The Sliced Wasserstein (SW) distance has emerged as a computationally attractive alternative to the Wasserstein distance by leveraging one-dimensional optimal transport along random projections. Standard estimators of the SW distance rely on Monte Carlo averages of one-dimensional Wasserstein distances computed via quantile functions, which require sorting projected samples and access to full datasets. In this work, we introduce a new class of estimators for the Sliced Wasserstein distance based on cumulative distribution functions (CDFs) of projected measures, that avoid sorting and scale via massive dataset parallelism. This class includes several estimators, some of them being indexed by hyperparameters controlling their variance or smoothness. We show that they are especially well suited to scenarios in which CDFs are more tractable than quantile functions, such as mixtures of Gaussians, and moreover that they are also naturally compatible with federated learning, since CDFs of projected data can be computed and aggregated locally without requiring the exchange of raw samples.

This paper examines how metric adjustments to Multidimensional Scaling (MDS) can enhance its effectiveness as a visual tool for pattern recognition. The distance under consideration, referred to as Max-D-SW, is an adjustment of the Max-Sliced Wasserstein distance. In contrast to the original formulation, which optimizes over single unit directions, Max-D-SW aggregates contributions over orthonormal bases. This modification provides a clear numerical advantage in MDS outcomes, particularly when applied to heavy-tailed distributions. We also establish sample-complexity bounds showing that Max-D-SW remains statistically tractable, with rates comparable to those of its max-sliced counterpart. Moreover, we show that a better sample complexity for a metric does not necessarily translate into better performance when the metric is used as an input for MDS.

Many modern generative modeling methods, including diffusion models, normalizing flows, and flow matching, estimate transport maps or plans between distributions without explicitly targeting an optimal transport (OT) map. In applications like generative modeling, the transport cost itself is irrelevant, and this makes it natural to target maps which are more tractable from either a statistical or computational standpoint. In this short note, we formalize the task of estimating any valid transport map in a rigorous minimax framework. One consequence of this framing is that it yields sample complexity lower bounds for any method whose learned object is evaluated as a transport map or plan, including flow matching and diffusion-based generative models, in settings where direct analysis would be challenging due to the analytic complexity of the methods and their target maps. We observe that, under standard, though strong, stability assumptions from the OT literature, estimating any valid transport map is statistically as hard as estimating the OT map. We complement these results with some examples showing that when these stability assumptions fail, alternative transport maps can be learned substantially more accurately than the OT map. Our minimax framing provides a rigorous foundation for understanding the statistical limits of modern transport-based generative methods and clarifies when targeting sub-optimal maps can provide real statistical advantages.

Optimal Transport (OT) has attracted significant interest in the machine learning community, not only for its ability to define meaningful distances between probability distributions - such as the Wasserstein distance - but also for its formulation of OT plans. Its computational complexity remains a bottleneck, though, and slicing techniques have been developed to scale OT to large datasets. Recently, a novel slicing scheme, dubbed min-SWGG, lifts a single one-dimensional plan back to the original multidimensional space, finally selecting the slice that yields the lowest Wasserstein distance as an approximation of the full OT plan. Despite its computational and theoretical advantages, min-SWGG inherits typical limitations of slicing methods: (i) the number of required slices grows exponentially with the data dimension, and (ii) it is constrained to linear projections. Here, we reformulate min-SWGG as a bilevel optimization problem and propose a differentiable approximation scheme to efficiently identify the optimal slice, even in high-dimensional settings. We furthermore define its generalized extension for accommodating to data living on manifolds. Finally, we demonstrate the practical value of our approach in various applications, including gradient flows on manifolds and highdimensional spaces, as well as a novel sliced OT-based conditional flow matching for image generation - where fast computation of transport plans is essential.

Yet, most existing approaches for hierarchical representation learning consider only one mode at a time. In this work, we propose an unsupervised method for jointly learning hierarchical representations of samples and features via TreeWasserstein Distance (TWD). Our method alternates between the two data modes. It first constructs a tree for one mode, then computes a TWD for the other mode based on that tree, and finally uses the resulting TWD to build the second mode's tree.

In this paper, we study the quantitative convergence of shallow neural networks trained via gradient descent to their associated Gaussian processes in the infinitewidth limit. While previous work has established qualitative convergence under broad settings, precise, finite-width estimates remain limited, particularly during training. We provide explicit upper bounds on the quadratic Wasserstein distance between the network output and its Gaussian approximation at any training time t 0, demonstrating polynomial decay with network width. Our results quantify how architectural parameters, such as width and input dimension, influence convergence, and how training dynamics affect the approximation error.

We propose SW-Guidance, a training-free approach for image generation conditioned on the color distribution of a reference image. While it is possible to generate an image with fixed colors by first creating an image from a text prompt and then applying a color style transfer method, this approach often results in semantically meaningless colors in the generated image. Our method solves this problem by modifying the sampling process of a diffusion model to incorporate the differentiable Sliced 1-Wasserstein distance between the color distribution of the generated image and the reference palette. Our method outperforms state-ofthe-art techniques for color-conditional generation in terms of color similarity to the reference, producing images that not only match the reference colors but also maintain semantic coherence with the original text prompt.

Optimal Transport (OT) has attracted significant interest in the machine learning community, not only for its ability to define meaningful distances between probability distributions -- such as the Wasserstein distance -- but also for its formulation of OT plans. Its computational complexity remains a bottleneck, though, and slicing techniques have been developed to scale OT to large datasets. Recently, a novel slicing scheme, dubbed min-SWGG, lifts a single one-dimensional plan back to the original multidimensional space, finally selecting the slice that yields the lowest Wasserstein distance as an approximation of the full OT plan. Despite its computational and theoretical advantages, min-SWGG inherits typical limitations of slicing methods: (i) the number of required slices grows exponentially with the data dimension, and (ii) it is constrained to linear projections. Here, we reformulate min-SWGG as a bilevel optimization problem and propose a differentiable approximation scheme to efficiently identify the optimal slice, even in high-dimensional settings. We furthermore define its generalized extension for accommodating data living on manifolds. Finally, we demonstrate the practical value of our approach in various applications, including gradient flows on manifolds and high-dimensional spaces, as well as a novel sliced OT-based conditional flow matching for image generation -- where fast computation of transport plans is essential.

Modeling the dynamics from sparsely time-resolved snapshot data is crucial for understanding complex cellular processes and behavior.

Score-based diffusion models have emerged as powerful tools in generative modeling, yet their theoretical foundations remain underexplored. In this work, we focus on the Wasserstein convergence analysis of score-based diffusion models. Specifically, we investigate the impact of various discretization schemes, including Euler discretization, exponential integrators, and midpoint randomization methods. Our analysis provides the first quantitative comparison of these discrete approximations, emphasizing their influence on convergence behavior. Furthermore, we explore scenarios where Hessian information is available and propose an accelerated sampler based on the local linearization method. We establish the first Wasserstein convergence analysis for such a Hessian-based method, showing that it achieves an improved convergence rate of order eO( d/ε), which significantly outperforms the standard rate eO(d/ε2)of vanilla diffusion models.