Optimal Transport

Many high-dimensional datasets concentrate near a low-dimensional structure embedded in the ambient space. Generative models for such data must control off-support mass while remaining computationally practical. Diffusion models use iterative denoising at inference, whereas standard normalizing flows require invertible, dimension-preserving maps. We propose MAGT (Manifold-Aligned Generative Transport), a direct transport from a low-dimensional base distribution to the data space. Its core objective compares the data and generator-induced scores at a selected Gaussian smoothing level. A posterior identity expresses this score through a latent conditional mean, which is approximated by self-normalized importance sampling over a finite anchor set. After training, generation requires one evaluation of the transport, whose image also carries an intrinsic density with respect to manifold volume. We establish a minimax-optimal Wasserstein convergence rate for an explicitly constructed localized spline-RePU coordinate transport estimator, and treat finite-anchor approximation separately. Experiments on synthetic, image, and tabular benchmarks compare fidelity, support alignment, and sampling cost with diffusion, flow-matching, and adversarial baselines.

This note gives a self-contained overview of some important properties of the Gromov-Wasserstein (GW) distance, compared with the standard linear optimal transport (OT) framework. More specifically, I explore the following questions: are GW optimal transport plans sparse? Under what conditions are they supported on a permutation? Do they satisfy a form of cyclical monotonicity? In particular, I present the conditionally negative semi-definite property and show that, when it holds, there are GW optimal plans that are sparse and supported on a permutation.

We study the slice-matching scheme, an efficient iterative method for distribution matching based on sliced optimal transport. We investigate convergence to the target distribution and derive quantitative non-asymptotic rates. To this end, we establish Lojasiewicz-type inequalities for the Sliced-Wasserstein objective. A key challenge is to control along the trajectory the constants in these inequalities. We show that this becomes tractable for Gaussian distributions. Specifically, eigenvalues are controlled when matching along random orthonormal bases at each iteration. We complement our theory with numerical experiments and illustrate the predicted dependence on dimension and step-size, as well as the stabilizing effect of orthonormal-basis sampling.

Understanding the distributional structure of high-dimensional datasets has become an important topic, yet direct visual characterization is difficult. In this work, we develop a geometric framework for characterizing the distributional structure of empirical datasets by quantifying their deviation from the Gaussian family under the geometry induced by optimal transport theory. Building on the cone structure of the relative translation invariant quadratic Wasserstein $(RW_2)$ space, we define two geometric quantities---the relative Wasserstein angle and the orthogonal projection distance---and show that they are well-defined because of the flat geometry of the filling cone between distributional rays. This formulation recasts the problem of measuring deviation from the Gaussian family as an orthogonal projection problem onto the Gaussian cone and reveals that the commonly used moment-matching Gaussian is, in general, not the $W_2$-nearest Gaussian to a non-Gaussian distribution. In one dimension, we derive closed-form expressions for the proposed quantities and extend closed-form expressions to several other location--scale families, including uniform, Laplace, and logistic distributions. In higher dimensions, we develop a numerical approximation method for the proposed quantities based on empirical optimal transport and covariance-shape optimization. Our experimental results show the empirical convergence and stability of the proposed methods and reveal that the $RW_2$ angle provides a robust and consistent measure of distributional non-Gaussianity. Moreover, these results provide empirical support for its potential use as an indicator of distributional heterogeneity.

We propose a framework for contextual distributionally robust optimization (DRO) that considers the causal and continuous structure of the underlying distribution, and we develop an interpretable and tractable decision rule. We first introduce the causal Sinkhorn discrepancy (CSD), an entropy-regularized causal Wasserstein distance that encourages continuous transport plans while preserving causal consistency. We then formulate a contextual DRO model with a CSD-based ambiguity set, termed Causal Sinkhorn DRO (Causal-SDRO), and derive its strong dual reformulation, where the worst-case distribution is characterized as a mixture of Gibbs distributions. To obtain an (infinite-dimensional) optimal policy, we propose a soft regression forest (SRF) decision rule: it preserves the interpretability of classical decision trees while being fully parametric, differentiable, and Lipschitz-smooth, enabling intrinsic interpretation from both global and local perspectives. To solve the Causal-SDRO with parametric decision rules, we develop an efficient stochastic compositional gradient algorithm that converges to an $\varepsilon$-stationary point at a rate of $\mathcal{O}(\varepsilon^{-4})$, matching that of standard stochastic gradient descent. Finally, we validate our method through numerical experiments on synthetic and real-world datasets, demonstrating its superior performance and interpretability.

Optimal transport (OT) and Gromov-Wasserstein (GW) alignment provide interpretable geometric frameworks for comparing, transforming, and aggregating heterogeneous datasets---tasks ubiquitous in data science and machine learning. Because these frameworks are computationally expensive, large-scale applications often rely on closed-form solutions for Gaussian distributions under quadratic cost. This work provides a comprehensive treatment of Gaussian, quadratic cost OT and inner product GW (IGW) alignment, closing several gaps in the literature to broaden applicability. First, we treat the open problem of IGW alignment between uncentered Gaussians on separable Hilbert spaces by giving an exact variational characterization through a quadratic optimization over isometries and co-isometries (orthogonal matrices in finite dimensions), for which we derive tight analytic upper and lower bounds. If at least one Gaussian measure is centered, the solution reduces to a fully closed-form expression, which we further extend to an analytic solution for the IGW barycenter between centered Gaussians. We also present a reduction of Gaussian multimarginal OT with pairwise quadratic costs to a tractable optimization problem and prove that every second-order stationary point of this problem is globally optimal. To demonstrate utility, we compare embedding distributions of already-trained language-model distillations and cluster synthetic users using covariance spectra of their text embeddings.

Conformal prediction (CP) constructs uncertainty sets for model outputs with finite-sample coverage guarantees. Yet ranking scores is straightforward only when they are scalar-valued, limiting CP to real-valued scores or ad-hoc one-dimensional reductions. Vector-valued scores arise naturally in multi-output regression and model aggregation, where each predictor in an ensemble provides its own score. Optimal transport (OT) defines vector ranks and center-outward multivariate quantile regions, though generally with asymptotic coverage guarantees. Applying a fixed transport map learned from calibration data to a new point introduces an uncontrolled approximation error. We restore finite-sample, distribution-free coverage by conformalizing vector-valued OT quantile regions. Each candidate's rank is defined by transporting the calibration scores augmented with that candidate's score, preserving the symmetry needed for validity. This appears to require a continuum of OT problems. However, we prove that the optimal assignment is piecewise constant across a fixed polyhedral partition of score space. This lets us characterize the entire prediction set in $O(n^3)$ time, matching the cost of a single assignment solve. It also addresses a limitation of prediction sets: they indicate which outcomes are plausible, but not their relative likelihood. In one dimension, conformal predictive distributions (CPDs) fill this gap by producing a predictive distribution with finite-sample calibration. Extending CPDs beyond one dimension remained an open problem. We construct, to our knowledge, the first multivariate CPDs with finite-sample calibration: a center-outward predictive distribution whose derived uncertainty regions have conformal coverage. We present both conservative and exact randomized versions; the latter generalizes the classical Dempster-Hill procedure.

We study the Sobolev IPM problem for probability measures supported on a graph metric space, where critic function is constrained to lie within the unit ball defined by Sobolev norm. Sobolev IPM is intrinsically coupled with $L^p$ geometric structure within its definition, limiting its ability to incorporate other prior geometry beyond the $L^p$ paradigm. Conversely, the classic optimal transport is flexible, easy to adapt to various geometric structures by simply changing its ground cost. An important example is Orlicz-Wasserstein (OW) which utilizes Orlicz geometric structure to generalize the $L^p$ within the standard $p$-order Wasserstein, and remarkably play a vital role to advance machine learning methodologies. Inspired by recent advantages of OW, in this work, we leverage a specific class of convex functions for Orlicz geometry to mitigate such limitation for Sobolev IPM, and propose the generalized Sobolev IPM (GSI). Our GSI approach encompasses Sobolev IPM as a special case while accommodating diverse geometric priors beyond $L^p$. It however brings up significant computational hurdles that compound those already notoriously inherent in Sobolev IPM. To address these challenges, we theoretically establish a novel connection between Orlicz-Sobolev norm and Musielak norm which facilitates a novel efficient regularization for GSI. By further exploiting the underlying graph structure, we show that the regularized GSI reduces to a simple univariate optimization problem, achieving notably computational efficiency, enabling its usage in practical applications. We empirically illustrate that the regularized GSI is several-order faster than the popular OW in computation, and its performances compare favorably to other transport baselines for comparing graph-based measures in document classification and topological data analysis.

Training time-series forecasting models requires aligning the conditional distribution of model forecasts with that of the label sequence. The standard direct forecast (DF) approach resorts to minimizing the conditional negative log-likelihood, typically estimated by the mean squared error. However, this estimation proves biased when the label sequence exhibits autocorrelation. In this paper, we propose DistDF, which achieves alignment by minimizing a distributional discrepancy between the conditional distributions of forecast and label sequences. Since such conditional discrepancies are difficult to estimate from finite time-series observations, we introduce a joint-distribution Wasserstein discrepancy for time-series forecasting, which provably upper bounds the conditional discrepancy of interest. The proposed discrepancy is tractable, differentiable, and readily compatible with gradient-based optimization. Extensive experiments show that DistDF improves diverse forecasting models and achieves leading performance. Code is available at https://anonymous.4open.science/r/DistDF-F66B.

Flow-matching models have recently emerged as a powerful framework for generative modeling, including 3D point cloud synthesis. However, their deployment is limited by the need for multiple sequential sampling steps at inference time. MeanFlow enables single-step generation by learning an average velocity field, but its quality depends heavily on the geometry of the training pairs. In this work, we propose an Optimal Transport-enhanced MeanFlow framework (OT-MF3D) for efficient and accurate 3D point cloud generation and completion. Beyond incorporating OT into MeanFlow, our method introduces a latent-conditioned training algorithm that decouples shape-level sampling from point-level matching. This enables exact point-level OT within each conditioned shape while performing shape-level transport in a compact latent space, avoiding the expensive nested OT computation in point-cloud space. Experiments on standard benchmarks show improved generation and completion quality over fast baselines, with substantially reduced training and inference costs compared with diffusion and other flow-based models.

The Expectation-Maximisation (EM) algorithm is a central tool in statistics and machine learning, widely used for latent-variable models such as Gaussian Mixture Models (GMMs). Despite its ubiquity, EM is typically treated as a non-differentiable black box, preventing its integration into modern learning pipelines where end-to-end gradient propagation is essential. In this work, we present and compare several differentiation strategies for EM, from full automatic differentiation to approximate methods, assessing their accuracy and computational efficiency. As a key application, we leverage this differentiable EM in the computation of the Mixture Wasserstein distance $\mathrm{MW}_2$ between GMMs, allowing $\mathrm{MW}_2$ to be used as a differentiable loss in imaging and machine learning tasks. To complement our practical use of $\mathrm{MW}_2$, we contribute a novel stability result which provides theoretical justification for the use of $\mathrm{MW}_2$ with EM, and also introduce a novel unbalanced variant of $\mathrm{MW}_2$. Numerical experiments on barycentre computation, colour and style transfer, image generation, and texture synthesis illustrate the versatility of the proposed approach in different settings.

We study inverse problems where an unknown potential is observed only through samples from the measure it induces by a convex variational principle. Such problems arise in learning costs, energies, and dynamics from distributional data, but the associated forward solution map is typically nonlinear and implicit. We show that its optimality gap nevertheless yields convex empirical objectives for finite-dimensional potential classes, and we introduce sharpened Fenchel--Young losses that add a data-dependent discrepancy inside the forward problem. This keeps the estimator calibrated while improving the local geometry of the loss. Our main stability theorem separates the inverse error analysis into measurement error, forward perturbation, and empirical curvature. We instantiate this principle for inverse entropic unbalanced optimal transport and for inverse Jordan--Kinderlehrer--Otto (JKO) learning from independent snapshot samples, obtaining high-probability parameter recovery bounds. JKO schemes discretize Wasserstein gradient flows through a sequence of variational problems over measures, making them a natural language for population dynamics observed through snapshots. In this JKO case, the sharpened objective reduces to an unbalanced transport problem, which also clarifies the connection between variational gap losses and quadratic iJKO$^\star$ surrogates. Numerical experiments illustrate the conditioning effect of sharpening and its benefits for sparse inverse-gradient-flow recovery.

Min cut is an important graph partitioning method. However, current solutions to the min cut problem suffer from slow speeds, difficulty in solving, and often converge to simple solutions. To address these issues, we relax the min cut problem into a double-bounded constraint and, for the first time, treat the min cut problem as a double-bounded nonlinear optimal transport problem. Additionally, we develop a method for solving double bounded nonlinear optimal transport based on the Frank-Wolfe method (abbreviated as DNF). We prove that for convex problems satisfying Lipschitz smoothness, the DNF method can achieve a convergence rate of $\mathcal{O}(\frac{1}{t})$. We apply DNF to size-constrained min-cut clustering and evaluate it on eight benchmark datasets. DNF achieves competitive clustering performance and matches or outperforms the compared baselines on several datasets and metrics.

We introduce a novel Wasserstein-1 ($W_1$) path-space divergence for stochastic and deterministic dynamics and establish a Wasserstein Uncertainty Propagation (WUP) theorem that bounds the $W_1$ distance between terminal distributions by the proposed divergence, equivalently characterized by a weighted $L^2$ discrepancy between the underlying drifts and the $W_1$ distance between their initial measures. A key ingredient is a probabilistic framework combining adjoint Feynman-Kac representations with synchronous coupling (and reflection coupling on bounded domains), yielding Wasserstein stability estimates beyond existing PDE- and Girsanov-based approaches. The framework accommodates time-varying and possibly degenerate diffusion coefficients, empirical and singular measures, and remains valid in the deterministic limit of flow matching. Unlike KL-based uncertainty quantification bounds, it does not require absolute continuity of path measures and therefore remains well-defined in singular settings. As consequences of the WUP theorem, we derive $W_1$ robustness and generalization bounds for score-based generative models and flow matching at both population and finite-sample levels. We further specialize the framework to group-symmetric targets, providing the first error analysis of equivariant flow-based models and the first quantitative comparison between data augmentation and equivariant inductive bias. Our analysis identifies a symmetry-aware Wasserstein path-space divergence that quantifies the model-form error induced by non-equivariant parametrizations. We prove that this error cannot be removed by additional data or training and vanishes only under equivariant architectures, establishing a precise theoretical advantage of equivariant inductive bias over data augmentation. Numerical experiments on group-symmetric Gaussian mixtures corroborate the theory.

Multi-view crowd localization and tracking aim to predict the ground-plane positions of all individuals in the scene and estimate their trajectories over time. Existing multi-view localization methods typically rely on Gaussian-smoothed density map supervision, whose inherent ambiguity in crowded regions may blur local peaks and degrade localization accuracy. In this work, we introduce a multi-view optimal transport framework that directly supervises point-level predictions on the ground plane, thereby avoiding the limitations of density map regression. Specifically, we propose a Mahalanobis distance-based multi-view optimal transport loss (M-MVOT), whose transport cost incorporates view ray direction and object-to-camera distance within a Mahalanobis formulation, with the direction guiding the ellipse orientation and the distance modulating the cost magnitude. For multi-view tracking, we first extend the proposed localization framework in a tracking-by-detection manner (MVOT-TBD). To further improve performance, we introduce a tracking-specific optimal transport loss (MVOT-Track) that explicitly supervises inter-frame association. Extensive experiments demonstrate that the proposed optimal transport-based framework consistently outperforms density map-based localization and tracking on multiple multi-view crowd datasets. Code is available at https://github.com/zqyq/MVOT-Track.

This paper considers the setting in which local Mixture-of-experts (MoE) models are fitted independently on distributed datasets and must be combined into a single global MoE model. Weighted averaging the local mixtures, whose experts are not aligned across machines, often yields a large mixture with many experts instead of $K$ and does not provide an estimator of the true model parameters. We propose to reduce this large mixture to a $K$-expert MoE by minimizing an expected transportation divergence, in which the transport plan depends on the covariates. A relaxation of this problem separates the experts from the gating network: the experts are computed by a majorization--minimization algorithm, which decreases the objective monotonically, and the gating network is then fitted to the gates implied by the optimal plan, with an explicit bound on the resulting loss. We show that the aggregated estimator is consistent whenever the local estimators are, without requiring the local models to have the same labeling across machines. The procedure only needs a single one-way communication round. Experiments on simulated and real data show accuracy close to centralized estimation at a fraction of its computation time. Our source code is available at https://github.com/nhat-thien/Distributed-Mixture-Of-Experts.

We study the approximation of probability measures in the Wasserstein-$p$ distance by structured classes of approximators, motivated by applications in imaging, machine learning, and physical measurement under sensor constraints. We obtain three sets of results. First, for measures with densities bounded away from zero on a bounded Lipschitz domain $Ω$, we prove that any approximation scheme for functions in $\mathrm{L}_p(Ω)$ transfers, with linear rate, to a corresponding approximation scheme for measures in $\mathrm{W}_p(Ω)$. The argument applies a theorem of Bogovskii on regularity of solutions to the continuity equation in the Benamou-Brenier formulation of optimal transport. We exhibit concrete approximation schemes (polynomials, shift-invariant spaces, cardinal interpolation with radial basis functions, kernel density estimators, and piecewise approximations on nonuniform Voronoi partitions) that fit the framework. As a matter of independent interest, we prove a negative Sobolev lower bound that generalizes existing bounds from $p=2$ to all $p\in(1,\infty)$. We also consider deterministic bounds for discrete approximations to arbitrary measures in terms of the mesh norm of a quasi-uniform set of points. We specialize these bounds to show that compactly supported measures admit a deterministic $N$-term approximation $μ_N$ such that $\mathrm{W}_p(μ,μ_N) = O(N^{-\frac{1}{d}})$ for all $d\geq 1$, which matches the asymptotic optimal quantizer rate. We also extend these results to non-compactly supported measures with appropriate tail decay.

We introduce LOT Wassmap, a computationally feasible algorithm to uncover low-dimensional structures in the Wasserstein space. The algorithm is motivated by the observation that many datasets are naturally interpreted as probability measures rather than points in $\mathbb{R}^n$, and that finding low-dimensional descriptions of such datasets requires manifold learning algorithms in the Wasserstein space. Most available algorithms are based on computing the pairwise Wasserstein distance matrix, which can be computationally challenging for large datasets in high dimensions. Our algorithm leverages approximation schemes such as Sinkhorn distances and linearized optimal transport to speed-up computations, and in particular, avoids computing a pairwise distance matrix. We provide guarantees on the embedding quality under such approximations, including when explicit descriptions of the probability measures are not available and one must deal with finite samples instead. Experiments demonstrate that LOT Wassmap attains correct embeddings and that the quality improves with increased sample size. We also show how LOT Wassmap significantly reduces the computational cost when compared to algorithms that depend on pairwise distance computations.

Recent studies revealed the mathematical connection between deep neural networks (DNNs) and dynamic systems. However, the specific dynamics that DNNs, especially deep residual networks (ResNets), tend to learn during training remain insufficiently characterized. To this end, we model the forward propagation of deep residual networks using continuity equations, in which the measure is conserved and infinite curves in the measure space connect the input distribution to the output one of a ResNet. We find ResNets with $L_2$ regularization attempt to learn the geodesic curve in the Wasserstein space, induced by the optimal transport map. Compared with plain networks, ResNets can better approximate the geodesic curve, which explains why ResNets can be optimized and generalize better. Numerical experiments show that the data tracks of a ResNet tend to be line-shaped in terms of the line-shape score, and the map learned by a ResNet is closer to the optimal transport map in terms of the optimal transport score. In a word, we conclude that ResNets learn the geodesic curve in the Wasserstein space and discretely engineer the data transformation in high-dimensional spaces.