Resolution limits of sparse coding in high dimensions

This paper addresses the problem of sparsity pattern detection for unknown κ-sparse n-dimensional signals observed through m noisy, random linear measurements. Sparsity pattern recovery arises in a number of settings including statistical model selection, pattern detection, and image acquisition. The main results in this paper are necessary and sufficient conditions for asymptotically-reliable sparsity pattern recovery in terms of the dimensions m, n and k as well as the signal-tonoise ratio (SNR) and the minimum-to-average ratio (MAR) of the nonzero entries of the signal. We show that m > 2κ log(n - κ)/(SNR ?MAR) is necessary for any algorithm to succeed, regardless of complexity; this matches a previous sufficient condition for maximum likelihood estimation within a constant factor under certain scalings of κ, SNR and MAR with n. We also show a sufficient condition for a computationally-trivial thresholding algorithm that is larger than the previous expression by only a factor of 4(1+SNR) and larger than the requirement for lasso by only a factor of 4/MAR. This provides insight on the precise value and limitations of convex programming-based algorithms.