Method of Alternating Proximations for Solving Linear Inverse Problems

SignaVimage reconstruction problems are typically posed as linear inverse problems with the objective of minimizing a least-squares (LS) loss function and a regularization term derived from a suitable prior. Several decades of work have gone into the design and efficient application of various penalty functions, whilst the design of the data-fidelity loss function has remained relatively unexplored. This paper proposes a distance-minimization approach to the design of the data-fidelity loss for solving linear inverse problems and discusses proximal algorithms for solving them. We discuss the implications of using the proposed formulation as opposed to the standard LS in terms of its effect on noise and convergence. We consider solving sparse signal recovery and image restoration tasks, in particular, image deblurring, and show that distance minimization techniques out-perform standard LS for the various models under consideration.