Simultaneous polynomial approximation and total variation denoising

This paper addresses the problem of smoothing data with additive step discontinuities. The problem formulation is based on least square polynomial approximation and total variation denoising. In earlier work, an ADMM algorithm was proposed to minimize a suitably defined sparsity-promoting cost function. In this paper, an algorithm is derived using the majorization-minimization optimization procedure. The new algorithm converges faster and, unlike the ADMM algorithm, has no parameters that need to be set. The proposed algorithm is formulated so as to utilize fast solvers for banded systems for high computational efficiency. This paper also gives optimality conditions so that the optimality of a result produced by the numerical algorithm can be readily validated.