Geometric Conditions for Subspace-Sparse Recovery

Given a dictionary Π and a signal ξ = Πx generated by a few linearly independent columns of Π, classical sparse recovery theory deals with the problem of uniquely recovering the sparse representation x of ξ. In this work, we consider the more general case where ξ lies in a low-dimensional subspace spanned by a few columns of Π, which are possibly linearly dependent. In this case, x may not unique, and the goal is to recover any subset of the columns of Π that spans the subspace containing ξ. We call such a representation x subspace-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.

[1]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.

[2]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[3]  Silouanos Brazitikos.  Geometry of Isotropic Convex Bodies , 2014 .

[4]  Huan Xu,et al.  Noisy Sparse Subspace Clustering , 2013, J. Mach. Learn. Res..

[5]  Emmanuel J. Candès,et al.  A Geometric Analysis of Subspace Clustering with Outliers , 2011, ArXiv.

[6]  René Vidal,et al.  Clustering disjoint subspaces via sparse representation , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[7]  Emmanuel J. Candès,et al.  Robust Subspace Clustering , 2013, ArXiv.

[8]  R. Vershynin.  Lectures in Geometric Functional Analysis , 2012 .

[9]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

[10]  Aswin C. Sankaranarayanan,et al.  Greedy feature selection for subspace clustering , 2013, J. Mach. Learn. Res..

[11]  Allen Y. Yang,et al.  Robust Face Recognition via Sparse Representation , 2009, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[12]  Zhang Yi,et al.  Scalable Sparse Subspace Clustering , 2013, 2013 IEEE Conference on Computer Vision and Pattern Recognition.

[13]  R. Vidal,et al.  Sparse Subspace Clustering: Algorithm, Theory, and Applications. , 2013, IEEE transactions on pattern analysis and machine intelligence.

[14]  Michael B. Wakin,et al.  Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property , 2009, IEEE Transactions on Information Theory.

[15]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[16]  René Vidal,et al.  Sparse subspace clustering , 2009, CVPR.

[17]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[18]  Hans-Peter Kriegel,et al.  Subspace clustering , 2012, WIREs Data Mining Knowl. Discov..

[19]  Anru Zhang,et al.  Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-Rank Matrices , 2013, IEEE Transactions on Information Theory.

[20]  René Vidal,et al.  Subspace-Sparse Representation , 2015, ArXiv.

[21]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.

[22]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[23]  Massimo Fornasier,et al.  Compressive Sensing , 2015, Handbook of Mathematical Methods in Imaging.