Implications of convergence rates in Sinkhorn balancing
A sequence of matrices \(\{A^{(t)}\}\), \(t=0,1,2,\ldots,\) is said to converge geometrically to a limit \(G\) if there exist positive constants \(M\) and \(\gamma\), where \(0<\gamma<1\), such that \(\| A^{(t)}- G\|\leq M\gamma^ t\) for some norm \(\|\cdot\|\), \(t=0,1,2,\ldots\). A nonnegative \(N\times N\) matrix \(A\) is said to have total support if there exists an \(N\times N\) doubly stochastic matrix \(B\) such that \(b_{ij}>0\) if and only if \(a_{ij}>0\) for all \(i,j=1,2,\ldots,N\). It is known that if \(A\) has total support then the sequence of matrices obtained beginning with \(A\) and alternately normalizing the column sums and then the row sums converges geometrically to a doubly stochastic limit. The author shows that if this iteration on \(A\) converges geometrically, then \(A\) has total support.
scientific article; zbMATH DE number 399294
scientific article