This paper concerns solving the sparse deconvolution and demixing problem using l1,2-minimization. We show that under a certain structured random model, robust and stable recovery is possible. The results extend results of Ling and Strohmer (Inverse Probl. 31, 115002 2015), and in particular theoretically explain certain experimental findings from that paper. Our results do not only apply to the deconvolution and demixing problem, but to recovery of column-sparse matrices in general.
However, as was pointed out in (Ling and Strohmer 2015 @cite_1 ), numerical experiments show that both @math - and @math -minimization perform significantly better than nuclear norm minimization when it comes to recovering matrices with the structure described above. This is not hard to argue heuristically: Assuming @math , the above approach tries to recover a @math -matrix of rank @math , which needs an order of @math measurements. On the other hand @math (or @math ) tries to recover an @math -column sparse (or @math -sparse) @math matrix , which only needs @math . For really small sparsities and moderate @math , @math can be smaller than @math . Therefore, the authors of the mentioned paper concentrate their efforts, as will we, on analysing the @math -minimization (and @math -minimization, respectively).
@cite_1
1604604340 2949351217 2136912397 2072356364
Matrix completion concerns the recovery of a low-rank matrix from a subset of its revealed entries, and nuclear norm minimization has emerged as an effective surrogate for this combinatorial problem. Here, we show that nuclear norm minimization can recover an arbitrary n×n matrix of rank r from O(nr log2(n)) revealed entries, provided that revealed entries are drawn proportionally to the local row and column coherences (closely related to leverage scores) of the underlying matrix. Our results are order-optimal up to logarithmic factors, and extend existing results for nuclear norm minimization which require strong incoherence conditions on the types of matrices that can be recovered, due to assumed uniformly distributed revealed entries. We further provide extensive numerical evidence that a proposed two-phase sampling algorithm can perform nearly as well as local-coherence sampling and without requiring a priori knowledge of the matrix coherence structure. Finally, we apply our results to quantify how weighted nuclear norm minimization can improve on unweighted minimization given an arbitrary set of sampled entries. The linearly constrained matrix rank minimization problem is widely applicable in many fields such as control, signal processing and system identification. The tightest convex relaxation of this problem is the linearly constrained nuclear norm minimization. Although the latter can be cast as a semidefinite programming problem, such an approach is computationally expensive to solve when the matrices are large. In this paper, we propose fixed point and Bregman iterative algorithms for solving the nuclear norm minimization problem and prove convergence of the first of these algorithms. By using a homotopy approach together with an approximate singular value decomposition procedure, we get a very fast, robust and powerful algorithm, which we call FPCA (Fixed Point Continuation with Approximate SVD), that can solve very large matrix rank minimization problems. Our numerical results on randomly generated and real matrix completion problems demonstrate that this algorithm is much faster and provides much better recoverability than semidefinite programming solvers such as SDPT3. For example, our algorithm can recover 1000 x 1000 matrices of rank 50 with a relative error of 1e-5 in about 3 minutes by sampling only 20 percent of the elements. We know of no other method that achieves as good recoverability. Numerical experiments on online recommendation, DNA microarray data set and image inpainting problems demonstrate the effectiveness of our algorithms. The problem of minimizing the rank of a matrix subject to affine constraints has applications in several areas including machine learning, and is known to be NP-hard. A tractable relaxation for this problem is nuclear norm (or trace norm) minimization, which is guaranteed to find the minimum rank matrix under suitable assumptions. In this paper, we propose a family of Iterative Reweighted Least Squares algorithms IRLS-p (with 0 ≤ p ≤ 1), as a computationally efficient way to improve over the performance of nuclear norm minimization. The algorithms can be viewed as (locally) minimizing certain smooth approximations to the rank function. When p = 1, we give theoretical guarantees similar to those for nuclear norm minimization, that is, recovery of low-rank matrices under certain assumptions on the operator defining the constraints. For p < 1, IRLS-p shows better empirical performance in terms of recovering low-rank matrices than nuclear norm minimization. We provide an efficient implementation for IRLS-p, and also present a related family of algorithms, sIRLS-p. These algorithms exhibit competitive run times and improved recovery when compared to existing algorithms for random instances of the matrix completion problem, as well as on the MovieLens movie recommendation data set. Low-rank matrix recovery from a corrupted observation has many applications in computer vision. Conventional methods address this problem by iterating between nuclear norm minimization and sparsity minimization. However, iterative nuclear norm minimization is computationally prohibitive for large-scale data (e.g., video) analysis. In this paper, we propose a Robust Orthogonal Subspace Learning (ROSL) method to achieve efficient low-rank recovery. Our intuition is a novel rank measure on the low-rank matrix that imposes the group sparsity of its coefficients under orthonormal subspace. We present an efficient sparse coding algorithm to minimize this rank measure and recover the low-rank matrix at quadratic complexity of the matrix size. We give theoretical proof to validate that this rank measure is lower bounded by nuclear norm and it has the same global minimum as the latter. To further accelerate ROSL to linear complexity, we also describe a faster version (ROSL+) empowered by random sampling. Our extensive experiments demonstrate that both ROSL and ROSL+ provide superior efficiency against the state-of-the-art methods at the same level of recovery accuracy.