Proximity function minimization using multiple Bregman projections, with applications to split feasibility and Kullback–Leibler distance minimization

Problems in signal detection and image recovery can sometimes be formulated as a convex feasibility problem (CFP) of finding a vector in the intersection of a finite family of closed convex sets. Algorithms for this purpose typically employ orthogonal or generalized projections onto the individual convex sets. The simultaneous multiprojection algorithm of Censor and Elfving for solving the CFP, in which different generalized projections may be used at the same time, has been shown to converge for the case of nonempty intersection; still open is the question of its convergence when the intersection of the closed convex sets is empty. Motivated by the geometric alternating minimization approach of Csiszár and Tusnády and the product space formulation of Pierra, we derive a new simultaneous multiprojection algorithm that employs generalized projections of Bregman to solve the convex feasibility problem or, in the inconsistent case, to minimize a proximity function that measures the average distance from a point to all convex sets. We assume that the Bregman distances involved are jointly convex, so that the proximity function itself is convex. When the intersection of the convex sets is empty, but the closure of the proximity function has a unique global minimizer, the sequence of iterates converges to this unique minimizer. Special cases of this algorithm include the ‘Expectation Maximization Maximum Likelihood’ method in emission tomography and a new convergence result for an algorithm that solves the split feasibility problem.