# Projecting onto rectangular matrices with prescribed row and column sums

arXiv: https://arxiv.org/abs/2105.12222 · HTML: https://mlsanity.com/paper/2105.12222/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 May 2021.

## Built on by (most cited first, 2 of 2)

- [A Fast and Accurate Splitting Method for Optimal Transport: Analysis and Implementation](https://mlsanity.com/paper/2110.11738/) (2021)
- [Bringing regularized optimal transport to lightspeed: a splitting method adapted for GPUs](https://mlsanity.com/paper/2305.18483/) (2023)

## Abstract (arXiv)

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.