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.
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. Let M be an n? × n matrix of rank r, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm, which we call OptSpace, that reconstructs M from |E| = O(rn) observed entries with relative root mean square error 1 2 RMSE ? C(?) (nr |E|)1 2 with probability larger than 1 - 1 n3. Further, if r = O(1) and M is sufficiently unstructured, then OptSpace reconstructs it exactly from |E| = O(n log n) entries with probability larger than 1 - 1 n3. This settles (in the case of bounded rank) a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|r log n), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices. This paper deals with the trace regression model where n entries or linear combinations of entries of an unknown m1×m2 matrix A0 corrupted by noise are observed. We propose a new nuclear norm penalized estimator of A0 and establish a general sharp oracle inequality for this estimator for arbitrary values of n,m1,m2 under the condition of isometry in expectation. Then this method is applied to the matrix completion problem. In this case, the estimator admits a simple explicit form and we prove that it satisfies oracle inequalities with faster rates of convergence than in the previous works. They are valid, in particular, in the high-dimensional setting m1m2 ≫ n. We show that the obtained rates are optimal up to logarithmic factors in a minimax sense and also derive, for any fixed matrix A0, a non-minimax lower bound on the rate of convergence of our estimator, which coincides with the upper bound up to a constant factor. Finally, we show that our procedure provides an exact recovery of the rank of A0 with probability close to 1. We also discuss the statistical learning setting where there is no underlying model determined by A0 and the aim is to find the best trace regression model approximating the data.