Filtering Random Graph Processes Over Random Time-Varying Graphs
For researchers in graph signal processing, this work provides the first theoretical analysis of graph filters under stochasticity and offers practical improvements for noise cancellation and computational efficiency.
This paper analyzes the statistical behavior of FIR and ARMA graph filters on random time-varying graphs and signals, showing that in expectation they behave as deterministic filters on the expected graph and signal. It proposes two methods exploiting randomness for improved noise cancellation and up to 4x complexity reduction with minimal performance loss.
Graph filters play a key role in processing the graph spectra of signals supported on the vertices of a graph. However, despite their widespread use, graph filters have been analyzed only in the deterministic setting, ignoring the impact of stochastic- ity in both the graph topology as well as the signal itself. To bridge this gap, we examine the statistical behavior of the two key filter types, finite impulse response (FIR) and autoregressive moving average (ARMA) graph filters, when operating on random time- varying graph signals (or random graph processes) over random time-varying graphs. Our analysis shows that (i) in expectation, the filters behave as the same deterministic filters operating on a deterministic graph, being the expected graph, having as input signal a deterministic signal, being the expected signal, and (ii) there are meaningful upper bounds for the variance of the filter output. We conclude the paper by proposing two novel ways of exploiting randomness to improve (joint graph-time) noise cancellation, as well as to reduce the computational complexity of graph filtering. As demonstrated by numerical results, these methods outperform the disjoint average and denoise algorithm, and yield a (up to) four times complexity redution, with very little difference from the optimal solution.