Geometry and Symmetry in Short-and-Sparse Deconvolution
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We study the Short-and-Sparse (SaS) deconvolution problem of recovering a short signal a_0 and a sparse signal x_0 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-p_0 short signal a_0 has shift coherence μ, and x_0 follows a random sparsity model with sparsity rate θ∈[c_1/p_0, c_2/p_0√(μ) + √(p_0)]·1/^2p_0. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.
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