Improving unbalanced downsampling via maximum spanning trees for graph signals

Xianwei Zheng, Yuan Yan Tang, Jiantao Zhou, Patrick Shen-Pei Wang

Abstract: The state-of-the-art downsampling method for graph signals has been constructed by using maximum spanning trees (MSTs) of the graphs. For the graph signals defined on unweighted densely connected graphs, such as social network data, the sampling rates via MST-based downsampling are not close to 1/2, leading to a unbalanced downsampling phenomenon on multi-level downsampling. The unbalance hinders the applications of MST-based downsampling on constructing graph signal multiscale transforms, such as graph wavelet decomposition and multiscale pyramid transform. In this paper, we propose a simple but efficient method to improve the performance of the MST-based method on downsampling balance. For every graph signal, we first propose an unbalance possibility to measure the unbalance of the MST-based downsampling. If the unbalance possibility is high, the downsampling will be conducted on an improved MST, which is constructed by rearranging the structure of the MST to reduce the downsampling unbalance. The experiment results on synthesis graph signal show that the proposed improved MST leads to balanced downsampling. That is, the sampling rates produced by the improved MST are closer to 1/2 in multi-level downsampling than the original MST-based method.