Minimum Majorization Decomposition

Abstract

A famous theorem of Birkhoff says that any doubly stochastic matrix D can be decomposed into a convex combination of permutation matrices R. The various decompositions correspond to probability distributions on the set of permutations that satisfy the linear constraints E[R] = D. This paper illustrates how to decompose D so that the resulting probability distribution is minimal in the sense that it does not majorize any other distribution satisfying these constraints.

Any distribution maximizing a strictly Schur concave function g under these linear constraints will be minimal in the above sense (Joe (1987)). In particular, for D in the relative interior of the convex hull of the permutation matrices, the probability functions p that maximize $g\left( p \right)\, = \, - \,\sum\nolimits_\pi {p\left( \pi \right)} \,$ log p(π), subject to E[R] = D, form an exponential family £ with sufficient statistic R.

This paper provides a theorem that characterizes the exponential family £ by a property called quasi-independence. Quasi-independence is defined in terms of the invariance of the product measure over Latin sets. The characterization suggests an algorithm for an explicit minimal decomposition of a doubly stochastic matrix.