An orthogonal partition selection strategy for the sampling of graph signals with successive local aggregations

Abstract

We study the sampling of graph signals with successive local aggregations, which computes measurements as the result of iterated applications of the aggregation operator observed at one or few nodes. It has been proved in the literature that using the first few observations at a single node, a perfect reconstruction can be achieved for bandlimited graph signals. Yet, it requires some restrictions to make the sampling matrix have a Vandermonde structure. Unfortunately, Vandermonde matrices are notoriously ill-conditioned resulting in severe instability even for moderately-sized graphs. Moreover, such a sampling strategy is not applicable for the sampling scheme at multiple nodes. In view of these, we re-establish a theory for the perfect reconstruction in this paper, starting with the single-node sampling scheme. To handle large-sized graphs well, we extend it to the multi-node sampling scheme. To do this, an orthogonal partition selection strategy (OPSS) is proposed to find a node set I with a minimum size such that the observations at I guarantee a perfect reconstruction for bandlimited graph signals, and two reconstruction methods are provided. When considering a noisy regime, a performance analysis of OPSS is provided, and thus yields an improved OPSS (IOPSS) which guarantees a stable reconstruction by presetting a threshold σts. Finally, several experiments are conducted to evaluate the performance of these techniques.