Robust Signal Processing Over Simplicial Complexes

The goal of this paper is to investigate the impact of perturbations of topological descriptors, such as graphs and simplicial complexes, on the robustness of filters acting on signals observed over such domains. Given a nominal graph that may undergo small perturbations of its edges, we design robust FIR filters using approximate closed form expressions for the perturbed eigendecomposition of the Laplacian matrix associated with the nominal graph. Then, we extend the analysis to simplicial complexes and show how the perturbation of a few triangles affect the homology class of the simplicial complex. Our small perturbation analysis of a second order simplicial complex yields approximate closed form expressions of the high order Laplacian eigenvalue/eigenvector perturbation, which are useful for the design of robust FIR filters acting on solenoidal signals. Numerical results assess the accuracy of the derived analysis and the effectiveness of the proposed method.