Entropic Proximal Operators for Nonnegative Trigonometric Polynomials

Signal processing applications of semidefinite optimization are often rooted in sum-of-squares representations of nonnegative trigonometric polynomials. Interior-point solvers for semidefinite optimization can handle constraints of this form with a per-iteration-complexity that is cubic in the degree of the trigonometric polynomial. The purpose of this paper is to discuss first-order methods with a lower complexity per iteration. The methods are based on generalized proximal operators defined in terms of the Itakura-Saito distance. This is the Bregman distance defined by the negative entropy function. The choice for the Itakura-Saito distance is motivated by the fact that the associated generalized projection on the set of normalized nonnegative trigonometric polynomials can be computed at a cost that is roughly quadratic in the degree of the polynomial. The generalized projection is the key operation in generalized proximal first-order methods that use Bregman distances instead of the squared Euclidean distance. The paper includes numerical results with Auslender and Teboulle's accelerated proximal gradient method for Bregman distances.