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Young-Heon Kim

Publications and source records attributed to Young-Heon Kim.

At least 19 recordsLinked to original sources
arXiv2026

Principal Curves In Metric Spaces And The Space Of Probability Measures

We introduce principal curves in Wasserstein space, and in general compact metric spaces. Our motivation for the Wasserstein case comes from optimal-transport-based trajectory inference, where a developing population of cells traces out a curve in Wasserstein space. Our framework enables new experimental procedures for collecting high-density time-courses of developing populations of cells: time-points can be processed in parallel (making it easier to collect more time-points). However, then the time of collection is unknown, and must be recovered by solving a seriation problem (or one-dimensional manifold learning problem). We propose an estimator based on Wasserstein principal curves, and prove it is consistent for recovering a curve of probability measures in Wasserstein space from empirical samples. This consistency theorem is obtained via a series of results regarding principal curves in compact metric spaces. In particular, we establish the validity of certain numerical discretization schemes for principal curves, which is a new result even in the Euclidean setting.

math.ST↗
arXiv2026

Mirror Langevin diffusions: Convergence rates and Markov chain approximations

Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $μ$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $μ$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $μ$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $μ$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincaré or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $μ$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $χ^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.

math.PR↗
arXiv2026

Another Look at Log-PCA for Probability Measures: A Dynamical Formulation and Statistical Convergence

This paper is concerned with learning principal variations of random probability measures on $\mathbb{R}^m$ under the Wasserstein geometry. We introduce a new dynamical formulation to interpret the log-PCA, a linearized principal geodesic analysis, as a variational approach. Our differentiable version, termed as the Wasserstein Tangential PCA (WT-PCA), captures the local principal modes of geodesic variations of a (weighted) probability measure on the Wasserstein space via its covariance operator at barycenter. Based on the dynamical perspective and leveraging parallel transport structure of the optimal transport problems, we derive a general statistical convergence rate of the empirical WT-PCA when estimated from data in terms of the 2-Wasserstein distance between the population and empirical barycenter reference measures.

stat.ML↗
arXiv2026

Quantitative Stability of Many-Marginal Schrodinger Bridge

In this paper, we explore quantitative stability of multi-marginal Schrödinger bridges with respect to the marginal constraints. We focus on the case where the number of marginal constraints is large (i.e. ``many-marginals"). When this number increases, we show that the Kullback--Leibler (KL) divergence between two multi-marginal Schrödinger bridges, as measures on the path space, can be asymptotically bounded by the terminal marginal KL divergence and a time-integrated squared discrepancy {that combines} Wasserstein-2 geodesic velocity fields with a log-density gradient term. Our stability upper bound is also asymptotically tight: it converges to zero as the number of marginal constraints increases with unperturbed marginal constraints. To the best of our knowledge, this is the first such stability result that addresses the many-marginal regime, giving error estimates that are asymptotically independent of the number of marginals. To achieve our result, the key step is to derive an asymptotic expansion (of order $k\ge 2$) of Schrödinger potentials with respect to a diminishing regularization coefficient. This result can also be applied to deriving asymptotic expansions of entropic Brenier maps in entropic optimal self-transport problems. As byproducts of our analyses, we also establish the asymptotic expansion of entropic optimal transport cost with respect to the diminishing regularization coefficient when two marginal constraints are sufficiently close. We also prove a stability property of the Schrödinger functional.

math.PR↗
arXiv2026

Extension of coupling via the Projection of Optimal Transport

In many statistical settings, two types of data are available: coupled data, which preserve the joint structure among variables but are limited in size due to cost or privacy constraints, and marginal data, which are available at larger scales but lack joint structure. Since standard methods require coupled data, marginal information is often discarded. We propose a fully nonparametric procedure that integrates decoupled marginal data with a limited amount of coupled data to improve the downstream analysis. The approach can be understood as an extension of coupling via projection in optimal transport. Specifically, the estimator is a solution for the optimal transport projection over the space of probability measures, which genuinely provides a natural geometric interpretation. Not only is its stability established, but its sample complexity is also derived using recent advances in statistical optimal transport. In addition to this, we present its explicit formula based on ``shadow," a notion introduced by Eckstein and Nutz. Furthermore, the estimator can be approximated in almost linear time and in parallel by entropic shadow, which demonstrates the theoretical and practical strengths of our methods. Lastly, we present experiments with real and synthetic data to justify the performance of our method.

stat.ME↗
arXiv2026

Safety-Guided Flow (SGF): A Unified Framework for Negative Guidance in Safe Generation

Safety mechanisms for diffusion and flow models have recently been developed along two distinct paths. In robot planning, control barrier functions are employed to guide generative trajectories away from obstacles at every denoising step by explicitly imposing geometric constraints. In parallel, recent data-driven, negative guidance approaches have been shown to suppress harmful content and promote diversity in generated samples. However, they rely on heuristics without clearly stating when safety guidance is actually necessary. In this paper, we first introduce a unified probabilistic framework using a Maximum Mean Discrepancy (MMD) potential for image generation tasks that recasts both Shielded Diffusion and Safe Denoiser as instances of our energy-based negative guidance against unsafe data samples. Furthermore, we leverage control-barrier functions analysis to justify the existence of a critical time window in which negative guidance must be strong; outside of this window, the guidance should decay to zero to ensure safe and high-quality generation. We evaluate our unified framework on several realistic safe generation scenarios, confirming that negative guidance should be applied in the early stages of the denoising process for successful safe generation.

cs.CV↗
arXiv2026

Uniqueness of the maximal solution of the supercooled Stefan problem in 1D

We prove uniqueness of the maximal weak solutions to the supercooled Stefan problem in 1 dimension. This follows by showing that in 1 dimension, the optimal solution of the corresponding free target optimal transport problem given in \cite{GeneralDimensions}, is independent of the choice of the cost function. Moreover, we show that the supercooled Stefan problem lacks monotonicity and $L^1$-Lipschitz stability, which are available in a similar problem considered in a previous paper \cite{freetarget}. However, in $1$ dimension, it has stability in the weak convergence of measures.

math.AP↗
arXiv2025

Wasserstein Mirror Gradient Flow as the limit of the Sinkhorn Algorithm

We prove that the sequence of marginals obtained from the iterations of the Sinkhorn algorithm or the iterative proportional fitting procedure (IPFP) on joint densities, converges to an absolutely continuous curve on the $2$-Wasserstein space, as the regularization parameter $\varepsilon$ goes to zero and the number of iterations is scaled as $1/\varepsilon$ (and other technical assumptions). This limit, which we call the Sinkhorn flow, is an example of a Wasserstein mirror gradient flow, a concept we introduce here inspired by the well-known Euclidean mirror gradient flows. In the case of Sinkhorn, the gradient is that of the relative entropy functional with respect to one of the marginals and the mirror is half of the squared Wasserstein distance functional from the other marginal. Interestingly, the norm of the velocity field of this flow can be interpreted as the metric derivative with respect to the linearized optimal transport (LOT) distance. An equivalent description of this flow is provided by the parabolic Monge-Ampère PDE whose connection to the Sinkhorn algorithm was noticed by Berman (2020). We derive conditions for exponential convergence for this limiting flow. We also construct a Mckean-Vlasov diffusion whose marginal distributions follow the Sinkhorn flow.

math.PR↗
arXiv2025

Structure of average distance minimizers in general dimensions

For a fixed, compactly supported probability measure $μ$ on the $d$-dimensional space $\mathbb{R}^d$, we consider the problem of minimizing the $p^{\mathrm{th}}$-power average distance functional over all compact, connected $Σ\subseteq \mathbb{R}^d$ with Hausdorff 1-measure $\mathcal{H}^1(Σ) \leq l$. This problem, known as the average distance problem, was first studied by Buttazzo, Oudet, and Stepanov in 2002, and has undergone a considerable amount of research since. We will provide a novel approach to studying this problem by analyzing it using the so-called \textit{barycentre field} considered previously by Hayase and two of the authors. This allows us to provide a complete topological description of minimizers in arbitrary dimensions when $p = 2$ and $p > \frac{1}{2}(3 + \sqrt{5}) \approx 2.618$, the first such result that includes the case when $d > 2$.

math.OC↗
arXiv2025

Monge-Kantorovich Fitting With Sobolev Budgets

Given $m < n$, we consider the problem of ``best'' approximating an $n\text{-d}$ probability measure $ρ$ via an $m\text{-d}$ measure $ν$ such that $\mathrm{supp}\ ν$ has bounded total ``complexity.'' When $ρ$ is concentrated near an $m\text{-d}$ set we may interpret this as a manifold learning problem with noisy data. However, we do not restrict our analysis to this case, as the more general formulation has broader applications. We quantify $ν$'s performance in approximating $ρ$ via the Monge-Kantorovich (also called Wasserstein) $p$-cost $\mathbb{W}_p^p(ρ, ν)$, and constrain the complexity by requiring $\mathrm{supp}\ ν$ to be coverable by an $f : \mathbb{R}^{m} \to \mathbb{R}^{n}$ whose $W^{k,q}$ Sobolev norm is bounded by $\ell \geq 0$. This allows us to reformulate the problem as minimizing a functional $\mathscr J_p(f)$ under the Sobolev ``budget'' $\ell$. This problem is closely related to (but distinct from) principal curves with length constraints when $m=1, k = 1$ and an unsupervised analogue of smoothing splines when $k > 1$. New challenges arise from the higher-order differentiability condition. We study the ``gradient'' of $\mathscr J_p$, which is given by a certain vector field that we call the barycenter field, and use it to prove a nontrivial (almost) strict monotonicity result. We also provide a natural discretization scheme and establish its consistency. We use this scheme as a toy model for a generative learning task, and by analogy, propose novel interpretations for the role regularization plays in improving training.

cs.LG↗
arXiv2025

Statistical inference of convex order by Wasserstein projection

Ranking distributions according to a stochastic order has wide applications in diverse areas. Although stochastic dominance has received much attention, convex order, particularly in general dimensions, has yet to be investigated from a statistical point of view. This article addresses this gap by introducing a simple statistical test for convex order based on the Wasserstein projection distance. This projection distance not only encodes whether two distributions are indeed in convex order, but also quantifies the deviation from the desired convex order and produces an optimal convex order approximation. Lipschitz stability of the backward and forward Wasserstein projection distance is proved, which leads to elegant consistency and concentration results of the estimator we employ as our test statistic. Combining these with state of the art results regarding the convergence rate of empirical distributions, we also derive upper bounds for the $p$-value and type I error of our test statistic, as well as upper bounds on the type II error for an appropriate class of strict alternatives. With proper choices of families of distributions, we further attain that the power of the proposed test increases to one as the number of samples grows to infinity. Lastly, we provide an efficient numerical scheme for our test statistic, by way of an entropic Frank-Wolfe algorithm. Experiments based on synthetic data sets illuminate the success of our approach.

stat.ME↗
arXiv2024

The Nonlocal Stefan Problem via a Martingale Transport

We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimization approach introduced for the local case, we construct global-time weak solutions and give a probabilistic interpretation for the solutions. An important ingredient in our analysis is a probabilistic interpretation of the enthalpy and temperature variables in terms of a particle system. Our approach in particular establishes the connection between the parabolic obstacle problem and the Stefan Problem for the nonlocal diffusions. For the melting problem, we show that our solution coincides with those studied in the literature, and obtain a new exponential convergence result.

math.AP↗
arXiv2024

Existence for the Supercooled Stefan Problem in General Dimensions

We prove the global-time existence of weak solutions to the supercooled Stefan problem. Our result holds in general space dimensions and with a general class of initial data. In addition, our solution is maximal in the sense of a certain stochastic order, among all comparable weak solutions starting from the same initial data. Our approach is based on a free target optimization problem for Brownian stopping times, where the main idea is to introduce a superharmonic cost function in the optimization problem. We will show that our choice of the cost function causes the target measure to accumulate near the prescribed domain boundary as much as possible. A central ingredient in our proof lies in the usage of dual problem: we prove the dual attainment and use the dual optimal solution to characterize the primal optimal solution. It follows in turn that the underlying particle dynamics yields a solution to the supercooled Stefan problem.

math.AP↗
arXiv2023

The Stefan problem and free targets of optimal Brownian martingale transport

We formulate and solve a free target optimal Brownian stopping problem from a given distribution while the target distribution is free and is conditioned to satisfy a given density height constraint. The free target optimization problem exhibits monotonicity, from which a remarkable universality follows, in the sense that the optimal target is independent of its Lagrangian cost type. In particular, the solutions to this optimization problem generate solutions to both unstable and stable type of the Stefan problem, where former stands for freezing of supercooled fluid $(St_1)$ and the latter for ice melting $(St_2)$. This unified approach to both types of Stefan problem is new. In particular we obtain global-time existence and weak-strong uniqueness for the ill-posed freezing problem $(St_1)$, for a given initial data and for a well-prepared class of initial domains generated from the initial data.

math.PR↗
arXiv2023

Towards a mathematical theory of trajectory inference

We devise a theoretical framework and a numerical method to infer trajectories of a stochastic process from samples of its temporal marginals. This problem arises in the analysis of single cell RNA-sequencing data, which provide high dimensional measurements of cell states but cannot track the trajectories of the cells over time. We prove that for a class of stochastic processes it is possible to recover the ground truth trajectories from limited samples of the temporal marginals at each time-point, and provide an efficient algorithm to do so in practice. The method we develop, Global Waddington-OT (gWOT), boils down to a smooth convex optimization problem posed globally over all time-points involving entropy-regularized optimal transport. We demonstrate that this problem can be solved efficiently in practice and yields good reconstructions, as we show on several synthetic and real datasets.

stat.ML↗
arXiv2021

Backward and Forward Wasserstein Projections in Stochastic Order

We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their characterizations require study on a case-by-case basis. Particular attention is given to convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. Our results extend to the forward convex order case the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to the early results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.

math.PR↗
arXiv2020

Optimal Stopping of Stochastic Transport Minimizing Submartingale Costs

Given a stochastic state process $(X_t)_t$ and a real-valued submartingale cost process $(S_t)_t$, we characterize optimal stopping times $τ$ that minimize the expectation of $S_τ$ while realizing given initial and target distributions $μ$ and $ν$, i.e., $X_0\sim μ$ and $X_τ\sim ν$. A dual optimization problem is considered and shown to be attained under suitable conditions. The optimal solution of the dual problem then provides a contact set, which characterizes the location where optimal stopping can occur. The optimal stopping time is uniquely determined as the first hitting time of this contact set provided we assume a natural structural assumption on the pair $(X_t, S_t)_t$, which generalizes the twist condition on the cost in optimal transport theory. This paper extends the Brownian motion settings studied in [15, 16] and deals with more general costs.

math.PR↗