Matrix completion focuses on recovering a matrix from a small subset of its observed elements, and has already gained cumulative attention in computer vision. Many previous approaches formulate this issue as a low-rank matrix approximation problem. Recently, a truncated nuclear norm has been presented as a surrogate of traditional nuclear norm, for better estimation to the rank of a matrix. The truncated nuclear norm regularization (TNNR) method is applicable in real-world scenarios. However, it is sensitive to the selection of the number of truncated singular values and requires numerous iterations to converge. Hereby, this paper proposes a revised approach called the double weighted truncated nuclear norm regularization (DW-TNNR), which assigns different weights to the rows and columns of a matrix separately, to accelerate the convergence with acceptable performance. The DW-TNNR is more robust to the number of truncated singular values than the TNNR. Instead of the iterative updating scheme in the second step of TNNR, this paper devises an efficient strategy that uses a gradient descent manner in a concise form, with a theoretical guarantee in optimization. Sufficient experiments conducted on real visual data prove that DW-TNNR has promising performance and holds the superiority in both speed and accuracy for matrix completion.
The TNNR was first presented by Hu @cite_15 , which approximated to the rank function better than the nuclear norm. In contrast to treating all singular values together, the TNNR leaves out the largest @math singular values, and tries to minimize the smallest @math singular values, where @math , @math are the dimensions of data, and @math is the number of truncated singular values. In advance, let us define where @math is the set of locations in @math with respect to the known elements and @math is the counterpart with respect to the missing elements. It indicates that @math is the complement of @math . Hence, the TNNR minimization model is formulated as where @math , @math , @math denotes the nuclear norm (sum of all singular values), and @math denotes the trace function. It can be solved by a two-step iterative fashion. Let @math as initialization. In Step 1 of the @math -th iteration, assume @math is the singular value decomposition (SVD) of @math , where @math and @math are the left and right orthogonal matrices. Thus, @math and @math are calculated directly through
@cite_15
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Recovering a large matrix from a small subset of its entries is a challenging problem arising in many real applications, such as image inpainting and recommender systems. Many existing approaches formulate this problem as a general low-rank matrix approximation problem. Since the rank operator is nonconvex and discontinuous, most of the recent theoretical studies use the nuclear norm as a convex relaxation. One major limitation of the existing approaches based on nuclear norm minimization is that all the singular values are simultaneously minimized, and thus the rank may not be well approximated in practice. In this paper, we propose to achieve a better approximation to the rank of matrix by truncated nuclear norm, which is given by the nuclear norm subtracted by the sum of the largest few singular values. In addition, we develop a novel matrix completion algorithm by minimizing the Truncated Nuclear Norm. We further develop three efficient iterative procedures, TNNR-ADMM, TNNR-APGL, and TNNR-ADMMAP, to solve the optimization problem. TNNR-ADMM utilizes the alternating direction method of multipliers (ADMM), while TNNR-AGPL applies the accelerated proximal gradient line search method (APGL) for the final optimization. For TNNR-ADMMAP, we make use of an adaptive penalty according to a novel update rule for ADMM to achieve a faster convergence rate. Our empirical study shows encouraging results of the proposed algorithms in comparison to the state-of-the-art matrix completion algorithms on both synthetic and real visual datasets. Rank minimization problem can be boiled down to either Nuclear Norm Minimization (NNM) or Weighted NNM (WNNM) problem. The problems related to NNM (or WNNM) can be solved iteratively by applying a closed-form proximal operator, called Singular Value Thresholding (SVT) (or Weighted SVT), but they suffer from high computational cost to compute a Singular Value Decomposition (SVD) at each iteration. In this paper, we propose an accurate and fast approximation method for SVT, called fast randomized SVT (FRSVT), where we avoid direct computation of SVD. The key idea is to extract an approximate basis for the range of a matrix from its compressed matrix. Given the basis, we compute the partial singular values of the original matrix from a small factored matrix. While the basis approximation is the bottleneck, our method is already severalfold faster than thin SVD. By adopting a range propagation technique, we can further avoid one of the bottleneck at each iteration. Our theoretical analysis provides a stepping stone between the approximation bound of SVD and its effect to NNM via SVT. Along with the analysis, our empirical results on both quantitative and qualitative studies show our approximation rarely harms the convergence behavior of the host algorithms. We apply it and validate the efficiency of our method on various vision problems, e.g. subspace clustering, weather artifact removal, simultaneous multi-image alignment and rectification. Given a matrix A e ℝ m ×n of rank r, and an integer k < r, the top k singular vectors provide the best rank-k approximation to A. When the columns of A have specific meaning, it is desirable to find (provably) "good" approximations to A k which use only a small number of columns in A. Proposed solutions to this problem have thus far focused on randomized algorithms. Our main result is a simple greedy deterministic algorithm with guarantees on the performance and the number of columns chosen. Specifically, our greedy algorithm chooses c columns from A with @math such that @math where C gr is the matrix composed of the c columns, @math is the pseudo-inverse of C gr ( @math is the best reconstruction of A from C gr), and ¼(A) is a measure of the coherence in the normalized columns of A. The running time of the algorithm is O(SVD(A k) + mnc) where SVD(A k) is the running time complexity of computing the first k singular vectors of A. To the best of our knowledge, this is the first deterministic algorithm with performance guarantees on the number of columns and a (1 + e) approximation ratio in Frobenius norm. The algorithm is quite simple and intuitive and is obtained by combining a generalization of the well known sparse approximation problem from information theory with an existence result on the possibility of sparse approximation. Tightening the analysis along either of these two dimensions would yield improved results. Minimization of the nuclear norm is often used as a surrogate, convex relaxation, for finding the minimum rank completion (recovery) of a partial matrix. The minimum nuclear norm problem can be solved as a trace minimization semidefinite programming problem, (SDP). The SDP and its dual are regular in the sense that they both satisfy strict feasibility. Interior point algorithms are the current methods of choice for these problems. This means that it is difficult to solve large scale problems and difficult to get high accuracy solutions. In this paper we take advantage of the structure at optimality for the minimum nuclear norm problem. We show that even though strict feasibility holds, the facial reduction framework can be successfully applied to obtain a proper face that contains the optimal set, and thus can dramatically reduce the size of the final nuclear norm problem while guaranteeing a low-rank solution. We include numerical tests for both exact and noisy cases. In all cases we assume that knowledge of a target rank is available.