Geometry and Symmetry in ShortandSparse Deconvolution
[edit]
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:35703580, 2019.
Abstract
We study the ShortandSparse (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 lengthp0 short signal a0 has shift coherence {\textmu}, and x0 follows a random sparsity model with sparsity rate $\theta$ $\in$ [c1/p0, c2/(p0\sqrt{\mu}+\sqrt{p0})] / (log^2(p0)) . Based on this geometry, we give a provable method that successfully solves SaS deconvolution with high probability.
Related Material


