Penalty decomposition methods for rank minimization

Abstract

In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems in which each subproblem is solved by a block coordinate descend method. Under some suitable assumptions, we show that any accumulation point of the sequence generated by our method when applied to the rank constrained minimization problem is a stationary point of a nonlinear reformulation of the problem. Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed.

High-Dimensional Uncertainty Quantification of Electronic and Photonic IC With Non-Gaussian Correlated Process Variations

“…The above problem can be solved by any 0 -minimization solver, such as COSAMP [55], difference-of-convex [56], and penalty decomposition [57]. The error bound is presented in the following theorem.…”

Error bounds for rank constrained optimization problems and applications

“…Hence, when the set Ω is characterizing some structure conflicted with low rank, such as the correlation or density matrix structure, the nuclear norm relaxation method will fail in yielding a low rank solution. In view of this, many researchers recently develop effective solution methods based on the sequential convex relaxation models arising from the penalty problems [7,12], the nonconvex surrogate problems [6,10,18,19], and the rank constrained optimization problem itself [16,26,28]. We notice that to measure the distance from any given point to the feasible set or the solution set plays a key role in the analysis of these methods.…”

Direct Matrix Factorization and Alignment Refinement: Application to Defect Detection

“…2, the error detection problem with l 0 -norm constraints, can also be solved effectively and efficiently by using [11]: compute a histogram of the entries of E, and then find a threshold value such that the histogram quantile (cumulative sum) above that threshold corresponds to γ out of mn pixels. Then we use that threshold to set other (small) entries (pixels) of E to zero.…”