We consider a problem of significant practical importance, namely, the reconstruction of a low-rank data matrix from a small subset of its entries. This problem appears in many areas such as collaborative filtering, computer vision and wireless sensor networks. In this paper, we focus on the matrix completion problem in the case when the observed samples are corrupted by noise. We compare the performance of three state-of-the-art matrix completion algorithms (OptSpace, ADMiRA and FPCA) on a single simulation platform and present numerical results. We show that in practice these efficient algorithms can be used to reconstruct real data matrices, as well as randomly generated matrices, accurately.
While most theoretical work focus on proving bounds for the exact matrix completion problem, a more interesting and practical problem is when the matrix @math is only approximately low rank or when the observation is corrupted by noise. The main focus of this is to design an algorithm to find an @math low-rank matrix @math that best approximates the original matrix @math and provide a bound on the root mean squared error (RMSE) given by, Cand e s and Plan introduced a generalization of the convex relaxation from @cite_7 to the noisy case, and provided a bound on the RMSE @cite_5 . More recently, a bound on the RMSE achieved by the algorithm with noisy observations was obtained in @cite_12 . This bound is order optimal in a number of situations and improves over the analogous result in @cite_5 . Detailed comparison of these two results are provided in Section .
@cite_5 @cite_12 @cite_7
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In this paper, we address compressed sensing of a low-rank matrix posing the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition providing an analogy between parsimonious representations of a sparse vector and a low-rank matrix and extending efficient greedy algorithms from the vector to the matrix case. In particular, we propose an efficient and guaranteed algorithm named atomic decomposition for minimum rank approximation (ADMiRA) that extends Needell and Tropp's compressive sampling matching pursuit (CoSaMP) algorithm from the sparse vector to the low-rank matrix case. The performance guarantee is given in terms of the rank-restricted isometry property (R-RIP) and bounds both the number of iterations and the error in the approximate solution for the general case of noisy measurements and approximately low-rank solution. With a sparse measurement operator as in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. Numerical experiments for the matrix completion problem show that, although the R-RIP is not satisfied in this case, ADMiRA is a competitive algorithm for matrix completion. 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. 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. We address the inverse problem that arises in compressed sensing of a low-rank matrix. Our approach is to pose the inverse problem as an approximation problem with a specified target rank of the solution. A simple search over the target rank then provides the minimum rank solution satisfying a prescribed data approximation bound. We propose an atomic decomposition that provides an analogy between parsimonious representations of a sparse vector and a low-rank matrix. Efficient greedy algorithms to solve the inverse problem for the vector case are extended to the matrix case through this atomic decomposition. In particular, we propose an efficient and guaranteed algorithm named ADMiRA that extends CoSaMP, its analogue for the vector case. The performance guarantee is given in terms of the rank-restricted isometry property and bounds both the number of iterations and the error in the approximate solution for the general case where the solution is approximately low-rank and the measurements are noisy. With a sparse measurement operator such as the one arising in the matrix completion problem, the computation in ADMiRA is linear in the number of measurements. The numerical experiments for the matrix completion problem show that, although the measurement operator in this case does not satisfy the rank-restricted isometry property, ADMiRA is a competitive algorithm for matrix completion.