Heat Kernel Smoothing Using Node-Invariant, Node-Variant and Edge-Variant Graph Filters

Abstract

In this paper, heat kernel smoothing (HKS) method is implemented by using polynomial graph filters. First, the discrete HKS is obtained from the heat equation by replacing continuous Laplacian operator with graph Laplacian matrix. The ideal transformation matrix of HKS is a matrix exponential which is only suitable for centralized implementation. Then, three distributed graph filters are presented to implement HKS including node-invariant graph filter, node-variant graph filter and edge-variant graph filter. The convex optimization method can be used to determine the optimal filter coefficients of these three kinds of graph filters. Next, these graph filters are compared in terms of design error, computational complexity and memory requirement. Finally, the effectiveness of HKS method is demonstrated by using temperature data denoising experiment of sensor network.