Improved thresholds for rank minimization

Nuclear norm minimization (NNM) has recently gained attention for its use in rank minimization problems. In this paper, we define weak, sectional and strong recovery for NNM to succeed at finding the low rank solution. We find tight conditions for these and analyze them for the case where the linear measurement operator consists of i.i.d. Gaussian entries. Finally we calculate the so called weak, sectional and strong thresholds for the success of nuclear norm minimization. To obtain our results, we generalize the notion of sign and support from sparse vectors to low rank matrices, and achieve a weak threshold which is much closer to the empirical phase transition curve of nuclear norm minimization than the existing bounds available in the literature.

[1]  J. Kuelbs.  Probability on Banach spaces , 1978 .

[2]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[3]  D. Donoho,et al.  Neighborliness of randomly projected simplices in high dimensions. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[4]  Pablo A. Parrilo,et al.  Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization , 2007, SIAM Rev..

[5]  Weiyu Xu,et al.  Null space conditions and thresholds for rank minimization , 2011, Math. Program..

[6]  Krishnamurthy Dvijotham,et al.  A nullspace analysis of the nuclear norm heuristic for rank minimization , 2010, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing.

[7]  D. Donoho,et al.  Thresholds for the Recovery of Sparse Solutions via L1 Minimization , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[8]  V. Girko.  An Introduction to Statistical Analysis of Random Arrays , 1998 .

[9]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[10]  Y. Gordon.  Some inequalities for Gaussian processes and applications , 1985 .

[11]  Weiyu Xu,et al.  Compressed sensing - probabilistic analysis of a null-space characterization , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.