Shifting Inequality and Recovery of Sparse Signals

Abstract

In this paper, we present a concise and coherent analysis of the constrained ℓ₁ minimization method for stable recovering of high-dimensional sparse signals both in the noiseless case and noisy case. The analysis is surprisingly simple and elementary, while leads to strong results. In particular, it is shown that the sparse recovery problem can be solved via ℓ₁ minimization under weaker conditions than what is known in the literature. A key technical tool is an elementary inequality, called Shifting Inequality, which, for a given nonnegative decreasing sequence, bounds the ℓ₂ norm of a subsequence in terms of the ℓ₁ norm of another subsequence by shifting the elements to the upper end.