This paper considers the classical error correcting problem which is frequently discussed in coding theory. We wish to recover an input vector f ∈ Rn from corrupted measurements y = Af + e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the ℓ1-minimization problem (‖x‖ℓ1:= i |xi|) min g∈R n ‖y − Ag‖ℓ1 provided that the support of the vector of errors is not too large, ‖e‖ℓ0: = |{i: ei ̸= 0} | ≤ ρ · m for some ρ> 0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work [5]. Finally, underlying the success of ℓ1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.

We derive performance limits for two closely related communication scenarios involving a wireless system with multiple-element transmitter antenna arrays: a point-to-point system with partial side information at the transmitter, and a broadcast system with multiple receivers. In both cases, ideal beamforming is impossible, leading to an inherently lower achievable performance as the quality of the side information degrades or as the number of receivers increases. Expected signal-tonoise ratio (SNR) and mutual information are both considered as performance measures. In the point-to-point case, we determine when the transmission strategy should use some form of beamforming and when it should not. We also show that, when properly chosen, even a small amount of side information can be quite valuable. For the broadcast scenario with an SNR criterion, we find the efficient frontier of operating points and show that even when the number of receivers is larger than the number of antenna array ...

We consider a spiked population model, proposed by Johnstone, whose population eigenvalues are all unit except for a few fixed eigenvalues. The question is to determine how the sample eigenvalues depend on the non-unit population ones when both sample size and population size become large. This paper completely determines the almost sure limits for a general class of samples. 1

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Variable selection plays an important role in high dimensional statistical modeling which nowa-days appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, estimation accuracy and computational cost are two top concerns. In a recent paper, Candes and Tao (2007) propose the Dantzig selector using L1 regularization and show that it achieves the ideal risk up to a logarithmic factor log p. Their innovative procedure and remarkable result are challenged when the dimensionality is ultra high as the factor log p can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method based on a correlation learning, called the Sure Independence Screening (SIS), to reduce dimensionality from high to a moderate scale that is below sample size. In a fairly general asymptotic framework, the SIS is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, an iterative SIS (ISIS) is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be ac-

We prove a lower bound of \Omega\Gamma m log m) for the size of any arithmetic circuit for the product of two matrices, over the real or complex numbers, as long as the circuit doesn't use products with field elements of absolute value larger than 1 (where m \Theta m is the size of each matrix). That is, our lower bound is super-linear in the number of inputs and is applied for circuits that use addition gates, product gates and products with field elements of absolute value up to 1.

Abstract. We prove an optimal estimate on the smallest singular value of a random subgaussian matrix, valid for all fixed dimensions. For an N × n matrix A with independent and identically distributed subgaussian entries, the smallest singular value of A is at least of the order √ N − √ n − 1 with high probability. A sharp estimate on the probability is also obtained. 1.

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