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

2026-05-19

AI Summary

This work addresses the limitations of traditional optimal transport, which enforces strict measure matching, and existing partial optimal transport approaches that only permit global or uniform rejection, thereby failing to accommodate structured rejection requirements informed by pointwise priors. The paper proposes Intent-Controlled Partial Optimal Transport (IC-POT), which introduces pointwise rejection costs for the first time, endowing each mass unit with independent decision-making capability. The formulation recasts the problem as a balanced Kantorovich problem over an augmented support set. By integrating dual optimization with support-set augmentation, IC-POT achieves structure-aware incomplete matching guided by local reliability. Experiments on positive-unlabeled learning, open partial domain adaptation, and multimodal satellite ocean observation demonstrate that IC-POT significantly outperforms baseline methods by effectively leveraging prior information to extract comparable signals.

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.

Problem

• partial optimal transport
• pointwise rejection
• side information
• structured matching
• unmatched mass

Innovation

• intent-controlled partial optimal transport
• pointwise rejection
• optimal transport
• side information
• dual formulation