Improved Structural Discovery and Representation Learning of Multi-Agent Data
Authors: Jennifer Hobbs
Matthew Holbrook
Nathan Frank
Long Sha
Patrick Lucey
Venue:  (2020)
Keywords: multi-agent
gaussian mixture
permutation learning
representation learning
group structure

Abstract: Central to all machine learning algorithms is data representation. For multi-agent systems, selecting a representation which adequately captures the interactions among agents is challenging due to the latent group structure which tends to vary depending on various contexts. However, in multi-agent systems with strong group structure, we can simultaneously learn this structure and map a set of agents to a consistently ordered representation for further learning. In this paper, we present a dynamic alignment method which provides a robust ordering of structured multi-agent data which allows for representation learning to occur in a fraction of the time of previous methods.  We demonstrate the value of this approach using a large amount of soccer tracking data from a professional league. 

Under review as a conference paper at ICLR 2020
IMPROVED STRUCTURAL DISCOVERY AND REPRE- SENTATION LEARNING OF MULTI-AGENT DATA
Anonymous authors
Paper under double-blind review
ABSTRACT
Central all machine learning algorithms is data representation. For multi-agent systems, selecting a representation which adequately captures the interactions among agents is challenging due the latent group structure which tends to vary depending on context. However, in multi-agent systems with strong group struc- ture, we can simultaneously learn this structure and map a set of agents sistently ordered representation for further learning. In this paper, we present a dynamic alignment method which provides robust ordering of structured multi- agent data enabling representation learning to occur in a fraction of the time of pre- vious methods. We demonstrate the value of this approach using a large amount of soccer tracking data from professional league.
con-
INTRODUCTION
The natural representation for many sources of unstructured data is intuitive to us as humans: for images, a 2D pixel representation; for speech, a spectrogram or linear filter-bank features; and for text, letters and characters. All of these possess fixed, rigid structure in space, time, or sequential ordering which are immediately amenable for further learning. For other unstructured data sources such point clouds, semantic graphs, and multi-agent trajectories, such an initial ordered structure does not naturally exist. These data sources are set or graph-like in nature and therefore the natural representation is unordered, posing significant challenge for many machine-learning techniques.
A domain where this is particularly pronounced is in the fine- mne-grained multi aagett player motions of team sport. Access to player tracking data changed how we understand and analyze sport (Miller
cet al. 2014: Franks et al., 20155 Wei et al., 2013) Cervone et al., 2014: Power et al. 2017: Sha al.
2016 2018: Yue et al. 2014). More relevantly, sport has risen to increasingly key space within
the machine learning community as an application to expand our understanding of adversarial multi-
agent motion, interaction, and representation (Lucey al. 2013 Le et al. 2017 Felsen al. 2018
Zheng et al. 2016)] Zhan et al. 2018 Yeh et al. 2019: Kurach et al. 2019
In sport there exists strong, complex group-structure which is less prevalent in other multi-agent systems such as pedestrian tracking. Specifically, the formation of a team captures not only the global shape and structure the group, but also enables the ordering of each agent according within the group structure. In this regard, sport possesses relational structure similar to that of faces and bodies, which can be represented based on fixed key-point ordering has allowed for cutting edge work across numerous tasks with a variety of approaches and architectures (Antonakos et al. 2015 Cootes al. 2001 Akhter al., 2012 Joo et al., 2015; Simon et al., 2017).
"role"
graph of key-points. In those domains, representation
Unlike for faces and bodies, the representation graph in sport is dynamic as players constantly move and switch positions. Thus dynamically discovering the appropriate representation of individual players according to their role in a formation affords us structural information while learning use- ful representation for subsequent tasks. This challenge was addressed by the original rose-based alignment of Lucey et al. (2013) and subsequently by /Billkoski et al. (2016) and Shaa et al. (0077). Role-based alignment allows us take unstructured multi-agent data, and reformat into consis- tent vector format that enables subsequent machine learning (Fig. [
Here formulate the role-based aligmment consisting of the phases of formation discovery and role assignment. Formation discovery uses unaligned data to learn an optimal formation template; during role assignment& bipartite mapping is applied between agents and roles in each frame to

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H
B
D
A
G
Figure 1:
A structure representation enables machine learning of multi-agent data. (Left) Data- points are colored according to the agent identity (letters denote agents given frame). (Right) By learning and aligning data to a formation template, represent agents in consistent vector form conducive to learning. Agents are now ordered by the role to which they are assigned.
produce "aligned data". A major limitation in past approaches the speed of the template discov- ery process. In this work we propose an improved approach to the above alignment methods which provides faster and more optimal template discovery for role-based representation learning. Impor- tantly, we seek to learn the same representation as Lucey et al. (2013), Bialkowski et al. (2016), and [Sha et al. (2017) by maximizing the same objective function of Bialkowski al. (2016) more effective manner. The reduced computational load enables on-the-fly fly-fly discovery of the forma- tion templates, new context-specific analysis, and rapid representation learning useful for modeling multi-agent spatiotemporal data.
Our main contributions are: the formulation of this problem as a three-step approach (formation discovery, role assignment, template-clusteringg the use of soft-assignment in the formation discov- ery phase thereby eliminating the costly hard-assignment step the Hungarian Algorithm (Kuhn) 1955). resetting training procedure based on the formation cigenvalues to prevent spurious op- tima, quantification of the impact of initialization convergence and stability, a restriction of the training data to key-frames for faster training with minimal impact on the learned representation, and a multi-agent clustering framework which captures the covariances across agents during the template-chustering phase.
BACKGROUND
2.1 REPRESENTING STRUCTURED MULTI-AGENT DATA
collection of agents is by nature a set and therefore no defined ordering exists a priori. To impose an arbitrary ordering introduces significant entropy into the system through the possible permuta- tions of agents in the imposed representation.
To circumvent this, some approaches in representing multi-agent tracking data in sport have included sorting the players based on an "anchor" agent (Mehrasa et al. 2018). This is limiting in that the optimal anchor task-specific, making the representation less generalizable. An "image-based representation (Yue et al., 2014 Zheng al. 2016 Miller et al. 2014) eliminates the need for an ordering, however, this representation is lossy, sparse, and high-dimensional.
The role-based alignment protocol for sport of of Lucey et al. (2013) used codebook of hand-crafted formation templates against which frame-leve'" samples were aligned. This work was extended by Bialkowski et al. (2016) which learned the template directly from the data. (hha et al.) (2017) further employed a hierarchical template learning framework, useful in both retrieval (Sha et al. [01 8) and
trajectory prediction (Felsen et al. 2018 Yeh et al.) (019). Le et al. (2017) similarly learned an agdnt-ordering directly from the data by learning separate oole-assignment and motion-prediction policies in an iterative and alternating fashion.
2.2
COOMUUTATION-EQUVVARII
APPROACHESS
Permutation -eqiiaaiionneuvvaiiiiinn approaches seek leverage network architectures which are insensitive to the ordering of the input data. Approaches using graph neural networks (GNN) (Kipf & Welling, 2016 Gilmer et al. 2017 Battaglia et al. 2018 2016 Hoshen (007)) have become very popular and shown tremendous promise. These approaches are particularly valuable for tasks (e.g. pedestrian tracking) which lack the strong coherent group structure of sport and therefore canmot leverage meth-
'Throughout we the term "frame" indicate single moment in time in reference to the data being obtained via optical tracking from video.
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Figure 2: An overview of the proposed method. The procedure consists of (1) Normalization, (2) Initialization, (3) Reshaping, (4) Formation Discovery, (5) Template Alignment, (6) Role Assign- ment, (7) Template Clustering. In the role assignment step, the template distributions are shown as unfilled thick ellipses and the observed distributions of the role-aligned data are shown as the textured ellipses.
ods such as role-an alignneeten.. Within sport, Kipf et al. (2018) used a GNN predict the trajectories of players while simultaneously learning the edge-weights of the graph. Yeh et al. (2019) demonstrated the advantages using GNNs forecast the future motion of players in sports, surpassing both the role (Bialkowski et al. 2016) and tree-based approaches (Sha et al. 2017) on most metrics.
The success of these approaches, however, does not negate the value of role-based alignment. The learned formation structure provides valuable insight into high-level organization of the group. Fur- thermore, many traditional machine learning techniques and common deep architectures require an ordered-agent representation. This is again similar to what is seen in the modeling of faces and bod- ies: great success has been achieved using geometric deep learning (Monti et al. 2017 Kipf et al. 2018), but approaches based fixed representation remain popular and effective (Tallor et al.
2017 Kanazawa et al. 2019 Arna et al. 2019 Walker et al. 2017 Rayat Imtiaz Hossain & Little
2018
Interesting work has also been done to learn permutations for self-supervised feature learning or ranking tasks (Adams & Zemel 2011; Mena al.) 2018 Cruz al. 2017). Central these ap- proaches is the process of Sinkhorn normalization (Sinkhorn & Knopp) 1967). which allows for soft-assignment during the training process and therefore flow of gradients. Exploring the appli- cation of Sinkhorn normalization to this task is beyond the scope of this current work, however, we provide additional context on this method in Section A.2
APPROACH
3.1 PROBLEM FORMULATION
Mathematically, the goal of the role-alignment procedure is to find the transformation A [U,,U2.... Un} x M + [R, R2,..., Rx] which maps the unstructured set U of N player trajectories to ordered set (i.e. vector) of K role-trajectories R. Each player trajectory is itself an ordered set of positions U, = [x 18 for an agent n E NJ and a frame s E []. We recognize M as the optimal permutation matrix which enables such ordering.
Thus our goal is to find the most probable set F of 2D probability density functions where
F = arg max P(F(R)
Generically N need not equal K as player may be sent off during game, hut for simplicity it is safe to assume N K in this work.
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N P(x) = Σ P(aln)P(n)) = N
N
1
Σ (2).
(2)
Bialkowski et al. (2016) transforms this equation into of entropy minimization where the goal is to minimize the amount of overlap (i.e. the KL (L-Divergence) between each role. The final optimiza- tion equation in terms of the total entropy H then becomes
arg min H(ln)).
(3)
1=1
See A.I I for additional details.
The authors then expectation maximization (EM) to approximate this solution and note similarity to k-means clustering. However, as they represent that data non-parametrically terms of per-role heat maps, hard assignment must be applied each iteration the distributions may be updated. Instead, we note that equation 2 describes the occupancy of space by any agent in any point in time mixture of conditional distributions across each of the N-roles. This is further equivalent to the sum over n--generating distributions. Thus if we model these generating distributions as d- dimensional Gaussian distributions, this reduces the template-diseovery process that of Gaussian Mixture Model.
3.2 TOY PROBLEM FORMULATION
Understanding the notion of independence under the different formulations of this problem is key. This may be better understood by considering a toy problem: imagine we have three independent 1D Gaussian distributions
wish sample S times from each. It is known that sample from each distribution in rounds, effectively generating the samples in "triplets", although the order within the triplets is random. We then seek to reassign the points back to their original distributions.
Following the approach of Bialkowski et al. (2166), the "structure" imposed by the triplet sampling enforced through the hard-assignment at each iteration. Recall, however, that the original dis- tributions were statistically independent; the triplet structure we wish to respect is imposed by the assignment step, not the underlying distributions.
Contrastingly, in our method the samples are treated as fully independent; had all samples been taken from the first distribution, followed by the second, followed by the third, the outcome the iilitribution-discvven
phase would be identical to that having sampled the data in rounds. Only after the three distributions are estimated would the assignment of each point in every triplet be assigned to the distribution which maximized the overall likelihood in that triplet.
Besides being more computationally effective (see Section 4.2), this allows us to find the true MLE of the distributions. Our method will always discover more optimal estimate of Eq. 2 This can understood considering how the assignment performed during optimization. For each triplet, the likelihood of assigning each point to each distribution is computed in both approaches. In our approach, this gives the likelihood under each mixture component. In the aasd-mssignment approach, however, if two (or more) points in triplet have their highest likelihood under the same component, the exclusionary assignment must result in a lower likelihood than assigning each point its preferred Gaussian.
Furthermore, in our approach, each sample contributes to every component of the mixture, thus the data under the mixture remains "fixed" during the optimization process. In contrast, as the hard- assignments are made, the samples contributing to each distribution changes each iteration. This, in combination with the sub-optimal likelihood above, effectively "breaks" the expectation maxi- mization step and can cause solutions to diverge oscillate, which is inconsistent with a maximum likelihood solution which must monotonically increase.
Thus our approach is computationally efficient, more intuitively captures the independence of the generating distributions versus the structure of the sampling, and ensures likelihood function that will converge under expectation maximization.
3.3 FORMATION-DISCOVER
Our procedure explained here is presented visually in Figure algorithmically in Algorithm A.3
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Data is normalized all teams attacking from left to right and have mean zero in each frame, thereby removing translational effects. Following the approach of Bialkowski et al. (2016). we initialize the cluster centers for formation-discover with the average player positions. The impact of this choice of initialization is explored in Section 4.5
We now structure all the data as single (SN) d vector where S is the total number of frames, N the total number of agents (10 outfielders in the case of soccer), and d the dimensionality of the data (2 here). The K-Means algorithm initialized with the player means calculated above and run to convergence; we find that running K-Means to convergence produces better results than running a fixed number of iterations as is commonly done for initialization. The cluster-centers of the last iteration are then used to initialize the subsequent mixture of Gaussians.
Mixture of Gaussians are known to suffer from component collapse and becoming trapped in patho- logical solutions. To combat this, we monitor the eigenvalues (A;) of each of the components throughout the EM process. If the eigenvalue ratio of any component becomes too large or too small, the next iteration runs Soft K-Means (i.e. mixture of Gaussians with spherical covariance) update instead of the full-covariance update. We find that the range < Å1 < 2 works well. In practice, we find this is often unnecessary when analyzing a single game as the player-initializatio provides the necessary stabilization, but becomes important for analysis over many teams/games where that initialization signal is weaker. We refer to this set of K distributions which maximizes the likelihood of the data the Formation, which we denote F²
Note that the formation is a set of distributions. To enforce an ordering. we must align to parent template, G", which is an ordered set of distributions. The specific ordering of this template unimportant so long as it is established and fixed. We align F* to G* by finding the Bhattacharyya distance (Bhattacharyya 1943) between each distribution F and G given by
DB = (MF. - uG;)"o (MF - MG;) + In
det
where o 0F*; t to
ydee /ge og,; det
2
create a K x K cost matrix and then use the Hungarian algorithm to find the best assignment. We have now produced our Témplate, T ordered set of distributions with established ordering that maximizes the likelihood of the data.
3.4 ROLE-ASSIGNMER
The process of role- assigmmetttt maps each player in each frame to a specific role with the restric- tion that only one player may occupy role in a given frame. We find the likelihood that each agent belongs to each of the discovered distributions in each frame which was already calculated during the formation -iisovvvrr step. This produces a N x K cost matrix in each frame; the Hun- garian algorithm is again used to make the optimal assignment. Thus have achieved the tasks of formation-discover
and role-assignment having had to apply the Hungarian algorithm on only a single pass of the data. We represent the aligned data as a x (dK) matrix R.
3.5 CLUSTERING MULTI-AGER DATA
With an established well-ordered representation, we are now able cluster the multi-agen data to discover sub-templates and perform other analysis. Sub-templates may be found either through flat or hierarchical clustering. Generically, we seek to find set of clusters C which partitions the data into distinct states according to:
arg min Σ Σ
- -(R
CLEC R,,R,EC.
For flat clustering, a d. -imeeinnsio K-Means model is fit to the data. To help initialize this clus- tering, we seed the model with the template means plus a small amount of noise. To determine the optimal number of clusters we use measure similar to Silhouette score (Rousseeuw 1987):
[P(R.) - 1kk ll [P(R.) Hel12
B(R) R Σ Σ
(5)
[P(RR) Panle
CLEC R.CC,
where μ4 is the mean of the cluster that example R, belongs to and junn is the mean of the closest neighbor cluster of example R, Equation 5 measures the dissimilarity between neighboring clusters
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and the compactness of the data within each cluster. By maximizing E we seek to capture the most discriminative clusters.
To learn tree of templates through hicrarchical clustering, we follow the method of (Sha et al. 2017) with minor modification on how the clusters and templates are initialized. Algorithm A.4 outlines this procedure.
RESULTS
4.1 DATASET
For this work, we used an entire season of player tracking data from a professional European soccer league, consisting of 380 games, 6 of which were omitted due to missing data. The data is collected from an in- venue optical tracking system which records the (æ, y) positions of the players at 10Hz. The data also contains single-frame frame event-labels (e.g. pass, shot, cross) in associated framess these events were used only to identify which frames contained the onset of an event which we call event- frames. Unless explicitly noted, the analysis used only event-frames for training, providing over 1.8million samples across the season.
4.2 RUN COMPLEXITY
Finding the optimal solution to K-Means is NP-hard, even for clusters. However, through stan- dard methods K-means clustering can achieve an average per-iteration complexity of (samples clusters démensions)) while Gaussian mixture models have a complexity of (samples clusters. dimensions" per iteration due to the additional calculation of the precision matrix (Lloyd 1982 Verbeek et 2003). Note that for all algorithms, samples becomes SN since each agent each frame contributes to the distributions. The Hungarian Algorithm has a complexity of elements per application.
In the original algorithm of Bialkowski et al. (2016), the cost matrix per frame is calculated in manner resembling that of the GMM, requiring the full distribution (i.e. mean and precision matrix) to be computed so the likelihoods may be calculated. However, the Hungarian Algorithm is then applicd across the N -aagents in each of the S-frames, This produces a per-iteration complexity of KSN) With K N, this simplifies to 5 (see Table I of Section A.5). In contrast, K-Means and GMM have a per-iteration complexity of SNN2 and & respectively. Therefore, for sport like soccer, N 10, causing the hard-assignment based algorithm to be 1000 times slower than the proposed approach.
4.3 COMPARISON OF DISCOVERED FF
As all of these methods are unsupervised, there is no notion of a "more accurate" formation. How- ever, as the goal is to find the FF which maximizes the likelihood of the data, for each team-game- period (TGP), we computed the formation via hard-assignment and our current method, and com- puted the per-sample average log-likelihood of the data under each method. Our method produced lower (i.e. more likely) log-likelihood for every formatnnn,ion, consistent with the theoretical guarantees (Figure A). In general, the difference between the two approaches was very small, an average difference log-likelihood of 0.028.
We compute the field area covered by a role as A T where A1 and A are the eigenvalues of VA,A, the covariance matrix for that role. On average, the field area covered by role under the current method 0.021m² smaller than the corresponding role under Bialkowski et al. (2016). This is consistent with the formations learned via the two methods being extremely similar; the average KL-Divergence (Kullback Leibler 195T) between corresponding roles under the two method is 1nits.
4.4 COMPRESSION EVALUATION
Template-basee bbsse alignment has been shown produce a compressed representation of multi-agent spatiotemporal data (Lucey et al., 2013). We repeat this analysis here in Figure 3 Similar to the approach in (Sha et al. 7), we evaluate the compressibility of the approach using clustering and principle component analysis (PCA). We randomly selected 500,000 frames from the larger data set. Frames were aligned according to Algorithm A.3 "Role_current" in Figure 3) via aard-assiggment

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Figure 3: (A) Difference in the per sample log-likelihood under the template learned calculated via hard- assigmmettt and the current method. All of the values are positive demonstrating the formation learned under the current method always captures the data better. (B) Template-basce alignment produces more compressed representation of the data than an identity-based representation. Left: reconstruction error as a function of the number of clusters. Right: variance accounted for as Role_current current corresponds to the method of the current work hard (often directly under Role curreen))) corresponds the assigmentt approach.
function of the number of eigenvectors used. In both Role
or left unordered ("Identity"). K-means clustering was applied both the original unaligned data and aligned data for varying values of K. The average within-cluster
(WCE) was calculated according to WCE R Ec, ER.EC, R - μ l122 where we again abuse C₃ to indicate the 1 cluster after K-means clustering. Similarly we run PCA on both the unaligned and aligned data and compute the variance explained by the eigenvectors: Variance Explained Σ A
where
A is the jth eigenvalue indicating the significance of the 1th eigenvector.
Role-based representation, regardless of the method used to compute it, is significantly more com- pressive than identity- based representation. The representation computed via the current method is slightly more compressive than under hard assignment: the per player reconstruction error over the range in Figure 3 on average 0.76m lower and the variance explained on average is 0.068 higher.
4.5 IMPACT OF INITIALIZATION AND KEY-FRAMES
The original cemplate-eeaninng procedure proposed initializing the algorithm with the distributions of each player, as players tend to spend much of their time in a specific role. In the subsequent work of Sha al. (017). random initialization at each layer was proposed.
To assess the impact of the player-mean initialization, we ran Alg. A.3 20 times per- rer- each of which contains about 1500 frames, and recorded the reconstruction error during the K-Means initialization phase of the algorithm. While the exact reconstruction error is sample specific, all samples showed the same trend Figure 4/ player-mean initialization begins with much lower reconstruction error and converges significantly more quickly, often within 10 steps. In contrast, the random initialization is much more variable, takes many more iterations to converge, and often does not converge to good solution.
The use of event-only "key frames" is also a key performance and stabilization measure. Limiting the data to event-only frames reduces the data by factor of ~ 10, producing speed-up of the same factor. This has minimal impact on the learned template as seen Figure 43. In most instances, the templates learned are almost identical: the average 1.2-distance between the center of two role distributions 0.24m and the average Bhattacharyya distance (Bhattacharyya 1943) between two role distributions is 0.078.
4.6 CONTEXT-SPECIHE FORMATIONS
Previously, due to the slowness of the hard-assignment approach, templates had to be learned as a part of preprocessing step before borage/analyissconsumpiioo Usually this would be done at the TeP-level, generating a total of "specialist" templates per game. In contrast, the proposed method allows templates to be computed "on the fly". For several thousand rows of data, the formation can be discovered and aligned in only a few seconds. This allows us to select data under interesting contexts and learn the template that best describes those scenarios across many games.

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A
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B
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Figure 4: Impact of initialization and key-frame selection. (A) Player-mean initialization (red) enables the K-Means initialization run convergence in fewer iterations than random (blue) initialization. (B) Learning the formation on event-only "key-frames" (thick line, no hashing) results formations which are very similar the formations learned on all data (thin line, hashing). but runs significantly faster due the reduced data size and is less prone to find spurious optima. Left: An average example showing the formations learned on the two sets data are very similar. Right: An unusual "bad" example showing more disagreement between the two data sets.
Counterattack
Maintenance
Trailing
Leading
Attacking
Home
Datanding
Away
Figure 5: Context-specific templates. (A) We trained distinct templates of a given team while attack- ing and defending against certain modes (aka. "styles") of play. Data is aggregated over multiple games across the season. (B) We trained distinct templates of given team while defending during the last 10 minutes of the games while trailing and leading, both home and away. Here we have added back in the average team (i.e. group) position show the overall positioning the pitch.
Figure shows two such analyses this method unlocks On the left (A) we examine the formation of team across entire season when they attacking in and defending against two very different "styles" of play (the very offensively aggressive counterattack, and a conservative "hold the ball" can examine how a team positions itself when
maintenance style) (Ruiz al.) 2017). Similarly, leading or trailing late in game both at home or away (B). In addition to learning the formation, we can add back in the overall group positioning to see where on the pitch the team attempts to position itself. Additionally, we can learn and align the unique formations of every team across an entire season in matter of minutes (see Figure 6in A.6). Other potential analyses could include computing the formation after substitutions or analyzing how teams perform when certain individuals occupy given role; such analyses are left future work.
SUMMARY
For multi aggett systems with a high degree structure such as that seen in team sport, we are able learn mapping which takes the of agents undue entropy from permutation. In this work we have shown an improved method for learning the group representation of structured multi-agent data which is significantly faster. Additionally, the monotonically decreasing nature of its objective function provides stability. Our approach ex- ploits the independence of the role-generating distributions during the template-learning phase and enforces the hard assignment of single agent to single role only during the final alignment step. This new approach, combination with smart choice of key-frame selection and initialization, allows for this representation to be learned over n3 times faster- factor of more than 1000 for sport like soccer. By learning this representation, we are able to perform season-wide contextual and on-the-fly fly representation learning which were previously computationally prohibitive.
ordered vector of agents without introducing
A live, on-the-fly demonstration he found at https:/ //www www.:dropbo.. www.dropbox.cco/.... yippliccaaaae ons mp 42d1 4p4421
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REFERENCES
Ryan Prescott Adams and Richard S Zemel. Ranking via sinkhorn propagation. arXiv preprint 1rXiv:1106.12255 2011.
Ijaz Akhter, Tomas Simon, Sohaib Khan, lain Matthews, and Yaser Sheikh. Bilinear spatiotemporal basis models. ACM Transactions on Graphics, ((()):7:1117:122 April 2012. doi: 10.1 2159516.2159523.
Epameinondas Antonakos, Joan Alabort-i Medina, and Stefanos Zafeiriou, Active pictorial struc- tures. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2015.
Anurag Arna, Carl Doersch, and Andrew Zisserman. Exploiting temporal context for 3d human pose estimation the wild. In Proceedings of the IEEE Conference on Computer Vision and Puttern Recognition, pp. 3395-3404, 2019.
Peter W. Battaglia, Razvan Pascanu, Matthew Lai, Danilo Jimenez Rezende, and Koray Kavukcuoglu. Interaction networks for learning about objects, relations and physics. CoRR, abs/1612.00222,, 2016. URL
htt.:/.orgabbbs/ll1.. 00222
abs
Peter W. Battaglia, Jessica B. Hamrick, Victor Bapst, Alvaro Sanchez-Gonzalez, Vinícius Flores Zambaldi, Mateusz Malinowski, Andrea Tacchetti, David Raposo, Adam Santoro, Ryan Faulkner, Çaglar Gülçehre, H. Francis Song, Andrew J. Ballard, Justin Gilmer, George E. Dahl, Ashish Vaswani, Kelsey R. Allen, Charles Nash. Victoria Langston, Chris Dyer, Nicolas Heess, Daan Wierstra, Pushmeet Kohli, Matthew Botvinick, Oriol Vinyals, Yujia Li, and Razvan Pascanu. Relational inductive biases, deep learning, and graph networks. CoRR, abs/1806.0122 2018.
URL http: http::://aviiv..ogaaa//////e/ wttpg/abs/1886... 01261
Anil Bhattacharyya. On measure of divergence between statistical populations defined by their probability distributions. Bull. Calcutta Math. Soc., 35:99-109, 1943.
Alina Bialkowski, Patrick Lucey, Peter Carr, lain Matthews, Sridha Sridharan, and Clinton Fookes. Discovering team structures in soccer from spatiotemporal data. IEEE Transactions on Knowl- edge and Data Engineering, 28(10)2259-22) 2605, 2016.
Dan Cervone, Alexander DAmour, Luke Bornn, and Kirk Goldsberry. Pointwise: Predicting points and valuing decisions in real time with nba optical tracking data. Proceedings of the Sth MIT Sloan Sports Analytics Conference, Boston, MA, USA, volume 28, pp. 3, 2014.
Timothy F Cootes, Gareth J Edwards, and Christopher J Taylor. Active appearance models. 1EEE Transactions on Pattern Analysis Machine Intelligence, (6):681-885 2001.
Rodrigo Santa Cruz, Basura Fernando, Anoop Cherian, and Stephen Gould. Deeppermnet: Visual permutation learning. CoRR, abs/ 1isss/2744,,,22, 2017.
02729
Panna Felsen, Patrick Lucey, and Sujoy Ganguly. Where will they go? predicting fine-grained adversarial multi-agent motion using conditional variational autoencoders. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 732-747, 2018
Alexander Franks, Andrew Miller, Luke Bornn, Kirk Goldsberry, et al. Characterizing the spatial structure of defensive skill in professional basketball. The Annals of Applied Statistics, 9(1):
94-121, 2015.
Justin Gilmer, Samuel Schoenholz, Patrick Riley, Oriol Vinyals, and George E. Dahl. Neural message passing for quantum chemistry, CoRR, abs/1704.01212, 2017. URL http::/aarxx axq/abs/170 01212
Yedid Hoshen. VAIN: attentional multi-agent predictive modeling. CoRR, abs/1706.00122,, 2017.
URL /ttp:///aivvvvv/. org/ abs/ 1706.0000 06122
Hanbyul Joo, Hao Liu, Lei Tan, Lin Gui, Bart Nabbe, lain Matthews, Takeo Kanade, Shohei Nobuhara, and Yaser Sheikh. Panoptic studio: massively multiview system for social motion capture. In The IEEE International Conference on Computer Vision (ICCV), 2015.

---

Under review as a conference paper at ICLR 2020
Angjoo Kanazawa, Jason Y. Zhang, Panna Felsen, and Jitendra Malik. Learning 3d human dynamics from video. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
Thomas Kipf, Ethan Fetaya, Cuan-Chich Wang, Max Welling, and Richard Zemel. Neural relational inference for interacting systems. arXiv preprint arXiv:1802.0468// 2018.
Thomas N. Kipf and Max Welling. Semi-supervised classification with graph convolutional net- 609. 02907
works. CoRR, Abs//660.02900) 2016. TRL.httpp//arxiv.ooo
Harold W Kuhn. The hungarian method for the assignment problem. Naval research logistics
quarterly, 2(1-2):83-97, 1955.
Solomon Kullback and Richard A Leibler. On information and sufficiency. The annals of mathe- matical stafisties, 22(1):79-86, 1951.
Karol Kurach, Anton Raichuk, Piotr Staczyk, Micha Zaje, Olivier Bachem, Lasse Espeholt, Carlos Riquelme, Damien Vincent, Marcin Michalski, Olivier Bousquet, and Sylvain Gelly. Google research football: A novel reinforcement learning environment, 2019.
Hoang M Le, Yisong Yue, Peter Carr, and Patrick Lucey. Coordinated multi-agent imitation learning- In Proceedings of the 34th International Conference on Machine Learning- 2003. JMLR. org, 2017.
Searning-Volume Volume 70, pp. 1995-
Stuart Lloyd. Least squares quantization in pem. JEEE transactions on information theory, 28(2): 129-137, 1982.
Patrick Lucey, Alina Bialkowski, Peter Carr, Stuart Morgan, lain Matthews, and Yaser Sheikh. Rep- resenting and discovering adversarial team behaviors using player roles. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2706-2713, 2013.
Nazanin Mchrasa, Yatao Zhong, Frederick Tung, Luke Bornn, and Greg Mori. Deep learning of player trajectory representations for team activity analysis. In MIT Sloan Sports Analytics Con- ference, 2018
Gonzalo Mena, David Belanger, Scott Linderman, and Jasper Snock. Learning latent permutations with gumbel-sinkhorn networks. arXiv preprint 1rX2v:/802.000 086608665, 2018.
Andrew Miller, Luke Bornn, Ryan Adams, and Kirk Goldsberry. Factorized point process inten- sities: A spatial analysis of professional basketball. In International Conference on Machine Learning. pp. 235-243, 2014.
Federico Monti, Davide Boscaini, Jonathan Masci, Emanuele Rodola, Jan Svoboda, and Michael M Bronstein. Geometric deep learning on graphs and manifolds using mixture model cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5115- 5124, 2017.
Paul Power, Hector Ruiz, Xinyu Wei, and Patrick Lucey. Not passes are created equal: Objec- tively measuring the risk and reward of passes in soccer from tracking data. pp. 1605-1613, 08 2017. ISBN 978-1-4503-4887-4, doi: 10. 00.114//09998333009800
Mir Rayat Imtiaz Hossain and James J Little. Exploiting temporal information for 3d human pose estimation. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 68-84, 2018.
Peter Rousseeuw, Silhouettes: a graphical aid to the interpretation and validation of cluster analy- sis. Journal of computational and applied mathematics, 20:53-65 16.53355-11 1987.
Hector Ruiz, Paul Power, Xinyu Wei, and Patrick Lucey. The leicester city fairytale?: Utilizing new soccer analytics tools to compare performance in the 15/16 & 16/17 epl seasons. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1991-2000 ACM, 2017.
10

---

Under review as a conference paper at ICLR 2020
Long Sha, Patrick Lucey, Yisong Yue, Peter Carr, Charlie Rohlf, and lain Matthews. Chalkboarding: A new spatiotemporal query paradigm for sports play retrieval. 03 2016. doi: 10.1145/285667. 2856772.
Long Sha, Patrick Lucey, Stephan Zheng, Taehwan Kim, Yisong Yue, and Sridha Sridharan. Fine-grained retrieval of sports plays using tree-based alignment of trajectories. arXiv preprint arXiv:1710.02255,, 2017.
Long Sha, Patrick Lucey, Yisong Yue, Xinyu Wei, Jennifer Hobbs, Charlie Rohlf, and Sridha Srid- haran. Interactive sports analytics: An intelligent interface for utilizing trajectories for interactive sports play retrieval and analyties. ACM Transactions on Computer- Humaan Interaction, 25:1-32, 04 2018. doi: 10.1 145/318596.
Tomas Simon, Hanbyul Joo, and Yaser Sheikh. Hand keypoint detection in single images using multiview bootstrapping. CVPR, 2017.
Richard Sinkhorn and Paul Knopp. Concerning nonnegative matrices and doubly stochastic matri- ces. Pacific Journal of Mathematics, 21(2):343-348, 1967.
Sarah Taylor, Taehwan Kim, Yisong Yue, Moshe Mahler, James Krahe, Anastasio Gareia Rodriguez, Jessica Hodgins, and lain Matthews. A deep learning approach for generalized speech animation. ACM Transactions on Graphics (TOG), 36(4):93, 2017.
Jakob J Verbeek, Nikos Vlassis, and Ben Kröse. Efficient greedy learning of gaussian mixture models. Neural computation, 1885)4499-485, 2003.
Jacob Walker, Kenneth Marino, Abhinav Gupta, and Martial Hebert. The pose knows: Video fore- casting by generating pose futures. In Proceedings of the IEEE International Conference on Computer Vision, pp. 3332-3341, 2017.
Xinyu Wei, Long Sha, Patrick Lucey, Stuart Morgan, and Sridha Sridharan. Large-scale analysis of formations in soccer. 2013 International Conference Digital Image Computing: Techniques and Applications (DICTA), pp. 1-8, 2013.
Raymond A. Yeh, Alexander G. Schwing, Jonathan Huang, and Kevin Murphy. Diverse gener- ation for multi-agent sports games. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
Yisong Yue, Patrick Lucey, Peter Carr, Alina Bialkowski, and lain Matthews. Learning fine-grained spatial models for dynamic sports play prediction. In 2014 IEEE International Conference on Data Mining, pp. 670-679. IEEE, 2014.
Eric Zhan, Stephan Zheng, Yisong Yue, Long Sha, and Patrick Lucey. Generative multi-agent behavioral cloning. arXiv, 2018.
Stephan Zheng, Yisong Yue, and Patrick Lucey. Generating long-term trajectories using deep hier- archical networks. In D. D. Lee, M. Sugiyama, U. V. Luxburg, I. Guyon, and R. Garnett (eds.), Advances in Neural Information Processing Systems 29, pp. 1543-1551 Curran Associates, Inc., 2016.

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Under review as a conference paper at ICLR 2020
A APPENDIX
A.1 SUMMARY OF THE FORMATION DISCOVERY ALGORITHM OF BIALKOWSKI ET AL.
(016))
To learn the per-role player distributions, the authors cast the problem as one of data paritioning where the goal is minimize the overlap of individual role P, and that of the team P(z) by placing penalty V, on that overlap as defined by
V -KL(P(()))(())))
(6)
where
(I(((=)II(=))) / P(x) log
P(x) Q(x)
dr.
Eq. can then be written
F arg max V
(8)
F
Substituting Eq.
6 into Eq. 8 yields
Σ P(n) / P(æ]nn) log P(æln)dr
V
/ P(æ((n) log P(æ)da.
Σ P(m)
(9)
n=l
Written in terms of the entropy H(x) = f log))) this simplifies
V= -H)) + H(aIn)
(10)
yielding the final equation of as given in Eq.
They approximate the solution by using expectation maximization (EM) summarized here:
Initialization: The data is normalized such that the average team position in each frame is placed at the origin. The formation is initialized by assigning player to single role for the entire game. This independent distributions describing the n different generating roles.
allows the construction
E-Step: An cost matrix is computed for each frame which is based on the hog-probability of each player being assigned to a particular role distribution.
M-Step:
The Hungarian Algorithm (Kuhn 1955 is used to (hard) assign each player to specific role in each frame. Once roles have been assigned for that iteration, the role distributions are recomputed.
Termination: The process repeats until convergence.
A.2 RELATION TO -INBHL-SN
In the original template discovery formulation, application of the Hungarian algorithm at each iter- ation made the strong requirement that single player be assigned to single cluster in each frame, that is, nx,1 = ...j Vi,j j E K, Va S.
Our method treats each agent in each frame as fully independent; if we examined at the assignment probabilities for given frame during training, the likelihood would exceed 1. Furthermore, during training, there are no restrictions on the values of each beyond the standard requirement that
ΣK मे 1. Thus at the end of the formation-learning step, there is requirement that each
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Th k The requirement that each role be occupied by only single agent in each frame and that weights are uniform is imposed only at the assignment step.
An approach based Sinkhorn normalization can be seen intermediate between these two paradigms, allowing for coft-assignment of every agent in a frame to every cluster during the learning process.
Thus during training, a player could still contribute to multiple clusters, but the weighting of that contribution is restricted by Sinkhorn normalization which requires the assignment matrix in each frame to be doubly stochastic. Following the notation of Cruz et al. (2017), rows R and columns C normalized according to
Qi
Qis
(Q) = Ek= Qik
Ci₂(Q)=
(11)
Σx= Qie
with the n iteration defined recursively as
ST) 1Q (CC
n-
(12)
otherwise.
Similar to our approach, Sinkhorn-based methods employ hard assignment via the Hungarian al- gorithm only after training completes. The additional benefit of this approach is that the Sinkhorn normalization function is differentiable and thus amenable to gradient-based methods.
However as our approach is based expectation maximization, the drawback to Sinkhorn- normalization is that the iterative normalization step is required in each frame at each EM iteration. Thus within the EM framework, it remains cost-prohibitive although gives final assignment results equivalent to both the original and current methods.
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A.3 OPTIMAL TEMPLATE LEARNING ALGORITHM
Algorithm Optimal Template Learning
Input:
U (U1, ....... Un} unordered player positions
G a parent/global template
Output:
= [R ,,...., Rx] player positions ordered by role
F the learned formation
T the alignment template
Normalization
1: normalize the positions in each frame so that the attacking team is going left to right
2: center-normalize the positional data according to M Unn -
Unn
Vn E N
3: format M according to f IRddN + RSNxd xd
Formation Discovery
4: conduct K-Means clustering for initialization: K-Means (M, Pinit
[M1, Mz,... ..... MN])
5:
function EIGENVAL UE REEESETIIGGGMM
6: while lower bound average gain < threshold do
7: if
Vn N then
Anz
8:
11
GMM Update
9: else
11 0, π
Soft K-Means Update
return F
Template Alignment
12:
function LLIGNTEMPLATES(FF. G')) create cost matrix s.t. Ci.j is the Mahalanobis distance between the 7th distribution in F and the jth distribution in G apply Hungarian algorithm to find optimal assignment of Fi to Gj
14:
return T
Role Assignment
16:
function APPLYALIGNMENT(RR T*)
17:
for in S do
18:
( (i*h-agent frame s), belongs to create cost matrix s.t. Cj, is the likelihood the R,,, the jthh distribution of T
19:
apply Hungarian algorithm find optimal assignment of R.,, to T
20:
return R
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Under review as a conference paper at ICLR 2020
A.4 TREE-
TREE-BASED BASED ALIGNMENT
This algorithm is adapted from Sha et al. 2017). The overall structure is the same with changes to the initializations captured in A.3
Algorithm Learning process of tree-based alignment
Input: R (U1, U2,..., UN} unordered player positions
G a parent/global template
T 0,C
Output: T,C function LEARNTREE(R)
2:
for each layer / do
3: for each node n
learn F=1 Tx R2 from Algorithm A.3 with =² =²= and R -R, the
4:
data contained in the parent node
5:
store [T 1:.... T 2 in 7'
6:
compute reconstruction loss with Eq. 4
7:
terminate when stropping criterion is met
8:
for each node n. do create K initialization vectors by adding small amounts of noise to the cluster means of T
9:
10:
conduct K-Means on Rt with different select cluster set C that maximize E' partition C to child nodes according C Store [C], C²2]
11:
12:
return T,C
A.5
RUN COMPLEXITY
Average per-iteration cun-complexity for various methods.
Table 1: Algorithm complexity per iteration
ALGORITHM COMPLEXITY
Hungarian EM
d
K-means
SN
GMM
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A.6 LEAGUE- LEAGUE-WIDE ANALYSIS
We formation discovery and alignment algorithm every team-game-ppee pereeeeerdd combination of an entire season. This analysis previously took upwards of 20 minutes to process game under the original approach, but now can be run in less than 10 seconds game.
Some teams consistently operate out of the same formation and therefore the templates and distri- bution centers are well isolated across the season (e.g. second row, far right). Others play different formations in different matches and therefore the game-to-game templates can vary dramatically (e.g. third row, second from the left; bottom row, second from the left). Our algorithm is able to learn these various templates and align them so that common, structured representation can be used across matches.
Figure 6: Discovered templates for each team across season of professional soccer. The global template learned by selecting data randomly across the season (all teams, all games). Each plot corresponds to team and a template is learned for each half of every game and aligned to the global template. The centroids of each role-distribution are plotted in black.
16


Reviews:
[{"review_id": "rJgy-CEe9S", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "After author rebuttal: \nThank you for your response to the review. I appreciate your point that sport applications can be interesting to this community, and your paper makes an important contribution in that direction. With that said, for a paper to be appealing to the ICLR audience, there should be more clear novelty in the method, and stronger experimental evaluation showing novel insights or out-performance over previous baselines. \n\n\n==================================================================\nThe paper addresses a problem in the context of understanding formations of sport player in a play field. The problem is that players form an ומ-ordered sets of entities and to analyze them, one would like to keep some canonical representation that allows us to compare formations across time. The idea of this paper is to assign roles to each players (name each member of the set) based on the distribution of their spatial locations on a field. Once this representation is learned, future work may use it for downstream problems, which are not discussed in this paper. \n\nGeneral: \n\nThe paper is very clearly written, the method makes sense and the related work are both well explained. \nI have two main concerns: First, the scope of the paper may be too narrow for a conference like ICLR, as the method part is fundamentally a clustering method, and the application is very specific. Second, I was expecting that the paper shows interesting results that can be obtained once the representation is learned, but this are lacking from the paper. \nBased on these two concerns, I dont think the paper would be a good fit for ICLR. \n\nTo make the paper more appealing to the ML community, my recommendation to the authors is to  show what can be achieved with the new representation, that was impossible with previous representations. For example, it is claimed that faster learning allows to compute it season wide and on-the-fly. What are useful tasks that your rep' can help with? How do these tasks perform with the new rep? \n\nOther comments and questions. \n\n-- What about other competing representations. Would it be useful to look at movement trajectories instead of distribution of isolated locations (cluster lines instead of points)? \n-- What happens when a player is replaced by another player of the same role. Wouldnt its distribution change? Dont you want your rep' to reflect such events? \n", "extra_scores": {"experience_assessment": "I have read many papers in this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "N/A", "review_assessment:_checking_correctness_of_derivations_and_theory": "N/A"}, "extra_text": {}}, {"review_id": "r1lT7803tH", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\n  *Synopsis*:\n  The paper proposes a new algorithm for template discovery and representation learning for unstructured multi-agent data with strong group roles (i.e. sports data). Their algorithm is several orders of magnitude faster because of a key assumption in the underlying structure of the position generating distributions of players conditioned on roles. They also formulate the problem in a novel way, splitting the structuring of the data into a three-step approach with each component requiring a different algorithm. This will allow the community to swap in new algorithms into the different components more simply than before. They finally provide some evidence of their algorithm working on a soccer dataset consisting of 380 games.\n\n  Main Contributions:\n  - A key insight into the underlying structure of the data, enabling an algorithm with orders of magnitude speed up.\n  - A new framing of the problem.\n\n  *Review*:\n\n  The paper is well written, and seems to be a nice improvement over prior art in terms of runtime. I am not very familiar with the surrounding literature, but had a few concerns over the current paper that I feel should be addressed. Overall, I think this paper could be accepted unless other reviewers more familiar with the literature find critical flaws. I would, however, like to see Q2 addressed by the authors.\n\n  Q1 It seems the key insight is that you are modeling the occupancy of space by an agent as a mixture of independent gaussian distributions. How reasonable is the assumption that each generating distribution conditioned on a role is independent of the others? Wouldn't a roles place on the field be dependent by the placement of other roles?\n\n  Q2 How does the performance (i.e. accuracy) of your algorithm compare to prior approaches (i.e. the Sha or Bialkowski)? Or really any baseline? It is hard to judge this method in isolation (other than the impressive computational gains), especially for someone unfamiliar with this literature. Is there a way you could include a comparison in the final draft? If not, why? \n      As a note: It seems you include some comparisons in section 4.3, 4.4, and 4.5, but it is unclear to me what the competitor is (i.e. A.3) and how it relates to the surrounding literature. If this is related to prior art, you should make it clear what algorithm this is in the main text, not relegating all discussion to the appendix.\n\n  C1: It would be nice to include table 1 (appendix) in the main paper, or you could include a simplification of the runtime complexity of the K-Means and GMM algorithms in terms you use for the Hungarian algorithm in the main text. This would make it easer to see the significant gains your approach (i.e. using k-means) gets over prior approaches.\n\n  *Minor Comments not taken into account in the review*\n  - You should be more clear what the Hungarian algorithm is. Specifically, the Kuhn 1955 should be cited when the algorithm is first mentioned.\n  - \"global shape and structure the group\" -> \"global shape and structure of the group\"\n  - \"'anchor' agent (Mehrasa et al.)\" -> year for citation\n\n----------\nPost Review\n\nI appreciate the author addressing my concerns and making clarifications here in discussion and in the text. I'm going to keep my score as a 6 primarily because I am not confident in the literature this paper is appealing to to truly advocate for its acceptance.\n\n", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}, {"review_id": "Bkx-lXf6KH", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "In this paper authors describe a way to quickly model interaction between many entities, in the form of mixture of Gaussians. They provide huge computational complexity improvements over Bialkowski et al. work, which allows them to analyse big datasets.\n\nPaper, in its current form, lacks quantitative evaluation, comparison to other methods (heavily cited in the paper), and any well established applications that can be quantified. Given that the method described is a composition of many known techniques, and some new tricks, it is hard to assess the value of these contributions without seeing more unified evaluation methodology. All the provided results (apart from computational complexity wrt. specific approach, which are quite amazing in isolation!) are qualitative and descriptive in nature, which together with the fact of being applied to a single, specific dataset, with makes it hard to judge any generality. This is a common problem with various \"structure discovery methods\" such as clustering techniques, but an important one to address.\n\nIt is also unclear to reviewer, whether the only (main?) claim authors want to make is that they provide speed-up to the existing solution. if this was the case, it is really hard to see how each of the proposed steps guarantees the same outputs as previous approaches, while at the same time only affecting speed. In particular authors make the claim of improvement stability, yet do not provide empirical evaluation showing these benefits over previous methods. \n\nPlease compare these to results reported e.g. in the cited work of Yeh et al. \"Diverse Generation for Multi-agent Sports Games\".\n\nOverall, in reviewer's opinion, results looks quite incremental, and are heavily concentrated around specific paper of Bialkowski et al., and as such might see limited interest in the ICLR community.\n\nMinor comments:\n- please unify use of quotations (authors use closing quotations for both opening and closing quotes)\n- is there an extra | symbol in eq. (4)?\n\n\nPlease note, that reviewer has little experience in this particular branch of the field, and thus some of the issues mentioned could come from misunderstanding the standards.", "extra_scores": {"experience_assessment": "I do not know much about this area.", "review_assessment:_thoroughness_in_paper_reading": "I read the paper at least twice and used my best judgement in assessing the paper.", "review_assessment:_checking_correctness_of_experiments": "I assessed the sensibility of the experiments.", "review_assessment:_checking_correctness_of_derivations_and_theory": "I assessed the sensibility of the derivations and theory."}, "extra_text": {}}]

Author rebuttal:

Meta-review: The work addresses the problem of inferring group structure from unstructured data in multi-agent learning settings, proposing a novel approach that has key computational / run time advantages over a prior approach. A key limitation raised by reviewers is the limited quantitative evaluation and comparison to previous approaches, as well as a resulting set of general insights into advantages of the proposed approach compared to prior work (beyond computational benefits). While some of the key limitations were addressed in the rebuttal, the contribution in its current form remains too narrow. The paper is not ready for publication at ICLR at this stage.
Decision: Reject