REVISITING MULTI-PERMUTATION EQUIVARIANCE THROUGH THE LENS OF IRREDUCIBLE REPRESENTATIONS

Yonatan Sverdlov, Ido Springer, Nadav Dym

Keywords: deep weight spaces, permutation equivariance, irredicible representations.

TL;DR: Characterization of all permutation equivariant linear mappings using irreducibles approach on several use-cases.

Abstract:

This paper explores the characterization of equivariant linear layers for representations of permutations and related groups. Unlike traditional approaches,
which address these problems using parameter-sharing, we consider an alternative
methodology based on irreducible representations and Schur’s lemma. Using this
methodology, we obtain an alternative derivation for existing models like DeepSets,
2-IGN graph equivariant networks, and Deep Weight Space (DWS) networks. The
derivation for DWS networks is significantly simpler than that of previous results.
Next, we extend our approach to unaligned symmetric sets, where equivariance
to the wreath product of groups is required. Previous works have addressed this
problem in a rather restrictive setting, in which almost all wreath equivariant layers
are Siamese. In contrast, we give a full characterization of layers in this case and
show that there is a vast number of additional non-Siamese layers in some settings.
We also show empirically that these additional non-Siamese layers can improve
performance in tasks like graph anomaly detection, weight space alignment, and
learning Wasserstein distances.

Primary Area: learning on graphs and other geometries & topologies

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Submission Number: 2593