Topological Slepians: Maximally Localized Representations of Signals Over Simplicial Complexes

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.

Dirac signal processing of higher-order topological signals

“…Higher-order networks and simplicial complexes not only reveal the topology of data [6,[8][9][10] but have also transformed our understanding of the interplay between structure and dynamics [1,3,11] in networked systems. In fact these higher order networks can support topological signals [12][13][14][15][16][17], i.e. variables not only associated to the nodes of a network but associated also to the links, to the (filled) triangles and in general to the higher dimensional simplices of simplicial complexes.…”