We consider the problem of recovering two unknown vectors, @math and @math , of length @math from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension @math and the other with dimension @math . Although the observed convolution is nonlinear in both @math and @math , it is linear in the rank-1 matrix formed by their outer product @math . This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for "generic" signals, the program can deconvolve @math and @math exactly when the maximum of @math and @math is almost on the order of @math . That is, we show that if @math is drawn from a random subspace of dimension @math , and @math is a vector in a subspace of dimension @math whose basis vectors are "spread out" in the frequency domain, then nuclear norm minimization recovers @math without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length @math which we code using a random @math coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length @math , then the receiver can recover both the channel response and the message when @math , to within constant and log factors.
While this paper is only concerned with recovery by nuclear norm minimization, other types of recovery techniques have proven effective both in theory and in practice; see for example @cite_4 @cite_12 @cite_14 . It is possible that the guarantees given in this paper could be extended to these other algorithms.
@cite_14 @cite_4 @cite_12
2136912397 2508366294 2747056667 2158121106
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. 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. The nonsmooth and nonconvex regularization has many applications in imaging science and machine learning research due to its excellent recovery performance. A proximal iteratively reweighted nuclear norm algorithm has been proposed for the nonsmooth and nonconvex matrix minimizations. In this paper, we aim to investigate the convergence of the algorithm. With the Kurdyka–Łojasiewicz property, we prove the algorithm globally converges to a critical point of the objective function. The numerical results presented in this paper coincide with our theoretical findings. This paper considers the recovery of a low-rank matrix from an observed version that simultaneously contains both 1) erasures, most entries are not observed, and 2) errors, values at a constant fraction of (unknown) locations are arbitrarily corrupted. We provide a new unified performance guarantee on when minimizing nuclear norm plus l1 norm succeeds in exact recovery. Our result allows for the simultaneous presence of random and deterministic components in both the error and erasure patterns. By specializing this one single result in different ways, we recover (up to poly-log factors) as corollaries all the existing results in exact matrix completion, and exact sparse and low-rank matrix decomposition. Our unified result also provides the first guarantees for 1) recovery when we observe a vanishing fraction of entries of a corrupted matrix, and 2) deterministic matrix completion.