Classifying Signals Under a Finite Abelian Group Action: The Finite Dimensional Setting

TL;DR

This paper extends algebraic signal classification methods from cyclic to all finite Abelian groups, introducing a flexible, computationally efficient transform that enhances robustness and broadens application scope.

Contribution

It generalizes previous algebraic transforms to all finite Abelian groups and introduces a new, more flexible transform avoiding high-power computations.

Findings

Constructed a Lipschitz translation invariant transform for all finite Abelian groups.

Proposed a new transform that is computationally less demanding.

Demonstrated local robustness of the new signal representations.

Abstract

Let $G$ be a finite group acting on $\mathbb{C}^N$. We study the problem of identifyng the class in $\mathbb{C}^N / G$ of a given signal: this encompasses several types of problems in signal processing. Some instances include certain generalizations of phase retrieval, image recognition, the analysis of textures, etc. In our previous work \cite{prev}, based on an algebraic approach, we constructed a Lipschitz translation invariant transform -- the case when $G$ is cyclic. Here, we extend our results to include all finite Abelian groups. Moreover, we show the existence of a new transform that avoids computing high powers of the moduli of the signal entries--which can be computationally taxing. The new transform does not enjoy the algebraic structure imposed in our earlier work and is thus more flexible. Other (even lower) dimensional representations are explored, however they only provide an almost everywhere (actually generic) recovery which is not a significant drawback for applications. Our constructions are locally robust and provide alternatives to other statistical and neural networks based methods.