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arXiv · 1206.0937

Detecting Activations over Graphs using Spanning Tree Wavelet Bases

James Sharpnack·Akshay Krishnamurthy·Aarti Singh
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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 $ε$-graphs.

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James Sharpnack, Akshay Krishnamurthy, Aarti Singh. 2012-07-12. Detecting Activations over Graphs using Spanning Tree Wavelet Bases. https://arxiv.org/abs/1206.0937

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