The Gromov–Wasserstein (GW) problem, a variant of the classical optimal transport (OT) problem, has attracted growing interest in the machine learning and data science communities due to its ability to quantify similarity between measures in different metric spaces. However, like the classical OT problem, GW imposes an equal mass constraint between measures, which restricts its application in many machine learning tasks. To address this limitation, the partial Gromov-Wasserstein (PGW) problem has been introduced. It relaxes the equal mass constraint, allowing the comparison of general positive Radon measures. Despite this, both GW and PGW face significant computational challenges due to their non-convex nature. To overcome these challenges, we propose the linear partial Gromov-Wasserstein (LPGW) embedding, a linearized embedding technique for the PGW problem. For $K$ different metric measure spaces, the pairwise computation of the PGW distance requires solving the PGW problem $\mathcal{O}(K^2)$ times. In contrast, the proposed linearization technique reduces this to $\mathcal{O}(K)$ times. Similar to the linearization technique for the classical OT problem, we prove that LPGW defines a valid metric for metric measure spaces. Finally, we demonstrate the effectiveness of LPGW in practical applications such as shape retrieval and learning with transport-based embeddings, showing that LPGW preserves the advantages of PGW in partial matching while significantly enhancing computational efficiency. The code is available at https://github.com/mint-vu/Linearized_Partial_Gromov_Wasserstein.

The Gromov-Wasserstein (GW) distance has gained increasing interest in the machine learning community in recent years, as it allows for the comparison of measures in different metric spaces. To overcome the limitations imposed by the equal mass requirements of the classical GW problem, researchers have begun exploring its application in unbalanced settings. However, Unbalanced GW (UGW) can only be regarded as a discrepancy rather than a rigorous metric/distance between two metric measure spaces (mm-spaces). In this paper, we propose a particular case of the UGW problem, termed Partial Gromov-Wasserstein (PGW). We establish that PGW is a well-defined metric between mm-spaces and discuss its theoretical properties, including the existence of a minimizer for the PGW problem and the relationship between PGW and GW, among others. We then propose two variants of the Frank-Wolfe algorithm for solving the PGW problem and show that they are mathematically and computationally equivalent. Moreover, based on our PGW metric, we introduce the analogous concept of barycenters for mm-spaces. Finally, we validate the effectiveness of our PGW metric and related solvers in applications such as shape matching, shape retrieval, and shape interpolation, comparing them against existing baselines. Our code is available at https://github.com/mint-vu/PGW_Metric.

Efficient comparison of spherical probability distributions becomes important in fields such as computer vision, geosciences, and medicine. Sliced optimal transport distances, such as spherical and stereographic spherical sliced Wasserstein distances, have recently been developed to address this need. These methods reduce the computational burden of optimal transport by slicing hyperspheres into one-dimensional projections, i.e., lines or circles. Concurrently, linear optimal transport has been proposed to embed distributions into $L^2$ spaces, where the $L^2$ distance approximates the optimal transport distance, thereby simplifying comparisons across multiple distributions. In this work, we introduce the Linear Spherical Sliced Optimal Transport (LSSOT) framework, which utilizes slicing to embed spherical distributions into $L^2$ spaces while preserving their intrinsic geometry, offering a computationally efficient metric for spherical probability measures. We establish the metricity of LSSOT and demonstrate its superior computational efficiency in applications such as cortical surface registration, 3D point cloud interpolation via gradient flow, and shape embedding. Our results demonstrate the significant computational benefits and high accuracy of LSSOT in these applications.

The optimal transport (OT) problem has gained significant traction in modern machine learning for its ability to: (1) provide versatile metrics, such as Wasserstein distances and their variants, and (2) determine optimal couplings between probability measures. To reduce the computational complexity of OT solvers, methods like entropic regularization and sliced optimal transport have been proposed. The sliced OT framework improves efficiency by comparing one-dimensional projections (slices) of high-dimensional distributions. However, despite their computational efficiency, sliced-Wasserstein approaches lack a transportation plan between the input measures, limiting their use in scenarios requiring explicit coupling. In this paper, we address two key questions: Can a transportation plan be constructed between two probability measures using the sliced transport framework? If so, can this plan be used to define a metric between the measures? We propose a ‘lifting’ operation to extend one-dimensional optimal transport plans back to the original space of the measures. By computing the expectation of these lifted plans, we derive a new transportation plan, termed expected sliced transport (EST) plans. We further prove that using the EST plan to weight the sum of the individual Euclidean costs $\|x - y\|^p$ for moving from $x$ to $y$ results in a valid metric between the input discrete probability measures. Finally, we demonstrate the connection between our approach and the recently proposed min-SWGG, along with illustrative numerical examples that support our theoretical findings.

Measuring similarities between different tasks is critical in a broad spectrum of machine learning problems, including transfer, multi-task, continual, and meta-learning. Most current approaches to measuring task similarities are architecture-dependent: (1) relying on pre-trained models, or (2) training networks on tasks and using forward transfer as a proxy for task similarity. In this paper, we leverage the optimal transport theory and define a novel task embedding for supervised classification that is model-agnostic, training-free, and capable of handling (partially) disjoint label sets. In short, given a dataset with ground-truth labels, we perform a label embedding through multi-dimensional scaling and concatenate dataset samples with their corresponding label embeddings. Then, we define the distance between two datasets as the 2-Wasserstein distance between their updated samples. Lastly, we leverage the 2-Wasserstein embedding framework to embed tasks into a vector space in which the Euclidean distance between the embedded points approximates the proposed 2-Wasserstein distance between tasks. We show that the proposed embedding leads to a significantly faster comparison of tasks compared to related approaches like the Optimal Transport Dataset Distance (OTDD). Furthermore, we demonstrate the effectiveness of our embedding through various numerical experiments and show statistically significant correlations between our proposed distance and the forward and backward transfer among tasks on a wide variety of image recognition datasets.

The optimal transport problem for measures supported on non-Euclidean spaces has recently gained ample interest in diverse applications involving representation learning. In this paper, we focus on circular probability measures, i.e., probability measures supported on the unit circle, and introduce a new computationally efficient metric for these measures, denoted as Linear Circular Optimal Transport (LCOT). The proposed metric comes with an explicit linear embedding that allows one to apply Machine Learning (ML) algorithms to the embedded measures and seamlessly modify the underlying metric for the ML algorithm to LCOT. We show that the proposed metric is rooted in the Circular Optimal Transport (COT) and can be considered the linearization of the COT metric with respect to a fixed reference measure. We provide a theoretical analysis of the proposed metric and derive the computational complexities for pairwise comparison of circular probability measures. Lastly, through a set of numerical experiments, we demonstrate the benefits of LCOT in learning representations from circular measures.

Comparing spherical probability distributions is of great interest in various fields, including geology, medical domains, computer vision, and deep representation learning. The utility of optimal transport-based distances, such as the Wasserstein distance, for comparing probability measures has spurred active research in developing computationally efficient variations of these distances for spherical probability measures. This paper introduces a high-speed and highly parallelizable distance for comparing spherical measures using the stereographic projection and the generalized Radon transform, which we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance. We carefully address the distance distortion caused by the stereographic projection and provide an extensive theoretical analysis of our proposed metric and its rotationally invariant variation. Finally, we evaluate the performance of the proposed metrics and compare them with recent baselines in terms of both speed and accuracy through a wide range of numerical studies, including gradient flows and self-supervised learning. Our code is available at https://github.com/mint-vu/s3wd.

Generative Adversarial Networks (GANs) are a popular method for learning distributions from data by modeling the target distribution as a function of a known distribution. The function, often referred to as the generator, is optimized to minimize a chosen distance measure between the generated and target distributions. One commonly used measure for this purpose is the Wasserstein distance. However, Wasserstein distance is hard to compute and optimize, and in practice entropic regularization techniques are used to facilitate its computation and improve numerical convergence. The influence of regularization on the learned solution, however, remains not well-understood. In this paper, we study how several popular entropic regularizations of Wasserstein distance impact the solution learned by a Wasserstein GAN in a simple benchmark setting where the generator is linear and the target distribution is high-dimensional Gaussian. We show that entropy regularization of Wasserstein distance promotes sparsification of the solution, while replacing the Wasserstein distance with the Sinkhorn divergence recovers the unregularized solution. The significant benefit of both regularization techniques is that they remove the curse of dimensionality suffered by Wasserstein distance. We show that in both cases the optimal generator can be learned to accuracy $\epsilon$ with $O(1/\epsilon^2)$ samples from the target distribution without requiring to constrain the discriminator. We thus conclude that these regularization techniques can improve the quality of the generator learned from empirical data in a way that is applicable for a large class of distributions.

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 problem in noisy point cloud registration and color adaptation. Our code is available at Github Link.

Optimal transport (OT) has gained popularity due to its various applications in fields such as machine learning, statistics, and signal processing. However, the balanced mass requirement limits its performance in practical problems. To address these limitations, variants of the OT problem, including unbalanced OT, Optimal partial transport (OPT), and Hellinger Kantorovich (HK), have been proposed. In this paper, we propose the Linear optimal partial transport (LOPT) embedding, which extends the (local) linearization technique on OT and HK to the OPT problem. The proposed embedding allows for faster computation of OPT distance between pairs of positive measures. Besides our theoretical contributions, we demonstrate the LOPT embedding technique in point-cloud interpolation and PCA analysis. Our code is available at https://github.com/Baio0/LinearOPT.

The optimal transport problem studies how to transport one measure to another in the most cost-effective way and has wide range of applications from economics to machine learning. In this paper, we introduce and study an information constrained variation of this problem. Our study yields a strengthening and generalization of Talagrand’s celebrated transportation cost inequality. Following Marton’s approach, we show that the new transportation cost inequality can be used to recover old and new concentration of measure results. Finally, we provide an application of this new inequality to network information theory. We show that it can be used to recover almost immediately a recent solution to a long-standing open problem posed by Cover regarding the capacity of the relay channel.

Generative Adversarial Networks (GANs) are a popular method for learning distributions from data by modeling the target distribution as a function of a known distribution. The function, often referred to as the generator, is optimized to minimize a chosen distance measure between the generated and target distributions. One commonly used measure for this purpose is the Wasserstein distance. However, Wasserstein distance is hard to compute and optimize, and in practice entropic regularization techniques are used to facilitate its computation and improve numerical convergence. The influence of regularization on the learned solution, however, remains not well-understood. In this paper, we study how several popular entropic regularizations of Wasserstein distance impact the solution learned by a Wasserstein GAN in a simple benchmark setting where the generator is linear and the target distribution is high-dimensional Gaussian. We show that entropy regularization of Wasserstein distance promotes sparsification of the solution, while replacing the Wasserstein distance with the Sinkhorn divergence recovers the unregularized solution. The significant benefit of both regularization techniques is that they remove the curse of dimensionality suffered by Wasserstein distance. We show that in both cases the optimal generator can be learned to accuracy ∊ with$O$(1/ ∊2) samples from the target distribution without requiring to constrain the discriminator. We thus conclude that these regularization techniques can improve the quality of the generator learned from empirical data in a way that is applicable for a large class of distributions.

The optimal transport problem studies how to transport one measure to another in the most cost-effective way and has wide range of applications from economics to machine learning. In this paper, we introduce and study an information constrained variation of this problem. Our study yields a strengthening and generalization of Talagrand's celebrated transportation cost inequality. Following Marton's approach, we show that the new transportation cost inequality can be used to recover old and new concentration of measure results. Finally, we provide an application of this inequality to network information theory. We show that it can be used to recover a recent solution to a long-standing open problem posed by Cover regarding the capacity of the relay channel.