arXiv subjects

Kengo Kato

Publications and source records attributed to Kengo Kato.

At least 19 recordsLinked to original sources

arXiv2026

Probabilistic Representation and Convergence of Gromov-Wasserstein Gradient Flows

Wasserstein gradient flows are intimately connected with evolution partial differential equations and diffusion processes. We take the first step in developing such connections for inner product Gromov--Wasserstein (IGW) gradient flows by studying the IGW gradient flow of the relative entropy $\mathsf{H}(\cdot\|\gamma)$ with respect to the standard Gaussian measure $\gamma$. We first show that $\mathsf{H}(\cdot\|\gamma)$ fails to be $\lambda$-convex along generalized or modified generalized IGW geodesics for any $\lambda \in \mathbb{R}$, and therefore falls outside the scope of the existing IGW gradient flow theory from Zhang et al. (2026). We bridge this gap by establishing a suitable \emph{local} convexity estimate that enables the construction of the gradient flow and its extension to the infinite time horizon. We then obtain increasingly explicit representations of the resulting dynamics. Starting from a partial integro-differential equation, we derive a nonlinear Fokker--Planck equation and show that its second-moment dynamics decouple from the law as they satisfy an autonomous matrix ODE. This reduces the IGW dynamics to a linear, time-inhomogeneous Fokker--Planck equation, yielding a probabilistic representation as the time-marginal flow of a linear stochastic differential equation resembling the Ornstein--Uhlenbeck process. Finally, we study its asymptotic behavior by establishing exponential convergence of the flow to $\gamma$ in relative entropy.

math.PR↗

arXiv2026

Sinkhorn algorithms for entropic vector quantile regression

Vector quantile regression (VQR) is an optimal transport (OT)-based framework that extends linear quantile regression to vector-valued response variables and can be formulated as an OT problem with a mean-independence constraint. In this paper, we study Sinkhorn-type algorithms for VQR with entropic regularization, building on the authors' previous work on its duality theory. We first study a direct adaptation of the classical Sinkhorn iteration based on solving the full Schr\"{o}dinger-type system characterizing the dual potentials, which requires solving an implicit functional equation at each iteration. Since the implicit equation does not admit a closed-form solution in general, we consider a modified algorithm that replaces the implicit update with a gradient ascent step, resulting in a computational scheme that does not involve any subroutine. The latter algorithm can be viewed as an implementable version of the former, idealized scheme that serves as a theoretical benchmark. For both algorithms, and for general compactly supported marginals, we establish linear convergence in both the dual objective value and the iterates. A key innovation in our analysis is the derivation of explicit quantitative bounds on the dual potentials and Sinkhorn iterates.

math.OC↗

arXiv2026

Entropic vector quantile regression: Duality and Gaussian case

Vector quantile regression (VQR) is an optimal transport (OT) problem subject to a mean-independence constraint that extends classical linear quantile regression to vector response variables. Motivated by computational considerations, prior work has considered entropic relaxation of VQR, but its fundamental structural and approximation properties are still much less understood than entropic OT. The goal of this paper is to address some of these gaps. First, we study duality theory for entropic VQR and establish strong duality and dual attainment for marginals with possibly unbounded supports. In addition, when all marginals are compactly supported, we show that dual potentials are real analytic. Second, building on our duality theory, when all marginals are Gaussian, we show that entropic VQR has a closed-form optimal solution, which is again Gaussian, and establish the precise approximation rate toward unregularized VQR.

math.ST↗

arXiv2025

Deviation Inequalities for R\'{e}nyi Divergence Estimators via Variational Expression

R\'enyi divergences play a pivotal role in information theory, statistics, and machine learning. While several estimators of these divergences have been proposed in the literature with their consistency properties established and minimax convergence rates quantified, existing accounts of probabilistic bounds governing the estimation error are relatively underdeveloped. Here, we make progress in this regard by establishing exponential deviation inequalities for smoothed plug-in estimators and neural estimators by relating the error to an appropriate empirical process and leveraging tools from empirical process theory. In particular, our approach does not require the underlying distributions to be compactly supported or have densities bounded away from zero, an assumption prevalent in existing results. The deviation inequality also leads to a one-sided concentration bound from the expectation, which is useful in random-coding arguments over continuous alphabets in information theory with potential applications to physical-layer security. As another concrete application, we consider a hypothesis testing framework for auditing R\'{e}nyi differential privacy using the neural estimator as a test statistic and obtain non-asymptotic performance guarantees for such a test.

cs.IT↗

arXiv2025

Convergence of empirical Gromov-Wasserstein distance

We study rates of convergence for estimation of the Gromov-Wasserstein (GW) distance. For two marginals supported on compact subsets of $\R^{d_x}$ and $\R^{d_y}$, respectively, with $\min \{ d_x,d_y \} > 4$, prior work established the rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ in $L^1$ for the plug-in empirical estimator based on $n$ i.i.d. samples. We extend this fundamental result to marginals with unbounded supports, assuming only finite polynomial moments. Our proof techniques for the upper bounds can be adapted to obtain sample complexity results for penalized Wasserstein alignment that encompasses the GW distance and Wasserstein Procrustes. Furthermore, we establish matching minimax lower bounds (up to logarithmic factors) for estimating the GW distance. Finally, we establish deviation inequalities for the error of empirical GW in cases where two marginals have compact supports, exponential tails, or finite polynomial moments. The deviation inequalities yield that the same rate $n^{-\frac{2}{\min\{d_x,d_y\}}}$ holds for empirical GW also with high probability.

math.ST↗

arXiv2024

Approximation rates of entropic maps in semidiscrete optimal transport

Entropic optimal transport offers a computationally tractable approximation to the classical problem. In this note, we study the approximation rate of the entropic optimal transport map (in approaching the Brenier map) when the regularization parameter $\varepsilon$ tends to zero in the semidiscrete setting, where the input measure is absolutely continuous while the output is finitely discrete. Previous work shows that the approximation rate is $O(\sqrt{\varepsilon})$ under the $L^2$-norm with respect to the input measure. In this work, we establish faster, $O(\varepsilon^2)$ rates up to polylogarithmic factors, under the dual Lipschitz norm, which is weaker than the $L^2$-norm. For the said dual norm, the $O(\varepsilon^2)$ rate is sharp. As a corollary, we derive a central limit theorem for the entropic estimator for the Brenier map in the dual Lipschitz space when the regularization parameter tends to zero as the sample size increases.

math.PR↗

arXiv2024

Limit Laws for Gromov-Wasserstein Alignment with Applications to Testing Graph Isomorphisms

The Gromov-Wasserstein (GW) distance enables comparing metric measure spaces based solely on their internal structure, making it invariant to isomorphic transformations. This property is particularly useful for comparing datasets that naturally admit isomorphic representations, such as unlabelled graphs or objects embedded in space. However, apart from the recently derived empirical convergence rates for the quadratic GW problem, a statistical theory for valid estimation and inference remains largely obscure. Pushing the frontier of statistical GW further, this work derives the first limit laws for the empirical GW distance across several settings of interest: (i)~discrete, (ii)~semi-discrete, and (iii)~general distributions under moment constraints under the entropically regularized GW distance. The derivations rely on a novel stability analysis of the GW functional in the marginal distributions. The limit laws then follow by an adaptation of the functional delta method. As asymptotic normality fails to hold in most cases, we establish the consistency of an efficient estimation procedure for the limiting law in the discrete case, bypassing the need for computationally intensive resampling methods. We apply these findings to testing whether collections of unlabelled graphs are generated from distributions that are isomorphic to each other.

math.ST↗

arXiv2024

Large deviations for dynamical Schr\"{o}dinger problems

We establish large deviations for dynamical Schr\"{o}dinger problems driven by perturbed Brownian motions when the noise parameter tends to zero. Our results show that Schr\"{o}dinger bridges charge exponentially small masses outside the support of the limiting law that agrees with the optimal solution to the dynamical Monge-Kantorovich optimal transport problem. Our proofs build on mixture representations of Schr\"{o}dinger bridges and establishing exponential continuity of Brownian bridges with respect to the initial and terminal points.

math.PR↗

arXiv2023

Entropic Gromov-Wasserstein Distances: Stability and Algorithms

The Gromov-Wasserstein (GW) distance quantifies discrepancy between metric measure spaces and provides a natural framework for aligning heterogeneous datasets. Alas, as exact computation of GW alignment is NP hard, entropic regularization provides an avenue towards a computationally tractable proxy. Leveraging a recently derived variational representation for the quadratic entropic GW (EGW) distance, this work derives the first efficient algorithms for solving the EGW problem subject to formal, non-asymptotic convergence guarantees. To that end, we derive smoothness and convexity properties of the objective in this variational problem, which enables its resolution by the accelerated gradient method. Our algorithms employs Sinkhorn's fixed point iterations to compute an approximate gradient, which we model as an inexact oracle. We furnish convergence rates towards local and even global solutions (the latter holds under a precise quantitative condition on the regularization parameter), characterize the effects of gradient inexactness, and prove that stationary points of the EGW problem converge towards a stationary point of the unregularized GW problem, in the limit of vanishing regularization. We provide numerical experiments that validate our theory and empirically demonstrate the state-of-the-art empirical performance of our algorithm.

math.OC↗

arXiv2023

Stability and statistical inference for semidiscrete optimal transport maps

We study statistical inference for the optimal transport (OT) map (also known as the Brenier map) from a known absolutely continuous reference distribution onto an unknown finitely discrete target distribution. We derive limit distributions for the $L^p$-error with arbitrary $p \in [1,\infty)$ and for linear functionals of the empirical OT map, together with their moment convergence. The former has a non-Gaussian limit, whose explicit density is derived, while the latter attains asymptotic normality. For both cases, we also establish consistency of the nonparametric bootstrap. The derivation of our limit theorems relies on new stability estimates of functionals of the OT map with respect to the dual potential vector, which may be of independent interest. We also discuss applications of our limit theorems to the construction of confidence sets for the OT map and inference for a maximum tail correlation. Finally, we show that, while the empirical OT map does not possess nontrivial weak limits in the $L^2$ space, it satisfies a central limit theorem in a dual H\"{o}lder space, and the Gaussian limit law attains the asymptotic efficiency bound.

math.ST↗

arXiv2022

Limit distribution theory for $f$-Divergences

$f$-divergences, which quantify discrepancy between probability distributions, are ubiquitous in information theory, machine learning, and statistics. While there are numerous methods for estimating $f$-divergences from data, a limit distribution theory, which quantifies fluctuations of the estimation error, is largely obscure. As limit theorems are pivotal for valid statistical inference, to close this gap, we develop a general methodology for deriving distributional limits for $f$-divergences based on the functional delta method and Hadamard directional differentiability. Focusing on four prominent $f$-divergences -- Kullback-Leibler divergence, $\chi^2$ divergence, squared Hellinger distance, and total variation distance -- we identify sufficient conditions on the population distributions for the existence of distributional limits and characterize the limiting variables. These results are used to derive one- and two-sample limit theorems for Gaussian-smoothed $f$-divergences, both under the null and the alternative. Finally, an application of the limit distribution theory to auditing differential privacy is proposed and analyzed for significance level and power against local alternatives.

math.ST↗

arXiv2022

Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances

Sliced Wasserstein distances preserve properties of classic Wasserstein distances while being more scalable for computation and estimation in high dimensions. The goal of this work is to quantify this scalability from three key aspects: (i) empirical convergence rates; (ii) robustness to data contamination; and (iii) efficient computational methods. For empirical convergence, we derive fast rates with explicit dependence of constants on dimension, subject to log-concavity of the population distributions. For robustness, we characterize minimax optimal, dimension-free robust estimation risks, and show an equivalence between robust sliced 1-Wasserstein estimation and robust mean estimation. This enables lifting statistical and algorithmic guarantees available for the latter to the sliced 1-Wasserstein setting. Moving on to computational aspects, we analyze the Monte Carlo estimator for the average-sliced distance, demonstrating that larger dimension can result in faster convergence of the numerical integration error. For the max-sliced distance, we focus on a subgradient-based local optimization algorithm that is frequently used in practice, albeit without formal guarantees, and establish an $O(ε^{-4})$ computational complexity bound for it. Our theory is validated by numerical experiments, which altogether provide a comprehensive quantitative account of the scalability question.

stat.ML↗

arXiv2022

Limit Theorems for Entropic Optimal Transport Maps and the Sinkhorn Divergence

We study limit theorems for entropic optimal transport (EOT) maps, dual potentials, and the Sinkhorn divergence. The key technical tool we use is a first and second-order Hadamard differentiability analysis of EOT potentials with respect to the marginal distributions, which may be of independent interest. Given the differentiability results, the functional delta method is used to obtain central limit theorems for empirical EOT potentials and maps. The second-order functional delta method is leveraged to establish the limit distribution of the empirical Sinkhorn divergence under the null. Building on the latter result, we further derive the null limit distribution of the Sinkhorn independence test statistic and characterize the correct order. Since our limit theorems follow from Hadamard differentiability of the relevant maps, as a byproduct, we also obtain bootstrap consistency and asymptotic efficiency of the empirical EOT map, potentials, and Sinkhorn divergence.

math.ST↗

arXiv2022

Statistical inference with regularized optimal transport

Optimal transport (OT) is a versatile framework for comparing probability measures, with many applications to statistics, machine learning, and applied mathematics. However, OT distances suffer from computational and statistical scalability issues to high dimensions, which motivated the study of regularized OT methods like slicing, smoothing, and entropic penalty. This work establishes a unified framework for deriving limit distributions of empirical regularized OT distances, semiparametric efficiency of the plug-in empirical estimator, and bootstrap consistency. We apply the unified framework to provide a comprehensive statistical treatment of: (i) average- and max-sliced $p$-Wasserstein distances, for which several gaps in existing literature are closed; (ii) smooth distances with compactly supported kernels, the analysis of which is motivated by computational considerations; and (iii) entropic OT, for which our method generalizes existing limit distribution results and establishes, for the first time, efficiency and bootstrap consistency. While our focus is on these three regularized OT distances as applications, the flexibility of the proposed framework renders it applicable to broad classes of functionals beyond these examples.

math.ST↗

arXiv2022

Improved Central Limit Theorem and bootstrap approximations in high dimensions

This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional econometric problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.

math.ST↗

arXiv2022

High-dimensional Data Bootstrap

This article reviews recent progress in high-dimensional bootstrap. We first review high-dimensional central limit theorems for distributions of sample mean vectors over the rectangles, bootstrap consistency results in high dimensions, and key techniques used to establish those results. We then review selected applications of high-dimensional bootstrap: construction of simultaneous confidence sets for high-dimensional vector parameters, multiple hypothesis testing via stepdown, post-selection inference, intersection bounds for partially identified parameters, and inference on best policies in policy evaluation. Finally, we also comment on a couple of future research directions.

math.ST↗

arXiv2022

Limit distribution theory for smooth $p$-Wasserstein distances

The Wasserstein distance is a metric on a space of probability measures that has seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked by the curse of dimensionality, whereby the number of data points needed to accurately estimate them grows exponentially with dimension. Gaussian smoothing was recently introduced as a means to alleviate the curse of dimensionality, giving rise to a parametric convergence rate in any dimension, while preserving the Wasserstein metric and topological structure. To facilitate valid statistical inference, in this work, we develop a comprehensive limit distribution theory for the empirical smooth Wasserstein distance. The limit distribution results leverage the functional delta method after embedding the domain of the Wasserstein distance into a certain dual Sobolev space, characterizing its Hadamard directional derivative for the dual Sobolev norm, and establishing weak convergence of the smooth empirical process in the dual space. To estimate the distributional limits, we also establish consistency of the nonparametric bootstrap. Finally, we use the limit distribution theory to study applications to generative modeling via minimum distance estimation with the smooth Wasserstein distance, showing asymptotic normality of optimal solutions for the quadratic cost.

math.PR↗

arXiv2022

Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications

The smooth 1-Wasserstein distance (SWD) $W_1^σ$ was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation while preserving the Wasserstein structure. Indeed, SWD exhibits parametric convergence rates and inherits the metric and topological structure of the classic Wasserstein distance. Motivated by the above, this work conducts a thorough statistical study of the SWD, including a high-dimensional limit distribution result for empirical $W_1^σ$, bootstrap consistency, concentration inequalities, and Berry-Esseen type bounds. The derived nondegenerate limit stands in sharp contrast with the classic empirical $W_1$, for which a similar result is known only in the one-dimensional case. We also explore asymptotics and characterize the limit distribution when the smoothing parameter $σ$ is scaled with $n$, converging to $0$ at a sufficiently slow rate. The dimensionality of the sampled distribution enters empirical SWD convergence bounds only through the prefactor (i.e., the constant). We provide a sharp characterization of this prefactor's dependence on the smoothing parameter and the intrinsic dimension. This result is then used to derive new empirical convergence rates for classic $W_1$ in terms of the intrinsic dimension. As applications of the limit distribution theory, we study two-sample testing and minimum distance estimation (MDE) under $W_1^σ$. We establish asymptotic validity of SWD testing, while for MDE, we prove measurability, almost sure convergence, and limit distributions for optimal estimators and their corresponding $W_1^σ$ error. Our results suggest that the SWD is well suited for high-dimensional statistical learning and inference.

math.ST↗