Take It or Leave It: Intent-Controlled Partial Optimal Transport

TL;DR

This paper introduces intent-controlled partial optimal transport (IC-POT), a flexible framework allowing pointwise rejection decisions based on side information, with theoretical foundations and practical applications in machine learning and geophysics.

Contribution

The paper proposes IC-POT, a novel generalization of partial optimal transport that incorporates pointwise rejection costs, supported by dual interpretation and solution via balanced Kantorovich OT.

Findings

Incorporating pointwise rejection improves performance in positive-unlabeled learning.

IC-POT effectively adapts to side information in open-partial domain adaptation.

Application to satellite ocean measurements demonstrates practical benefits of IC-POT.

Abstract

While optimal transport (OT) enforces a rigid constraint by requiring two measures to be matched exactly, partial optimal transport relaxes this requirement by allowing mass to remain unmatched through a global budget, scalar rebate, or uniform rejection rule. However, many applications call for more structured, pointwise rejection mechanisms, where the decision to leave mass unmatched depends on side-specific reliability, support geometry, or external information about which components should participate in the comparison. We introduce \emph{intent-controlled partial optimal transport} (IC-POT), a targeted generalization of partial transport that replaces the global rejection paradigm with pointwise rejection costs over both measures. We show that the resulting optimization problem admits a dual interpretation in terms of local acceptance thresholds and can be solved by recasting it as a balanced Kantorovich OT problem on an augmented support. Beyond theoretical analysis, we demonstrate the practical relevance of IC-POT in settings where rejection is driven by side information. In positive-unlabeled learning and open-partial domain adaptation, incorporating pointwise rejection rules that encode statistical structure improves fixed baseline pipelines. Finally, we motivate the use of IC-POT with a geophysical practical case: multi-modal satellite ocean measurements, for which physical and sensors priors naturally inform the rejection mechanism and define the retrieved comparable signal information.