Algebraic Convolutional Filters on Lie Group Algebras

TL;DR

This paper introduces a novel algebraic convolutional filter on Lie group algebras that simplifies group convolutional neural networks by eliminating the need for lifting, and demonstrates its stability and effectiveness on datasets with SO(3) symmetry.

Contribution

It proposes a new convolutional filter directly on Lie group algebras, removing the costly lifting step in group CNNs, and establishes its stability through multigraph signal processing.

Findings

Effective on datasets with SO(3) symmetry

Demonstrates stability of the proposed filter

Reduces computational complexity compared to existing methods

Abstract

Group convolutional neural networks are a useful tool for utilizing symmetries known to be in a signal; however, they require that the signal is defined on the group itself. Existing approaches either work directly with group signals, or they impose a lifting step with heuristics to compute the convolution which can be computationally costly. Taking an algebraic signal processing perspective, we propose a novel convolutional filter from the Lie group algebra directly, thereby removing the need to lift altogether. Furthermore, we establish stability of the filter by drawing connections to multigraph signal processing. The proposed filter is evaluated on a classification problem on two datasets with $SO(3)$ group symmetries.