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Nearoptimal and computationally efficient detectors for weak and sparse graphstructured patterns
"... Abstract—In this paper, we review our recent work on detecting weak patterns that are sparse and localized on a graph. This problem is relevant to many applications including detecting anomalies in sensor and computer networks, brain activity, coexpressions in gene networks, disease outbreaks etc. ..."
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Abstract—In this paper, we review our recent work on detecting weak patterns that are sparse and localized on a graph. This problem is relevant to many applications including detecting anomalies in sensor and computer networks, brain activity, coexpressions in gene networks, disease outbreaks etc. We characterize such a class of weak and sparse graphstructured patterns by small subsets of weakly activated nodes with a low cut in an underlying known graph. On one hand, the combinatorial nature of this class renders traditional detectors such as GLRT (aka scan statistic) computationally intractable for general graphs. On the other hand, attempts to develop feasible detectors such as fast subset scanning or averaging/thresholding sacrifice statistical efficiency. We describe and compare three detectors for weak graphstructured patterns that are developed using tools from graph theory, optimization and machine learning. These detectors are computationally efficient, applicable to graphs and patterns with general structures and come with precise theoretical guarantees, often achieving nearoptimal statistical performance. Index Terms—graph patterns, structured sparsity, detection I.
Detecting anomalous activity on networks with the graph Fourier scan statistic. arXiv: 1311.7217
, 2014
"... We consider the problem of deciding, based on a single noisy measurement at each vertex of a given graph, whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation. This problem is relevant to several applications such as surveil ..."
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We consider the problem of deciding, based on a single noisy measurement at each vertex of a given graph, whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation. This problem is relevant to several applications such as surveillance, disease outbreak detection, biomedical imaging, environmental monitoring, etc. Since the activations in these problems often tend to be localized to small groups of vertices in the graphs, we model such activity by a class of signals that are supported over a (possibly disconnected) cluster with low cut size relative to its size. We analyze the corresponding generalized likelihood ratio (GLR) statistics and relate it to the problem of finding a sparsest cut in the graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the graph Fourier scan statistic, and analyze its properties. We show how its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on a few significant graph topologies. Finally, we demonstrate theoretically and with simulations that the graph Fourier scan statistic can outperform näıve testing procedures based on global averaging and vertexwise thresholding. We also demonstrate the usefulness of the GFSS by analyzing groundwater Arsenic concentrations from a U.S. Geological Survey dataset. 1