Paper Title
Theoretical Stability Analysis Of Basis Pursuit Denoising

Abstract
Basis Pursuit Denoising (BPDN) is a popular tool for sparse signal recovery with noisy interference by relaxing Basis Pursuit (BP) with error tolerance. In general, the error tolerance set to be the energy of noise is expected to outperform that set to be 0. However, in fact, under some conditions, the error between BPDN solution and ground truth probably becomes larger along with the increase of error tolerance. In this paper, we want to explore under what conditions, the error tolerance set to be the energy of noise results in the better performance than that set to be 0 under Gaussian random noise. We start this issue from discussing objective values of BPDN in both cases and show that objective values are directly related to performance especially when signal dimension is 2. We leave the case with larger signal dimension for future work. Our analyses are based on an innovative methodology, called conic geometry. Simulations validate our theoretical analyses.