Geometry and Symmetry in Short-and-Sparse Deconvolution

Han-Wen Kuo, Yuqian Zhang, Yenson Lau, John Wright

Abstract: We study the Short-and-Sparse (SaS) deconvolution problem of recovering a short signal a0 and a sparse signal x0 from their convolution. We propose a method based on nonconvex optimization, which under certain conditions recovers the target short and sparse signals, up to a signed shift symmetry which is intrinsic to this model. This symmetry plays a central role in shaping the optimization landscape for deconvolution. We give a regional analysis, which characterizes this landscape geometrically, on a union of subspaces. Our geometric characterization holds when the length-p0 short signal a0 has shift coherence μ, and x0 follows a random sparsity model with sparsity rate <= []. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.