For example, the phase transitionsįor ℓ 1 minimization used in recovering sparse signals and nuclear norm minimization used in recovering low-rank matrix have been well Number of measurements is smaller than the threshold, the convex optimization will fail to recover the underlying structured signals with high probability.Ī series of works studying convex geometry for linear inverse problems have made substantial progress in theoretically characterizing the phase transition phenomenon for convex optimizations in Numerical results empirically show that these convex optimization based signal recovery algorithms often exhibitĪ phase transition phenomenon: when the number of measurements exceeds a certain threshold, the convex optimization can correctly recover the structured signals with high probability when the Minimization, to efficiently recover the signalįrom a much smaller number of measurements than the ambient signal dimension. Is to exploit the sparse structures inherent to the underlying signal, and design sparsity-promoting convex optimization programs, such as ℓ 1 For example, in compressed sensing, the main idea In the last decade, using convex optimization to recover parsimoniously-modeled signal or data from a limited number of samples has attracted significant research interests inĬompressed sensing, machine learning and statistics.
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