On Learning Sets of Symmetric Elements

TL;DR

This paper introduces Deep Sets for Symmetric Elements (DSS) layers, a novel neural network architecture that effectively learns from sets of symmetric elements, improving expressiveness and performance across various applications.

Contribution

The paper characterizes the space of linear layers equivariant to element symmetries and reordering, and introduces DSS layers that are universal approximators for invariant and equivariant functions.

Findings

DSS layers are more expressive than Siamese networks.

DSS layers improve performance on image, graph, and point-cloud tasks.

Networks with DSS layers are straightforward to implement.

Abstract

Learning from unordered sets is a fundamental learning setup, recently attracting increasing attention. Research in this area has focused on the case where elements of the set are represented by feature vectors, and far less emphasis has been given to the common case where set elements themselves adhere to their own symmetries. That case is relevant to numerous applications, from deblurring image bursts to multi-view 3D shape recognition and reconstruction. In this paper, we present a principled approach to learning sets of general symmetric elements. We first characterize the space of linear layers that are equivariant both to element reordering and to the inherent symmetries of elements, like translation in the case of images. We further show that networks that are composed of these layers, called Deep Sets for Symmetric Elements (DSS) layers, are universal approximators of both invariant and equivariant functions, and that these networks are strictly more expressive than Siamese networks. DSS layers are also straightforward to implement. Finally, we show that they improve over existing set-learning architectures in a series of experiments with images, graphs, and point-clouds.