Geometric Conditions for Subspace-Sparse Recovery
Abstract
Given a dictionary \Pi and a signal ξ= \Pi \mathbf x generated by a few \textitlinearly independent columns of \Pi, classical sparse recovery theory deals with the problem of uniquely recovering the sparse representation \mathbf x of ξ. In this work, we consider the more general case where ξlies in a low-dimensional subspace spanned by a few columns of \Pi, which are possibly \textitlinearly dependent. In this case, \mathbf x may not unique, and the goal is to recover any subset of the columns of \Pi that spans the subspace containing ξ. We call such a representation \mathbf x \textitsubspace-sparse. We study conditions under which existing pursuit methods recover a subspace-sparse representation. Such conditions reveal important geometric insights and have implications for the theory of classical sparse recovery as well as subspace clustering.
BibTeX
@InProceedings{pmlr-v37-you15,
title = {Geometric Conditions for Subspace-Sparse Recovery},
author = {You, Chong and Vidal, Rene},
booktitle = {Proceedings of the 32nd International Conference on Machine Learning},
pages = {1585--1593},
year = {2015},
editor = {Bach, Francis and Blei, David},
volume = {37},
series = {Proceedings of Machine Learning Research},
address = {Lille, France},
month = {07--09 Jul},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v37/you15.pdf},
url = {https://proceedings.mlr.press/v37/you15.html},
abstract = {Given a dictionary \Pi and a signal ξ= \Pi \mathbf x generated by a few \textitlinearly independent columns of \Pi, classical sparse recovery theory deals with the problem of uniquely recovering the sparse representation \mathbf x of ξ. In this work, we consider the more general case where ξlies in a low-dimensional subspace spanned by a few columns of \Pi, which are possibly \textitlinearly dependent. In this case, \mathbf x may not unique, and the goal is to recover any subset of the columns of \Pi that spans the subspace containing ξ. We call such a representation \mathbf x \textitsubspace-sparse. We study conditions under which existing pursuit methods recover a subspace-sparse representation. Such conditions reveal important geometric insights and have implications for the theory of classical sparse recovery as well as subspace clustering.}
}