Node-Asynchronous Implementation of Filter Banks on Graphs

Filter banks on graphs are shown to be useful for analyzing data defined over networks, as they decompose a graph signal into components with low variation and high variation. Based on recent node-asynchronous implementation of graph filters, this study proposes an asynchronous implementation of filter banks on graphs. In the proposed algorithm nodes follow a randomized collect-compute-broadcast scheme: if a node is in the passive stage it collects the data sent by its incoming neighbors and stores only the most recent data. When a node gets into the active stage at a random time instance, it does the necessary filtering computations locally, and broadcasts a state vector to its outgoing neighbors. When the underlying filters (of the filter bank) are rational functions with the same denominator, the proposed filter bank implementation does not require additional communication between the neighboring nodes. However, computations done by a node increase linearly with the number of filters in the bank. It is also proven that the proposed asynchronous implementation converges to the desired output of the filter bank in the mean-squared sense under mild stability conditions. The convergence is verified also with numerical experiments.