On the dynamics of ℓ1 decoding: A microscopic approach

Abstract

ℓ1 minimization, also called Basis Pursuit, has been known to have strong sparse recovery performance both theoretically and empirically. Previously known analytical approaches for ℓ1 minimization have limitations in deriving custom stability performance bounds for signals with sparsity (the number of nonzero elements) level beyond the ℓ1 weak recovery threshold [6]. In this paper, instead of focusing on the static decoding results of ℓ1 minimization, we develop a microscopic analytical approach by studying the dynamics of ℓ1 minimization. This approach can give useful characterizations of ℓ1 decoding results and lead to new performance bounds on ℓ1 decoding error. Contrary to known stability results for ℓ1 minimization below the ℓ1 weak threshold, we prove that ℓ1 minimization decoding errors can experience an explosive growth in terms of the signal tail immediately beyond the ℓ1 minimization weak threshold. This new analytical approach is motivated by the applications of analyzing the emerging iterative reweighted ℓ1 minimization algorithms.