Distributed Denoising over Simplicial Complexes using Chebyshev Polynomial Approximation

Abstract

In this work, we focus on denoising smooth signals supported on simplicial complexes in a distributed manner. We assume that the simplicial signals are dominantly smooth on either the lower or upper Laplacian matrices, which are used to compose the so-called Hodge Laplacian matrix. This corresponds to denoising non-harmonic signals on simplicial complexes. We pose the denoising problem as a convex optimization problem, where we assign different weights to the quadratic regularizers related to the upper and lower Hodge Laplacian matrices and express the optimal solution as a sum of simplicial complex operators related to the two Laplacian matrices. We then use the recursive relation of the Chebyshev polynomial to implement these operators in a distributed manner. We demonstrate the efficacy of the developed framework on synthetic and real-world datasets.