Open-Set Domain Adaptation via Free Boundary Optimal Transport With Support Control

Optimal transport (OT) has proven highly successful in various machine learning tasks, primarily by measuring distributional differences. As a distance-driven method, OT heavily relies on the underlying distance structure of the sample space but lacks the ability to incorporate label-related information. Consequently, it often underperforms and suffers from significant degradation in real-world scenarios where the data structure is not guaranteed. This limitation becomes particularly critical in open set domain adaptation (OSDA), which demands clear discrimination between known classes (shared between the source domain and the target domain) and unknown classes (unique to the target domain). Thus, it is essential to explore a controllable transport mechanism with class information for learning complex clusters structure. In this article, we propose the controllable partial OT (POT) from a free boundary formulation, which provide a solution to control the supports of transport plan by an analytic parameter. We provide a theoretical justification that the proposed support control mechanism facilitates class-conditional distribution alignment, and we further establish the equivalence between free boundary OT and conditional OT. Intuitively, learning the transport supports reduces to a binary decision problem-whether to transport or not-which can be naturally interpreted as a way to quantify interclass and intraclass transport behaviors. To support this quantification, we establish the existence of an optimal parameter associated with the two bounds that characterize intraclass and interclass transport. Methodologically, a two-point metric module is developed to form a significant gap between interclass and intraclass transport, which ensure the empirical validity of our method. Extensive experiments validate our theoretical findings and confirm the effectiveness of our proposed model.