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Covariance estimation becomes challenging in the regime where the number p of variables outstrips the number n of samples available to construct the estimate. One way to circumvent this problem is to assume that the covariance matrix is nearly sparse and to focus on estimating only the significant entries. To analyse this approach, Levina & Vershynin (2011, Probab. Theory Related Fields) introduce a formalism called masked covariance estimation, where each entry of the sample covariance estimator is reweighted to reflect an a priori assessment of its importance. This paper provides a short analysis of the masked sample covariance estimator by means of a matrix concentration inequality. The main result applies to general distributions with at least four moments. Specialized to the case of a Gaussian distribution, the theory offers qualitative improvements over earlier work. For example, the new results show that n=O(B log2 p) samples suffice to estimate a banded covariance matrix with bandwidth B up to a relative spectral-norm error, in contrast to the sample complexity n=O(B log5 p) obtained by Levina and Vershynin. [ABSTRACT FROM PUBLISHER]

Recent research indicates that many convex optimization problems with random constraints exhibit a phase transition as the number of constraints increases. For example, this phenomenon emerges in the ℓ1 minimization method for identifying a sparse vector from random linear measurements. Indeed, the ℓ1 approach succeeds with high probability when the number of measurements exceeds a threshold that depends on the sparsity level; otherwise, it fails with high probability. This paper provides the first rigorous analysis that explains why phase transitions are ubiquitous in random convex optimization problems. It also describes tools for making reliable predictions about the quantitative aspects of the transition, including the location and the width of the transition region. These techniques apply to regularized linear inverse problems with random measurements, to demixing problems under a random incoherence model, and also to cone programs with random affine constraints. The applied results depend on foundational research in conic geometry. This paper introduces a summary parameter, called the statistical dimension, that canonically extends the dimension of a linear subspace to the class of convex cones. The main technical result demonstrates that the sequence of intrinsic volumes of a convex cone concentrates sharply around the statistical dimension. This fact leads to accurate bounds on the probability that a randomly rotated cone shares a ray with a fixed cone. [ABSTRACT FROM AUTHOR]