Sparse Partial Optimal Transport via Quadratic Regularization

TL;DR

This paper introduces a quadratic regularization approach for Partial Optimal Transport that promotes sparsity in transport plans, improving applicability in real-world scenarios where sparse solutions are preferred.

Contribution

The paper proposes a novel quadratic regularized formulation for Partial Optimal Transport that induces sparsity, offering an alternative to entropic regularization methods.

Findings

Quadratic regularization yields sparser transport plans.

QPT outperforms entropic methods in applications requiring sparsity.

Experimental results on synthetic and real datasets validate the approach.

Abstract

Partial Optimal Transport (POT) has recently emerged as a central tool in various Machine Learning (ML) applications. It lifts the stringent assumption of the conventional Optimal Transport (OT) that input measures are of equal masses, which is often not guaranteed in real-world datasets, and thus offers greater flexibility by permitting transport between unbalanced input measures. Nevertheless, existing major solvers for POT commonly rely on entropic regularization for acceleration and thus return dense transport plans, hindering the adoption of POT in various applications that favor sparsity. In this paper, as an alternative approach to the entropic POT formulation in the literature, we propose a novel formulation of POT with quadratic regularization, hence termed quadratic regularized POT (QPOT), which induces sparsity to the transport plan and consequently facilitates the adoption of POT in many applications with sparsity requirements. Extensive experiments on synthetic and CIFAR-10 datasets, as well as real-world applications such as color transfer and domain adaptations, consistently demonstrate the improved sparsity and favorable performance of our proposed QPOT formulation.