End-to-end backpropagation is the standard training method of neural networks.

End-to-end backpropagation is the standard training method of neural networks.

The optimal transport (OT) map is a geometry-driven transformation between high-dimensional probability distributions which underpins a wide range of tasks in statistics, applied probability, and machine learning. However, existing statistical theory for OT map estimation is quite restricted, hinging on Brenier's theorem (quadratic cost, absolutely continuous source) to guarantee existence and uniqueness of a deterministic OT map, on which various additional regularity assumptions are imposed to obtain quantitative error bounds. In many real-world problems these conditions fail or cannot be certified, in which case optimal transportation is possible only via stochastic maps that can split mass. To broaden the scope of map estimation theory to such settings, this work introduces a novel metric for evaluating the transportation quality of stochastic maps. Under this metric, we develop computationally efficient map estimators with near-optimal finite-sample risk bounds, subject to easy-to-verify minimal assumptions. Our analysis further accommodates common forms of adversarial sample contamination, yielding estimators with robust estimation guarantees. Empirical experiments are provided which validate our theory and demonstrate the utility of the proposed framework in settings where existing theory fails. These contributions constitute the first general-purpose theory for map estimation, compatible with a wide spectrum of real-world applications where optimal transport may be intrinsically stochastic.

We revisit the signal denoising problem through the lens of optimal transport: the goal is to recover an unknown scalar signal distribution $X \sim P$ from noisy observations $Y = X + σZ$, with $Z$ being standard Gaussian independent of $X$ and $σ>0$ a known noise level. Let $Q$ denote the distribution of $Y$. We introduce a hierarchy of denoisers $T_0, T_1, \ldots, T_\infty : \mathbb{R} \to \mathbb{R}$ that are agnostic to the signal distribution $P$, depending only on higher-order score functions of $Q$. Each denoiser $T_K$ is progressively refined using the $(2K-1)$-th order score function of $Q$ at noise resolution $σ^{2K}$, achieving better denoising quality measured by the Wasserstein metric $W(T_K \sharp Q, P)$. The limiting denoiser $T_\infty$ identifies the optimal transport map with $T_\infty \sharp Q = P$. We provide a complete characterization of the combinatorial structure underlying this hierarchy through Bell polynomial recursions, revealing how higher-order score functions encode the optimal transport map for signal denoising. We study two estimation strategies with convergence rates for higher-order scores from i.i.d. samples drawn from $Q$: (i) plug-in estimation via Gaussian kernel smoothing, and (ii) direct estimation via higher-order score matching. This hierarchy of agnostic denoisers opens new perspectives in signal denoising and empirical Bayes.

Color harmonization adjusts the colors of an inserted object so that it perceptually matches the surrounding image, resulting in a seamless composite. The harmonization problem naturally arises in augmented reality (AR), yet harmonization algorithms are not currently integrated into AR pipelines because real-time solutions are scarce. In this work, we address color harmonization for AR by proposing a lightweight approach that supports on-device inference. For this, we leverage classical optimal transport theory by training a compact encoder to predict the Monge-Kantorovich transport map. We benchmark our MKL-Harmonizer algorithm against state-of-the-art methods and demonstrate that for real composite AR images our method achieves the best aggregated score. We release our dedicated AR dataset of composite images with pixel-accurate masks and data-gathering toolkit to support further data acquisition by researchers.

Over the past few years, numerous computational models have been developed to solve Optimal Transport (OT) in a stochastic setting, where distributions are represented by samples and where the goal is to find the closest map to the ground truth OT map, unknown in practical settings. So far, no quantitative criterion has yet been put forward to tune the parameters of these models and select maps that best approximate the ground truth. To perform this task, we propose to leverage the Brenier formulation of OT. Theoretically, we show that this formulation guarantees that, up to sharp a distortion parameter depending on the smoothness/strong convexity and a statistical deviation term, the selected map achieves the lowest quadratic error to the ground truth. This criterion, estimated via convex optimization, enables parameter tuning and model selection among entropic regularization of OT, input convex neural networks and smooth and strongly convex nearest-Brenier (SSNB) models. We also use this criterion to question the use of OT in Domain-Adaptation (DA). In a standard DA experiment, it enables us to identify the potential that is closest to the true OT map between the source and the target. Y et, we observe that this selected potential is far from being the one that performs best for the downstream transfer classification task.

Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets.

OT maps can be obtained analytically. This strategy yields pairs of continuous benchmark measures in high-dimensional spaces such as spaces of images. We thoroughly evaluate existing optimal transport solvers using these benchmark measures.