Department of Mathematics

I will talk about the ratio of the $L_1 $ and $L_2 $ norms, denoted as $L_1/L_2$, to promote sparsity.  Due to the  non-convexity and non-linearity, there has been little attention to this scale-invariant model. Compared to popular models in the literature such as the $L_p$ model for $p\in(0,1)$ and the transformed $L_1$ (TL1), this ratio model is parameter free. Theoretically, we present a strong null space property (sNSP) and prove that any sparse vector is a local minimizer of the $L_1/L_2 $ model provided with this sNSP condition. We then focus on a variant of the $L_1/L_2$ model to apply on the gradient. This gradient model is analogous to total variation, which is the $L_1$ norm on the gradient. We discuss an iteratively reweighed algorithm to minimize the proposed model with guaranteed convergence. Experiments on the MRI reconstruction and limited-angle CT reconstruction show that our approach outperforms the state-of-the-art methods.