Social trust prediction addresses the significant problem of exploring interactions among users in social networks. Naturally, this problem can be formulated in the matrix completion framework, with each entry indicating the trustness or distrustness. However, there are two challenges for the social trust problem: 1) the observed data are with sign (1-bit) measurements; 2) they are typically sampled non-uniformly. Most of the previous matrix completion methods do not well handle the two issues. Motivated by the recent progress of max-norm, we propose to solve the problem with a 1-bit max-norm constrained formulation. Since max-norm is not easy to optimize, we utilize a reformulation of max-norm which facilitates an efficient projected gradient decent algorithm. We demonstrate the superiority of our formulation on two benchmark datasets.
The problem of matrix completion is to recover a low-rank matrix from a subset of entries @cite_25 , which is given by: where @math is the data matrix, @math is the recovered matrix, and @math is the index set of observed entries. Th optimization problem is not only NP-hard, but requires double exponential time complexity with the number of samples @cite_23 . To solve the above problem, one alternative is to use nuclear norm as a relaxation to the rank function: where @math denotes the sum of singular values of matrix @math . @cite_19 developed a first-order procedure to solve the convex problem ), namely singular value thresholding (SVT). @cite_3 minimized the rank minimization by the singular value projection (SVP) algorithm. @cite_24 solved the problem by first trimming each row and column with too few entries, then compute the truncated SVD of the trimmed matrix. Under certain conditions, it showed accurate recovery on the order of @math samples ( @math is the number of samples, @math is the rank of recovered matrix). With the rapid development of matrix completion problem, some more efficient methods have been proposed @cite_16 @cite_15 @cite_10 @cite_27 .
@cite_3 @cite_24 @cite_19 @cite_27 @cite_23 @cite_15 @cite_16 @cite_10 @cite_25
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This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible, but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n x n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog(n). This paper is concerned with the problem of recovering an unknown matrix from a small fraction of its entries. This is known as the matrix completion problem, and comes up in a great number of applications, including the famous Netflix Prize and other similar questions in collaborative filtering. In general, accurate recovery of a matrix from a small number of entries is impossible; but the knowledge that the unknown matrix has low rank radically changes this premise, making the search for solutions meaningful. This paper presents optimality results quantifying the minimum number of entries needed to recover a matrix of rank r exactly by any method whatsoever (the information theoretic limit). More importantly, the paper shows that, under certain incoherence assumptions on the singular vectors of the matrix, recovery is possible by solving a convenient convex program as soon as the number of entries is on the order of the information theoretic limit (up to logarithmic factors). This convex program simply finds, among all matrices consistent with the observed entries, that with minimum nuclear norm. As an example, we show that on the order of nr log(n) samples are needed to recover a random n x n matrix of rank r by any method, and to be sure, nuclear norm minimization succeeds as soon as the number of entries is of the form nr polylog(n). Matrix completion is a rank minimization problem to recover a low-rank data matrix from a small subset of its entries. Since the matrix rank is nonconvex and discrete, many existing approaches approximate the matrix rank as the nuclear norm. However, the truncated nuclear norm is known to be a better approximation to the matrix rank than the nuclear norm, exploiting a priori target rank information about the problem in rank minimization. In this paper, we propose a computationally efficient truncated nuclear norm minimization algorithm for matrix completion, which we call TNNM-ALM. We reformulate the original optimization problem by introducing slack variables and considering noise in the observation. The central contribution of this paper is to solve it efficiently via the augmented Lagrange multiplier (ALM) method, where the optimization variables are updated by closed-form solutions. We apply the proposed TNNM-ALM algorithm to ghost-free high dynamic range imaging by exploiting the low-rank structure of irradiance maps from low dynamic range images. Experimental results on both synthetic and real visual data show that the proposed algorithm achieves significantly lower reconstruction errors and superior robustness against noise than the conventional approaches, while providing substantial improvement in speed, thereby applicable to a wide range of imaging applications. The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions—a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale problems, we propose a low-rank factorization model and construct a nonlinear successive over-relaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Extensive numerical experiments show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclear-norm minimization algorithms. In addition, convergence of this nonlinear SOR algorithm to a stationary point is analyzed. 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. Recovering low-rank and sparse matrices from incomplete or corrupted observations is an important problem in statistics, machine learning, computer vision, as well as signal and image processing. In theory, this problem can be solved by the natural convex joint mixed relaxations (i.e., l 1 -norm and trace norm) under certain conditions. However, all current provable algorithms suffer from superlinear per-iteration cost, which severely limits their applicability to large-scale problems. In this paper, we propose a scalable, provable and structured robust bilinear factorization (RBF) method to recover low-rank and sparse matrices from missing and grossly corrupted data, i.e., robust matrix completion (RMC), or incomplete and grossly corrupted measurements, i.e., compressive principal component pursuit (CPCP). Specifically, we first present two small-scale matrix trace norm regularized bilinear factorization models for RMC and CPCP problems, in which repetitively calculating SVD of a large-scale matrix is replaced by updating two much smaller factor matrices. Then, we apply the alternating direction method of multipliers (ADMM) to efficiently solve the RMC problems. Finally, we provide the convergence analysis of our algorithm, and extend it to address general CPCP problems. Experimental results verified both the efficiency and effectiveness of our method compared with the state-of-the-art methods. 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 finding the missing values of a matrix given a few of its entries, called matrix completion, has gathered a lot of attention in the recent years. Although the problem under the standard low rank assumption is NP-hard, Cand es and Recht showed that it can be exactly relaxed if the number of observed entries is sufficiently large. In this work, we introduce a novel matrix completion model that makes use of proximity information about rows and columns by assuming they form communities. This assumption makes sense in several real-world problems like in recommender systems, where there are communities of people sharing preferences, while products form clusters that receive similar ratings. Our main goal is thus to find a low-rank solution that is structured by the proximities of rows and columns encoded by graphs. We borrow ideas from manifold learning to constrain our solution to be smooth on these graphs, in order to implicitly force row and column proximities. Our matrix recovery model is formulated as a convex non-smooth optimization problem, for which a well-posed iterative scheme is provided. We study and evaluate the proposed matrix completion on synthetic and real data, showing that the proposed structured low-rank recovery model outperforms the standard matrix completion model in many situations. We use convex relaxation techniques to provide a sequence of regularized low-rank solutions for large-scale matrix completion problems. Using the nuclear norm as a regularizer, we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm. Our algorithm SOFT-IMPUTE iteratively replaces the missing elements with those obtained from a soft-thresholded SVD. With warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter. The computationally intensive part of our algorithm is in computing a low-rank SVD of a dense matrix. Exploiting the problem structure, we show that the task can be performed with a complexity of order linear in the matrix dimensions. Our semidefinite-programming algorithm is readily scalable to large matrices; for example SOFT-IMPUTE takes a few hours to compute low-rank approximations of a 106 X 106 incomplete matrix with 107 observed entries, and fits a rank-95 approximation to the full Netflix training set in 3.3 hours. Our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state-of-the-art techniques.