Limit points of the iterative scaling procedure

TL;DR

This paper investigates the limit points of the iterative scaling procedure, providing an efficient algorithm to identify these points and improve convergence analysis for matrices with specified row and column sums.

Contribution

It introduces an efficient algorithm to determine the limit points of ISP, enhancing understanding of its convergence behavior in general cases.

Findings

The sequence of matrices has at most two limit points.

The new algorithm efficiently finds these limit points.

Convergence can be slow when the limit points are distinct.

Abstract

The iterative scaling procedure (ISP) is an algorithm which computes a sequence of matrices, starting from some given matrix. The objective is to find a matrix 'proportional' to the given matrix, having given row and column sums. In many cases, for example if the initial matrix is strictly positive, the sequence is convergent. In the general case, it is known that the sequence has at most two limit points. When these are distinct, convergence can be slow. We give an efficient algorithm which finds these limit points, invoking the ISP only on instances for which the procedure is convergent.