On the Dynamics of l1 Decoding: A Microscopic Approach

l(1) minimization, also called Basis Pursuit, has been known to have strong sparse recovery performance both theoretically and empirically. Previously known analytical approaches for l(1) minimization have limitations in deriving custom stability performance bounds for signals with sparsity ( the number of nonzero elements) level beyond the l(1) weak recovery threshold [ 6]. In this paper, instead of focusing on the static decoding results of l(1) minimization, we develop a microscopic analytical approach by studying the dynamics of l(1) minimization. This approach can give useful characterizations of l(1) decoding results and lead to new performance bounds on l(1) decoding error. Contrary to known stability results for l(1) minimization below the l(1) weak threshold, we prove that l(1) minimization decoding errors can experience an explosive growth in terms of the signal tail immediately beyond the l(1) minimization weak threshold. This new analytical approach is motivated by the applications of analyzing the emerging iterative reweighted l(1) minimization algorithms.