Decentralised signal processing on graphs via matrix inverse approximation

In the processing of signals defined over graph domains, it is highly desirable to have algorithms that can be implemented in a decentralised manner, whereby each node only needs to exchange information within a localized subgraph of nodes. The most common example is when the subgraph consist of immediate neighbors and this allows for distributed processing. Many graph signal operators, in their original forms, are not amenable to distributed processing. To overcome this deficiency, a polynomial approximation is usually applied to the original operator to yield a distributed operator which is a matrix polynomial of the graph shift matrix. This approach however is only applicable when the original operator is a function of the graph shift matrix. In this paper, we propose a generalized approach to approximate graph signal operators that are not necessary functions of the shift matrix. The key idea here is to restrict the approximated matrix inverse to have small geodesic-width so that multiplication with the small geodesic-width matrix can be implemented in a decentralised manner. Furthermore, to increase the accuracy of the approximated operator, an iterative algorithm which also has the decentralised property and low computational complexity, is proposed. We apply the proposed approach to signal inpainting and signal reconstruction in filter banks. Numerical results verify the effectiveness of the proposed algorithm. The proposed algorithms can also be used to efficiently solve linear system of equations.