Minimax Lower Bounds for Noisy Matrix Completion Under Sparse Factor Models

This paper examines fundamental error characteristics for a general class of matrix completion problems, where matrix of interest is a product of two a priori unknown matrices, one of which is sparse, and the observations are noisy. Our main contributions come in the form of minimax lower bounds for the expected per-element squared error for these problems under several noise/corruption models; specifically, we analyze scenarios where the corruptions are characterized by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations. Our results establish that the error bounds derived in (Soni et al., 2014) for complexity-regularized maximum likelihood estimators achieve, up to multiplicative constant and logarithmic factors, the minimax error rates in each of these noise scenarios, provided the sparse factor exhibits linear sparsity.