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.... It is the mathematical analysis which reveals the correctness of this value. In fact, the proof of the theorem gives a whole range of correct values, and we have selected a sufficiently simple value in that range. Another comment is that one can obtain results with larger probabilities of success, i.e. of the form 1−O(n−β) (or 1−O(n−β(1) )) for β > 0 at the expense of reducing the value of ρr. 1.5 Connections with prior work and innovations The last year or two have seen the rapid development of a scientific literature concerned with the matrix completion problem introduced in [8], see also [7,10,22,23,29] and the references therein. In a nutshell, the matrix completion problem is that of recovering a low-rank matrix from only a small fraction of its entries, and by extension, from a small number of linear functionals. Although other methods have been proposed [29], the method of choice is to use convex optimization [7,10,22,23,45]: among all the matrices consistent with the data, simply find that with minimum nuclear norm. The papers cited above all prove the mathematical validity of this approach, and our mathematical analysis borrows ideas from this literature, and especially from those pion...

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...ent in experimental quantum-state tomography—a fact which initially motivated this work. C. Intuition The basic intuition underlying our results differs little from previous approaches [2], [3], [1], =-=[4]-=-. For the sake of being self-contained, we still give a brief non-technical account of some aspects we find essential. (Technical differences to existing publications are outlined in the next section....

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...low-rank matrix solutions to underdetermined linear equations [12, 13]. One of the most striking results concerns the exact completion of low-rank matrices from only a small fraction of their entries =-=[13, 14, 15, 16]-=-. 2 There, a similar convex relaxation is employed, replacing the highly non-convex matrix rank with the nuclear norm (or sum of singular values). The robust PCA problem outlined above combines aspect...

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...ions are often noisy, or even grossly corrupted, and observations may be missing. For small noise (e.g., Gaussian) a reasonable strategy is simply to relax the equality constraint in (4), similar to (=-=Candes & Plan, 2009-=-). If we imagine instead that a fraction of the data vectors are grossly corrupted, a more reasonable objective might be min Z,E ||Z||∗ + λ||E||2,1, (5) s.t., X = XZ + E, where ||E||2,1 = ∑ √∑ n n j=1...

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...r data. So the subspace methods have been gaining much attention in recent years. For example, the widely used Principal Component Analysis (PCA) method and the recently established matrix completion =-=[6]-=- and recovery [7] methods are essentially based on the hypothesis that the data is approximately drawn from a low-rank subspace. However, a given dataset can seldom be well described by a single subsp...

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... A. However, the restricted isometry property in the usual sense, i.e., ”in probability”, cf., e.g., [22], does not hold for matrix completion, since for n < m1m2 there trivially exists a matrix of rank 1 in the null space of the sampling operator. NUCLEAR NORM PENALIZATION AND OPTIMAL RATES 3 One can also consider more general matrix measurement models in which, for a given orthonormal basis in the space of matrices, a random sample of Fourier coefficients of the target matrix A0 is observed subject to a random noise. For more discussion on matrix completion with other types of sampling, see [8, 10, 11, 15, 17] and references therein. Example 2. Column masks. Assume that the design matrices Xi are i.i.d. replications of a random matrix X, which has only one nonzero column. For instance, let the distribution of X be such that all the columns have equal probability to be non-zero, and the random entries of nonzero column x(j) are such that E(x(j)x>(j)) is the identity matrix. Then ‖A‖2L2(Π) = ‖A‖ 2 2/m2, ∀A ∈ Rm1×m2 , so that condition (1.4) is satisfied with µ = √ m2. More generally, in view of application to multi-task learning, cf. [23], one can be interested in considering non-identically distribu...