Topological Slepians: Maximally Localized Representations of Signals over Simplicial Complexes

TL;DR

This paper introduces topological Slepians, a new class of signals over simplicial complexes that are optimally localized in both topological and frequency domains, enabling improved sparse representation and denoising.

Contribution

It proposes a novel construction of topological Slepians as eigenvectors of localization operators and develops dictionaries that form non-degenerate frames for signals on topological spaces.

Findings

Effective in sparse signal representation

Improves denoising of edge flows

Provides a principled way to build localized dictionaries

Abstract

This paper introduces topological Slepians, i.e., a novel class of signals defined over topological spaces (e.g., simplicial complexes) that are maximally concentrated on the topological domain (e.g., over a set of nodes, edges, triangles, etc.) and perfectly localized on the dual domain (e.g., a set of frequencies). These signals are obtained as the principal eigenvectors of a matrix built from proper localization operators acting over topology and frequency domains. Then, we suggest a principled procedure to build dictionaries of topological Slepians, which theoretically provide non-degenerate frames. Finally, we evaluate the effectiveness of the proposed topological Slepian dictionary in two applications, i.e., sparse signal representation and denoising of edge flows.