Noncommutative Sinkhorn theorem and generalizations

Our topic traces back to DEMING AND STEPHAN [1] who sought to reconcile an empirical joint distribution of two random variables with a priori known marginals for each and proposed an iterative scheme for determining a solution. Thus, in the case of finite probability spaces, the problem data consist of two probability vectors p0 and p1 together with the empirical joint probability matrix P = [p(i, j)], i.e., a matrix P such that p(i, j) ≥ 0 and ∑ i,j p(i, j) = 1. Then the task is to suitably modify P into P̂ = [p(i, j)] which is consistent with the marginals, i.e., ∑