Sinkhorn algorithms and linear programming solvers for optimal partial transport problems

TL;DR

This paper introduces a generalized framework for optimal partial transport problems, extending classical OT by incorporating function-based mass modifications, and develops specialized Sinkhorn and linear programming algorithms for these new formulations.

Contribution

It proposes a novel generalization of optimal partial transport problems with function-based mass terms and provides tailored Sinkhorn and linear programming algorithms.

Findings

Developed a dual formulation for the generalized OPT problems.

Designed Sinkhorn algorithms adapted to the generalized setting.

Created a linear programming solver for the new OPT formulations.

Abstract

In this note, we generalize the classical optimal partial transport (OPT) problem by modifying the mass destruction/creation term to function-based terms, introducing what we term ``generalized optimal partial transport'' problems. We then discuss the dual formulation of these problems and the associated Sinkhorn solver. Finally, we explore how these new OPT problems relate to classical optimal transport (OT) problems and introduce a linear programming solver tailored for these generalized scenarios.