Sinkhorn Divergence

Teacher-free one-step generative models such as W-Flow learn directly from distributional objectives, with the Sinkhorn divergence providing the strongest empirical performance among the energies studied in W-Flow. However, Sinkhorn divergence is based on balanced optimal transport, whose exact marginal constraints can induce undesirable transport under finite minibatch fluctuations. We introduce UW-Flow, which replaces the balanced Sinkhorn energy with a debiased unbalanced Sinkhorn divergence and derives the corresponding Wasserstein velocity. The resulting dynamics retain W-Flow's cross-minus-self structure, recover the balanced limit, and preserve target identifiability under suitable assumptions. We further characterize the finite-batch velocity error of balanced and unbalanced transport by comparing each minibatch field with its population counterpart. Our analysis identifies conditions under which marginal relaxation yields smaller minibatch-to-population velocity error, and shows that for separated matched modes, balanced sensitivity grows with inter-mode separation while the corresponding Unbalanced Optimal Transport (UOT) sensitivity remains controlled by local transport geometry. Controlled experiments on fixed-support and continuous two-mode distributions support these predictions. On MNIST, CIFAR-10, and ImageNet-100, UW-Flow improves generation under matched training budgets, with the largest gains in unconditional generation and consistent improvements in the class-conditional setting.

In this paper, we propose a local Sinkhorn divergence framework for conditional distribution reconstruction of multidimensional random fields. By utilizing the debiased Sinkhorn divergence, our proposed approach develops a differentiable and computationally efficient local distribution matching objective to train stochastic neural networks (SNNs). Furthermore, we establish theoretical generalization error estimates for our local Sinkhorn divergence framework, which explicitly characterizes the trade-off between approximation bias and statistical efficiency controlled by the regularization parameter and reveals how our proposed local Sinkhorn divergence loss function can be efficiently applied to learning multidimensional random field models. The proposed framework provides a scalable alternative to exact local optimal transport for conditional distribution reconstruction, offering a practical compromise between geometric fidelity, statistical efficiency, and computational scalability for uncertainty quantification and probabilistic scientific machine learning. Through various numerical examples, we compare our proposed local Sinkhorn divergence framework with other loss functions to train SNNs and with other machine-learning-based uncertainty quantification frameworks, demonstrating that the proposed local Sinkhorn divergence framework achieves an effective balance between reconstruction accuracy and computational efficiency while maintaining good scalability for multidimensional stochastic systems.

Generative models are increasingly adopted in distributionally robust optimization (DRO), but existing approaches trade off model compatibility and adversarial structure: methods that accept arbitrary samplers do not restrict worst-case laws to a generator family, while generator-parameterized adversaries rely on model-specific access such as likelihoods, scores, or training data. We propose Generative Distributionally Robust Optimization (GDRO), a principled framework that accepts any sampleable conditional generator as the nominal model and restricts worst-case laws to a chosen conditional generator family. The key is the sampler-Sinkhorn pairing: samplers represent the conditional laws exactly, while Sinkhorn divergence compares their induced distributions without likelihood access and can be estimated from samples alone. The resulting population problem admits a direct finite-sample approximation and differentiable primal-dual implementation at the active decision context. For Lipschitz losses, the population Sinkhorn radius bounds downstream degradation. Across explicit and implicit generators, our method reduces rare-context inventory regret by 60% and SocialGAN navigation collisions by 50% relative to nominal decisions.

When analyzing a manifold learning algorithm for data lying on a smooth, compact, connected Riemannian submanifold $(\mathcal{M}, g)$ of $\mathbb{R}^d$, a key estimate for the geodesic distance $d_g$ is that there exists $K > 0$ such that $0 \leq d_g(p, q)^2 - \|p-q\|^2 \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We observe that more generally, when $\mathcal{M}$ is equipped with a smooth symmetric divergence $D$ satisfying a non-degeneracy condition and $g$ is given by $g_p := \frac{1}{2}\mathrm{Hess}_p(D(p, \cdot))$ for all $p \in \mathcal{M}$, there exists $K > 0$ such that $\left| D(p, q) - d_g(p, q)^2 \right| \leq K d_g(p, q)^4$ for all $p, q \in \mathcal{M}$. We demonstrate that this is sufficient for the pointwise convergence of graph Laplacians constructed with $D$ and discuss examples where $D$ is given by the Sinkhorn divergence on a family of probability measures parametrized by a manifold.

Optimal transport provides a powerful framework for comparing measures while respecting the geometry of their support, but comes with an expensive computational cost, hindering its potential application to real world use cases. On manifolds, convolutional algorithms based on the heat kernel have been proposed to alleviate this cost, but their theoretical properties remain largely unexplored. We establish that the heat kernel cost converges to the optimal transport cost as time vanishes in the balanced and unbalanced cases. In the specific case of the 2-sphere $\mathbb{S}^2$, we ensure that the associated Sinkhorn divergences retains the desirable geometric and analytic properties of classical optimal transport discrepancies. Moreover, we leverage the harmonic structure of the sphere to derive a fast Sinkhorn algorithm, requiring only $\mathcal{O}(n)$ memory and $\mathcal{O}(n^{3/2})$ time per iteration, with fully dense GPU-friendly operations. We validate its computational efficiency on synthetic data, and discuss its potential use in the evaluation of global climate models, providing both spatial and seasonal insights into models performances.

We introduce the Sinkhorn treatment effect, an entropic optimal transport measure of divergence between counterfactual distributions. Unlike classical quantities such as the average treatment effect, this measure captures differences across entire distributions. We analyze this divergence as a statistical functional and show it can be written as a smooth transformation of counterfactual mean embeddings with an appropriate kernel. This characterization allows us to establish first-order pathwise differentiability in general, and second-order pathwise differentiability under the null hypothesis of equal counterfactual distributions. Leveraging this smoothness, we construct debiased estimators and use them to obtain asymptotically valid tests for distributional treatment effects with a fixed entropic regularization parameter. Because the power of the test depends on this unknown parameter, we further propose an aggregated test that combines evidence across a grid of regularization choices. Experiments on simulated and image data demonstrate the practical advantages of our estimator and testing procedure.

Recently, Deng et al. (2026) proposed Generative Modeling via Drifting (GMD), a novel framework for generative tasks. This note presents an analysis of GMD through the lens of Wasserstein Gradient Flows (WGF), i.e., the path of steepest descent for a functional in the space of probability measures, equipped with the geometry of optimal transport. Unlike previous WGF-based contributions, GMD can be thought of as directly targeting a fixed point of a specific WGF flow. We demonstrate three main results: first, that one algorithm proposed by Deng et al. (2026) corresponds to finding the limiting point of a WGF on the KL divergence, with Parzen smoothing on the densities. Second, that the algorithm actually implemented by Deng et al. (2026) corresponds to a different procedure, which bears some resemblance to the fixed point of a WGF on the Sinkhorn divergence, but lacks certain desirable properties of the latter. Third, the same same idea can be extended to the limiting point of other WGFs, including the Maximum Mean Discrepancy (MMD), the sliced Wasserstein distance, and GAN critic functions.

We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.

We establish a theoretical link between the recently proposed "drifting" generative dynamics and gradient flows induced by the Sinkhorn divergence. In a particle discretization, the drift field admits a cross-minus-self decomposition: an attractive term toward the target distribution and a repulsive/self-correction term toward the current model, both expressed via one-sided normalized Gibbs kernels. We show that Sinkhorn divergence yields an analogous cross-minus-self structure, but with each term defined by entropic optimal-transport couplings obtained through two-sided Sinkhorn scaling (i.e., enforcing both marginals). This provides a precise sense in which drifting acts as a surrogate for a Sinkhorn-divergence gradient flow, interpolating between one-sided normalization and full two-sided Sinkhorn scaling. Crucially, this connection resolves an identifiability gap in prior drifting formulations: leveraging the definiteness of the Sinkhorn divergence, we show that zero drift (equilibrium of the dynamics) implies that the model and target measures match. Experiments show that Sinkhorn drifting reduces sensitivity to kernel temperature and improves one-step generative quality, trading off additional training time for a more stable optimization, without altering the inference procedure used by drift methods. These theoretical gains translate to strong low-temperature improvements in practice: on FFHQ-ALAE at the lowest temperature setting we evaluate, Sinkhorn drifting reduces mean FID from 187.7 to 37.1 and mean latent EMD from 453.3 to 144.4, while on MNIST it preserves full class coverage across the temperature sweep. Project page: https://mint-vu.github.io/SinkhornDrifting/