A Simplified Approach to Recovery Conditions for Low Rank Matrices

Recovering sparse vectors and low-rank matrices from noisy linear measurements has been the focus of much recent research. Various reconstruction algorithms have been studied, including $\ell_1$ and nuclear norm minimization as well as $\ell_p$ minimization with $p<1$. These algorithms are known to succeed if certain conditions on the measurement map are satisfied. Proofs of robust recovery for matrices have so far been much more involved than in the vector case. In this paper, we show how several robust classes of recovery conditions can be extended from vectors to matrices in a simple and transparent way, leading to the best known restricted isometry and nullspace conditions for matrix recovery. Our results rely on the ability to "vectorize" matrices through the use of a key singular value inequality.Comment: 6 pages, This is a modified version of a paper submitted to ISIT 2011; Proc. Intl. Symp. Info. Theory (ISIT), Aug 201

Iterative Log Thresholding

Sparse reconstruction approaches using the re-weighted l1-penalty have been shown, both empirically and theoretically, to provide a significant improvement in recovering sparse signals in comparison to the l1-relaxation. However, numerical optimization of such penalties involves solving problems with l1-norms in the objective many times. Using the direct link of reweighted l1-penalties to the concave log-regularizer for sparsity, we derive a simple prox-like algorithm for the log-regularized formulation. The proximal splitting step of the algorithm has a closed form solution, and we call the algorithm 'log-thresholding' in analogy to soft thresholding for the l1-penalty. We establish convergence results, and demonstrate that log-thresholding provides more accurate sparse reconstructions compared to both soft and hard thresholding. Furthermore, the approach can be directly extended to optimization over matrices with penalty for rank (i.e. the nuclear norm penalty and its re-weigthed version), where we suggest a singular-value log-thresholding approach.Comment: 5 pages, 4 figure

Schatten-pp Quasi-Norm Regularized Matrix Optimization via Iterative Reweighted Singular Value Minimization

In this paper we study general Schatten-$p$ quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized $vector$ minimization problem are equivalent to those of an SPQN regularized $matrix$ minimization reformulation. We also show that any local minimizer of the SPQN regularized matrix minimization problems must be a first-order stationary point. Moreover, we derive lower bounds for nonzero singular values of the first-order stationary points and hence also of the local minimizers of the SPQN regularized matrix minimization problems. The iterative reweighted singular value minimization (IRSVM) methods are then proposed to solve these problems, whose subproblems are shown to have a closed-form solution. In contrast to the analogous methods for the SPQN regularized $vector$ minimization problems, the convergence analysis of these methods is significantly more challenging. We develop a novel approach to establishing the convergence of these methods, which makes use of the expression of a specific solution of their subproblems and avoids the intricate issue of finding the explicit expression for the Clarke subdifferential of the objective of their subproblems. In particular, we show that any accumulation point of the sequence generated by the IRSVM methods is a first-order stationary point of the problems. Our computational results demonstrate that the IRSVM methods generally outperform some recently developed state-of-the-art methods in terms of solution quality and/or speed.Comment: This paper has been withdrawn by the author due to major revision and correction