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 × 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 nrpolylog(n).

On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.

This paper surveys recent work in applying ideas from graphical models and message passing algorithms to solve large scale regularized regression problems. In particular, the focus is on compressed sensing reconstruction via ℓ1 penalized least-squares (known as LASSO or BPDN). We discuss how to derive fast approximate message passing algorithms to solve this problem. Surprisingly, the analysis of such algorithms allows to prove exact high-dimensional limit results for the LASSO risk. This paper will appear as a chapter in a book on ‘Compressed Sensing ’ edited by Yonina Eldar and Gitta Kutynok. 1

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 × 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 nrpolylog(n).

of the rapidly developing field of compressed sensing, and is already changing the way engineers think about data acquisition, hence this special issue and others, see [2] for example. Concretely, if a signal has a sparse frequency spectrum and we only have information about a few time or space samples, then one can invoke linear programming to interpolate the signal exactly. One can of course exchange time (or space) and frequency, and recover sparse signals from just a few of their Fourier coefficients as well. Imagine now that we only observe a few entries of a data matrix. Then is it possible to accurately—or even exactly—guess the entries that we have not seen? For example, suppose we observe a few movie ratings from a large data matrix in which rows are users and columns are movies (we can only observe a few ratings because each user is typically rating a few movies as opposed to the tens of thousands of movies which are available). Can we predict the rating a user would hypothetically assign to a movie he/she has not seen? In general, everybody would agree that recovering a data matrix from a subset of its entries is impossible. However, if the unknown matrix is known to have low rank or approximately low rank, then accurate and even exact recovery is possible by nuclear norm minimization [8], [12]. This revelation, which to some extent is inspired by the great body of work in compressed sensing, is the subject of this paper. From now on, we will refer to the problem of inferring the many missing entries as the matrix completion problem. By extension, inferring a matrix from just a few linear functionals will be called the the low-rank matrix recovery problem. Now just as sparse signal recovery is arguably of paramount importance these days, we do believe that matrix completion and, in general, low-rank matrix recovery is just as important, and will become increasingly studied in years to come. For now, we give a few examples of applications in which these problems do come up. • Collaborative filtering. In a few words, collaborative filtering is the task of making automatic predictions about the interests of a user by collecting taste information from many users [21]. Perhaps the most well-known implementation of collaborating filterarXiv:0903.3131v1