Neural Fourier Transform: A General Approach to Equivariant Representation Learning

TL;DR

Neural Fourier Transform (NFT) offers a universal framework for learning equivariant representations by identifying linear group actions without prior assumptions, supported by theoretical foundations and experimental validation.

Contribution

NFT introduces a general method to learn latent linear group actions without explicit knowledge of the data's symmetry group, bridging theory and practical applications.

Findings

NFT's theoretical foundation links linear equivariant features to invariant kernels.

Experimental results demonstrate NFT's effectiveness across different scenarios.

NFT generalizes symmetry learning beyond architectural assumptions.

Abstract

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.