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Demixing and Blind Deconvolution of Graph-Diffused Sparse Signals

This paper generalizes the classical joint problem of signal demixing and blind deconvolution to the realm of graphs. We investigate a setup where a single observation formed by the sum of multiple graph signals is available. The main assumption is that each individual signal is generated by an originally sparse input diffused through the graph via the application of a graph filter. In this context, we address the related problems of: 1) separating the individual graph signals, 2) identifying the unknown input supports, and 3) estimating the coefficients of the diffusing graph filters. We first consider the case where each signal – prior to mixing – is diffused in a different graph. We then particularize the results for the more challenging case where all the signals are diffused in the same graph. The corresponding demixing and blind graph-signal deconvolution problems are formulated, convex relaxations are presented, and recovery conditions are discussed. Numerical experiments in both the single and multiple graph cases show the capabilities of demixing in synthetic and biology-inspired graphs.