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This paper examines a general class of noisy matrix completion tasks, where the goal is to estimate a matrix from observations obtained at a subset of its entries, each of which is subject to random noise or corruption. Our specific focus is on settings where the matrix to be estimated is well-approximated by a product of two (a priori unknown) matrices, one of which is sparse. Such structural models—referred to here as sparse factor models—have been widely used, for example, in subspace clustering applications, as well as in contemporary sparse modeling and dictionary learning tasks. Our main theoretical contributions are estimation error bounds for sparsity-regularized maximum likelihood estimators for the problems of this form, which are applicable to a number of different observation noise or corruption models. Several specific implications are examined, including scenarios where observations are corrupted by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly quantized (e.g., 1 b) observations. We also propose a simple algorithmic approach based on the alternating direction method of multipliers for these tasks, and provide experimental evidence to support our error analyses. [ABSTRACT FROM AUTHOR]

This paper studies the partial estimation of Gaussian graphical models from high-dimensional empirical observations. We derive a convex formulation for this problem using \ell1-regularized maximum-likelihood estimation, which can be solved via a smoothing approximation algorithm. Statistical estimation performance can be established for our method. The proposed approach has competitive empirical performance compared with existing methods, as demonstrated by various experiments on synthetic and real data sets. [ABSTRACT FROM AUTHOR]