Detecting Activations over Graphs using Spanning Tree Wavelet Bases

TL;DR

This paper develops a graph signal detection method using spanning tree wavelet bases, providing theoretical guarantees and near-optimal performance for various graph types under Gaussian noise.

Contribution

Introduces a novel spanning tree wavelet basis for graph signal detection, with provable guarantees and improved performance over existing methods.

Findings

The method can distinguish signals in low SNR regimes.

Uniform spanning tree basis yields stronger theoretical guarantees.

Near-optimal detection performance on specific graph classes.

Abstract

We consider the detection of activations over graphs under Gaussian noise, where signals are piece-wise constant over the graph. Despite the wide applicability of such a detection algorithm, there has been little success in the development of computationally feasible methods with proveable theoretical guarantees for general graph topologies. We cast this as a hypothesis testing problem, and first provide a universal necessary condition for asymptotic distinguishability of the null and alternative hypotheses. We then introduce the spanning tree wavelet basis over graphs, a localized basis that reflects the topology of the graph, and prove that for any spanning tree, this approach can distinguish null from alternative in a low signal-to-noise regime. Lastly, we improve on this result and show that using the uniform spanning tree in the basis construction yields a randomized test with stronger theoretical guarantees that in many cases matches our necessary conditions. Specifically, we obtain near-optimal performance in edge transitive graphs, $k$-nearest neighbor graphs, and $\epsilon$-graphs.