Implications of convergence rates in Sinkhorn balancing

Abstract Let D N be the set N × N stochastic matrices without zero columns. Starting with a matrix A(0) ϵ D N, Sinkhorn balancing is the iteration of alternately normalizing the column and row sums of A(0). It has been shown that if A(0) has total support then the iteration converges geometrically to a doubly stochastic limit. We show that the converse is true: geometric convergence implies total support.