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 context. 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 multiagent data enabling 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.
*Synopsis*:
  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.
  Main Contributions:
  - A key insight into the underlying structure of the data, enabling an algorithm with orders of magnitude speed up.
  - A new framing of the problem.
  *Review*:
  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.
  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?
  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? 
      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.
  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.
  *Minor Comments not taken into account in the review*
  - You should be more clear what the Hungarian algorithm is. Specifically, the Kuhn 1955 should be cited when the algorithm is first mentioned.
  - "global shape and structure the group" -> "global shape and structure of the group"
  - "'anchor' agent (Mehrasa et al.)" -> year for citation
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Post Review
I 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.