Geometry and Symmetry in Short-and-Sparse Deconvolution
Han-Wen Kuo, Yenson Lau, Yuqian Zhang, 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 {\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.
BibTeX
@InProceedings{pmlr-v97-kuo19a,
title = {Geometry and Symmetry in Short-and-Sparse Deconvolution},
author = {Kuo, Han-Wen and Lau, Yenson and Zhang, Yuqian and Wright, John},
booktitle = {Proceedings of the 36th International Conference on Machine Learning},
pages = {3570--3580},
year = {2019},
editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan},
volume = {97},
series = {Proceedings of Machine Learning Research},
month = {09--15 Jun},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v97/kuo19a/kuo19a.pdf},
url = {https://proceedings.mlr.press/v97/kuo19a.html},
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 {\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.}
}