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.

Vision-language models (VLMs), such as CLIP, have demonstrated impressive zero-shot capabilities for various downstream tasks. Their performance can be further enhanced through few-shot prompt tuning methods. However, current studies evaluate the performance of learned prompts separately on base and new classes. This evaluation lacks practicality for real-world applications since downstream tasks cannot determine whether the data belongs to base or new classes in advance. In this paper, we explore a problem setting called Open-world Prompt Tuning (OPT), which involves tuning prompts on base classes and evaluating on a combination of base and new classes. By introducing Decomposed Prompt Tuning framework (DePT), we theoretically demonstrate that OPT can be solved by incorporating out-of-distribution detection into prompt tuning, thereby enhancing the base-to-new discriminability. Based on DePT, we present a novel prompt tuning approach, namely, Decomposed Context Optimization (DeCoOp), which introduces new-class detectors and sub-classifiers to further enhance the base-class and new-class discriminability. Experimental results on 11 benchmark datasets validate the effectiveness of DePT and demonstrate that DeCoOp outperforms current state-of-the-art methods, providing a significant 2% average accuracy improvement.

Optimal transport (OT) has become exceedingly popular in machine learning, data science, and computer vision. The core assumption in the OT problem is the equal total amount of mass in source and target measures, which limits its application. Optimal Partial Transport (OPT) is a recently proposed solution to this limitation. Similar to the OT problem, the computation of OPT relies on solving a linear programming problem (often in high dimensions), which can become computationally prohibitive. In this paper, we propose an efficient algorithm for calculating the OPT problem between two non-negative measures in one dimension. Next, following the idea of sliced OT distances, we utilize slicing to define the sliced OPT distance. Finally, we demonstrate the computational and accuracy benefits of the sliced OPT-based method in various numerical experiments. In particular, we show an application of our proposed Sliced-OPT in noisy point cloud registration.

Optimal transport and its related problems, including optimal partial transport, have proven to be valuable tools in machine learning for computing meaningful distances between probability or positive measures. This success has led to a growing interest in defining transport-based distances that allow for comparing signed measures and, more generally, multi-channeled signals. Transport $\mathrm{L}^{p}$ distances are notable extensions of the optimal transport framework to signed and possibly multi-channeled signals. In this paper, we introduce partial transport $\mathrm{L}^{p}$ distances as a new family of metrics for comparing generic signals, benefiting from the robustness of partial transport distances. We provide theoretical background such as the existence of optimal plans and the behavior of the distance in various limits. Furthermore, we introduce the sliced variation of these distances, which allows for rapid comparison of generic signals. Finally, we demonstrate the application of the proposed distances in signal class separability and nearest neighbor classification.