Fast Signal Recovery From Saturated Measurements by Linear Loss and Nonconvex Penalties

Sign information is the key for overcoming the inevitable saturation error in compressive sensing system, which causes loss of information and may result in great bias. For sparse signal recovery from saturation, we propose to use linear loss to improve the effectiveness from the existing methods that utilize hard constraints/hinge loss for sign consistency. Due to the use of linear loss, analytical solution in the update progress is obtained and some nonconvex penalties are applicable, e.g., minimax concave penalty, <inline-formula><tex-math notation="LaTeX">$\ell _0$ </tex-math></inline-formula> norm, and sorted <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math> </inline-formula> norm. Theoretical analysis reveals that the estimation error can still be bounded. Generally, with linear loss and nonconvex penalties, the recovery performance can be significantly improved and the computational time is also largely saved, which is verified by the numerical experiments.

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