Bias Reduction in Compressed Sensing

Sparsity and rank functions are important ways of regularizing under-determined linear systems. Optimization of the resulting formulations is made difficult since both these penalties are non-convex and discontinuous. The most common remedy is to instead use the $\ell^1$- and nuclear-norms. While these are convex and can therefore be reliably optimized they suffer from a shrinking bias that degrades the solution quality in the presence of noise. In this paper we combine recently developed bias free non-convex alternatives with the nuclear- and $\ell^1-$penalties. This reduces bias and still enables reliable optimization properties. We develop an efficient minimization scheme using derived proximal operators and evaluate the method on several real and synthetic computer vision applications with promising results.