Geometry and Symmetry in Short-and-Sparse Deconvolution

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 θ ∈ [c₁/p₀, c₂p₀√μ + √p₀]·1/log²p₀. Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.