Topological Signal Processing over Weighted Simplicial Complexes

TL;DR

This paper introduces topological signal processing tools for weighted simplicial complexes, leveraging weighted Hodge Laplacian theory to jointly learn weights and filters for complex signals, enhancing data feature extraction.

Contribution

It presents novel methods for joint weight and filter learning over weighted simplicial complexes using weighted Hodge Laplacian theory.

Findings

Effective procedures for joint weight and filter learning.

Numerical validation demonstrates improved signal analysis.

Enhanced extraction of higher-order data features.

Abstract

Weighing the topological domain over which data can be represented and analysed is a key strategy in many signal processing and machine learning applications, enabling the extraction and exploitation of meaningful data features and their (higher order) relationships. Our goal in this paper is to present topological signal processing tools for weighted simplicial complexes. Specifically, relying on the weighted Hodge Laplacian theory, we propose efficient strategies to jointly learn the weights of the complex and the filters for the solenoidal, irrotational and harmonic components of the signals defined over the complex. We numerically asses the effectiveness of the proposed procedures.