Some Notes on Alternating Optimization

Two new theorems that give very general local and global convergence and rate of convergence results which hold for all partitionings of x are state (without proofs).

Abstract

Let f : ℜs ↦ ℜ be a real-valued scalar field, and let x = (x1,…, xs)T ∈ ℜs be partitioned into t subsets of non-overlapping variables as x = (X1,…,Xt )T, with Xi ∈ ℜp 1, for i = 1,…, t, ∈t i=1pi = s Alternating optimization (AO) is an iterative procedure for minimizing (or maximizing) the function f(x) = f(X1,X2,…,Xt) jointly over all variables by alternating restricted minimizations over the individual subsets of variables X1,…,Xt. AO is the basis for the c-means clustering algorithms (t=2), many forms of vector quantization (t = 2, 3 and 4), and the expectation-maximization (EM) algorithm (t = 4) for normal mixture decomposition. First we review where and how AO fits into the overall optimization landscape. Then we discuss the important theoretical issues connected with the AO approach. Finally, we state (without proofs) two new theorems that give very general local and global convergence and rate of convergence results which hold for all partitionings of x.