We consider a problem of considerable practical interest: the recovery of a data matrix from a sampling of its entries. Suppose that we observe m entries selected uniformly at random from a matrix M. Can we complete the matrix and recover the entries that we have not seen? We show that one can perfectly recover most low-rank matrices from what appears to be an incomplete set of entries. We prove that if the number m of sampled entries obeys m ≥ C n 1.2 r log n for some positive numerical constant C, then with very high probability, most n × n matrices of rank r can be perfectly recovered by solving a simple convex optimization program. This program finds the matrix with minimum nuclear norm that fits the data. The condition above assumes that the rank is not too large. However, if one replaces the 1.2 exponent with 1.25, then the result holds for all values of the rank. Similar results hold for arbitrary rectangular matrices as well. Our results are connected with the recent literature on compressed sensing, and show that objects other than signals and images can be perfectly reconstructed from very limited information.

The low-rank matrix approximation problem involves finding of a rank k version of a m × n matrix A, labeled Ak, such that Ak is as ”close ” as possible to the best SVD approximation version of A at the same rank level. Previous approaches approximate matrix A by non-uniformly adaptive sampling some columns (or rows) of A, hoping that this subset of columns contain enough information about A. The sub-matrix is then used for the approximation process. However, these approaches are often computationally intensive due to the complexity in the adaptive sampling. In this paper, we propose a fast and efficient algorithm which at first pre-processes matrix A in order to spread out information (energy) of every columns (or rows) of A, then randomly selects some of its columns (or rows). Finally, a rank-k approximation is generated from the row space of these selected sets. The preprocessing step is performed by uniformly randomizing signs of entries of A and transforming all columns of A by an orthonormal matrix F with existing fast implementation (e.g. Hadamard, FFT, DCT...). Our main contribution is summarized as follows. 1) We show that by uniformly selecting at random d rows of the preprocessed matrix with d = O 1

Abstract. Let Fn denote the free group with n generators g1, g2,..., gn. Let λ stand for the left regular representation of Fn and let τ be the standard trace associated to λ. Given any positive integer d, we study the operator space structure of the subspace Wp(n, d) of Lp(τ) generated by the family of operators λ(gi1gi2 · · · gi) with 1 ≤ ik ≤ n. Moreover, our description of this d operator space holds up to a constant which does not depend on n or p, so that our result remains valid for infinitely many generators. We also consider the subspace of Lp(τ) generated by the image under λ of the set of reduced words of length d. Our result extends to any exponent 1 ≤ p ≤ ∞ a previous result of Buchholz for the space W∞(n, d). The main application is a certain interpolation theorem, valid for any degree d (extending a result of the second author restricted to d = 1). In the simplest case d = 2, our theorem can be stated as follows: consider the space Kp formed of all block matrices a = (aij) with entries in the Schatten class Sp, such that a is in Sp relative to ℓ2 ⊗ ℓ2 and moreover such that ( ∑ ij a ∗ ijaij) 1/2 and ( ∑ ij aija ∗ ij)1/2 both belong to Sp. We equip Kp with the maximum of the three corresponding norms. Then, for 2 ≤ p ≤ ∞ we have Kp ≃ (K2, K∞)θ with 1/p = (1 − θ)/2.

Low-rank matrix approximations, such as the truncated singular value decomposition and the rank-revealing QR decomposition, play a central role in data analysis and scientific computing. This work surveys recent research which demonstrates that randomization offers a powerful tool for performing low-rank matrix approximation. These techniques exploit modern computational architectures more fully than classical methods and open the possibility of dealing with truly massive data sets. In particular, these techniques offer a route toward principal component analysis (PCA) for petascale data. This paper presents a modular framework for constructing randomized algorithms that compute partial matrix decompositions. These methods use random sampling to identify a subspace that captures most of the action of a matrix. The input matrix is then compressed—either explicitly or implicitly—to this subspace, and the reduced matrix is manipulated deterministically to obtain the desired low-rank factorization. In many cases, this approach beats its classical competitors in terms of accuracy, speed, and robustness. These claims are supported by extensive numerical experiments and a detailed error analysis. The specific benefits of randomized techniques depend on the computational environment. Consider

These notes give a mathematical introduction to compressive sensing focusing on recovery using ℓ1-minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using ℓ1-minimization.

Let A denote the reduced amalgamated free product of a family

Abstract. This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison–Singer problem. The result shows that every unitnorm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systematic use of symmetrization and Khintchine inequalities to estimate the norm of some random matrices. 1.

Abstract. This paper presents an improved analysis of a structured dimension-reduction map called the subsampled randomized Hadamard transform. This argument demonstrates that the map preserves the Euclidean geometry of an entire subspace of vectors. The new proof is much simpler than previous approaches, and it offers—for the first time—optimal constants in the estimate on the number of dimensions required for the embedding. 1.

Abstract—Compressed sensing seeks to recover a sparse vector from a small number of linear and non-adaptive measurements. While most work so far focuses on Gaussian or Bernoulli random measurements we investigate the use of partial random circulant and Toeplitz matrices in connection with recovery by ℓ1-minization. In contrast to recent work in this direction we allow the use of an arbitrary subset of rows of a circulant and Toeplitz matrix. Our recovery result predicts that the necessary number of measurements to ensure sparse reconstruction by ℓ1-minimization with random partial circulant or Toeplitz matrices scales linearly in the sparsity up to a log-factor in the ambient dimension. This represents a significant improvement over previous recovery results for such matrices. As a main tool for the proofs we use a new version of the non-commutative Khintchine inequality. I.

to the non-asymptotic analysis of random