervised methods train models using experimental measurements or labels from databases of clinical variants. Standard machine learning tools including linear regression, random forests, and support-vector machines can be used [37]. Models have been designed specifically for proteins, using feature engineering such as Envision [38] and PolyPhen-2 [39], ensemble methods such as Revel [40], MPC [41], CADD [42], and M-CAP [43], language models such as UniRep [21, 16] and ESM [12], and other representation learning approaches [44, 45].
+
+Unsupervised mutation effect predictors work by inferring the likelihood of a mutation from the evolutionary landscape of the original protein. A density model fit to related sequences is used for scoring. SIFT [46] is a first order approach using a position-specific-scoring-matrix. EVMutation [4] extends this to a second-order approach by training a Potts model on the MSA. DeepSequence [20] includes higher-order interactions by training a VAE on the MSA instead, using the ELBO to score mutations. Riesselman et al. [47] proposes using an autoregressive model that does not require the sequences to be aligned. Laine et al. [48] uses an evolutionary tree structure inferred from the MSA to compare mutations.
+
+Hsu et al. [15] show that unsupervised mutational effect predictors can be extended to perform supervised predictions, with better unsupervised predictors generally resulting in better supervised predictors. This suggests improving unsupervised prediction can drive progress in both settings. Concurrent with our work, Hie et al. [49] use open-source protein language models ESM-1b and TAPE to predict the direction of evolution in protein fitness landscapes.
+
+# 7 Discussion
+
+Advances in language modeling at scale are bringing the goal of a general purpose model for proteins closer to realization. This line of work aspires to a model that learns to read and write biology in its native language, that can be directly applied across a range of protein understanding and design tasks. For scalability, learning from sequences is important: while there are no central databases of high-throughput functional measurements, and few compilations exist, billions of sequences are available to learn from in sequence databases [50, 51]. Sequences give an unparalleled view into the vast diversity and complexity of molecular parts invented by nature through billions of years of evolution.
+
+Unsupervised structure [52–54, 28, 55, 56] and function [3, 4] learning methods first effectively realized the idea that biological properties could be read directly from sequences without supervision from experimental measurements. However these methods are not general purpose in the sense that a specialized model must be trained for every protein for which a prediction is to be made. We show that the same performance can be realized by a general purpose model that has been trained across many diverse protein families. Similar to observations on the learning of tertiary protein structure in large language models [12, 26], we find that increasing the scale of models leads to improvements in function learning. The understanding of mutational landscapes in the models correlates with the molecular basis of function in proteins, capturing binding sites and amino acid preferences that are determined by the folded structure.
+
+Zero-shot transfer is an interesting capability of large scale language models, and represents a major point of departure from the unsupervised learning methods that are the basis for current state-of-theart inference of protein structure and function. The capability for zero-shot transfer implies that a model can be trained once and then applied to perform inference for many tasks. It is also a window into deeper questions about the forms of generalization that are possible in learning from sequences. Reading structural and functional design principles from sequences is a necessary capability for writing new biologically active sequences. Generalization in the zero-shot setting suggests the potential for large language models to capture knowledge that can be transferred to generating new functional proteins.
+
+# Acknowledgments and Disclosure of Funding
+
+We thank Naman Goyal, Chloe Hsu, Zeming Lin, Ishan Misra, Myle Ott, Sergey Ovchinnikov, and Ethan Perez for valuable input on the paper. Roshan Rao was supported by funding from Facebook, Berkeley Deep Drive, and DARPA XAI.
+
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+        "type": "text",
+        "text": "Language models enable zero-shot prediction of the effects of mutations on protein function ",
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+        "text": "Joshua Meier 1 2 Roshan Rao 3 Robert Verkuil 1 Jason Liu 1 Tom Sercu 1 Alexander Rives 1 2 ",
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+        "text": "Modeling the effect of sequence variation on function is a fundamental problem for understanding and designing proteins. Since evolution encodes information about function into patterns in protein sequences, unsupervised models of variant effects can be learned from sequence data. The approach to date has been to fit a model to a family of related sequences. The conventional setting is limited, since a new model must be trained for each prediction task. We show that using only zero-shot inference, without any supervision from experimental data or additional training, protein language models capture the functional effects of sequence variation, performing at state-of-the-art. ",
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+    {
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+        "text": "1 Introduction ",
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+    },
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+        "text": "Proteins have a myriad of diverse functions that underlie the complexity of life. Protein sequences encode function via structure through the spontaneous folding of the sequence into the three dimensional structure of the protein [1]. The effects of sequence mutations on function form a landscape that reveals how function constrains sequence. Alterations at some sites in a protein sequence cannot be tolerated because they are essential to the protein’s function. Other sites evolve together because the structure and function is determined by them collectively. Mutations can enhance the activity of a protein, attenuate it, or leave it unchanged. ",
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+        "text": "The functional effect of sequence variations can be measured through deep mutational scanning experiments [2]. Consisting of thousands to hundreds of thousands of measurements of protein function, deep mutational scans give insight into the intrinsic constraints on a protein’s structure and function. Due to the cost and difficulty of implementing such experiments, compilations of deep mutational scanning data include experiments on a few dozens of proteins at most, relative to the tens of thousands of proteins encoded in the human genome, and the millions more across the tree of life that we would like to understand. ",
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+    },
+    {
+        "type": "text",
+        "text": "A model that learns the landscape linking sequence to function can provide insight into function without having to do experiments. Unsupervised models of mutational effects can be learned from sequences [3, 4]. Statistical patterns in a family of evolutionarily related protein sequences contain information about structure and function [5–7]. This is because the properties of a protein act as constraints on the selection of sequences through evolution [8]. ",
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+        "type": "text",
+        "text": "In the natural language modeling community, there has been interest in zero-shot transfer of models to new tasks. Massive language models can solve tasks they haven’t been directly trained on [9–11]. Recently protein language models have achieved state-of-the-art in various structure prediction tasks [12–14]. Work to date has mainly focused on transfer in the classical representation learning setting, using pre-trained features with supervision on the downstream task. ",
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+        "img_path": "images/1652ff3fe8fd19a92d3d1daa55b292f281bfda542c9b42a504809d8893cff617.jpg",
+        "image_caption": [
+            "Figure 1: Depiction of a mutational effect prediction task. The objective is to score the effect of sequence mutations on the function of a protein. Deep mutational scanning experiments provide ground truth experimental measurements of the protein’s function (fluorescence activity in the example here) for a large set of single mutations or combinations of mutations. For each protein, the prediction task is to score each possible mutation and rank its relative activity. Predictions for single substitutions can be described in a score matrix. The columns are the positions in the sequence. The rows are the possible variations at each position. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this work we show that language models trained on large and diverse protein sequence databases can predict experimental measurements of protein function without further supervision. Prior work has focused on transferring the representations using supervision from experimental data [15, 16]. We find that language models can transfer to predict functional measurements without supervision. Language models perform zero-shot and few-shot prediction of mutational effects across a variety of proteins with widely differing functions. We perform experiments with state-of-the-art protein language models ESM-1b [12] and MSA Transformer [13]. We introduce a new protein language model, ESM-1v, with zero-shot performance comparable to state-of-the-art mutational effect predictors. Performance can be further improved by fine-tuning the model with sequences from the protein family. Predictions capture the functional landscape of the protein, correlate with amino acid conservation patterns in the core and surface, and identify residues responsible for binding and activity. ",
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+    {
+        "type": "text",
+        "text": "2 Zero-shot transfer ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Zero-shot learning has classically described the extension of a classifier to a new set of classes that have not been seen in training [17]. In natural language processing this idea has been extended to describe the transfer of models to entirely new tasks without further training. Proposed as zero-data learning by Larochelle et al. [18], this perspective on transfer has been at the center of recent work understanding the generalization capabilities of large language models [9–11, 19]. The distinction from representation learning is that the models are used directly without additional supervision for the task. This means that the tasks must be learned purely from pre-training. ",
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+    },
+    {
+        "type": "text",
+        "text": "In this work we take a similar perspective on zero-shot transfer to that of GPT-3, described in Brown et al. [10]. We define zero-shot transfer to be transfer of a model to a new task without any further supervision to specialize the model to the task. We also consider the closely related idea of few-shot transfer. Here as in Brown et al. [10] we define the few-shot setting to be one in which a few positive examples are given to the model as inputs at inference time. As in the zero-shot setting, no gradient updates are performed to specialize the model. Similar to Brown et al. [10], the claim is not one of out-of-distribution generalization. The assumption is that in the pre-training stage, the model learns information relevant to the tasks to which it will later be transferred. In the case of protein language models, the pre-training dataset includes sequences from across evolution, which implies the model may see examples of sequences from protein families on which it will be evaluated. The essential departure from the standard approach in computational biology is that the model is general purpose and can be applied across a variety of tasks without specialization. ",
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+    },
+    {
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+        "img_path": "images/bbf328ffdfa3e742bec3e891fde1e83564ab08c3637186440dcae4002cccba89.jpg",
+        "image_caption": [
+            "Figure 2: Steps involved in variant effect prediction methods. Compared with EVMutation [4] and DeepSequence [20], MSA Transformer and ESM-1v require no task-specific model training for inference. Moreover, ESM-1v does not require MSA generation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Measurements of function, a property of central importance to the understanding and design of proteins, are a practical ground for studying the generalization capability of protein language models. Deep mutational scanning experiments measure the effects of thousands to hundreds of thousands of mutations on a single protein, and have been performed on a variety of proteins having different functions and using various forms of experimental measurement. We study zero-shot and few-shot transfer of protein language models to function prediction using this data. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Supervised methods trained with data from experimental measurements [15, 16], and unsupervised methods trained only on sequences [3, 4] have been developed for prediction of mutational effects. Unsupervised mutational effect predictors are trained as task specific models on sequences from an individual protein family. In this view every protein is an independent prediction task where the objective is to score the effect of mutations on the protein’s function. While mutational effect predictors trained on multiple sequence alignments (MSAs) are typically described as unsupervised, they can also be seen as weakly supervised. Hsu et al. [15] observe that such models have weak supervision on the task through the MSA, which describes the fitness landscape of the protein through positive examples. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If protein language models can learn the information necessary to solve a task from pre-training, then they can be applied directly to new instances of the task, without specialization. This would mean that in practice a single general purpose model can be trained once and then applied to a variety of possible tasks. Thus zero-shot and few-shot transfer represent fundamentally new unsupervised learning capabilities that protein language models can bring to the computational biology toolkit. ",
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+    {
+        "type": "text",
+        "text": "3 Method ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Protein language models trained with the masked language modeling objective are supervised to output the probability that an amino acid occurs at a position in a protein given the surrounding context. We use this capability to score sequence variations. For a given mutation we can consider the amino acid in the wildtype protein as a reference state, comparing the probability assigned to the mutated amino acid with the probability assigned to the wildtype. ",
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+    {
+        "type": "table",
+        "img_path": "images/efa663e8bf8cca512e2f4cf01f9cb70084a34280053c0e3acbc5ee05735878d9.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>Full</td><td>Test</td></tr><tr><td>PSSM EVMutation (published)</td><td>0.460 0.508</td><td>0.460 0.495</td></tr><tr><td>EVMutation (replicated)</td><td>0.511</td><td>0.498</td></tr><tr><td>DeepSequence (published)</td><td>0.514</td><td>0.499</td></tr><tr><td>DeepSequence (replicated)</td><td>0.520</td><td>0.506</td></tr><tr><td>MSA Transformer</td><td>0.542</td><td>0.524</td></tr><tr><td>ESM-1v (zero shot) ESM-1v (+further training)</td><td>0.509</td><td>0.482</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Comparison of protein language models to state-of-the-art methods. Average |Spearman $\\rho |$ on full and test sets. DeepSequence and ESM-1v models are each ensembles of 5 models. MSA Transformer is a single model, but is ensembled across 5 random samples of the MSA. ",
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+    },
+    {
+        "type": "text",
+        "text": "We score mutations using the log odds ratio at the mutated position, assuming an additive model when multiple mutations $T$ exist in the same sequence: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c0c7a776b2bea09152f1876e1742fe7059f6d26963e729955d09c8c4e7f53359.jpg",
+        "text": "$$\n\\sum _ { t \\in T } \\log p ( x _ { t } = x _ { t } ^ { m t } | x _ { \\backslash T } ) - \\log p ( x _ { t } = x _ { t } ^ { w t } | x _ { \\backslash T } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here the sum is over the mutated positions, and the sequence input to the model is masked at every mutated position. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 Zero-shot and few-shot transfer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In the zero-shot setting, inference is performed directly on the sequence to be evaluated. Since the MSA Transformer can take multiple sequences as input at inference time, we use this model in the few-shot setting, where additional sequences from the protein family are provided along with the sequence to be evaluated. In both the zero-shot and few-shot settings, only forward passes of the models are performed during inference; no gradient updates are taken. Fig. 2 illustrates the approach in comparison to the current practice of fitting a new model for each task. ",
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+    {
+        "type": "text",
+        "text": "3.2 Inference efficiency ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Inference with ESM-1v is more efficient than current state-of-the-art methods. This is a result of two important differences: (i) the effect of mutations can be inferred directly without training a task-specific model; (ii) fitness landscapes can be predicted with a single forward pass. Time requirements are summarized in Fig. 7. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.3 Scoring with MSA Transformer ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We score mutations with MSA Transformer using the log odds ratio and additive model in Eq. (1). However, since MSA Transformer uses a set of sequences for inference, we input the sequence to be evaluated as the first sequence, and provide additional sequences from the MSA as context. Masking and scoring are performed on the first sequence only. ",
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+    {
+        "type": "text",
+        "text": "4 Results ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 Experimental setup ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Prediction Models We compare to state-of-the-art unsupervised variant prediction methods, EVMutation [4] and DeepSequence [20]. We also examine performance of a variety of protein language models that have been recently introduced in the literature. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The position specific scoring matrix (PSSM), EVmutation [4], and DeepSequence [20] methods are all MSA based. The PSSM treats each position in the sequence independently, factorizing the likelihood into one term per sequence position. EVmutation is a Potts model, which adds pairwise terms modeling the interactions between positions. DeepSequence introduces a latent code, allowing potential higher-order interactions between positions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/43c45825cfe5cd5eb060a1a70bb3776601e4c547fd35142ac8a655968e6b084e.jpg",
+        "image_caption": [
+            "Figure 3: Per task performance. Comparison across 41 deep mutational scanning datasets. Points are |Spearman $\\rho |$ on each dataset, error bars show standard deviation of 20 bootstrapped samples. Validation proteins are shown to the left of the dividing line and test proteins to the right. In 17 out of the 41 tasks, ESM-1v zero-shot has a higher |Spearman $\\rho |$ than DeepSequence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "UniRep [21], TAPE [22], ProtBERT-BFD [14], ESM-1b [12], and ESM-1v (introduced here), are all single-sequence language models trained on large databases of unaligned and unrelated protein sequences (e.g. Pfam [23] or UniRef [24]). With the exception of UniRep, which is trained using next token prediction, all models are trained with masked language modeling [25]. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Finally, the MSA Transformer [13] is a combination of both approaches; it is trained on a large database of MSAs using masked language modeling and takes an MSA as input during inference. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "ESM-1v We train ESM-1v, a 650M parameter transformer language model for prediction of variant effects, on 98 million diverse protein sequences across evolution. The model is trained only on sequences, without any supervision from experimental measurements of function. We use Uniref90 2020-03 [24], employing the ESM-1b architecture and masked language modeling approach of Rives et al. [12]. The model attains a perplexity of 7.29 on a set of held-out Uniref90 sequences (Table 10). We train five models with different seeds to produce an ensemble. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Models are evaluated on a set of 41 deep mutational scans collected by Riesselman et al. [20], which comprise a variety of tasks assessing a diverse set of proteins. Across tasks, the experiments differ in the functions tested and in the measurements performed. We treat each deep mutational scanning dataset as a separate prediction task, scoring each of the variants in the dataset with the model. The tasks are split into a validation set of ten mutational scanning datasets and a test set consisting of the remaining datasets. We evaluate performance by comparing the scores with the experimental measurements using Spearman rank correlation. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Comparisons Since the published versions of EVMutation and DeepSequence use MSAs generated from an earlier version of Uniref100, we generate new MSAs using EVMutation methodology and the version of Uniref100 concurrent with our pretraining dataset. We train replications of EVMutation and DeepSequence using their open source code. The same MSAs are also used in few-shot experiments with MSA Transformer and unsupervised fine-tuning experiments with ESM-1v. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Language models enable zero-shot and few-shot prediction of the effects of mutations ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "ESM-1v and MSA Transformer models make state-of-the-art predictions. Table 1 compares overall performance of the models across the 41 mutational scanning datasets. Fig. 3 presents a comparison between ESM-1v and DeepSequence on each of the tasks. Zero-shot inference with ESM-1v has a better correlation with experimental measurements than DeepSequence on 17 of the 41 datasets. The two methods are not statistically distinguishable via a paired $t$ -test. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/5457182653c2263c8f454555d6e60164a38fb0d2afde0a1c6ba2b1c8126edb94.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Models</td><td>Full</td><td>Test</td></tr><tr><td>UniRep</td><td>0.156</td><td>0.151</td></tr><tr><td>TAPE</td><td>0.171</td><td>0.175</td></tr><tr><td>ProtBERT-BFD</td><td>0.428</td><td>0.399</td></tr><tr><td>ESM-1b</td><td>0.459</td><td>0.424</td></tr><tr><td>ESM-1vt</td><td>0.484</td><td>0.457</td></tr><tr><td>ESM-1v*</td><td>0.509</td><td>0.482</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Zero-shot performance. Average |Spearman $\\rho |$ on full and test sets. †Average performance of five ESM-1v models. ?Ensemble of the five ESM-1v models. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 2 compares protein language models in the zero-shot setting. ESM-1v outperforms existing protein language models TAPE [22], UniRep [21], ProtBERT-BFD [14], and ESM-1b [12]. Fig. 8 breaks down performance across each of the tasks. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Pre-training data We examine the effect of the clustering level of pre-training data. Fig. 4 compares models pre-trained on datasets clustered at increasing sequence identity thresholds. ESM1b is trained on sequences clustered at a $50 \\%$ identity threshold. Improvements are seen using a $70 \\%$ threshold with greatest improvement at $90 \\%$ . Uniref100 performance appears to deteriorate early in training despite being the largest of the datasets. These results establish a link between model performance and the data distribution, highlighting the importance of training data in the design of protein language models. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Scoring methods We compare four scoring methods on the validation set - masked marginals, wildtype marginals, mutant marginals, and psuedolikelihood. Table 5 shows that the masked marginal approach described in Eq. (1) outperforms other scoring methods, including ones in which the likelihood changes at non-mutated positions are considered. The scoring methods are described in detail in Appendix A. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Parameter count Previous work with protein language models has established a link between model scale and learning of protein structure [12, 26]. We examine zero-shot transfer performance as a function of parameter count. We train models using the same width, depth, and learning rate as described in Henighan et al. [27], observing improvements with scale (Fig. 9). These findings suggest that continued scaling of the models will further improve results. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.3 MSA Transformer ",
+        "text_level": 1,
+        "bbox": [
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+            623,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We examine how the sequences provided to MSA Transformer affect few-shot transfer. Table 8 compares sequence selection methods that vary the diversity of the sequences. Providing a more diverse set of sequences improves few-shot performance. Selecting a set of sequences to maximize diversity outperforms selecting a diversity minimizing set of sequences. Random sampling performs even better, and sampling sequences according to sequence weights [28] performs best. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We also vary the number of sequences used for inference. Fig. 11 shows few-shot performance as a function of the number of sequences given as input. The model performs well using only a few sequences, but performs best with 384 total sequences. In the main tables we report results sampling 384 sequences using sequence reweighting and ensembling predictions over five different subsamples from the MSA. ",
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+    {
+        "type": "text",
+        "text": "4.4 Unsupervised fine-tuning on MSAs ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "While ESM-1v performs well when evaluated in the zero-shot setting, we explore whether results can be improved by fine-tuning on the MSA. Fine-tuning on MSAs has been used in previous work [21, 16] as a stage in transfer learning to specialize a pre-trained model to a protein family, before applying supervision with labeled data. Here we consider using the fine-tuned model to make unsupervised predictions directly, without adding supervision from experimental data. ",
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+    {
+        "type": "image",
+        "img_path": "images/8091e2784327233652acd1ed4fe80d706fc49d6d0f362f8f9713999d7747cccc.jpg",
+        "image_caption": [
+            "Figure 4: Comparison of pre-training datasets. Average |Spearman $\\rho |$ on the single-mutation validation set. While a $50 \\%$ clustering threshold was used for ESM-1b, training with $90 \\%$ clustering results in a significant improvement on variant prediction tasks. Notably, models trained on Uniref100, the largest dataset in this figure, appear to deteriorate early in training. These results establish a link between model performance and the data distribution, and highlight the importance of training data in the design of protein language models. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "We observe that naively fine-tuning the model on the MSA results in rapid overfitting and poor performance on the prediction tasks (Fig. 12). While we experiment with a variety of approaches to freezing parameters during fine-tuning, detailed in Appendix B, none produce significant improvements. We find that an approach using pre-training sequences to regularize the fine-tuning performs well and enables training of all parameters without overfitting (Fig. 13). Spiked fine-tuning improves average absolute Spearman rho on the full dataset from 0.510 for zero-shot evaluation to 0.537 with fine-tuning. ",
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+        "text": "5 Analysis of models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Protein structure and function ESM-1v probabilities reflect the functional properties of sites within the protein. We use the entropy of the model’s predictions for a position as a measure of its estimation of conservation. The lowest entropy predictions cluster at binding sites. Fig. 14 compares the distribution of the model’s entropy between binding sites and non-binding sites. A significant difference is observed between the entropy assignment to binding and non-binding site residues. Fig. 5 visualizes the side chains of the 10 lowest entropy residues as predicted by the model on the crystal structure of DNA methyltransferase M.HaeIII interacting with its DNA substrate. In the crystal structure a cytosine of the substrate is inserted into the active site of the enzyme. The low entropy residues cluster in the active site and interact with the cytosine. Additional examples are visualized in Fig. 18. ",
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+    {
+        "type": "text",
+        "text": "The model probabilities also correspond to structure. Fig. 15 compares the entropy assigned to sites that are buried in the core of the protein vs. exposed on the surface. The model assigns significantly lower entropy to sites that are in the core of the protein, consistent with the idea that tight packing in the core places greater constraints on the selection of residues. Fig. 5B visualizes the entropy assigned by the model to each position overlayed on the structure of Indole-3-glycerolphosphate Synthase, a TIM barrel protein. Higher entropy is assigned to residues having outward facing side chains on the alpha helices, while lower entropy is assigned to the inward facing positions. Fig. 17 compares the probability assigned to hydrophobic, polar, and charged amino acids for buried sites vs. non-buried sites. The model prefers hydrophobic residues in the core and hydrophilic residues on the surface. The model probabilities closely match the empirical probabilities and those from the PSSM. Fig. 5C visualizes probability assigned to hydrophobic amino acids on the structure of ",
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+        "img_path": "images/2ea85c03a08acec02f38e3c0f7439ab7a8794f0fc9dd070ba947f7ef79ed915f.jpg",
+        "image_caption": [
+            "Figure 5: ESM-1v reflects the molecular basis of function in proteins. (A) DNA methylase HaeIII (pdbid: 1DCT [29]). Side chains for the top 10 positions with lowest prediction entropy shown in blue. Low-entropy positions cluster in the active site. (B) TIM Barrel (pdbid: 1IGS [30]) with residues colored by entropy. The model’s predictions for residues on the surface have highest entropy (red) while those in the core have lower entropy (blue). Notably, residues on the alpha helices show a clear gradient from high to low entropy as residues transition from surface-facing to core-facing. (C) Sucrose-specific Porin (pdbid: 1A0T [31]), a transmembrane protein. The model predicts a hydrophobic band where the protein is embedded in the membrane. "
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+    {
+        "type": "text",
+        "text": "Sucrose-specific Porin, a transmembrane protein. The model predicts a hydrophobic band in the center where the protein embeds in the membrane. ",
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+        "type": "text",
+        "text": "Calibration We evaluate model calibration using 15008 sequences with length $< 1 0 2 4$ from the trRosetta [32] dataset. ESM-1v probabilities for each amino acid at each position are calculated with the masked marginal probability in Eq. (1). Fig. 6 shows that the model is generally well calibrated for all amino acids except Methionine. ESM-1v always predicts Methionine as the first position in the sequence since full protein sequences always start with it, so care must be used when applying the model to subsequences. When excepting the first residue, the model achieves an average calibration error (defined for the multi-class setting in Appendix D.4) of 0.006. ",
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+    {
+        "type": "text",
+        "text": "We also explore the relationship between conservation (entropy of the PSSM) and the model’s predicted entropy. Fig. 16 shows that these are well correlated (Pearson’s $r = 0 . 4 4$ ), suggesting the model is able to identify conserved positions. ",
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+        "type": "text",
+        "text": "6 Related Work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "6.1 Protein language models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the past few years, a number of groups have developed language models for protein sequences [21, 33, 14, 34, 35, 22, 13, 12]. These models have been used for many tasks, including supervised low-N function prediction [16, 12], remote homology detection [22, 12], and protein generation [35]. The approach to the tasks typically involves transfer learning, where a pretrained language model is fine-tuned for a particular problem. Vig et al. [36] and Rao et al. [26] found that transformer attention corresponds to known biological properties such as structure and binding sites and can be used to predict contacts. ",
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+    {
+        "type": "image",
+        "img_path": "images/22719229610aa8a6b0b3b3a8d8d5b6b256a8c4636893a0c752af2b26b6e02432.jpg",
+        "image_caption": [
+            "Figure 6: Calibration plot for ESM-1v predictions on each of the 20 naturally occurring amino acids on the trRosetta dataset. The multi-class classification is converted into a set of 20 oneversus-all classifications for the purpose of this analysis. Left and right plots show calibration of all positions and positions excluding the first residue, respectively. Since full sequences always start with Methionine, the model overwhelmingly predicts it in the first position. When evaluating the model on subsequences, such as those in the trRosetta dataset, this causes a miscalibration at the first residue. Including the first residue, the model has an average calibration error (ACE) of 0.011 in the first case and 0.006 in the second. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "6.2 Mutation effect prediction ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Supervised and unsupervised methods have been developed for prediction of mutational effects. Supervised methods train models using experimental measurements or labels from databases of clinical variants. Standard machine learning tools including linear regression, random forests, and support-vector machines can be used [37]. Models have been designed specifically for proteins, using feature engineering such as Envision [38] and PolyPhen-2 [39], ensemble methods such as Revel [40], MPC [41], CADD [42], and M-CAP [43], language models such as UniRep [21, 16] and ESM [12], and other representation learning approaches [44, 45]. ",
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+    {
+        "type": "text",
+        "text": "Unsupervised mutation effect predictors work by inferring the likelihood of a mutation from the evolutionary landscape of the original protein. A density model fit to related sequences is used for scoring. SIFT [46] is a first order approach using a position-specific-scoring-matrix. EVMutation [4] extends this to a second-order approach by training a Potts model on the MSA. DeepSequence [20] includes higher-order interactions by training a VAE on the MSA instead, using the ELBO to score mutations. Riesselman et al. [47] proposes using an autoregressive model that does not require the sequences to be aligned. Laine et al. [48] uses an evolutionary tree structure inferred from the MSA to compare mutations. ",
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+    {
+        "type": "text",
+        "text": "Hsu et al. [15] show that unsupervised mutational effect predictors can be extended to perform supervised predictions, with better unsupervised predictors generally resulting in better supervised predictors. This suggests improving unsupervised prediction can drive progress in both settings. Concurrent with our work, Hie et al. [49] use open-source protein language models ESM-1b and TAPE to predict the direction of evolution in protein fitness landscapes. ",
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+    {
+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Advances in language modeling at scale are bringing the goal of a general purpose model for proteins closer to realization. This line of work aspires to a model that learns to read and write biology in its native language, that can be directly applied across a range of protein understanding and design tasks. For scalability, learning from sequences is important: while there are no central databases of high-throughput functional measurements, and few compilations exist, billions of sequences are available to learn from in sequence databases [50, 51]. Sequences give an unparalleled view into the vast diversity and complexity of molecular parts invented by nature through billions of years of evolution. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Unsupervised structure [52–54, 28, 55, 56] and function [3, 4] learning methods first effectively realized the idea that biological properties could be read directly from sequences without supervision from experimental measurements. However these methods are not general purpose in the sense that a specialized model must be trained for every protein for which a prediction is to be made. We show that the same performance can be realized by a general purpose model that has been trained across many diverse protein families. Similar to observations on the learning of tertiary protein structure in large language models [12, 26], we find that increasing the scale of models leads to improvements in function learning. The understanding of mutational landscapes in the models correlates with the molecular basis of function in proteins, capturing binding sites and amino acid preferences that are determined by the folded structure. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Zero-shot transfer is an interesting capability of large scale language models, and represents a major point of departure from the unsupervised learning methods that are the basis for current state-of-theart inference of protein structure and function. The capability for zero-shot transfer implies that a model can be trained once and then applied to perform inference for many tasks. It is also a window into deeper questions about the forms of generalization that are possible in learning from sequences. Reading structural and functional design principles from sequences is a necessary capability for writing new biologically active sequences. Generalization in the zero-shot setting suggests the potential for large language models to capture knowledge that can be transferred to generating new functional proteins. ",
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+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We thank Naman Goyal, Chloe Hsu, Zeming Lin, Ishan Misra, Myle Ott, Sergey Ovchinnikov, and Ethan Perez for valuable input on the paper. Roshan Rao was supported by funding from Facebook, Berkeley Deep Drive, and DARPA XAI. ",
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+    {
+        "type": "text",
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+]

parse/train/v2tmeZVV9-c/v2tmeZVV9-c.md

@@ -0,0 +1,325 @@
+# ACCURATELY SOLVING ROD DYNAMICS WITH GRAPH LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Iterative solvers are widely used to accurately simulate physical systems. These solvers require initial guesses to generate a sequence of improving approximate solutions. In this contribution, we introduce a novel method to accelerate iterative solvers for rod dynamics with graph networks (GNs) by predicting the initial guesses to reduce the number of iterations. Unlike existing methods that aim to learn physical systems in an end-to-end manner, our approach guarantees longterm stability and therefore leads to more accurate solutions. Furthermore, our method improves the run time performance of traditional iterative solvers for rod dynamics. To explore our method we make use of position-based dynamics (PBD) as a common solver for physical systems and evaluate it by simulating the dynamics of elastic rods. Our approach is able to generalize across different initial conditions, discretizations, and realistic material properties. We demonstrate that it also performs well when taking discontinuous effects into account such as collisions between individual rods. Finally, to illustrate the scalability of our approach, we simulate complex 3D tree models composed of over a thousand individual branch segments swaying in wind fields.
+
+# 1 INTRODUCTION
+
+The numeric simulation of a dynamic system commonly comprises the derivation of the mathematical model given by the underlying differential equations and their integration forward in time. In the context of physics-based systems, the mathematical model is usually based on first principles and depending on the properties of the simulated system, the numerical integration of a complex system can be very resource demanding (Nealen et al., 2006), e.g., hindering interactive applications. Enabled by the success of deep neural networks to serve as effective function approximators, researchers recently started investigating the applicability of neural networks for simulating dynamic systems. While many physical phenomena can well be described within fixed spatial domains (e.g., in fluid dynamics) that can be learned with convolutional neural network (CNN) architectures (Chu & Thuerey, 2017; Guo et al., 2016; Tompson et al., 2016; Xiao et al., 2020), a large range of physical systems can more naturally be represented as graphs. Examples include systems based on connected particles (Muller et al., 2007), coupled oscillators (Michels & Desbrun, 2015; Michels et al., 2014), ¨ or finite elements (Nealen et al., 2006). Existing methods enable learning these systems often in an end-to-end manner and with a focus on replacing the entire or a part of the integration procedure. A number of methods show initial success in approximating physical systems; however, they often fail to reliably simulate the state of a system over longer time horizons if significant disadvantages are not accepted, such as the use of large datasets containing long-term simulations and the employment of specific memory structures (Sanchez-Gonzalez et al., 2020).
+
+In this paper, we aim to improve the performance of iterative solvers for physical systems with graph networks (GN). An iterative solver requires an initial guess, and based on it generates a sequence of improving approximate solutions. The initial guess can be computed by simply using values obtained in the previous iteration or by solving a few steps of a similar but simpler physical system. The performance of an iterative solver is significantly influenced by the calculation of the initial guess, which we aim to replace with the prediction of a GN. To demonstrate our approach, we use a position-based dynamics (PBD) solver that approximates physical phenomena by using sets of connected vertices (Bender et al., 2017; 2014b; Macklin et al., 2016; Muller et al., 2007). To ¨ simulate a physical system, PBD first computes updated locations of vertices using symplectic Euler integration to then correct the initial position estimates so as to satisfy a set of predefined constraints. The correction step is known as constraint projection and is commonly solved iteratively. The explicit forward integration for predicting the system’s updated state has negligible cost, whereas the projection step is computationally expensive. Our goal is to employ a GN to predict the outcome of the constraint projection step as an initial guess. This way, our approach inherits the long-term stability of a classic PBD solver, while providing better run-time performance.
+
+![](images/66d9183e801abcbe38f01980eb6475c1e896e2ffa8061ae8d4129f00236e3887.jpg)  
+Figure 1: Renderings taken from real-time simulations of the elastic deformation of a helix falling down on the ground plane (left) and two rods colliding with each other (right).
+
+To showcase the capabilities of our combined PBD solver, we aim to simulate the physically plausible mechanics of elastic rods. Rods play an important role for a variety of application domains, ranging from surgical simulation of sutures (Feess et al., 2016), catheters, and tendons (Pai, 2002), to human hair (Michels et al., 2015) and vegetation (Pirk et al., 2017) in animated movies. Furthermore, approaches exist to realistically simulate rods as sets of connected vertices accurately capturing their mechanical properties (Bergou et al., 2008; Kugelstadt & Schoemer, 2016; Michels et al., 2015; Pai, 2002). Our approach is able to generalize across different initial conditions, rod discretizations, and realistic material parameters such as Young’s modulus and torsional modulus (Deul et al., 2018). Moreover, we demonstrate that our approach can handle discontinuous collisions between individual rods. Figure 1 shows examples of elastic rod deformations of a helix falling down (left) and two colliding rods (right). Finally, we show that the data-driven prediction of the initial guesses of the constraint projection leads to a decreased number of required iterations, which – in turn – results in a significant increase of performance compared to canonical initial guesses.
+
+In summary, our contributions are: (1) we show how to accelerate iterative solvers with GNs; (2) we show that our network-enabled solver ensures long-term stability required for simulating physical systems; (3) we showcase the effectiveness of our method by realistically simulating elastic rods; (4) we demonstrate accuracy and generalizability of our approach by simulating different scenarios and various mechanical properties of rods including collisions and complex topologies (dynamic tree simulations).
+
+# 2 RELATED WORK
+
+In the following we provide an overview of the related work, which spans from data-driven physics simulations and graph learning to position-based dynamics and elastic rods.
+
+Data-driven Physics Simulations. It has been recognized that neural networks can be used as effective function approximators for physical and dynamic systems. To this end, early approaches focus on emulating the dynamics of physics through learned controllers (Grzeszczuk et al., 1998) or by designing subspace integrators (Barbic & James, 2005). Today, a range of approaches exist that enable ˇ learning ordinary and partial differential equations (Lagaris et al., 1998; Raissi et al., 2019; Raissi & Karniadakis, 2018), for example, to transform them into optimization problems (Dissanayake & Phan-Thien, 1994), to accelerate their computation (Mishra, 2018; Sirignano & Spiliopoulos, 2018), or to solve for advection and diffusion in complex geometries (Berg & Nystrom, 2018). ¨ Other methods focus on specific data-driven solutions for non-linear elasticity (Ibanez et al., 2017), ˜ for approximating Maxwell’s equation in photonic simulations (Trivedi et al., 2019), or for animating cloth (Wang et al., 2011), partially focusing on interactive applications (Holden et al., 2019). More recently, research on data-driven approaches for modeling the intricacies of fluid dynamics has gained momentum (Ladicky et al., 2015; Ummenhofer et al., 2020). Due to fixed-size spatial rep- ´ resentation of Eulerian fluid solvers, a number of approaches rely on CNN-type architectures (Chu & Thuerey, 2017; Guo et al., 2016; Tompson et al., 2016; Xiao et al., 2020). Furthermore, it has been shown that data-driven approaches can even be used to approximate the temporal evolution of fluid flows (Wiewel et al., 2018), to compute liquid splashing (Um et al., 2017), artistic styletransfer (Kim et al., 2020), or to derive fluid dynamics from reduced sets of parameters (Kim et al., 2019).
+
+Graph-based Learning. Graphs have proven to be a powerful representation for learning a wide range of tasks (Battaglia et al., 2018; Scarselli et al., 2009). In particular, it has been shown that graphs enable learning knowledge representations (Kipf et al., 2018), message passing (Gilmer et al., 2017), or to encode long-range dependencies, e.g., as found in video processing (Wang et al., 2017). A variety of methods uses graph-based representations to learn properties of dynamic physical systems, e.g. for climate prediction (Seo & Liu, 2019), with an emphasis on individual objects (Chang et al., 2016) and their relations (Sanchez-Gonzalez et al., 2018), for partially observable systems (Li et al., 2018), the prevalent interactions within physical systems (Kipf et al., 2018), hierarchically-organized particle systems (Mrowca et al., 2018), or – more generally – physical simulation (Sanchez-Gonzalez et al., 2019; 2020). While many of the existing approaches learn the time integration of physical systems in an end-to-end manner, we use a graph network to predict the outcome of a PBD solver for rod dynamics to enable more efficient computations.
+
+![](images/43792af9bed5d0f27d102a89cfeb714f437c3c4218295dc2fb7b707853997e9d.jpg)  
+Figure 2: Illustration of the discretization of a single rod using several rod segments arranged along its centerline (left). Each rod segment is described by its position and orientation within the generalized coordinates $\mathbf { p } _ { i }$ . The Lagrange multipliers $\lambda _ { i }$ represent the interaction between rod segments. The forward integration path is illustrated in red (middle) and constraint projection in green (right).
+
+Position-based Dynamics and Elastic Rods. PBD is a robust and efficient approach for simulating position changes of connected sets of vertices (Bender et al., 2017; 2014b; Macklin et al., 2016; Muller et al., 2007). Compared to forced-based methods that compute the force directly, the inter- ¨ action between different vertices in PBD is realized by a constraint projection step in an iterative manner. To avoid the dependency of the system’s stiffness on the number of iterations and the time step size, an extended position-based dynamics approach was introduced (XPBD) (Macklin et al., 2016). A number of methods exist that model the dynamic properties of rods that can even simulate more complicated rod mechanics (Pai, 2002). Moreover, particle systems were employed to simulate the dynamics of rods (Michels et al., 2017) and, in particular, for the physically accurate simulation of thin fibers (Michels et al., 2015) such as present in human hair or textiles. On a different trajectory, it has been recognized that rods can be simulated based on PBD (Umetani et al., 2014). The initial formulation was improved (Kugelstadt & Schoemer, 2016) by including the orientation of rod segments in the system’s state to account for torsion effects. Later, the XPBD framework was utilized (Deul et al., 2018) to address the non-physical influence of iteration numbers and steps sizes, which enables the more accurate simulation of elastic rods.
+
+# 3 METHODOLOGY
+
+We propose a novel approach to simulate the temporal evolution of a dynamic system which consists of elastic rods. Each rod is discretized using several rod segments arranged along its centerline (Figure 2). For each rod segment, its state is described by its position, orientation, velocity and angular velocity. The state of the system is given as the set of the individual states of all rod segments. The simulation is carried out by employing PBD (Muller et al., 2007) directly manipulating the sys- ¨ tem’s state. Orientations are represented as quaternions allowing for a convenient implementation of bending and twisting effects (Kugelstadt & Schoemer, 2016). Extended PBD (XPBD) (Macklin et al., 2016) is implemented to avoid that the rods’ stiffnesses depends on the time step size and the number of iterations (Deul et al., 2018).
+
+The generalized coordinates of a rod segment $i$ at time $t$ is given by $\mathbf { p } _ { i , t } \in \mathbb { R } ^ { 3 } \times \mathbb { H }$ , which includes its position $\mathbf { x } _ { i , t } \in \mathbb { R } ^ { 3 }$ given in Cartesian coordinates and its orientation described by a quaternion $\mathbf { q } _ { i , t } \in \mathbb { H }$ . Correspondingly, $\boldsymbol { v } _ { i , t } \in \mathbb { R } ^ { 6 }$ refers to the generalized velocity of the rod segment, which includes velocity and angular velocity. The system is continuously updated during the simulation by applying corrections $\bar { \Delta } \mathbf { p } _ { i } = ( \Delta \mathbf { x } _ { i } , \Delta \phi _ { i } ) ^ { \mathsf { T } } \in \mathbb { R } ^ { 6 }$ with position shifts $\Delta \mathbf { x } _ { i } \in \mathbb { R } ^ { 3 }$ and orientation shifts $\bar { \Delta { \phi _ { i } } } \in \mathbb { R } ^ { 3 }$ representing the integration of the angular velocity.1
+
+A single time integration step is presented in Algorithm 1. In the beginning (lines 1 to 4), generalized velocity and generalized coordinates are updated by employing a symplectic Euler integration step. In this regard, $\mathbf { a } _ { \mathrm { e x t } }$ denotes the generalized acceleration due to the external net force, e.g., given by gravity. XPBD (Macklin et al., 2016) employs the Lagrange multiplier $\boldsymbol { \lambda }$ which is initialized as zero (line 5) along with the integrated generalized coordinates $\mathbf { p } ^ { * }$ . Collision detection results are stored in $\mathbf { C o l l } _ { \mathrm { r - r } }$ and $\mathbf { C o l l } _ { \mathrm { r - p } }$ (line 6), where $\mathbf { C o l l } _ { \mathrm { r - r } }$ includes all the pairs of two rod segments that potentially collide with each other and $\mathbf { C o l l } _ { \mathrm { r - p } }$ includes information of all rod segments that potentially collide with another object such as a plane. Within several solver iterations, we alternate between rod constraint projection and the collision constrain projection (lines 7 to 15). The rod constraints include shear-stretch and bend-twist constraints representing the corresponding elastic energy. The Lagrange multiplier represents the interaction between rod segments. Figure 2 illustrated the discretization for a single rod into several interacting segments. The correction values $\Delta \mathbf { p }$ and $\Delta \lambda$ in line 9 are computed by constraint projection (Deul et al., 2018; Kugelstadt & Schoemer, 2016). The generalized coordinates and Lagrange multipliers are updated for each rod (lines 8 to 12), and rodrod and rod-plane collisions are addressed to update the generalized coordinates. For details about the collision projection procedure, we refer to Macklin et al. (2014). For the non-collision case, the steps within line 6 and 13 are not needed.
+
+Algorithm 1 Numerical integration procedure updating $\mathbf { p } _ { i , t } \mapsto \mathbf { p } _ { i , t + \Delta t }$ and $\begin{array} { r } { \pmb { v } _ { i , t } \mapsto \pmb { v } _ { i , t + \Delta t } } \end{array}$ .
+
+1: for all rod segments do   
+2: $\pmb { v } _ { i } ^ { * } \gets \pmb { v } _ { i , t } + \Delta t \mathbf { a } _ { \mathrm { e x t } }$   
+3: $\mathbf { p } _ { i } ^ { * } \gets \mathbf { p } _ { i , t } + \Delta t \ \mathbf { H } ( \mathbf { q } _ { i , t } ) \mathbf { \sigma } _ { i } ^ { * }$ with $\mathbf { H } ( \mathbf { q } _ { i , t } ) : = \left[ \mathbf { 1 } _ { 3 \times 3 } , \mathbf { 0 } _ { 3 \times 3 } ; \mathbf { 0 } _ { 4 \times 3 } , \mathbf { G } ( \mathbf { q } _ { i , t } ) \right]$   
+4: end for   
+5: $\pmb { \lambda } ^ { 0 }  \mathbf { 0 } , \mathbf { p } ^ { 0 }  \mathbf { p } ^ { * }$   
+6: $\mathrm { ( C o l l _ { r \mathrm { - } r } , C o l l _ { r \mathrm { - } p } )  }$ generateCollisionConstraints $( \mathbf { p } , \mathbf { p } ^ { * } )$   
+7: for $j  0$ to number of required solver iterations do   
+8: for all rods do   
+9: $\begin{array} { l } { ( \Delta \mathbf { p } , \Delta \pmb { \lambda } )  \mathsf { r o d C } } \\ { \lambda ^ { j + 1 }  \lambda ^ { j } + \Delta \pmb { \lambda } } \\ { \mathbf { p } ^ { j + 1 }  \mathbf { p } ^ { j } + \Delta \mathbf { p } } \end{array}$ onstraintProjection $( \mathbf { p } ^ { j } , \lambda ^ { j } )$   
+10:   
+11:   
+12: end for   
+13: $\mathbf { p } ^ { j + 1 } $ updateCollisionConstraintProjection $( \mathbf { p } ^ { j + 1 } , \mathbf { C o l l } _ { \mathrm { r - r } } , \mathbf { C o l l } _ { \mathrm { r - p } } )$   
+14: $j \gets j + 1$   
+15: end for   
+16: for all rod segments do   
+17: $\begin{array} { r l } & { \mathbf { p } _ { i , t + \Delta t }  \mathbf { p } _ { i } ^ { j } } \\ & { \mathbf { \delta } \mathbf { { \sigma } } \mathbf { { v } } _ { i , t + \Delta t }  \mathbf { H } ^ { \top } ( \mathbf { q } _ { i , t } ) ( \mathbf { p } _ { i , t + \Delta t } - \mathbf { p } _ { i , t } ) / \Delta t } \end{array}$   
+18:   
+19: end for
+
+![](images/1568f99e63d0de48aa59add5ada4bf6c639c6b83fad06ae80808b98bf25bb205.jpg)  
+Figure 3: Illustration of our approach incorporating a network which consists of $M$ graph networks (GN-blocks) into the position-based dynamics framework.
+
+The most expensive part of Algorithm 1 involves the computation of the corrections of generalized coordinates and Lagrange multipliers (line 9). This projection step requires the solution of a linear system which is a linearization of a non-linear one so that the matrix depends on the system’s state making it impossible to precompute its inverse. Instead, a new system at every point in time is solved iteratively using the conjugated gradient (CG) solver. Such iterative solvers are widely used in the context of physical simulations and regularly described as the de facto standard (Barrett et al., 1994; Saad, 2003) since they often show superior performance and usually scale well allowing for exploiting parallel hardware. However, we would like to point out that also highly efficient direct solvers can be found in the literature (Deul et al., 2018).
+
+Instead of fully replacing the projection step in an end-to-end learning manner, we follow the strategy of accelerating it by first computing a guess
+
+$$
+( \Delta \mathbf { p } ^ { 0 } , \Delta \lambda ^ { 0 } ) \gets \mathsf { c o r r e c t i o n } \mathsf { G u e s s } ( \mathbf { p } ^ { j } ) ,
+$$
+
+for the iterative procedure (line 9)
+
+$$
+( \Delta { \bf p } , \Delta \lambda ) \gets \mathrm { r o d } { \sf C o n s t r a i n t P r o j e c t i o n } ( { \bf p } ^ { j } , \lambda ^ { j } , \Delta { \bf p } ^ { 0 } , \Delta \lambda ^ { 0 } ) .
+$$
+
+A neural network is employed to compute the initial guess in Eq. (1) for the constraint projection. The motivation for this approach is to reduce the number of iterations required for the convergence of the CG solver which solves the linear system in Eq. (2) compared to the canonical initialization with zeros. We obtain our final framework by replacing line 9 in Algorithm 1 with Eq. (1) and Eq. (2) as illustrated in Figure 3, which is inherently as accurate as the traditional PBD method. We name the data-driven part COPINGNet (“COnstraint Projection INitial Guess Network”) which learns to compute the correction guess.
+
+# 3.1 GRAPH ENCODING
+
+COPINGNet is a graph network based architecture which we apply in order to compute initial guesses for $\Delta \mathbf { p }$ and $\Delta \lambda$ . In this regard, we need to incorporate the rods’ state into a graph description (Battaglia et al., 2018). A graph $G = ( V , E , U )$ usually consists of nodes (or vertices) $V$ , edges $E$ as well as global features $U$ . However, in our framework, global features $U$ are not used. For example, gravity could be a potentially meaningful global feature, but it also can be easily included as an external acceleration. Hence, $U = \emptyset$ and the graph can just be represented as $G = G ( V , E )$ . In our case, the rods’ segments within the scene are represented by the graph’s nodes while the interactions between the rods’ segments are represented by the edges.
+
+COPINGNet provides a graph-to-graph mapping: $\mathbb { G } ^ { \mathrm { { i n } } } \xrightarrow { } \mathbb { G } ^ { \circ \mathbf { u t } }$ , from an input graph $G _ { \mathrm { i n } } \in \mathbb { G } ^ { \mathrm { i n } }$ to an output graph $G _ { \mathsf { o u t } } \in \mathbb { G } ^ { \mathsf { o u t } }$ . Nodes and edges of both graphs are equipped with specific features. In the case of the input graph, the node features describe the state of the rods’ segments, i.e.
+
+in which the positions are denoted with $\mathbf { x } _ { i } ~ \in ~ \mathbb { R } ^ { 3 }$ , the orientations with $\mathbf { q } _ { i } ~ \in ~ \mathbb { H }$ , the radii with $r _ { i } ~ \in \mathbb { R } ^ { > 0 }$ , the densities with $\rho _ { i } \in \mathbb { R } ^ { > 0 }$ , and the segment lengths with $\bar { \ell _ { i } } \ \in \ \mathbb { R } ^ { > 0 }$ . Moreover, a segment position indicator $\alpha _ { i } \in [ 0 , 1 ] \ \subset \ \mathbb { R }$ is included corresponding to a parameterization by arc length.2 Binary segment flag $\mathsf { \bar { f } } _ { 0 _ { i } } \in \{ 0 , 1 \}$ , “left” end flag $f _ { 1 _ { i } } \in \{ 0 , 1 \}$ and “right” end flag $f _ { 2 _ { i } } \in \{ 0 , 1 \}$ are set to zero if the specific segment respectively the left or right segment of the rod is fixed and to one otherwise. The nodes of $G _ { \mathrm { i n } }$ are given as the set of $V _ { \mathfrak { i n } } = \mathbf { \bar { \cup } } _ { i = 1 } ^ { n } \bar { \left\{ \mathbf { v } _ { \mathfrak { i n } , i } \right\} }$ , in which $n = | V _ { \mathsf { i n } } |$ denotes the number of segments in the scene. The nodes of $G _ { \mathsf { o u t } }$ contain the correction values of the generalized coordinates, i.e.
+
+While rod segments are represented as node features, we represent constraints between rod segments as edge features: $\mathbf { e } _ { \mathsf { i n } , i } = ( \omega _ { i } , _ { \mathsf { e } } ^ { } , T _ { i } ^ { \mathsf { T } } ) ^ { \mathsf { T } } \in \mathbb { E } ^ { \mathsf { i n } } \subseteq \mathbb { R } ^ { 5 }$ ,
+
+in which the (rest) Darboux vector $\omega \in \mathbb { R } ^ { 3 }$ describes the static angle of two rod segments, and Young’s modulus $Y ~ \in ~ \mathbb { R } ^ { > 0 }$ and torsion modulus $T ~ \in ~ \mathbb { R } ^ { > 0 }$ are corresponding to extension, bending, and twist constraint parameters. The set of edges of the input graph is then given by $E _ { \mathsf { i n } } = \mathsf { \bar { U } } _ { i = 1 } ^ { m } \{ \mathbf { e _ { i \mathsf { n } , i } } \}$ , in which $m = \vert E _ { \mathrm { i n } } \vert$ denotes the number of interactions between different segments. The correction of the Lagrange multiplier $\Delta \lambda _ { i } \in \mathbb { R } ^ { 6 }$ is stored in the output edges:
+
+The set of output edges is then given by $E _ { \mathsf { o u t } } = \cup _ { i = 1 } ^ { m } \{ \mathbf { e } _ { \mathsf { o u t } , i } \}$
+
+The connectivity information $C$ of each graph is stored in two vectors $\mathbf { c } _ { \mathsf { s d } }$ and $\mathbf { c } _ { \mathrm { r v } }$ containing the sender node index and the receiver node index of each corresponding edge. This concludes the specification of the input graph $G _ { \mathsf { i n } } = G ( V _ { \mathsf { i n } } , E _ { \mathsf { i n } } )$ and the output graph $\begin{array} { r } { G _ { \mathrm { o u t } } = G ( V _ { \mathrm { o u t } } , E _ { \mathrm { o u t } } ) } \end{array}$ with connectivity information $C = ( \mathbf { c } _ { \mathsf { s d } } , \mathbf { c } _ { \mathsf { r v } } )$ .
+
+# 3.2 NETWORK STRUCTURE
+
+In the following, we describe the structure of COPINGNet after we formalized its input and output in the previous section. As illustrated in Figure 3, the network consists of an encoder network, multiple stacks of GN-blocks, and a decoder network. The graph network from Battaglia et al. (2018) is used as a basic element and denoted as a GN-block. Residual connection between blocks could improve performance of neural networks in both CNN (He et al., 2015), and graph neural network [Li et al. (2019)]. As in related work (Sanchez-Gonzalez et al., 2020), we employ the residual connection between the GN-blocks, but our encoder/decoder network directly deals with the graph. The encoder network performs a mapping: $\mathbb { G } ^ { \mathsf { i n } } \to \mathbb { G } ^ { \mathsf { i a t e n t } }$ and is implemented using two multi-layer perceptrons (MLPs), $\mathsf { \bar { M } L P _ { e d g e } : \mathbb { E } ^ { i n } } \overset { \cdot } {  } \mathbb { E } ^ { | \mathrm { a t e n t } } \subseteq \mathbb { R } ^ { l }$ and $\mathsf { M L P } _ { \mathsf { n o d e } } \colon \mathbb { V } ^ { \mathsf { i n } } \to \mathbb { V } ^ { \mathsf { l a t e n t } } \subseteq \mathbb { R } ^ { l }$ , in which $l$ denotes the latent size. They work separately and thus $E _ { \mathsf { e n } } = \mathsf { M L P } _ { \mathsf { e d g e } } ( E _ { \mathsf { i n } } )$ and $V _ { \mathsf { e n } } = \mathsf { M L P } _ { \mathsf { n o d e } } ( V _ { \mathsf { i n } } )$ . Edge features $E _ { \mathrm { i n } }$ from the input are constant for a rod during the simulation and this results in constant encoded edge features $E _ { \mathrm { e n } }$ , which could be recorded after the first run and used afterwards during inference. The edge feature $\mathbf { e } _ { \mathrm { i n } , i }$ contains the material parameters which could vary by different orders of magnitude. Hence, we normalize Young’s modulus and torsion modulus before feeding them into the network. After encoding, the graph $G _ { \mathsf { e n } } ( V _ { \mathsf { e n } } , E _ { \mathsf { e n } } ) \ \in \ \mathbb { G } ^ { | \mathsf { a t e n t } }$ is passed to several GN-blocks with residual connections. Each GN-block also contains two MLPs. However, they use message passing taking advantage of neighbourhood nodes/edges information (Battaglia et al.,
+
+Under review as a conference paper at ICLR 2021   
+
+<table><tr><td>Train/Val 256/100</td><td>#Steps 50</td><td>#Nodes N Ud(10,55)</td><td>Young&#x27;sModulus Y 10° Pa,a ~U(4.0,6.0)</td><td>Initial Angle0 U(0°,45.0°)</td><td>RodLength l U(1.0 m,5.0 m)</td></tr><tr><td>Train/Val</td><td>#Steps</td><td>#Nodes N</td><td>Torsion Modulus T</td><td>Helix Radius/Height</td><td>Winding Number</td></tr><tr><td>256/100</td><td>50</td><td>Ud(45,105)</td><td>10° Pa,a ~U(4.0,6.0)</td><td>u(0.4 m,0.6 m)/u(0.4 m,0.6 m)</td><td>U(2.0,3.0)</td></tr></table>
+
+Table 1: Specification of training and validation datasets for the two scenarios of an initially straight bending rod (top) and an elastic helix (bottom). The datasets are comprised of a number of data points (left) each describing the rod’s dynamics within $t \in [ 0 \mathrm { ~ s } , 5 0 \Delta t ]$ discretized with a time step size of $\Delta t = 0 . 0 2 { \mathrm { ~ s } }$ . The number of nodes $N$ is sampled from a discrete uniform distribution $\mathcal { U } _ { \mathrm { d } }$ while the remaining parameters are sampled from a continuous uniform distribution $\mathcal { U }$ . We trained our network for 5 hours for the bending rod scenario and 6 hours for the helix case.
+
+![](images/2c001d770b6a2025e2608f89b2ea7d64d5e863ed2a8335a6ed9184e85bda896c.jpg)  
+Figure 4: Illustration of the ratio of COPINGNet’s inference time $t _ { \mathrm { i n f e r } }$ and the vanilla CG solver’s run time $t _ { C G }$ (purple curves; right vertical axis) for the initially straight bending rod (left) and the elastic helix (right) simulations. The black curves show the CG iteration number ratio while the red curves show the total speedup of the constraint projection when taking into account COPINGNet’s inference time (left vertical axis). The pink curves show the total speedup of the entire simulations. The orange and green dashed line indicate the lower and upper boundaries of the total number of nodes used in the training data. We can observe a speedup even for rods and helices with greater node number than the ones used in the the training dataset. The result is averaged from 50 simulations each running 100 steps.
+
+2018). A number of $M$ GN-blocks enable the use of neighbourhood information with distances smaller or equal to $M$ . The graph network performs a mapping within the latent space: $\mathbb { G } ^ { \mathsf { l a t e n t } } \to$ $\mathbb { G } ^ { | \mathsf { a t e n t } }$ , and after $M$ GN-blocks, we obtain $\bar { G } _ { \mathsf { e n } } ^ { \prime } ( V _ { \mathsf { e n } } ^ { \prime } , E _ { \mathsf { e n } } ^ { \prime } ) \in \mathbb { G } ^ { \mathsf { l a t e n t } }$ . The decoder network performs a mapping: $\mathbb { G } ^ { \mathsf { l a t e n t } } \to \mathbb { G } ^ { \mathsf { o u t } }$ , which has a similar structure as the encoder network. Two MLPs $\mathsf { M L P _ { e d g e } : \mathbb { E } ^ { l a t e n t } \to \mathbb { E } ^ { o u t } }$ and $\mathsf { M L P _ { n o d e } : \mathbb { V } ^ { l a t e n t } \to \mathbb { V } ^ { o u t } }$ compute $\begin{array} { r } { E _ { \mathsf { o u t } } = \mathsf { M L P _ { e d g e } } ( E _ { \mathsf { e n } } ^ { ' } ) } \end{array}$ and $V _ { \mathrm { o u t } } = \mathsf { M L P } _ { \mathsf { n o d e } } ( V _ { \mathrm { e n } } ^ { ' } )$ . A tanh-function at the end of ${ \mathsf { M L P } } _ { \mathsf { e d g e } }$ and ${ \mathsf { M L P } } _ { \mathsf { n o d e } }$ is used restricting the output to the interval $[ - 1 , 1 ] \subset \mathbb { R }$ . COPINGNet learns the relative correction values. The generated dataset is normalized, and the maximum correction value of the generalized coordinates and the Lagrange multipliers will be recorded as norms. The final correction value is achieved by multiplying the relative correction values and the norms. This normalization process damps the noise caused by the network and leads to a more stable performance. For simplicity, all the MLPs in different blocks have the same number of layers, and the same width as latent size $l$ within the latent layers. The input and output sizes of each MLP are consistent with the corresponding node/edge feature dimensions. The loss $L$ is computed from both parts, nodes and edges,
+
+$$
+L : = \mathsf { M S E } ( V _ { \mathrm { o u t } } , \tilde { V } _ { \mathrm { o u t } } ) + \mathsf { M S E } ( E _ { \mathrm { o u t } } , \tilde { E } _ { \mathrm { o u t } } ) ,
+$$
+
+in which $( V _ { \mathrm { o u t } } , E _ { \mathrm { o u t } } )$ is the output graph’s ground truth in contrast to COPINGNet’s prediction $( \tilde { V } _ { \mathrm { o u t } } , \tilde { E } _ { \mathrm { o u t } } )$ . The mean squared error between $\chi$ and $\tilde { \chi }$ is denoted with ${ \mathsf { M S E } } ( \chi , \tilde { \chi } )$ .
+
+For evaluation purposes, we also incorporated the $k$ -nearest neighbor $k$ -NN) algorithm as described in the supplementary material setting $k = 3$ .
+
+# 4 EVALUATION
+
+We generate training and validation datasets based on two scenarios: an initially straight bending rod and an elastic helix each fixed at one end oscillating under the influence of gravity. The specification of these datasets is provided in Table 1. The PBD code is written in $\mathrm { C } { + } { + }$ , while the COPINGNet is implemented in PyTorch. The training is performed on an NVIDIA
+
+![](images/ed1e9b6d007b1d7eedb0fe3c233c6801ad15aea379f4a368495e2ea2be90dc0b.jpg)  
+Figure 5: Realistic biomechanical simulation of a 3D tree model composed of over $1 k$ nodes swaying in a wind field. Our GN approach performs correctly even under a large number of rod segments while increasing performance of the original PBD method.
+
+![](images/62c93181cb7245b24d09430f33cbbbb1481298dad4a26237055e6d5ad4b12b0b.jpg)  
+Figure 6: The dashed horizontal lines show the benchmark CG solver’s constant performance for an initially straight bending rod simulation (left) and an elastic helix simulation (right). The parameters $\phi _ { 0 } = 0 ^ { \circ }$ , $N = 3 0$ , $\ell = 3 . 0 \mathrm { m }$ , $Y = 1 . 0 \cdot 1 0 ^ { 5 }$ Pa (left), and $\mathsf { H R } = 0 . 5 \ \mathrm { m }$ , $\mathsf { H } \mathsf { H } = 0 . 5 \ \mathrm { m }$ , $\mathsf { H W } = 2 . 5$ , $G = 1 . 0 \cdot 1 0 ^ { 5 }$ Pa, $N = 6 0$ (right) are applied. We are comparing our COPINGNet-assisted PBD approach (orange) to a simplified version in which the $k$ -NN (green) is used to predict the initial guess. Moreover (left), we added a performance measurement in which we restricted the solution to be in a two-dimensional plane (black). The results are averages over 20 simulations and smoothed with a window of 10 frames. The dashed vertical lines mark the number of 50 frames contained in the training data set. For the helix simulation we observe a speed up of approximately $50 \%$ even beyond the training data (50 time frames). For the bending rod simulation we observe a speedup that further decreases with increasing time frame number.
+
+Discretization. Our approach addresses the acceleration of the most expensive part within PBD by providing an accurate initial guess of the constraint projection. We measure the system’s complexity by the number of nodes in a rod. Figure 4 shows the ratio of COPINGNet’s inference run time (black and red curves) compared to the run time of the vanilla CG solver (purple curves) for different numbers of nodes. The increasing black and red curves indicate that the speedup of COPINGNet is more significant with greater number of nodes. With only a few nodes the CG solver performs better due to the inference overhead. Once the number of nodes is increasing, a significant speedup can be obtained of up to $5 0 \%$ for the constraint projection. Surprisingly, our approach also performs well when going far beyond the sampling range (orange to green dashed lines). Since the constraint projection is the most time-consuming part of the entire simulation, the speedup converges to that of the whole simulation (pink curves) with increasing number of nodes as shown in Figure 4.
+
+Temporal Evolution. In addition to the complexity analysis, we also analyze the required number of CG iterations for the vanilla constraint projection compared to the one accelerated using COPINGNet over time as shown in Figure 6. We obtain a significant total speedup compared to the CG solver (dashed blue line). As stated in Table 1, our training data contains dynamical simulations of 50 time steps. In this range we observe the highest speedup. As was the case for the complexity analysis, we again obtain a significant speedup for simulations beyond 50 time steps. This performance gain is more present for the helix (green curve) compared to the rod simulation (orange curve).
+
+Long-term Stability. In case the constraint projection is completely replaced by COPINGNet (endto-end approach), stability of the PBD method decreases as error accumulation takes place. This is illustrated in Figure 7 showing the temporal evolution of the relative change of the total rod length. An initially straight rod bending under the influence of gravity is simulated using the parameters $\phi _ { 0 } = 0 ^ { \circ }$ , $N = 3 0$ , $\ell = 4 . 0 \mathrm { m }$ , and varying Young’s modulus $Y$ . In this scenario the rod length is expected to stay constant during the mechanical simulation. In case COPINGNet is used in an endto-end manner (colored lines), where the whole constraint projection step is replaced, we observe that the rod length changes incorrectly. In fact, the divergence of using COPINGNet to replace the constraint projection step increases exponentially beyond the range of the training data (50 steps). To the contrary, when COPINGNet is used to only estimate the initial guess of the constraint projection (black lines), no rod deformation takes place even beyond the training data range.
+
+Collisions. Figure 1 illustrates a collision of an elastic helix with the ground plane and a collision between two rods. Collision detection is efficiently implemented using the hierarchical spatial hashing scheme according to Eitz & Gu (2007). Rod-rod collisions are then treated with line-to-line distance constraints, and collisions with the ground plane using half-space constraints. Moreover, frictional effects are implemented according to Macklin et al. (2014). This approach allows us to handle discontinuous events such as collisions between individual rods and other objects. Moreover, for this experiment we use the neural networks trained on the rod and helix simulations. Employing COPINGNet trained on the helix data to simulate the collision of the ground plane $( \mathsf { H } \mathsf { R } = \mathsf { H } \mathsf { H } = 0 . 5 \mathrm { ~ m ~ }$ , $\mathsf { H W } = 2 . 5$ , $G = 1 . { \overset { \vartriangle } { 0 } } \cdot 1 0 ^ { 6 } { \overset { \vartriangle } { \ v u } }$ Pa, $N = 5 0$ ), we measure a total speedup of approx. $1 0 \%$ . In the case of two colliding rods $\langle \phi _ { 0 } = 0 ^ { \circ }$ , $N \in \{ 2 0 , 3 0 \}$ , $\ell \in \{ 4 . 0 \mathrm { m } , 4 . 5 \mathrm { m } \}$ , $Y = 1 . 0 \cdot 1 0 ^ { 6 } \ : \mathrm { P a } )$ , we obtain a speedup of approx. $6 \%$ .
+
+Complex Scenarios. Our method is also capable to deal with complex scenarios such as 3D tree models swaying in wind fields as shown in Figure 5. We represent trees using the extended Cosserat rod theory introduced in Pirk et al. (2017). This method allows us to simulate realistic biomechanics of thousands of rod segments at interactive rates. We generated 100 different tree topologies using 70 individual rods (average node number: 1056) and simulate the swaying motions of these tree models with vanilla PBD to generate the training dataset. For the evaluation, 30 different tree topologies have been generated for each of the following four experiments using 10 rods (204 nodes), 20 rods (355 nodes), 40 rods (654 nodes), and 70 rods (1061 nodes). We were able to significantly improve the runtime performance of the method by $1 7 . 0 \%$ (10 rods), $1 5 . 8 \%$ (20 rods), $1 3 . { \bar { 0 \% } }$ (40 rods), and $1 1 . 1 \%$ (70 rods). This takes into account the inference time introduced by the neural network.
+
+![](images/a880abe5ec3310b65f98fc0d9a3fc219a05368751e37b593573a311958b42906.jpg)  
+Figure 7: Illustration of the relative change of the total rod length $( l / l _ { 0 } )$ for different values of Young’s moduli $( Y )$ . Colored lines show different results for COPINGNet replacing the constraint projection. The thick black line represent the result for COPINGNet replacing only the initial guess. This indicates that the end-to-end approach does not generalize past the 50 time steps used in preparing the training data.
+
+Generalization. Figure 7 and Figure 10 (supplementary material) indicate that using COPINGNet in an end-to-end manner does not generalize beyond the training data. Specifically, the end-to-end setup diverges in terms of rod geometry and segment position from the correct solution. This effect increases significantly beyond the training data range. Although, we only use COPINGNet as a benchmark for this evaluation, other GNs are expected to perform similarly. Common workarounds to increase the stability of dynamical systems with neural networks are temporal skip-connections, recurrent training, and data augmentation. However, these approaches focus on runtime speed and memory performance rather than stability (Holden et al., 2019). Interestingly, replacing just the initial guess with a GN does not deform the rod or introduces discontinuities at any observed stage of the bending simulation. This means that employing GNs can result in a performance increase without a loss of stability. Furthermore, a GN that only provides initial guesses seems to also generalize to other scenarios. We observed performance improvements for the collision and complex tree case, although the network was never trained on these different rod discretizations and topologies. This indicates that our method is capable to generalize within a specific physical scenario and to a lesser extent to other scenarios.
+
+# 5 CONCLUSION
+
+We discovered that applying GNs for replacing the initial guess has fundamental advantages over end-to-end approaches. First, our network-enabled solver ensures long-term stability inherited from traditional solvers for physical systems, while improving runtime performance. Second, our approach is able to generalize across different initial conditions, rod discretizations, and material parameters, and it handles discontinuous effects such as collisions. While end-to-end approaches offer more significant speedups, our method is superior in cases where stability is an essential requirement. Our approach to accelerate iterative solvers with GNs opens multiple avenues for future work. For one, it would be interesting to explore mechanical systems describing general deformable (e.g. textiles) or volumetric objects, which have been simulated with PBD. Second, our approach can be applied to other iterative methods, such as in finite elements analysis or in the context of linear complementarity problems (LCP). This would allow us to accelerate physical simulations, when iterative solvers are applied, without compromising stability.
+
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+
+# SUPPLEMENTARY MATERIAL
+
+DYNAMIC RESULTS
+
+![](images/0b597f1dd7e6c0c813a7a05d75e5ba2bca3c7d5c214676eb06c33d3baf6e7b1a.jpg)  
+Figure 8: Illustration of the temporal evolution of scenarios (a) to (d) for the initially straight bending rod, and (e) to (f) for the elastic helix. The different cases represent various material property configurations.
+
+In Figure 8 we show the temporal evolution of different scenarios (a) to (d) for the initially straight bending rod, and (e) to (f) for the elastic helix. The default parameters of the initially straight bending rod are $\phi _ { 0 } ~ = ~ 0 ^ { \circ }$ , $N \ = \ 3 0$ , $\ell ~ = ~ 4 . 0 ~ \mathrm { m }$ , and $\dot { Y } ~ = ~ 1 . 0 \cdot 1 0 ^ { 5 } ~ \mathrm { P a }$ . In (a), we modify $\phi _ { 0 } ~ \in ~ \{ 0 . 0 ^ { \circ } , 1 5 . 0 ^ { \circ } , 3 0 . 0 ^ { \circ } , 4 5 . 0 ^ { \circ } \}$ . In (b), we modify $N ~ \in ~ \{ 1 0 , 2 0 , 4 0 , 6 0 \}$ . In (c), we modify $\ell \in \{ 1 . 0 \mathrm { m } , 2 . 0 \mathrm { m } , 3 . 0 \mathrm { m } , 4 . 0 \mathrm { m } ,$ $5 . 0 \mathrm { m } , 6 . 0 \mathrm { m } \}$ . In (d), we modify $Y ~ \in ~ \{ 2 . 0 ~ \cdot$ $1 0 ^ { 4 } \mathrm { P a } , 5 . 0 \cdot 1 \dot { 0 } ^ { 4 } \mathrm { P a } , 1 . \dot { 0 } \cdot 1 0 ^ { 5 } \mathrm { P a } , 2 . 0 \cdot 1 0 ^ { 5 } \mathrm { P a } , 5 . 0 \cdot 1 0 ^ { 5 } \mathrm { P a } , 1 . \dot { 0 } \cdot 1 0 ^ { 6 } \mathrm { P a }$ . The default parameters of the elastic helix are $\mathsf { H } \mathsf { R } \ = \ 0 . 5 \ \mathrm { m }$ (helix radius), $\mathsf { H } \mathsf { H } \ = \ 0 . 5 \ \mathrm { m }$ (helix height), $\mathsf { H W } = 2 . 5$ (winding number), $T \ = \ 1 . 0 \cdot 1 0 ^ { 5 } \mathrm { { \ P a } }$ , and $N ~ = ~ 6 0$ . In (e), we modify $( { \mathsf { H } } { \mathsf { R } } , { \mathsf { H } } { \mathsf { H } } , { \mathsf { H } } { \mathsf { W } } ) \ \in$ $\{ ( 0 . 4 \mathrm { m } , 0 . 4 \mathrm { m } , 2 . 0 \mathrm { m } )$ , $( 0 . 4 \mathrm { m } , 0 . 4 \mathrm { m } , 3 . 0 \mathrm { m } )$ , $( 0 . 5 \mathrm { m } , 0 . 5 \mathrm { m } , 2 . 0 \mathrm { m } )$ , $( 0 . 5 \mathrm { m } , 0 . 5 \mathrm { m } , 3 . 0 \mathrm { m } ) \}$ . In (f), we modify $N \in \{ 3 0 , 6 0 , 8 0 , 1 0 0 \}$ . In (g), we modify $G \in \{ 2 . 0 \cdot 1 0 ^ { 4 } \mathrm { { P a } , 5 . 0 \cdot 1 0 ^ { 4 } \mathrm { { P a } , 1 . 0 \cdot 1 0 ^ { 5 } \mathrm { { P a } , 1 . 0 \cdot } } } $ $1 0 ^ { 6 } \mathrm { P a } \}$ . For each experiment, the rod colors indicate the corresponding parameters in the following (ascending) order: blue, orange, green, red, purple, brown.
+
+Dynamic results of our graph learning assisted simulations can be found in the video
+
+GraphLearningPhysicalSystems.mp4 (2:20 min, 20.6 MB, $1 9 2 0 \times 1 0 8 0$ , H.264)
+
+provided as additional supplementary material. These results are obtained by employing three GN blocks, two MLP layers, and a MLP width of 32. Parameters have been used which are are not covered within the training dataset.
+
+Moreover, to further demonstrate general applicability, Figure 9 shows an additional simulation of a tightening knot for which COPINGNet-based PBD shows a total speedup of $1 0 . 4 \%$ compared to vanilla PBD. COPINGNet is trained using a dataset containing 20 simulations of the same knot scenario with different discretizations over 100 frames. It can clearly be observed that COPINGNet generalizes beyond the training data.
+
+![](images/6752cbda86a8300a84044c3b5b500309708fa2e4efd317ebbb30644ecd988263.jpg)  
+Figure 9: Illustration of the temporal evolution of a knot scenario. The rod is fixed at both ends and the knot is pulled tight. The predicted results by end-to-end COPINGNet-based learning are shown in red while the results computed with COPINGNet-assisted PBD are shown in black. The visualization of the last frame contains a close-up to demonstrate that the knot structure is still preserved. The rod parameters are $N = 1 1 0$ , $R = 0 . 0 1 { \mathrm { m } }$ , $\ell = 1 1 . 0 \mathrm { m }$ , and $Y = T = 1 . 0 \cdot 1 0 ^ { 4 } \mathrm { P a }$ .
+
+Figure 10 illustrates the temporal evolution of a bending rod and elastic helix scenarios using different approaches. For the bending rod case, the parameters $\phi _ { 0 } = 0 ^ { \circ }$ , $N = 3 0$ , $\ell = 4 . 0 ~ \mathrm { m }$ , and $Y = 1 . { \overset { \underset { \textstyle \star } { } } { 0 } } \cdot 1 0 ^ { 5 }$ Pa are used. In the helix case, $\mathsf { H R } = 0 . 5 \ : \mathrm { m }$ (helix radius), $\mathsf { H H } = 0 . 5 \ : \mathrm { m }$ (helix height), $\mathsf { H W } = 2 . 5$ (winding number), $T = 1 . 0 \cdot 1 0 ^ { 5 } ~ \mathrm { P }$ a, and $N = 6 0$ were applied. The temporal evolution of the positions’ normalized root mean square error (NRMSE) is shown in Figure 11.
+
+![](images/b3bcd2519c63fdba03741161d6ac7115740fe4407ef7cdce2603dadb5e9a83d0.jpg)  
+Figure 10: Illustration of the temporal evolution of bending rod (upper row) and elastic helix (lower row) scenarios simulated using our COPINGNet-assisted PBD approach (blue curves), vanilla PBD (orange dotted curves), $k$ -NN-based end-to-end learning (purple curves) and GN-based end-to-end learning (green curves). While the COPINGNet-assisted approach and the vanilla PBD solver allow us to simulate both cases in a stable manner, both end-to-end learning approaches diverge with increasing time due to error accumulation.
+
+![](images/7af2ff4964c586a7b2b5eec02829443f3cf76cb9efa13d93aaa05b1cafb91ed6.jpg)  
+Figure 11: Illustration of the positions’ normalized root mean square error (NRMSE) compared to vanilla PBD using our COPINGNet-assisted PBD approach (blue curves), $k$ -NN-based end-toend learning (purple curves), and GN-based end-to-end learning (green curves) for the bending rod (left) and the elastic helix (right) scenarios. While the COPINGNet-assisted PBD approach converges to an almost correct solution, both end-to-end learning approaches show large error rates with increasing time frame number.
+
+# ARCHITECTURE
+
+The evaluation of COPINGNet’s architecture, introduced in Section 3.2, is presented in Figure 12. The performance is studied for different numbers of GN-blocks and MLP layers, and different MLP widths (latent sizes). The measurements demonstrate that a larger number of GN-blocks usually increases the performance while the performance improvement is not longer significant for more than four GN-blocks. This is plausible and can be considered analogously to the size of the stencil of a numerical integrator: a larger number of GN-blocks means that a specific node can take advantage of information gained from neighbourhoods further away. Correspondingly, a larger stencil can do so as well. However, increasing the size of the stencil usually does not result in more accurate results once the critical size is reached. This can be observed here as well. In contrast, it can be clearly observed that deeper MLP networks do not improve the performance. This is consistent with other research on graph networks (Sanchez-Gonzalez et al., 2020). However, increasing the MLP width can increase the network’s ability to generalize. It is shown that the highest performance is achieved with the largest MLP width, especially in the case of the helix scenarios. Since increasing the number of GN-blocks and MLP width will lead to longer inference time, we compromise by choosing medium numbers. For further evaluations, we employ three GN-blocks, two MLP layers, and a MLP width of 32.
+
+![](images/f63ca5cbc4111c5cfe5f36c2ca8872b766dd5b0361615dbc9bd1b72442468255.jpg)  
+Figure 12: Illustration of the evaluation of COPINGNet’s architecture (blue: benchmark architecture). The result is averaged from 50 test simulations each running for 100 time steps.
+
+$k$ -NEAREST NEIGHBOR ALGORITHM
+
+For evaluation purposes, we also incorporated the $k$ -nearest neighbor ( $k$ -NN) algorithm into our framework to compute an initial guess for the constraint projection step.
+
+Consider a $k$ -NN data point $( \bf x _ { i } , \bf y _ { i } )$ . Its components contain node and edge feature information:
+
+$$
+\begin{array} { r } { \mathbf { x } _ { i } = ( \mathbf { v } _ { \mathrm { i n } , i } , \mathbf { e } _ { \mathrm { i n } , i _ { k } } , \ldots ) ^ { \mathsf { T } } , \mathbf { y } _ { i } = ( \mathbf { v } _ { \mathrm { o u t } , i } , \mathbf { e } _ { \mathrm { o u t } , i _ { k } } , \ldots ) ^ { \mathsf { T } } , k \in \mathbb { N } _ { 0 } . } \end{array}
+$$
+
+Simulation data is collected to form a dataset $\left\{ \left( \mathbf { x } _ { i } , \mathbf { y } _ { i } \right) \right\}$ . During inference, the nearest $k$ data points of $\mathbf { x } _ { i } ^ { * }$ are picked and linear interpolations are performed to obtain the predicted $\mathbf { y } _ { i } ^ { * }$ .
+
+For each input node feature $\mathbf { v } _ { \mathrm { i n } , i } ^ { * }$ , we obtain the corresponding output node feature $\tilde { \mathbf { v } } _ { \mathrm { o u t } , i } = \mathbf { v } _ { \mathrm { o u t } , i } ^ { * }$ The output edge feature is averaged from the information of the corresponding component $\mathbf { y } _ { j } ^ { * }$ :
+
+$$
+\tilde { \mathbf { e } } _ { \mathrm { o u t } , i } = \frac { 1 } { M } \sum _ { j } \mathbf { e } _ { \mathrm { o u t } , j _ { k } } ^ { * } ,
+$$
+
+in which $M$ denotes the number of nodes.
+
+The $k$ -NN algorithm is applied as an end-to-end method completely replacing constraint projection or to predict an initial guess.

parse/train/v2tmeZVV9-c/v2tmeZVV9-c_content_list.json

@@ -0,0 +1,1672 @@
+[
+    {
+        "type": "text",
+        "text": "ACCURATELY SOLVING ROD DYNAMICS WITH GRAPH LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            55,
+            656,
+            103
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            127,
+            398,
+            155
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            152,
+            544,
+            167
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Iterative solvers are widely used to accurately simulate physical systems. These solvers require initial guesses to generate a sequence of improving approximate solutions. In this contribution, we introduce a novel method to accelerate iterative solvers for rod dynamics with graph networks (GNs) by predicting the initial guesses to reduce the number of iterations. Unlike existing methods that aim to learn physical systems in an end-to-end manner, our approach guarantees longterm stability and therefore leads to more accurate solutions. Furthermore, our method improves the run time performance of traditional iterative solvers for rod dynamics. To explore our method we make use of position-based dynamics (PBD) as a common solver for physical systems and evaluate it by simulating the dynamics of elastic rods. Our approach is able to generalize across different initial conditions, discretizations, and realistic material properties. We demonstrate that it also performs well when taking discontinuous effects into account such as collisions between individual rods. Finally, to illustrate the scalability of our approach, we simulate complex 3D tree models composed of over a thousand individual branch segments swaying in wind fields. ",
+        "bbox": [
+            233,
+            183,
+            764,
+            404
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            436,
+            336,
+            452
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The numeric simulation of a dynamic system commonly comprises the derivation of the mathematical model given by the underlying differential equations and their integration forward in time. In the context of physics-based systems, the mathematical model is usually based on first principles and depending on the properties of the simulated system, the numerical integration of a complex system can be very resource demanding (Nealen et al., 2006), e.g., hindering interactive applications. Enabled by the success of deep neural networks to serve as effective function approximators, researchers recently started investigating the applicability of neural networks for simulating dynamic systems. While many physical phenomena can well be described within fixed spatial domains (e.g., in fluid dynamics) that can be learned with convolutional neural network (CNN) architectures (Chu & Thuerey, 2017; Guo et al., 2016; Tompson et al., 2016; Xiao et al., 2020), a large range of physical systems can more naturally be represented as graphs. Examples include systems based on connected particles (Muller et al., 2007), coupled oscillators (Michels & Desbrun, 2015; Michels et al., 2014), ¨ or finite elements (Nealen et al., 2006). Existing methods enable learning these systems often in an end-to-end manner and with a focus on replacing the entire or a part of the integration procedure. A number of methods show initial success in approximating physical systems; however, they often fail to reliably simulate the state of a system over longer time horizons if significant disadvantages are not accepted, such as the use of large datasets containing long-term simulations and the employment of specific memory structures (Sanchez-Gonzalez et al., 2020). ",
+        "bbox": [
+            174,
+            459,
+            825,
+            708
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we aim to improve the performance of iterative solvers for physical systems with graph networks (GN). An iterative solver requires an initial guess, and based on it generates a sequence of improving approximate solutions. The initial guess can be computed by simply using values obtained in the previous iteration or by solving a few steps of a similar but simpler physical system. The performance of an iterative solver is significantly influenced by the calculation of the initial guess, which we aim to replace with the prediction of a GN. To demonstrate our approach, we use a position-based dynamics (PBD) solver that approximates physical phenomena by using sets of connected vertices (Bender et al., 2017; 2014b; Macklin et al., 2016; Muller et al., 2007). To ¨ simulate a physical system, PBD first computes updated locations of vertices using symplectic Euler integration to then correct the initial position estimates so as to satisfy a set of predefined constraints. The correction step is known as constraint projection and is commonly solved iteratively. The explicit forward integration for predicting the system’s updated state has negligible cost, whereas the projection step is computationally expensive. Our goal is to employ a GN to predict the outcome of the constraint projection step as an initial guess. This way, our approach inherits the long-term stability of a classic PBD solver, while providing better run-time performance. ",
+        "bbox": [
+            174,
+            715,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/66d9183e801abcbe38f01980eb6475c1e896e2ffa8061ae8d4129f00236e3887.jpg",
+        "image_caption": [
+            "Figure 1: Renderings taken from real-time simulations of the elastic deformation of a helix falling down on the ground plane (left) and two rods colliding with each other (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            50,
+            823,
+            117
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To showcase the capabilities of our combined PBD solver, we aim to simulate the physically plausible mechanics of elastic rods. Rods play an important role for a variety of application domains, ranging from surgical simulation of sutures (Feess et al., 2016), catheters, and tendons (Pai, 2002), to human hair (Michels et al., 2015) and vegetation (Pirk et al., 2017) in animated movies. Furthermore, approaches exist to realistically simulate rods as sets of connected vertices accurately capturing their mechanical properties (Bergou et al., 2008; Kugelstadt & Schoemer, 2016; Michels et al., 2015; Pai, 2002). Our approach is able to generalize across different initial conditions, rod discretizations, and realistic material parameters such as Young’s modulus and torsional modulus (Deul et al., 2018). Moreover, we demonstrate that our approach can handle discontinuous collisions between individual rods. Figure 1 shows examples of elastic rod deformations of a helix falling down (left) and two colliding rods (right). Finally, we show that the data-driven prediction of the initial guesses of the constraint projection leads to a decreased number of required iterations, which – in turn – results in a significant increase of performance compared to canonical initial guesses. ",
+        "bbox": [
+            174,
+            156,
+            825,
+            337
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In summary, our contributions are: (1) we show how to accelerate iterative solvers with GNs; (2) we show that our network-enabled solver ensures long-term stability required for simulating physical systems; (3) we showcase the effectiveness of our method by realistically simulating elastic rods; (4) we demonstrate accuracy and generalizability of our approach by simulating different scenarios and various mechanical properties of rods including collisions and complex topologies (dynamic tree simulations). ",
+        "bbox": [
+            174,
+            344,
+            825,
+            428
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            434,
+            346,
+            449
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the following we provide an overview of the related work, which spans from data-driven physics simulations and graph learning to position-based dynamics and elastic rods. ",
+        "bbox": [
+            174,
+            452,
+            821,
+            479
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Data-driven Physics Simulations. It has been recognized that neural networks can be used as effective function approximators for physical and dynamic systems. To this end, early approaches focus on emulating the dynamics of physics through learned controllers (Grzeszczuk et al., 1998) or by designing subspace integrators (Barbic & James, 2005). Today, a range of approaches exist that enable ˇ learning ordinary and partial differential equations (Lagaris et al., 1998; Raissi et al., 2019; Raissi & Karniadakis, 2018), for example, to transform them into optimization problems (Dissanayake & Phan-Thien, 1994), to accelerate their computation (Mishra, 2018; Sirignano & Spiliopoulos, 2018), or to solve for advection and diffusion in complex geometries (Berg & Nystrom, 2018). ¨ Other methods focus on specific data-driven solutions for non-linear elasticity (Ibanez et al., 2017), ˜ for approximating Maxwell’s equation in photonic simulations (Trivedi et al., 2019), or for animating cloth (Wang et al., 2011), partially focusing on interactive applications (Holden et al., 2019). More recently, research on data-driven approaches for modeling the intricacies of fluid dynamics has gained momentum (Ladicky et al., 2015; Ummenhofer et al., 2020). Due to fixed-size spatial rep- ´ resentation of Eulerian fluid solvers, a number of approaches rely on CNN-type architectures (Chu & Thuerey, 2017; Guo et al., 2016; Tompson et al., 2016; Xiao et al., 2020). Furthermore, it has been shown that data-driven approaches can even be used to approximate the temporal evolution of fluid flows (Wiewel et al., 2018), to compute liquid splashing (Um et al., 2017), artistic styletransfer (Kim et al., 2020), or to derive fluid dynamics from reduced sets of parameters (Kim et al., 2019). ",
+        "bbox": [
+            174,
+            486,
+            825,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Graph-based Learning. Graphs have proven to be a powerful representation for learning a wide range of tasks (Battaglia et al., 2018; Scarselli et al., 2009). In particular, it has been shown that graphs enable learning knowledge representations (Kipf et al., 2018), message passing (Gilmer et al., 2017), or to encode long-range dependencies, e.g., as found in video processing (Wang et al., 2017). A variety of methods uses graph-based representations to learn properties of dynamic physical systems, e.g. for climate prediction (Seo & Liu, 2019), with an emphasis on individual objects (Chang et al., 2016) and their relations (Sanchez-Gonzalez et al., 2018), for partially observable systems (Li et al., 2018), the prevalent interactions within physical systems (Kipf et al., 2018), hierarchically-organized particle systems (Mrowca et al., 2018), or – more generally – physical simulation (Sanchez-Gonzalez et al., 2019; 2020). While many of the existing approaches learn the time integration of physical systems in an end-to-end manner, we use a graph network to predict the outcome of a PBD solver for rod dynamics to enable more efficient computations. ",
+        "bbox": [
+            173,
+            757,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/43792af9bed5d0f27d102a89cfeb714f437c3c4218295dc2fb7b707853997e9d.jpg",
+        "image_caption": [
+            "Figure 2: Illustration of the discretization of a single rod using several rod segments arranged along its centerline (left). Each rod segment is described by its position and orientation within the generalized coordinates $\\mathbf { p } _ { i }$ . The Lagrange multipliers $\\lambda _ { i }$ represent the interaction between rod segments. The forward integration path is illustrated in red (middle) and constraint projection in green (right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            51,
+            821,
+            133
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Position-based Dynamics and Elastic Rods. PBD is a robust and efficient approach for simulating position changes of connected sets of vertices (Bender et al., 2017; 2014b; Macklin et al., 2016; Muller et al., 2007). Compared to forced-based methods that compute the force directly, the inter- ¨ action between different vertices in PBD is realized by a constraint projection step in an iterative manner. To avoid the dependency of the system’s stiffness on the number of iterations and the time step size, an extended position-based dynamics approach was introduced (XPBD) (Macklin et al., 2016). A number of methods exist that model the dynamic properties of rods that can even simulate more complicated rod mechanics (Pai, 2002). Moreover, particle systems were employed to simulate the dynamics of rods (Michels et al., 2017) and, in particular, for the physically accurate simulation of thin fibers (Michels et al., 2015) such as present in human hair or textiles. On a different trajectory, it has been recognized that rods can be simulated based on PBD (Umetani et al., 2014). The initial formulation was improved (Kugelstadt & Schoemer, 2016) by including the orientation of rod segments in the system’s state to account for torsion effects. Later, the XPBD framework was utilized (Deul et al., 2018) to address the non-physical influence of iteration numbers and steps sizes, which enables the more accurate simulation of elastic rods. ",
+        "bbox": [
+            174,
+            199,
+            825,
+            406
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            421,
+            341,
+            436
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We propose a novel approach to simulate the temporal evolution of a dynamic system which consists of elastic rods. Each rod is discretized using several rod segments arranged along its centerline (Figure 2). For each rod segment, its state is described by its position, orientation, velocity and angular velocity. The state of the system is given as the set of the individual states of all rod segments. The simulation is carried out by employing PBD (Muller et al., 2007) directly manipulating the sys- ¨ tem’s state. Orientations are represented as quaternions allowing for a convenient implementation of bending and twisting effects (Kugelstadt & Schoemer, 2016). Extended PBD (XPBD) (Macklin et al., 2016) is implemented to avoid that the rods’ stiffnesses depends on the time step size and the number of iterations (Deul et al., 2018). ",
+        "bbox": [
+            173,
+            441,
+            825,
+            565
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The generalized coordinates of a rod segment $i$ at time $t$ is given by $\\mathbf { p } _ { i , t } \\in \\mathbb { R } ^ { 3 } \\times \\mathbb { H }$ , which includes its position $\\mathbf { x } _ { i , t } \\in \\mathbb { R } ^ { 3 }$ given in Cartesian coordinates and its orientation described by a quaternion $\\mathbf { q } _ { i , t } \\in \\mathbb { H }$ . Correspondingly, $\\boldsymbol { v } _ { i , t } \\in \\mathbb { R } ^ { 6 }$ refers to the generalized velocity of the rod segment, which includes velocity and angular velocity. The system is continuously updated during the simulation by applying corrections $\\bar { \\Delta } \\mathbf { p } _ { i } = ( \\Delta \\mathbf { x } _ { i } , \\Delta \\phi _ { i } ) ^ { \\mathsf { T } } \\in \\mathbb { R } ^ { 6 }$ with position shifts $\\Delta \\mathbf { x } _ { i } \\in \\mathbb { R } ^ { 3 }$ and orientation shifts $\\bar { \\Delta { \\phi _ { i } } } \\in \\mathbb { R } ^ { 3 }$ representing the integration of the angular velocity.1 ",
+        "bbox": [
+            174,
+            566,
+            825,
+            651
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A single time integration step is presented in Algorithm 1. In the beginning (lines 1 to 4), generalized velocity and generalized coordinates are updated by employing a symplectic Euler integration step. In this regard, $\\mathbf { a } _ { \\mathrm { e x t } }$ denotes the generalized acceleration due to the external net force, e.g., given by gravity. XPBD (Macklin et al., 2016) employs the Lagrange multiplier $\\boldsymbol { \\lambda }$ which is initialized as zero (line 5) along with the integrated generalized coordinates $\\mathbf { p } ^ { * }$ . Collision detection results are stored in $\\mathbf { C o l l } _ { \\mathrm { r - r } }$ and $\\mathbf { C o l l } _ { \\mathrm { r - p } }$ (line 6), where $\\mathbf { C o l l } _ { \\mathrm { r - r } }$ includes all the pairs of two rod segments that potentially collide with each other and $\\mathbf { C o l l } _ { \\mathrm { r - p } }$ includes information of all rod segments that potentially collide with another object such as a plane. Within several solver iterations, we alternate between rod constraint projection and the collision constrain projection (lines 7 to 15). The rod constraints include shear-stretch and bend-twist constraints representing the corresponding elastic energy. The Lagrange multiplier represents the interaction between rod segments. Figure 2 illustrated the discretization for a single rod into several interacting segments. The correction values $\\Delta \\mathbf { p }$ and $\\Delta \\lambda$ in line 9 are computed by constraint projection (Deul et al., 2018; Kugelstadt & Schoemer, 2016). The generalized coordinates and Lagrange multipliers are updated for each rod (lines 8 to 12), and rodrod and rod-plane collisions are addressed to update the generalized coordinates. For details about the collision projection procedure, we refer to Macklin et al. (2014). For the non-collision case, the steps within line 6 and 13 are not needed. ",
+        "bbox": [
+            173,
+            652,
+            825,
+            886
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Numerical integration procedure updating $\\mathbf { p } _ { i , t } \\mapsto \\mathbf { p } _ { i , t + \\Delta t }$ and $\\begin{array} { r } { \\pmb { v } _ { i , t } \\mapsto \\pmb { v } _ { i , t + \\Delta t } } \\end{array}$ . ",
+        "bbox": [
+            173,
+            103,
+            779,
+            119
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "1: for all rod segments do   \n2: $\\pmb { v } _ { i } ^ { * } \\gets \\pmb { v } _ { i , t } + \\Delta t \\mathbf { a } _ { \\mathrm { e x t } }$   \n3: $\\mathbf { p } _ { i } ^ { * } \\gets \\mathbf { p } _ { i , t } + \\Delta t \\ \\mathbf { H } ( \\mathbf { q } _ { i , t } ) \\mathbf { \\sigma } _ { i } ^ { * }$ with $\\mathbf { H } ( \\mathbf { q } _ { i , t } ) : = \\left[ \\mathbf { 1 } _ { 3 \\times 3 } , \\mathbf { 0 } _ { 3 \\times 3 } ; \\mathbf { 0 } _ { 4 \\times 3 } , \\mathbf { G } ( \\mathbf { q } _ { i , t } ) \\right]$   \n4: end for   \n5: $\\pmb { \\lambda } ^ { 0 }  \\mathbf { 0 } , \\mathbf { p } ^ { 0 }  \\mathbf { p } ^ { * }$   \n6: $\\mathrm { ( C o l l _ { r \\mathrm { - } r } , C o l l _ { r \\mathrm { - } p } )  }$ generateCollisionConstraints $( \\mathbf { p } , \\mathbf { p } ^ { * } )$   \n7: for $j  0$ to number of required solver iterations do   \n8: for all rods do   \n9: $\\begin{array} { l } { ( \\Delta \\mathbf { p } , \\Delta \\pmb { \\lambda } )  \\mathsf { r o d C } } \\\\ { \\lambda ^ { j + 1 }  \\lambda ^ { j } + \\Delta \\pmb { \\lambda } } \\\\ { \\mathbf { p } ^ { j + 1 }  \\mathbf { p } ^ { j } + \\Delta \\mathbf { p } } \\end{array}$ onstraintProjection $( \\mathbf { p } ^ { j } , \\lambda ^ { j } )$   \n10:   \n11:   \n12: end for   \n13: $\\mathbf { p } ^ { j + 1 } $ updateCollisionConstraintProjection $( \\mathbf { p } ^ { j + 1 } , \\mathbf { C o l l } _ { \\mathrm { r - r } } , \\mathbf { C o l l } _ { \\mathrm { r - p } } )$   \n14: $j \\gets j + 1$   \n15: end for   \n16: for all rod segments do   \n17: $\\begin{array} { r l } & { \\mathbf { p } _ { i , t + \\Delta t }  \\mathbf { p } _ { i } ^ { j } } \\\\ & { \\mathbf { \\delta } \\mathbf { { \\sigma } } \\mathbf { { v } } _ { i , t + \\Delta t }  \\mathbf { H } ^ { \\top } ( \\mathbf { q } _ { i , t } ) ( \\mathbf { p } _ { i , t + \\Delta t } - \\mathbf { p } _ { i , t } ) / \\Delta t } \\end{array}$   \n18:   \n19: end for ",
+        "bbox": [
+            174,
+            121,
+            771,
+            359
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/1568f99e63d0de48aa59add5ada4bf6c639c6b83fad06ae80808b98bf25bb205.jpg",
+        "image_caption": [
+            "Figure 3: Illustration of our approach incorporating a network which consists of $M$ graph networks (GN-blocks) into the position-based dynamics framework. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            367,
+            823,
+            512
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The most expensive part of Algorithm 1 involves the computation of the corrections of generalized coordinates and Lagrange multipliers (line 9). This projection step requires the solution of a linear system which is a linearization of a non-linear one so that the matrix depends on the system’s state making it impossible to precompute its inverse. Instead, a new system at every point in time is solved iteratively using the conjugated gradient (CG) solver. Such iterative solvers are widely used in the context of physical simulations and regularly described as the de facto standard (Barrett et al., 1994; Saad, 2003) since they often show superior performance and usually scale well allowing for exploiting parallel hardware. However, we would like to point out that also highly efficient direct solvers can be found in the literature (Deul et al., 2018). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Instead of fully replacing the projection step in an end-to-end learning manner, we follow the strategy of accelerating it by first computing a guess ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/82df0370a7624d0375b2a60ca1a031889150f92f2dfc0e759f6ec8859b7d7991.jpg",
+        "text": "$$\n( \\Delta \\mathbf { p } ^ { 0 } , \\Delta \\lambda ^ { 0 } ) \\gets \\mathsf { c o r r e c t i o n } \\mathsf { G u e s s } ( \\mathbf { p } ^ { j } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "for the iterative procedure (line 9) ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fe78d41afdd3dc7225498e866362b5cbf6ba935df53cd97090affc3ef9c80f7c.jpg",
+        "text": "$$\n( \\Delta { \\bf p } , \\Delta \\lambda ) \\gets \\mathrm { r o d } { \\sf C o n s t r a i n t P r o j e c t i o n } ( { \\bf p } ^ { j } , \\lambda ^ { j } , \\Delta { \\bf p } ^ { 0 } , \\Delta \\lambda ^ { 0 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "A neural network is employed to compute the initial guess in Eq. (1) for the constraint projection. The motivation for this approach is to reduce the number of iterations required for the convergence of the CG solver which solves the linear system in Eq. (2) compared to the canonical initialization with zeros. We obtain our final framework by replacing line 9 in Algorithm 1 with Eq. (1) and Eq. (2) as illustrated in Figure 3, which is inherently as accurate as the traditional PBD method. We name the data-driven part COPINGNet (“COnstraint Projection INitial Guess Network”) which learns to compute the correction guess. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.1 GRAPH ENCODING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "COPINGNet is a graph network based architecture which we apply in order to compute initial guesses for $\\Delta \\mathbf { p }$ and $\\Delta \\lambda$ . In this regard, we need to incorporate the rods’ state into a graph description (Battaglia et al., 2018). A graph $G = ( V , E , U )$ usually consists of nodes (or vertices) $V$ , edges $E$ as well as global features $U$ . However, in our framework, global features $U$ are not used. For example, gravity could be a potentially meaningful global feature, but it also can be easily included as an external acceleration. Hence, $U = \\emptyset$ and the graph can just be represented as $G = G ( V , E )$ . In our case, the rods’ segments within the scene are represented by the graph’s nodes while the interactions between the rods’ segments are represented by the edges. ",
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+    {
+        "type": "text",
+        "text": "COPINGNet provides a graph-to-graph mapping: $\\mathbb { G } ^ { \\mathrm { { i n } } } \\xrightarrow { } \\mathbb { G } ^ { \\circ \\mathbf { u t } }$ , from an input graph $G _ { \\mathrm { i n } } \\in \\mathbb { G } ^ { \\mathrm { i n } }$ to an output graph $G _ { \\mathsf { o u t } } \\in \\mathbb { G } ^ { \\mathsf { o u t } }$ . Nodes and edges of both graphs are equipped with specific features. In the case of the input graph, the node features describe the state of the rods’ segments, i.e. ",
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+    },
+    {
+        "type": "text",
+        "text": "in which the positions are denoted with $\\mathbf { x } _ { i } ~ \\in ~ \\mathbb { R } ^ { 3 }$ , the orientations with $\\mathbf { q } _ { i } ~ \\in ~ \\mathbb { H }$ , the radii with $r _ { i } ~ \\in \\mathbb { R } ^ { > 0 }$ , the densities with $\\rho _ { i } \\in \\mathbb { R } ^ { > 0 }$ , and the segment lengths with $\\bar { \\ell _ { i } } \\ \\in \\ \\mathbb { R } ^ { > 0 }$ . Moreover, a segment position indicator $\\alpha _ { i } \\in [ 0 , 1 ] \\ \\subset \\ \\mathbb { R }$ is included corresponding to a parameterization by arc length.2 Binary segment flag $\\mathsf { \\bar { f } } _ { 0 _ { i } } \\in \\{ 0 , 1 \\}$ , “left” end flag $f _ { 1 _ { i } } \\in \\{ 0 , 1 \\}$ and “right” end flag $f _ { 2 _ { i } } \\in \\{ 0 , 1 \\}$ are set to zero if the specific segment respectively the left or right segment of the rod is fixed and to one otherwise. The nodes of $G _ { \\mathrm { i n } }$ are given as the set of $V _ { \\mathfrak { i n } } = \\mathbf { \\bar { \\cup } } _ { i = 1 } ^ { n } \\bar { \\left\\{ \\mathbf { v } _ { \\mathfrak { i n } , i } \\right\\} }$ , in which $n = | V _ { \\mathsf { i n } } |$ denotes the number of segments in the scene. The nodes of $G _ { \\mathsf { o u t } }$ contain the correction values of the generalized coordinates, i.e. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "While rod segments are represented as node features, we represent constraints between rod segments as edge features: $\\mathbf { e } _ { \\mathsf { i n } , i } = ( \\omega _ { i } , _ { \\mathsf { e } } ^ { } , T _ { i } ^ { \\mathsf { T } } ) ^ { \\mathsf { T } } \\in \\mathbb { E } ^ { \\mathsf { i n } } \\subseteq \\mathbb { R } ^ { 5 }$ , ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "in which the (rest) Darboux vector $\\omega \\in \\mathbb { R } ^ { 3 }$ describes the static angle of two rod segments, and Young’s modulus $Y ~ \\in ~ \\mathbb { R } ^ { > 0 }$ and torsion modulus $T ~ \\in ~ \\mathbb { R } ^ { > 0 }$ are corresponding to extension, bending, and twist constraint parameters. The set of edges of the input graph is then given by $E _ { \\mathsf { i n } } = \\mathsf { \\bar { U } } _ { i = 1 } ^ { m } \\{ \\mathbf { e _ { i \\mathsf { n } , i } } \\}$ , in which $m = \\vert E _ { \\mathrm { i n } } \\vert$ denotes the number of interactions between different segments. The correction of the Lagrange multiplier $\\Delta \\lambda _ { i } \\in \\mathbb { R } ^ { 6 }$ is stored in the output edges: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The set of output edges is then given by $E _ { \\mathsf { o u t } } = \\cup _ { i = 1 } ^ { m } \\{ \\mathbf { e } _ { \\mathsf { o u t } , i } \\}$ ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The connectivity information $C$ of each graph is stored in two vectors $\\mathbf { c } _ { \\mathsf { s d } }$ and $\\mathbf { c } _ { \\mathrm { r v } }$ containing the sender node index and the receiver node index of each corresponding edge. This concludes the specification of the input graph $G _ { \\mathsf { i n } } = G ( V _ { \\mathsf { i n } } , E _ { \\mathsf { i n } } )$ and the output graph $\\begin{array} { r } { G _ { \\mathrm { o u t } } = G ( V _ { \\mathrm { o u t } } , E _ { \\mathrm { o u t } } ) } \\end{array}$ with connectivity information $C = ( \\mathbf { c } _ { \\mathsf { s d } } , \\mathbf { c } _ { \\mathsf { r v } } )$ . ",
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+    {
+        "type": "text",
+        "text": "3.2 NETWORK STRUCTURE ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In the following, we describe the structure of COPINGNet after we formalized its input and output in the previous section. As illustrated in Figure 3, the network consists of an encoder network, multiple stacks of GN-blocks, and a decoder network. The graph network from Battaglia et al. (2018) is used as a basic element and denoted as a GN-block. Residual connection between blocks could improve performance of neural networks in both CNN (He et al., 2015), and graph neural network [Li et al. (2019)]. As in related work (Sanchez-Gonzalez et al., 2020), we employ the residual connection between the GN-blocks, but our encoder/decoder network directly deals with the graph. The encoder network performs a mapping: $\\mathbb { G } ^ { \\mathsf { i n } } \\to \\mathbb { G } ^ { \\mathsf { i a t e n t } }$ and is implemented using two multi-layer perceptrons (MLPs), $\\mathsf { \\bar { M } L P _ { e d g e } : \\mathbb { E } ^ { i n } } \\overset { \\cdot } {  } \\mathbb { E } ^ { | \\mathrm { a t e n t } } \\subseteq \\mathbb { R } ^ { l }$ and $\\mathsf { M L P } _ { \\mathsf { n o d e } } \\colon \\mathbb { V } ^ { \\mathsf { i n } } \\to \\mathbb { V } ^ { \\mathsf { l a t e n t } } \\subseteq \\mathbb { R } ^ { l }$ , in which $l$ denotes the latent size. They work separately and thus $E _ { \\mathsf { e n } } = \\mathsf { M L P } _ { \\mathsf { e d g e } } ( E _ { \\mathsf { i n } } )$ and $V _ { \\mathsf { e n } } = \\mathsf { M L P } _ { \\mathsf { n o d e } } ( V _ { \\mathsf { i n } } )$ . Edge features $E _ { \\mathrm { i n } }$ from the input are constant for a rod during the simulation and this results in constant encoded edge features $E _ { \\mathrm { e n } }$ , which could be recorded after the first run and used afterwards during inference. The edge feature $\\mathbf { e } _ { \\mathrm { i n } , i }$ contains the material parameters which could vary by different orders of magnitude. Hence, we normalize Young’s modulus and torsion modulus before feeding them into the network. After encoding, the graph $G _ { \\mathsf { e n } } ( V _ { \\mathsf { e n } } , E _ { \\mathsf { e n } } ) \\ \\in \\ \\mathbb { G } ^ { | \\mathsf { a t e n t } }$ is passed to several GN-blocks with residual connections. Each GN-block also contains two MLPs. However, they use message passing taking advantage of neighbourhood nodes/edges information (Battaglia et al., ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/23efbca533370192dfbc94a158adf9f744347a0d1fb867b07fb5ed8f7905d3de.jpg",
+        "table_caption": [
+            "Under review as a conference paper at ICLR 2021 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Train/Val 256/100</td><td>#Steps 50</td><td>#Nodes N Ud(10,55)</td><td>Young&#x27;sModulus Y 10° Pa,a ~U(4.0,6.0)</td><td>Initial Angle0 U(0°,45.0°)</td><td>RodLength l U(1.0 m,5.0 m)</td></tr><tr><td>Train/Val</td><td>#Steps</td><td>#Nodes N</td><td>Torsion Modulus T</td><td>Helix Radius/Height</td><td>Winding Number</td></tr><tr><td>256/100</td><td>50</td><td>Ud(45,105)</td><td>10° Pa,a ~U(4.0,6.0)</td><td>u(0.4 m,0.6 m)/u(0.4 m,0.6 m)</td><td>U(2.0,3.0)</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 1: Specification of training and validation datasets for the two scenarios of an initially straight bending rod (top) and an elastic helix (bottom). The datasets are comprised of a number of data points (left) each describing the rod’s dynamics within $t \\in [ 0 \\mathrm { ~ s } , 5 0 \\Delta t ]$ discretized with a time step size of $\\Delta t = 0 . 0 2 { \\mathrm { ~ s } }$ . The number of nodes $N$ is sampled from a discrete uniform distribution $\\mathcal { U } _ { \\mathrm { d } }$ while the remaining parameters are sampled from a continuous uniform distribution $\\mathcal { U }$ . We trained our network for 5 hours for the bending rod scenario and 6 hours for the helix case. ",
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+    {
+        "type": "image",
+        "img_path": "images/2c001d770b6a2025e2608f89b2ea7d64d5e863ed2a8335a6ed9184e85bda896c.jpg",
+        "image_caption": [
+            "Figure 4: Illustration of the ratio of COPINGNet’s inference time $t _ { \\mathrm { i n f e r } }$ and the vanilla CG solver’s run time $t _ { C G }$ (purple curves; right vertical axis) for the initially straight bending rod (left) and the elastic helix (right) simulations. The black curves show the CG iteration number ratio while the red curves show the total speedup of the constraint projection when taking into account COPINGNet’s inference time (left vertical axis). The pink curves show the total speedup of the entire simulations. The orange and green dashed line indicate the lower and upper boundaries of the total number of nodes used in the training data. We can observe a speedup even for rods and helices with greater node number than the ones used in the the training dataset. The result is averaged from 50 simulations each running 100 steps. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "2018). A number of $M$ GN-blocks enable the use of neighbourhood information with distances smaller or equal to $M$ . The graph network performs a mapping within the latent space: $\\mathbb { G } ^ { \\mathsf { l a t e n t } } \\to$ $\\mathbb { G } ^ { | \\mathsf { a t e n t } }$ , and after $M$ GN-blocks, we obtain $\\bar { G } _ { \\mathsf { e n } } ^ { \\prime } ( V _ { \\mathsf { e n } } ^ { \\prime } , E _ { \\mathsf { e n } } ^ { \\prime } ) \\in \\mathbb { G } ^ { \\mathsf { l a t e n t } }$ . The decoder network performs a mapping: $\\mathbb { G } ^ { \\mathsf { l a t e n t } } \\to \\mathbb { G } ^ { \\mathsf { o u t } }$ , which has a similar structure as the encoder network. Two MLPs $\\mathsf { M L P _ { e d g e } : \\mathbb { E } ^ { l a t e n t } \\to \\mathbb { E } ^ { o u t } }$ and $\\mathsf { M L P _ { n o d e } : \\mathbb { V } ^ { l a t e n t } \\to \\mathbb { V } ^ { o u t } }$ compute $\\begin{array} { r } { E _ { \\mathsf { o u t } } = \\mathsf { M L P _ { e d g e } } ( E _ { \\mathsf { e n } } ^ { ' } ) } \\end{array}$ and $V _ { \\mathrm { o u t } } = \\mathsf { M L P } _ { \\mathsf { n o d e } } ( V _ { \\mathrm { e n } } ^ { ' } )$ . A tanh-function at the end of ${ \\mathsf { M L P } } _ { \\mathsf { e d g e } }$ and ${ \\mathsf { M L P } } _ { \\mathsf { n o d e } }$ is used restricting the output to the interval $[ - 1 , 1 ] \\subset \\mathbb { R }$ . COPINGNet learns the relative correction values. The generated dataset is normalized, and the maximum correction value of the generalized coordinates and the Lagrange multipliers will be recorded as norms. The final correction value is achieved by multiplying the relative correction values and the norms. This normalization process damps the noise caused by the network and leads to a more stable performance. For simplicity, all the MLPs in different blocks have the same number of layers, and the same width as latent size $l$ within the latent layers. The input and output sizes of each MLP are consistent with the corresponding node/edge feature dimensions. The loss $L$ is computed from both parts, nodes and edges, ",
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1756e570a68102c82c60790ea88b5e96bd53889cddbbe4c0386a22844d1dc451.jpg",
+        "text": "$$\nL : = \\mathsf { M S E } ( V _ { \\mathrm { o u t } } , \\tilde { V } _ { \\mathrm { o u t } } ) + \\mathsf { M S E } ( E _ { \\mathrm { o u t } } , \\tilde { E } _ { \\mathrm { o u t } } ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "in which $( V _ { \\mathrm { o u t } } , E _ { \\mathrm { o u t } } )$ is the output graph’s ground truth in contrast to COPINGNet’s prediction $( \\tilde { V } _ { \\mathrm { o u t } } , \\tilde { E } _ { \\mathrm { o u t } } )$ . The mean squared error between $\\chi$ and $\\tilde { \\chi }$ is denoted with ${ \\mathsf { M S E } } ( \\chi , \\tilde { \\chi } )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For evaluation purposes, we also incorporated the $k$ -nearest neighbor $k$ -NN) algorithm as described in the supplementary material setting $k = 3$ . ",
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+    {
+        "type": "text",
+        "text": "4 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We generate training and validation datasets based on two scenarios: an initially straight bending rod and an elastic helix each fixed at one end oscillating under the influence of gravity. The specification of these datasets is provided in Table 1. The PBD code is written in $\\mathrm { C } { + } { + }$ , while the COPINGNet is implemented in PyTorch. The training is performed on an NVIDIA\rR $\\mathrm { T e s l a } ^ { \\textregistered }$ V100 GPU. The training time varies from 8 to 30 hours for different architecture parameters. A constant learning rate of $\\eta = 0 . 0 0 1$ was used and a mean square loss function was employed. Our approach generalizes across different initial conditions, geometries, discretizations, and material parameters. In the supplementary material we show that it is possible to robustly generate various dynamical results (Figure 9). For a discussion on the network architecture please see Figure 11 (supplementary material). ",
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/ed1e9b6d007b1d7eedb0fe3c233c6801ad15aea379f4a368495e2ea2be90dc0b.jpg",
+        "image_caption": [
+            "Figure 5: Realistic biomechanical simulation of a 3D tree model composed of over $1 k$ nodes swaying in a wind field. Our GN approach performs correctly even under a large number of rod segments while increasing performance of the original PBD method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
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+        "image_caption": [
+            "Figure 6: The dashed horizontal lines show the benchmark CG solver’s constant performance for an initially straight bending rod simulation (left) and an elastic helix simulation (right). The parameters $\\phi _ { 0 } = 0 ^ { \\circ }$ , $N = 3 0$ , $\\ell = 3 . 0 \\mathrm { m }$ , $Y = 1 . 0 \\cdot 1 0 ^ { 5 }$ Pa (left), and $\\mathsf { H R } = 0 . 5 \\ \\mathrm { m }$ , $\\mathsf { H } \\mathsf { H } = 0 . 5 \\ \\mathrm { m }$ , $\\mathsf { H W } = 2 . 5$ , $G = 1 . 0 \\cdot 1 0 ^ { 5 }$ Pa, $N = 6 0$ (right) are applied. We are comparing our COPINGNet-assisted PBD approach (orange) to a simplified version in which the $k$ -NN (green) is used to predict the initial guess. Moreover (left), we added a performance measurement in which we restricted the solution to be in a two-dimensional plane (black). The results are averages over 20 simulations and smoothed with a window of 10 frames. The dashed vertical lines mark the number of 50 frames contained in the training data set. For the helix simulation we observe a speed up of approximately $50 \\%$ even beyond the training data (50 time frames). For the bending rod simulation we observe a speedup that further decreases with increasing time frame number. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Discretization. Our approach addresses the acceleration of the most expensive part within PBD by providing an accurate initial guess of the constraint projection. We measure the system’s complexity by the number of nodes in a rod. Figure 4 shows the ratio of COPINGNet’s inference run time (black and red curves) compared to the run time of the vanilla CG solver (purple curves) for different numbers of nodes. The increasing black and red curves indicate that the speedup of COPINGNet is more significant with greater number of nodes. With only a few nodes the CG solver performs better due to the inference overhead. Once the number of nodes is increasing, a significant speedup can be obtained of up to $5 0 \\%$ for the constraint projection. Surprisingly, our approach also performs well when going far beyond the sampling range (orange to green dashed lines). Since the constraint projection is the most time-consuming part of the entire simulation, the speedup converges to that of the whole simulation (pink curves) with increasing number of nodes as shown in Figure 4. ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Temporal Evolution. In addition to the complexity analysis, we also analyze the required number of CG iterations for the vanilla constraint projection compared to the one accelerated using COPINGNet over time as shown in Figure 6. We obtain a significant total speedup compared to the CG solver (dashed blue line). As stated in Table 1, our training data contains dynamical simulations of 50 time steps. In this range we observe the highest speedup. As was the case for the complexity analysis, we again obtain a significant speedup for simulations beyond 50 time steps. This performance gain is more present for the helix (green curve) compared to the rod simulation (orange curve). ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Long-term Stability. In case the constraint projection is completely replaced by COPINGNet (endto-end approach), stability of the PBD method decreases as error accumulation takes place. This is illustrated in Figure 7 showing the temporal evolution of the relative change of the total rod length. An initially straight rod bending under the influence of gravity is simulated using the parameters $\\phi _ { 0 } = 0 ^ { \\circ }$ , $N = 3 0$ , $\\ell = 4 . 0 \\mathrm { m }$ , and varying Young’s modulus $Y$ . In this scenario the rod length is expected to stay constant during the mechanical simulation. In case COPINGNet is used in an endto-end manner (colored lines), where the whole constraint projection step is replaced, we observe that the rod length changes incorrectly. In fact, the divergence of using COPINGNet to replace the constraint projection step increases exponentially beyond the range of the training data (50 steps). To the contrary, when COPINGNet is used to only estimate the initial guess of the constraint projection (black lines), no rod deformation takes place even beyond the training data range. ",
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+    {
+        "type": "text",
+        "text": "Collisions. Figure 1 illustrates a collision of an elastic helix with the ground plane and a collision between two rods. Collision detection is efficiently implemented using the hierarchical spatial hashing scheme according to Eitz & Gu (2007). Rod-rod collisions are then treated with line-to-line distance constraints, and collisions with the ground plane using half-space constraints. Moreover, frictional effects are implemented according to Macklin et al. (2014). This approach allows us to handle discontinuous events such as collisions between individual rods and other objects. Moreover, for this experiment we use the neural networks trained on the rod and helix simulations. Employing COPINGNet trained on the helix data to simulate the collision of the ground plane $( \\mathsf { H } \\mathsf { R } = \\mathsf { H } \\mathsf { H } = 0 . 5 \\mathrm { ~ m ~ }$ , $\\mathsf { H W } = 2 . 5$ , $G = 1 . { \\overset { \\vartriangle } { 0 } } \\cdot 1 0 ^ { 6 } { \\overset { \\vartriangle } { \\ v u } }$ Pa, $N = 5 0$ ), we measure a total speedup of approx. $1 0 \\%$ . In the case of two colliding rods $\\langle \\phi _ { 0 } = 0 ^ { \\circ }$ , $N \\in \\{ 2 0 , 3 0 \\}$ , $\\ell \\in \\{ 4 . 0 \\mathrm { m } , 4 . 5 \\mathrm { m } \\}$ , $Y = 1 . 0 \\cdot 1 0 ^ { 6 } \\ : \\mathrm { P a } )$ , we obtain a speedup of approx. $6 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "Complex Scenarios. Our method is also capable to deal with complex scenarios such as 3D tree models swaying in wind fields as shown in Figure 5. We represent trees using the extended Cosserat rod theory introduced in Pirk et al. (2017). This method allows us to simulate realistic biomechanics of thousands of rod segments at interactive rates. We generated 100 different tree topologies using 70 individual rods (average node number: 1056) and simulate the swaying motions of these tree models with vanilla PBD to generate the training dataset. For the evaluation, 30 different tree topologies have been generated for each of the following four experiments using 10 rods (204 nodes), 20 rods (355 nodes), 40 rods (654 nodes), and 70 rods (1061 nodes). We were able to significantly improve the runtime performance of the method by $1 7 . 0 \\%$ (10 rods), $1 5 . 8 \\%$ (20 rods), $1 3 . { \\bar { 0 \\% } }$ (40 rods), and $1 1 . 1 \\%$ (70 rods). This takes into account the inference time introduced by the neural network. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/a880abe5ec3310b65f98fc0d9a3fc219a05368751e37b593573a311958b42906.jpg",
+        "image_caption": [
+            "Figure 7: Illustration of the relative change of the total rod length $( l / l _ { 0 } )$ for different values of Young’s moduli $( Y )$ . Colored lines show different results for COPINGNet replacing the constraint projection. The thick black line represent the result for COPINGNet replacing only the initial guess. This indicates that the end-to-end approach does not generalize past the 50 time steps used in preparing the training data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Generalization. Figure 7 and Figure 10 (supplementary material) indicate that using COPINGNet in an end-to-end manner does not generalize beyond the training data. Specifically, the end-to-end setup diverges in terms of rod geometry and segment position from the correct solution. This effect increases significantly beyond the training data range. Although, we only use COPINGNet as a benchmark for this evaluation, other GNs are expected to perform similarly. Common workarounds to increase the stability of dynamical systems with neural networks are temporal skip-connections, recurrent training, and data augmentation. However, these approaches focus on runtime speed and memory performance rather than stability (Holden et al., 2019). Interestingly, replacing just the initial guess with a GN does not deform the rod or introduces discontinuities at any observed stage of the bending simulation. This means that employing GNs can result in a performance increase without a loss of stability. Furthermore, a GN that only provides initial guesses seems to also generalize to other scenarios. We observed performance improvements for the collision and complex tree case, although the network was never trained on these different rod discretizations and topologies. This indicates that our method is capable to generalize within a specific physical scenario and to a lesser extent to other scenarios. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We discovered that applying GNs for replacing the initial guess has fundamental advantages over end-to-end approaches. First, our network-enabled solver ensures long-term stability inherited from traditional solvers for physical systems, while improving runtime performance. Second, our approach is able to generalize across different initial conditions, rod discretizations, and material parameters, and it handles discontinuous effects such as collisions. While end-to-end approaches offer more significant speedups, our method is superior in cases where stability is an essential requirement. Our approach to accelerate iterative solvers with GNs opens multiple avenues for future work. For one, it would be interesting to explore mechanical systems describing general deformable (e.g. textiles) or volumetric objects, which have been simulated with PBD. Second, our approach can be applied to other iterative methods, such as in finite elements analysis or in the context of linear complementarity problems (LCP). This would allow us to accelerate physical simulations, when iterative solvers are applied, without compromising stability. ",
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+    {
+        "type": "text",
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+        "text": "Steffen Wiewel, Moritz Becher, and Nils Thuerey. Latent-space physics: Towards learning the temporal evolution of fluid flow. CoRR, abs/1802.10123, 2018. ",
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+        "text": "Xiangyun Xiao, Yanqing Zhou, Hui Wang, and Xubo Yang. A novel cnn-based poisson solver for fluid simulation. IEEE Transactions on Visualization and Computer Graphics, 26(3):1454–1465, 2020. ",
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+    {
+        "type": "text",
+        "text": "SUPPLEMENTARY MATERIAL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            418,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "DYNAMIC RESULTS ",
+        "bbox": [
+            174,
+            145,
+            312,
+            160
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/0b597f1dd7e6c0c813a7a05d75e5ba2bca3c7d5c214676eb06c33d3baf6e7b1a.jpg",
+        "image_caption": [
+            "Figure 8: Illustration of the temporal evolution of scenarios (a) to (d) for the initially straight bending rod, and (e) to (f) for the elastic helix. The different cases represent various material property configurations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            202,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8 we show the temporal evolution of different scenarios (a) to (d) for the initially straight bending rod, and (e) to (f) for the elastic helix. The default parameters of the initially straight bending rod are $\\phi _ { 0 } ~ = ~ 0 ^ { \\circ }$ , $N \\ = \\ 3 0$ , $\\ell ~ = ~ 4 . 0 ~ \\mathrm { m }$ , and $\\dot { Y } ~ = ~ 1 . 0 \\cdot 1 0 ^ { 5 } ~ \\mathrm { P a }$ . In (a), we modify $\\phi _ { 0 } ~ \\in ~ \\{ 0 . 0 ^ { \\circ } , 1 5 . 0 ^ { \\circ } , 3 0 . 0 ^ { \\circ } , 4 5 . 0 ^ { \\circ } \\}$ . In (b), we modify $N ~ \\in ~ \\{ 1 0 , 2 0 , 4 0 , 6 0 \\}$ . In (c), we modify $\\ell \\in \\{ 1 . 0 \\mathrm { m } , 2 . 0 \\mathrm { m } , 3 . 0 \\mathrm { m } , 4 . 0 \\mathrm { m } ,$ $5 . 0 \\mathrm { m } , 6 . 0 \\mathrm { m } \\}$ . In (d), we modify $Y ~ \\in ~ \\{ 2 . 0 ~ \\cdot$ $1 0 ^ { 4 } \\mathrm { P a } , 5 . 0 \\cdot 1 \\dot { 0 } ^ { 4 } \\mathrm { P a } , 1 . \\dot { 0 } \\cdot 1 0 ^ { 5 } \\mathrm { P a } , 2 . 0 \\cdot 1 0 ^ { 5 } \\mathrm { P a } , 5 . 0 \\cdot 1 0 ^ { 5 } \\mathrm { P a } , 1 . \\dot { 0 } \\cdot 1 0 ^ { 6 } \\mathrm { P a }$ . The default parameters of the elastic helix are $\\mathsf { H } \\mathsf { R } \\ = \\ 0 . 5 \\ \\mathrm { m }$ (helix radius), $\\mathsf { H } \\mathsf { H } \\ = \\ 0 . 5 \\ \\mathrm { m }$ (helix height), $\\mathsf { H W } = 2 . 5$ (winding number), $T \\ = \\ 1 . 0 \\cdot 1 0 ^ { 5 } \\mathrm { { \\ P a } }$ , and $N ~ = ~ 6 0$ . In (e), we modify $( { \\mathsf { H } } { \\mathsf { R } } , { \\mathsf { H } } { \\mathsf { H } } , { \\mathsf { H } } { \\mathsf { W } } ) \\ \\in$ $\\{ ( 0 . 4 \\mathrm { m } , 0 . 4 \\mathrm { m } , 2 . 0 \\mathrm { m } )$ , $( 0 . 4 \\mathrm { m } , 0 . 4 \\mathrm { m } , 3 . 0 \\mathrm { m } )$ , $( 0 . 5 \\mathrm { m } , 0 . 5 \\mathrm { m } , 2 . 0 \\mathrm { m } )$ , $( 0 . 5 \\mathrm { m } , 0 . 5 \\mathrm { m } , 3 . 0 \\mathrm { m } ) \\}$ . In (f), we modify $N \\in \\{ 3 0 , 6 0 , 8 0 , 1 0 0 \\}$ . In (g), we modify $G \\in \\{ 2 . 0 \\cdot 1 0 ^ { 4 } \\mathrm { { P a } , 5 . 0 \\cdot 1 0 ^ { 4 } \\mathrm { { P a } , 1 . 0 \\cdot 1 0 ^ { 5 } \\mathrm { { P a } , 1 . 0 \\cdot } } } $ $1 0 ^ { 6 } \\mathrm { P a } \\}$ . For each experiment, the rod colors indicate the corresponding parameters in the following (ascending) order: blue, orange, green, red, purple, brown. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Dynamic results of our graph learning assisted simulations can be found in the video ",
+        "bbox": [
+            176,
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+            728,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "GraphLearningPhysicalSystems.mp4 (2:20 min, 20.6 MB, $1 9 2 0 \\times 1 0 8 0$ , H.264) ",
+        "bbox": [
+            225,
+            299,
+            763,
+            314
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "provided as additional supplementary material. These results are obtained by employing three GN blocks, two MLP layers, and a MLP width of 32. Parameters have been used which are are not covered within the training dataset. ",
+        "bbox": [
+            174,
+            321,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Moreover, to further demonstrate general applicability, Figure 9 shows an additional simulation of a tightening knot for which COPINGNet-based PBD shows a total speedup of $1 0 . 4 \\%$ compared to vanilla PBD. COPINGNet is trained using a dataset containing 20 simulations of the same knot scenario with different discretizations over 100 frames. It can clearly be observed that COPINGNet generalizes beyond the training data. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/6752cbda86a8300a84044c3b5b500309708fa2e4efd317ebbb30644ecd988263.jpg",
+        "image_caption": [
+            "Figure 9: Illustration of the temporal evolution of a knot scenario. The rod is fixed at both ends and the knot is pulled tight. The predicted results by end-to-end COPINGNet-based learning are shown in red while the results computed with COPINGNet-assisted PBD are shown in black. The visualization of the last frame contains a close-up to demonstrate that the knot structure is still preserved. The rod parameters are $N = 1 1 0$ , $R = 0 . 0 1 { \\mathrm { m } }$ , $\\ell = 1 1 . 0 \\mathrm { m }$ , and $Y = T = 1 . 0 \\cdot 1 0 ^ { 4 } \\mathrm { P a }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            457,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Figure 10 illustrates the temporal evolution of a bending rod and elastic helix scenarios using different approaches. For the bending rod case, the parameters $\\phi _ { 0 } = 0 ^ { \\circ }$ , $N = 3 0$ , $\\ell = 4 . 0 ~ \\mathrm { m }$ , and $Y = 1 . { \\overset { \\underset { \\textstyle \\star } { } } { 0 } } \\cdot 1 0 ^ { 5 }$ Pa are used. In the helix case, $\\mathsf { H R } = 0 . 5 \\ : \\mathrm { m }$ (helix radius), $\\mathsf { H H } = 0 . 5 \\ : \\mathrm { m }$ (helix height), $\\mathsf { H W } = 2 . 5$ (winding number), $T = 1 . 0 \\cdot 1 0 ^ { 5 } ~ \\mathrm { P }$ a, and $N = 6 0$ were applied. The temporal evolution of the positions’ normalized root mean square error (NRMSE) is shown in Figure 11. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/b3bcd2519c63fdba03741161d6ac7115740fe4407ef7cdce2603dadb5e9a83d0.jpg",
+        "image_caption": [
+            "Figure 10: Illustration of the temporal evolution of bending rod (upper row) and elastic helix (lower row) scenarios simulated using our COPINGNet-assisted PBD approach (blue curves), vanilla PBD (orange dotted curves), $k$ -NN-based end-to-end learning (purple curves) and GN-based end-to-end learning (green curves). While the COPINGNet-assisted approach and the vanilla PBD solver allow us to simulate both cases in a stable manner, both end-to-end learning approaches diverge with increasing time due to error accumulation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/7af2ff4964c586a7b2b5eec02829443f3cf76cb9efa13d93aaa05b1cafb91ed6.jpg",
+        "image_caption": [
+            "Figure 11: Illustration of the positions’ normalized root mean square error (NRMSE) compared to vanilla PBD using our COPINGNet-assisted PBD approach (blue curves), $k$ -NN-based end-toend learning (purple curves), and GN-based end-to-end learning (green curves) for the bending rod (left) and the elastic helix (right) scenarios. While the COPINGNet-assisted PBD approach converges to an almost correct solution, both end-to-end learning approaches show large error rates with increasing time frame number. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            801,
+            287,
+            814
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The evaluation of COPINGNet’s architecture, introduced in Section 3.2, is presented in Figure 12. The performance is studied for different numbers of GN-blocks and MLP layers, and different MLP widths (latent sizes). The measurements demonstrate that a larger number of GN-blocks usually increases the performance while the performance improvement is not longer significant for more than four GN-blocks. This is plausible and can be considered analogously to the size of the stencil of a numerical integrator: a larger number of GN-blocks means that a specific node can take advantage of information gained from neighbourhoods further away. Correspondingly, a larger stencil can do so as well. However, increasing the size of the stencil usually does not result in more accurate results once the critical size is reached. This can be observed here as well. In contrast, it can be clearly observed that deeper MLP networks do not improve the performance. This is consistent with other research on graph networks (Sanchez-Gonzalez et al., 2020). However, increasing the MLP width can increase the network’s ability to generalize. It is shown that the highest performance is achieved with the largest MLP width, especially in the case of the helix scenarios. Since increasing the number of GN-blocks and MLP width will lead to longer inference time, we compromise by choosing medium numbers. For further evaluations, we employ three GN-blocks, two MLP layers, and a MLP width of 32. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/f63ca5cbc4111c5cfe5f36c2ca8872b766dd5b0361615dbc9bd1b72442468255.jpg",
+        "image_caption": [
+            "Figure 12: Illustration of the evaluation of COPINGNet’s architecture (blue: benchmark architecture). The result is averaged from 50 test simulations each running for 100 time steps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "$k$ -NEAREST NEIGHBOR ALGORITHM ",
+        "bbox": [
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+            460,
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+            474
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For evaluation purposes, we also incorporated the $k$ -nearest neighbor ( $k$ -NN) algorithm into our framework to compute an initial guess for the constraint projection step. ",
+        "bbox": [
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Consider a $k$ -NN data point $( \\bf x _ { i } , \\bf y _ { i } )$ . Its components contain node and edge feature information: ",
+        "bbox": [
+            171,
+            521,
+            803,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/336c32ab166769c9d02c1ab101ea8a10b5a902b8ff3ae4779dabf04b64e7260c.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbf { x } _ { i } = ( \\mathbf { v } _ { \\mathrm { i n } , i } , \\mathbf { e } _ { \\mathrm { i n } , i _ { k } } , \\ldots ) ^ { \\mathsf { T } } , \\mathbf { y } _ { i } = ( \\mathbf { v } _ { \\mathrm { o u t } , i } , \\mathbf { e } _ { \\mathrm { o u t } , i _ { k } } , \\ldots ) ^ { \\mathsf { T } } , k \\in \\mathbb { N } _ { 0 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Simulation data is collected to form a dataset $\\left\\{ \\left( \\mathbf { x } _ { i } , \\mathbf { y } _ { i } \\right) \\right\\}$ . During inference, the nearest $k$ data points of $\\mathbf { x } _ { i } ^ { * }$ are picked and linear interpolations are performed to obtain the predicted $\\mathbf { y } _ { i } ^ { * }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For each input node feature $\\mathbf { v } _ { \\mathrm { i n } , i } ^ { * }$ , we obtain the corresponding output node feature $\\tilde { \\mathbf { v } } _ { \\mathrm { o u t } , i } = \\mathbf { v } _ { \\mathrm { o u t } , i } ^ { * }$ The output edge feature is averaged from the information of the corresponding component $\\mathbf { y } _ { j } ^ { * }$ : ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b61be40b5e6aa1416e0f3ad10516f14aae2b5ee7b039f7ca8fb451ff11edc10.jpg",
+        "text": "$$\n\\tilde { \\mathbf { e } } _ { \\mathrm { o u t } , i } = \\frac { 1 } { M } \\sum _ { j } \\mathbf { e } _ { \\mathrm { o u t } , j _ { k } } ^ { * } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "in which $M$ denotes the number of nodes. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The $k$ -NN algorithm is applied as an end-to-end method completely replacing constraint projection or to predict an initial guess. ",
+        "bbox": [
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+        ],
+        "page_idx": 15
+    }
+]