Distribution-aware $\ell_1$ Analysis Minimization

TL;DR

This paper introduces an analytical approach to select optimal weights for weighted  analysis minimization, leveraging known distribution information to reduce measurement requirements in sparse signal recovery.

Contribution

It provides a method to analytically determine near-optimal weights based on distribution parameters, improving measurement efficiency in analysis-sparse recovery.

Findings

Significantly fewer measurements needed with the proposed weighting scheme.

Method works effectively in both noiseless and noisy environments.

Numerical results confirm the superiority over regular  analysis minimization.

Abstract

This work is about recovering an analysis-sparse vector, i.e. sparse vector in some transform domain, from under-sampled measurements. In real-world applications, there often exist random analysis-sparse vectors whose distribution in the analysis domain are known. To exploit this information, a weighted $\ell_1$ analysis minimization is often considered. The task of choosing the weights in this case is however challenging and non-trivial. In this work, we provide an analytical method to choose the suitable weights. Specifically, we first obtain a tight upper-bound expression for the expected number of required measurements. This bound depends on two critical parameters: support distribution and expected sign of the analysis domain which are both accessible in advance. Then, we calculate the near-optimal weights by minimizing this expression with respect to the weights. Our strategy works for both noiseless and noisy settings. Numerical results demonstrate the superiority of our proposed method. Specifically, the weighted $\ell_1$ analysis minimization with our near-optimal weighting design considerably needs fewer measurements than its regular $\ell_1$ analysis counterpart.