Title: Robust Filter Design for Graph Signals

Abstract: Our goal in this paper is the robust design of filters acting on signals observed over graphs subject to small perturbations of their edges. The focus is on developing a method to identify spectral and polynomial graph filters that can adapt to the perturbations in the underlying graph structure while ensuring the filters adhere to the desired spectral mask. To address this, we propose a novel approach that leverages approximate closed-form expressions for the perturbed eigendecomposition of the Laplacian matrix associated with the nominal topology. Furthermore, when dealing with noisy input signals for graph filters, we propose a strategy for designing FIR filters that jointly minimize the approximation error with respect to the ideal filter and the estimation error of the output, ensuring robustness against both graph perturbations and noise. Numerical results validate the effectiveness of our proposed strategies, highlighting their capability to efficiently manage perturbations and noise.