Young‐Heon Kim

Rentian Yao, Young-Heon Kim, Geoffrey Schiebinger

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.

Forest Kobayashi, Jonathan Hayase, Young-Heon Kim

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.

Peng Xu, Changbo Zhu, Young-Heon Kim et al.

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.

Hugo Lavenant, Stephen Zhang, Young-Heon Kim et al.

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.