Projecting onto rectangular matrices with prescribed row and column sums

TL;DR

This paper extends Romero's projection formula to rectangular matrices with scaled row and column sums, using Moore-Penrose inverses, applicable even to Hilbert-Schmidt operators, and includes numerical experiments.

Contribution

It provides a new formula for projecting onto matrices with scaled sums, generalizing previous results and applicable to broader operator classes.

Findings

Derived a new projection formula for scaled row and column sums

Applicable to Hilbert-Schmidt operators without consistency assumptions

Numerical experiments demonstrate the effectiveness of the new projection

Abstract

In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero's formula have been rediscovered by Khoury and by Glunt, Hayden, and Reams, for bistochastic (square) matrices in 1998. These results have found various generalizations and applications. In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore-Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert-Schmidt operators and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.