On the convergence of Block Majorization-Minimization algorithms on the Grassmann Manifold

Abstract

The Majorization-Minimization (MM) framework is widely used to derive efficient algorithms for specific problems that require the optimization of a cost function (which can be convex or not). It is based on a sequential optimization of a surrogate function over closed convex sets. A natural extension of this framework incorporates ideas of Block Coordinate Descent (BCD) algorithms into the MM framework, also known as block MM. The rationale behind the block extension is to partition the optimization variables into several independent blocks, to obtain a surrogate for each block, and to optimize the surrogate of each block cyclically. The advantage of the block MM is that the construction and successive optimization of the surrogate functions is potentially easier than with the non-block alternative. The purpose of this letter is to exploit the geometrical properties of the Grassmann manifold (a non-convex set) for the purpose of extending classical convergence proofs of the block MM when at least one of the blocks is constrained in this manifold.