Topological Slepians

This repo contains the code used for implementing the numerical results in the paper:

"Topological Slepians: maximally localized representations of signals over simplicial complexes"

C. Battiloro, P. Di Lorenzo, S. Barbarossa

DIET Department, Sapienza University of Rome, Rome, Italy

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.

Summary

The code is ready to run (with saving directory to be specified on local machines). For any questions, comments or suggestions, please e-mail Claudio Battiloro at claudio.battiloro@uniroma1.it. The results showed in the sparse representation curve are in sparsity.csv (that it the output of simplicial_slepians_experiments.py. The NMSE vs SNS curve of the denoising task can be obtained via the denoising_curve_script_vs_SNR.m that takes as inputs the denoising results (i.e. the outputs of simplicial_slepians_experiments_denois.py).

Thanks to Mitch Roddenberry (Rice ECE) for sharing the code of his Hodgelets paper; this code and the experimental set-up are built on top of it.

Important files description

1. simplicial_slepians_experiments.py: This python script computes the results of the representation task.

2. simplicial_slepians_experiments_denois.py: This python script computes the results of the denoising task.

3. lib.py: This python script contains the implementation of Topological Slepians and competitors.