# Accelerated Schemes for the $L_1/L_2$ Minimization

## TL;DR

This paper introduces accelerated algorithms for $L_1/L_2$ minimization in sparse recovery, demonstrating efficiency and effectiveness, especially with high dynamic range signals, and providing empirical insights into exact $L_1$ recovery.

## Contribution

It proposes three new numerical algorithms for $L_1/L_2$ minimization, including two adaptive schemes that reduce computation time and analyze their convergence.

## Key findings

- Algorithms are comparable to state-of-the-art methods.
- Adaptive schemes work well with high dynamic range signals.
- Empirical evidence suggests conditions for exact $L_1$ recovery.

## Abstract

In this paper, we consider the $L_1/L_2 $ minimization for sparse recovery and study its relationship with the $L_1$-$ \alpha L_2 $ model. Based on this relationship, we propose three numerical algorithms to minimize this ratio model, two of which work as adaptive schemes and greatly reduce the computation time. Focusing on two adaptive schemes, we discuss their connection to existing approaches and analyze their convergence. The experimental results demonstrate the proposed approaches are comparable to the state-of-the-art methods in sparse recovery and work particularly well when the ground-truth signal has a high dynamic range. Lastly, we reveal some empirical evidence on the exact $L_1$ recovery under various combinations of sparsity, coherence, and dynamic ranges, which calls for theoretical justification in the future.