  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.
