We propose a new regularized optimal transport (OT) formulation, termed sliced-regularized optimal transport (SROT). Unlike entropic OT (EOT), which regularizes the transport plan toward an independent coupling, SROT regularizes it toward a smoothened sliced OT (SOT) plan. To the best of our knowledge, SROT is the first approach to leverage a version of SOT plan as a reference to improve classical OT. We provide a formal definition of SROT, derive its dual formulation, and provide a post-Bayesian interpretation of SROT. We then develop a Sinkhorn-style algorithm for efficient computation, retaining the same scalability advantages as EOT. By incorporating a scalable SOT plan as a prior, SROT yields more accurate approximations of the exact OT plan than EOT under the same level of regularization. Moreover, the resulting transport plan improves upon the reference SOT plan itself. We further introduce the corresponding OT divergence induced by SROT, named SROT divergence, and analyze its topological and computational properties. Finally, we validate our approach through experiments on synthetic datasets and color transfer tasks, demonstrating that SROT is better than both EOT and SOT in approximating exact OT. Additional experiments on gradient flows further highlight the advantages of SROT divergence.

Self-supervised learning on graphs aims to learn graph representations in an unsupervised manner. While graph contrastive learning (GCL - relying on graph augmentation for creating perturbation views of anchor graphs and maximizing/minimizing similarity for positive/negative pairs) is a popular self-supervised method, it faces challenges in finding label-invariant augmented graphs and determining the exact extent of similarity between sample pairs to be achieved. In this work, we propose an alternative self-supervised solution that (i) goes beyond the label invariance assumption without distinguishing between positive/negative samples, (ii) can calibrate the encoder for preserving not only the structural information inside the graph, but the matching information between different graphs, (iii) learns isometric embeddings that preserve the distance between graphs, a by-product of our objective. Motivated by optimal transport theory, this scheme relies on an observation that the optimal transport plans between node representations at the output space, which measure the matching probability between two distributions, should be consistent with the plans between the corresponding graphs at the input space. The experimental findings include: (i) The plan alignment strategy significantly outperforms the counterpart using the transport distance; (ii) The proposed model shows superior performance using only node attributes as calibration signals, without relying on edge information; (iii) Our model maintains robust results even under high perturbation rates; (iv) Extensive experiments on various benchmarks validate the effectiveness of the proposed method.

We can see that our proposed model can effectively reduce the number of tasks with classification rates of less than 60%. To be our best knowledge, those novel tasks performed poorly by few-shot learning methods usually have the relatively large domain differences with all base classes, where the importance of each base class for novel sample might be similar. Different from Free-lunch, which only selects topw base classes to estimate the distribution of novel sample and might omit some relevant information, we utilizes all base classes by introducing the adaptive weight information over all base classes for each novel sample. It indicates that our proposed H-OT can effectively enhance distribution calibration method when there is a big domain difference between base and novel classes.

Few-shot classification aims to learn a classifier to recognize unseen classes during training, where the learned model can easily become over-fitted based on the biased distribution formed by only a few training examples. A recent solution to this problem is calibrating the distribution of these few sample classes by transferring statistics from the base classes with sufficient examples, where how to decide the transfer weights from base classes to novel classes is the key. However, principled approaches for learning the transfer weights have not been carefully studied. To this end, we propose a novel distribution calibration method by learning the adaptive weight matrix between novel samples and base classes, which is built upon a hierarchical Optimal Transport (H-OT) framework. By minimizing the high-level OT distance between novel samples and base classes, we can view the learned transport plan as the adaptive weight information for transferring the statistics of base classes. The learning of the cost function between a base class and novel class in the high-level OT leads to the introduction of the lowlevel OT, which considers the weights of all the data samples in the base class. Experiments on standard benchmarks demonstrate that our proposed plug-andplay model outperforms competing approaches and owns desired cross-domain generalization ability, proving the effectiveness of the learned adaptive weights. 1

Comparison of statistical distances can also enable distribution testing, quantification of distribution shifts, and provide methods to correct for distribution shift through domainadaptation[12]. Optimal transport theory provides a rich set of tools for comparing distributions inWasserstein Distance.

However, the generating process can be different in several aspects, such as the conditions and devices used for acquisition, different pre-processing, different compressions, etc. Domain adaptation