Expressive Sign Equivariant Networks for Spectral Geometric Learning

Derek Lim, Joshua Robinson, Stefanie Jegelka, Haggai Maron

Keywords: Eigenvectors, spectral, geometry, universal approximation, graph, equivariance, invariance

TL;DR: We prove that a novel type of equivariance to eigenvector symmetries is useful in several applications, and develop provably expressive networks with this equivariance.

Abstract: Recent work has shown the utility of developing machine learning models that respect the structure and symmetries of eigenvectors. These works promote sign invariance, since for any eigenvector v the negation -v is also an eigenvector. However, we show that sign invariance is theoretically limited for tasks such as building orthogonally equivariant models and learning node positional encodings for link prediction in graphs. In this work, we demonstrate the benefits of sign equivariance for these tasks. To obtain these benefits, we develop novel sign equivariant neural network architectures. Our models are based on a new analytic characterization of sign equivariant polynomials and thus inherit provable expressiveness properties. Controlled synthetic experiments show that our networks can achieve the theoretically predicted benefits of sign equivariant models.

Submission Number: 7752