arXiv subjects

Jonathan Niles-Weed

Publications and source records attributed to Jonathan Niles-Weed.

At least 19 recordsLinked to original sources

arXiv2026

Nearly sharp comparison results for sliced and max-sliced Wasserstein distances

We prove new comparison results between the Wasserstein distance and its sliced and max-sliced counterparts. First, we show that the H\"older exponent~$\frac{2}{d+2}$ obtained by Bobkov and G\"otze for the max-sliced 1-Wasserstein distance on the unit ball is optimal for every $d \geq 2$, settling a question raised in their work. Second, we show that sharper comparisons are possible under stronger structural assumptions: if $\nu$ is a discrete measure and the optimal coupling between $\mu$ and $\nu$ transports each point to a nearest atom of $\nu$, then $W_p(\mu, \nu) \leq C \sqrt{d}\, K \, \mathrm{SW}_{p,1}(\mu, \nu)$ for a universal constant $C$, where the complexity parameter $K$ is always at most the number of atoms $N$ and can be substantially smaller. This complements a similar bound due to Park and Slep\v{c}ev. An analogous bound holds for the sliced Wasserstein distance based on $k$-dimensional projections. Finally, using a construction from geometric discrepancy theory due to Chen and Travaglini, we prove that the linear dependence on $K$ in this bound cannot be improved, up to polylogarithmic factors.

math.PR↗

arXiv2026

Elementary Symmetric Polynomial Inequalities for Centered Vectors and Matrices

We prove new inequalities for elementary symmetric polynomials (ESPs) for vectors that sum to zero, and for square matrices with zero row and column sums. We apply these results to obtain a unified upper bound on the mean-field approximation guarantee for permutation mixtures, as well as a sharp $\chi^2$ version of the de Finetti theorem for finite sequences over a small alphabet. The main proof ideas were developed by the GPT-5.5 Pro model.

math.CO↗

arXiv2026

Delocalization of bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin

Unadjusted samplers such as unadjusted Hamiltonian Monte Carlo and underdamped Langevin are well-known to be biased. Metropolis--Hastings adjustment has been conventionally incorporated into Hamiltonian Monte Carlo to eliminate the bias. However, this adjustment can significantly increase the iteration complexity due to the small step size required for reasonable Metropolis acceptance rates. In this work, we extend the \emph{delocalization of bias} phenomenon, previously established for the overdamped Langevin algorithm, to these two unadjusted algorithms. We show that to control the $W_2$ bias of any $K$-dimensional marginal of a high-dimensional distribution, $O(\sqrt{K})$ integration steps suffice up to $\log d$ terms, assuming either weak or sparse interactions among variables. The discrete-time integrators here introduce technical difficulties beyond those of the overdamped setting, which we address through a broadly applicable matrix-polynomial framework that characterizes their propagators. Our result for the underdamped Langevin algorithm is valid for all large friction parameters, implying that the Leimkuhler-Matthews integrator for the overdamped Langevin dynamics also exhibits delocalization of bias.

stat.CO↗

arXiv2026

Wasserstein Least Squares: A Canonical Regression Method for Probability Distributions

We perform a mathematical and statistical analysis of the Wasserstein least squares problem, a regression method for vector-valued covariates and distribution-valued responses. Our proposal contrasts with other distributional regression methods by having a direct interpretation in terms of random variables, as a nonparametric analogue of the classic random-effects model. On the mathematical side, we use a strategy of Lavenant (2024) to show that Wasserstein least squares is the canonical extension of Euclidean least squares to the space of probability distributions from the perspective of convex analysis; this viewpoint gives rise to multimarginal and dual formulations of the Wasserstein least squares problem, extending a similar theory for Wasserstein barycenters. We perform a statistical analysis of the Wasserstein least squares problem under the template deformation model, showing, surprisingly, that estimation is possible at the n^{-1/2} rate. As a special case, we obtain improved rates of estimation for Wasserstein barycenters, which are an exponential improvement over those established by Ahidar-Coutrix, Le Gouic and Paris (2020). Finally, we propose a heuristic particle method for Wasserstein least squares and use it to conduct a novel analysis of large-scale demographic data from the RAND Health and Retirement Study.

math.ST↗

arXiv2026

The Mean-Field Limit of Online Stochastic Vector Balancing

We study an online vector balancing problem, in which $n$ independent Gaussian random vectors $\boldsymbol{\zeta}(1),\dots,\boldsymbol{\zeta}(n) \sim \mathcal{N}(0, I_n)$, each of dimension $n$, arrive one at a time. The goal is to choose signs $\varepsilon(1),\dots,\varepsilon(n) \in \{\pm 1\}$ with $\varepsilon(k)$ depending only on $\boldsymbol{\zeta}(1),\dots,\boldsymbol{\zeta}(k)$, so as to minimize the expected $\ell^{\infty}$ norm of the signed sum $\frac{1}{\sqrt{n}}\sum_{k = 1}^n \varepsilon(k) \boldsymbol{\zeta}(k)$. Prior work showed that the optimal value $V^n$ is $O(1)$, at least for Rademacher $\boldsymbol{\zeta}(k)$'s, by constructing specific algorithms. Our main contribution is to determine the exact limit $V^{\infty} = \lim_{n\to\infty} V^n$ as the value of a nonstandard stochastic control problem of mean-field type: find the narrowest terminal interval into which a Brownian motion can be adaptively steered under a uniform-in-time $L^2$ constraint on the drift. The proof of the lower bound $V^{\infty} \leq \liminf_{n \to \infty} V^n$ uses probabilistic compactness arguments, and is very flexible. In fact, we show that the lower bound is universal, in that it holds as long as the entries of the $\boldsymbol{\zeta}(k)$ vectors are i.i.d. with mean zero, variance 1, and finite fourth moment. The proof of the upper bound $\limsup_{n \to \infty} V^n \leq V^{\infty}$ is more delicate, relying on dynamic programming principles and a priori bounds obtained from a coupling procedure involving the F\"ollmer drift, which makes explicit use of the Gaussian structure. In addition to our main convergence result, we provide some analysis and asymptotics for the limiting mean-field control problem.

math.PR↗

arXiv2026

Reweighted information inequalities

We establish a variant of the log-Sobolev and transport-information inequalities for mixture distributions. If a probability measure $\pi$ can be decomposed into components that individually satisfy such inequalities, then any measure $\mu$ close to $\pi$ in relative Fisher information is close in relative entropy or transport distance to a reweighted version of $\pi$ with the same mixture components but possibly different weights. This provides a user-friendly interpretation of Fisher information bounds for non-log-concave measures and explains phenomena observed in the analysis of Langevin Monte Carlo for multimodal distributions.

cs.IT↗

arXiv2025

Besting Good--Turing: Optimality of Non-Parametric Maximum Likelihood for Distribution Estimation

When faced with a small sample from a large universe of possible outcomes, scientists often turn to the venerable Good--Turing estimator. Despite its pedigree, however, this estimator comes with considerable drawbacks, such as the need to hand-tune smoothing parameters and the lack of a precise optimality guarantee. We introduce a parameter-free estimator that bests Good--Turing in both theory and practice. Our method marries two classic ideas, namely Robbins's empirical Bayes and Kiefer--Wolfowitz non-parametric maximum likelihood estimation (NPMLE), to learn an implicit prior from data and then convert it into probability estimates. We prove that the resulting estimator attains the optimal instance-wise risk up to logarithmic factors in the competitive framework of Orlitsky and Suresh, and that the Good--Turing estimator is strictly suboptimal in the same framework. Our simulations on synthetic data and experiments with English corpora and U.S. Census data show that our estimator consistently outperforms both the Good--Turing estimator and explicit Bayes procedures.

math.ST↗

arXiv2025

Beyond entropic regularization: Debiased Gaussian estimators for discrete optimal transport and general linear programs

This work proposes new estimators for discrete optimal transport plans that enjoy Gaussian limits centered at the true solution. This behavior stands in stark contrast with the performance of existing estimators, including those based on entropic regularization, which are asymptotically biased and only satisfy a CLT centered at a regularized version of the population-level plan. We develop a new regularization approach based on a different class of penalty functions, which can be viewed as the duals of those previously considered in the literature. The key feature of these penalty schemes it that they give rise to preliminary estimates that are asymptotically linear in the penalization strength. Our final estimator is obtained by constructing an appropriate linear combination of two penalized solutions corresponding to two different tuning parameters so that the bias introduced by the penalization cancels out. Unlike classical debiasing procedures, therefore, our proposal entirely avoids the delicate problem of estimating and then subtracting the estimated bias term. Our proofs, which apply beyond the case of optimal transport, are based on a novel asymptotic analysis of penalization schemes for linear programs. As a corollary of our results, we obtain the consistency of the naive bootstrap for fully data-driven inference on the true optimal solution. Simulation results and two data analyses support strongly the benefits of our approach relative to existing techniques.

math.ST↗

arXiv2025

Trajectory Inference with Smooth Schr\"odinger Bridges

Motivated by applications in trajectory inference and particle tracking, we introduce Smooth Schr\"odinger Bridges. Our proposal generalizes prior work by allowing the reference process in the Schr\"odinger Bridge problem to be a smooth Gaussian process, leading to more regular and interpretable trajectories in applications. Though na\"ively smoothing the reference process leads to a computationally intractable problem, we identify a class of processes (including the Mat\'ern processes) for which the resulting Smooth Schr\"odinger Bridge problem can be lifted to a simpler problem on phase space, which can be solved in polynomial time. We develop a practical approximation of this algorithm that outperforms existing methods on numerous simulated and real single-cell RNAseq datasets. The code can be found at https://github.com/WanliHongC/Smooth_SB

stat.ML↗

arXiv2024

Conditional simulation via entropic optimal transport: Toward non-parametric estimation of conditional Brenier maps

Conditional simulation is a fundamental task in statistical modeling: Generate samples from the conditionals given finitely many data points from a joint distribution. One promising approach is to construct conditional Brenier maps, where the components of the map pushforward a reference distribution to conditionals of the target. While many estimators exist, few, if any, come with statistical or algorithmic guarantees. To this end, we propose a non-parametric estimator for conditional Brenier maps based on the computational scalability of \emph{entropic} optimal transport. Our estimator leverages a result of Carlier et al. (2010), which shows that optimal transport maps under a rescaled quadratic cost asymptotically converge to conditional Brenier maps; our estimator is precisely the entropic analogues of these converging maps. We provide heuristic justifications for choosing the scaling parameter in the cost as a function of the number of samples by fully characterizing the Gaussian setting. We conclude by comparing the performance of the estimator to other machine learning and non-parametric approaches on benchmark datasets and Bayesian inference problems.

stat.ML↗

arXiv2024

Central Limit Theorems for Smooth Optimal Transport Maps

One of the central objects in the theory of optimal transport is the Brenier map: the unique monotone transformation which pushes forward an absolutely continuous probability law onto any other given law. A line of recent work has analyzed $L^2$ convergence rates of plugin estimators of Brenier maps, which are defined as the Brenier map between density estimators of the underlying distributions. In this work, we show that such estimators satisfy a pointwise central limit theorem when the underlying laws are supported on the flat torus of dimension $d \geq 3$. We also derive a negative result, showing that these estimators do not converge weakly in $L^2$ when the dimension is sufficiently large. Our proofs hinge upon a quantitative linearization of the Monge-Ampère equation, which may be of independent interest. This result allows us to reduce our problem to that of deriving limit laws for the solution of a uniformly elliptic partial differential equation with a stochastic right-hand side, subject to periodic boundary conditions.

math.PR↗

arXiv2024

The EM-algorithm and the Method of Moments in Softmax Mixture Models

Softmax Mixture Models (SMMs) are discrete $K$-component mixture models for the probabilities of selecting one of $p$ candidate feature vectors $X_1,\ldots,X_p\in\mathbb{R}^L$ in heterogeneous populations and are widely used in econometrics and scientific applications. Related softmax mixture mechanisms also appear in modern LLM architectures. We provide a theoretical and methodological study of SMMs, focusing on the Expectation-Maximization (EM) algorithm and the Method of Moments (MoM). We show that EM recovers the mixture atoms at the parametric rate, up to logarithmic factors, after $\mathcal{O}(\log N)$ iterations, provided atom separation is at least of order $\log K$. This improves on separation conditions in existing analyses of EM for high-dimensional Gaussian mixtures. We also develop MoM procedures for parameter and subspace estimation. Although MoM parameter estimates converge more slowly than EM and can deteriorate with $K$, they provide provable warm starts for EM and are useful for small $K$. For general $K$, we estimate the atom subspace via MoM and recommend running EM from multiple random initializations within this subspace. Finally, as $p\to\infty$, we show that SMMs approximate mixtures of exponential tilts of the feature distribution, yielding asymptotic identifiability.

stat.ML↗

arXiv2024

Estimation and inference for the Wasserstein distance between mixing measures in topic models

The Wasserstein distance between mixing measures has come to occupy a central place in the statistical analysis of mixture models. This work proposes a new canonical interpretation of this distance and provides tools to perform inference on the Wasserstein distance between mixing measures in topic models. We consider the general setting of an identifiable mixture model consisting of mixtures of distributions from a set $\mathcal{A}$ equipped with an arbitrary metric $d$, and show that the Wasserstein distance between mixing measures is uniquely characterized as the most discriminative convex extension of the metric $d$ to the set of mixtures of elements of $\mathcal{A}$. The Wasserstein distance between mixing measures has been widely used in the study of such models, but without axiomatic justification. Our results establish this metric to be a canonical choice. Specializing our results to topic models, we consider estimation and inference of this distance. Though upper bounds for its estimation have been recently established elsewhere, we prove the first minimax lower bounds for the estimation of the Wasserstein distance in topic models. We also establish fully data-driven inferential tools for the Wasserstein distance in the topic model context. Our results apply to potentially sparse mixtures of high-dimensional discrete probability distributions. These results allow us to obtain the first asymptotically valid confidence intervals for the Wasserstein distance in topic models.

math.ST↗

arXiv2024

Plug-in estimation of Schrödinger bridges

We propose a procedure for estimating the Schrödinger bridge between two probability distributions. Unlike existing approaches, our method does not require iteratively simulating forward and backward diffusions or training neural networks to fit unknown drifts. Instead, we show that the potentials obtained from solving the static entropic optimal transport problem between the source and target samples can be modified to yield a natural plug-in estimator of the time-dependent drift that defines the bridge between two measures. Under minimal assumptions, we show that our proposal, which we call the \emph{Sinkhorn bridge}, provably estimates the Schrödinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure. Our approach combines results from the areas of sampling, and theoretical and statistical entropic optimal transport.

stat.ML↗

arXiv2024

Convergence of Unadjusted Langevin in High Dimensions: Delocalization of Bias

The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension $d$ of the problem increases, the number of iterations required to ensure convergence within a desired error in the $W_2$ metric scales in proportion to $d$ or $\sqrt{d}$. In this paper, we argue that, despite this poor scaling of the $W_2$ error for the full set of variables, the behavior for a small number of variables can be significantly better: a number of iterations proportional to $K$, up to logarithmic terms in $d$, often suffices for the algorithm to converge to within a desired $W_2$ error for all $K$-marginals. We refer to this effect as delocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel $W_{2,\ell^\infty}$ metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.

stat.ML↗

arXiv2024

Approximate independence of permutation mixtures

We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call \emph{permutation mixtures}) and their i.i.d. counterparts. Our results are based on a novel method for controlling $\chi^2$ divergences between exchangeable mixtures, which is tighter than the existing methods of moments or cumulants. At a technical level, a key innovation in our proofs is a new Maclaurin-type inequality for elementary symmetric polynomials of variables that sum to zero and an upper bound on permanents of doubly-stochastic positive semidefinite matrices. We obtain as a corollary a new de Finetti-style theorem (in the language of Diaconis and Freedman, 1987), as well as several new statistical results, including a differential privacy guarantee for the ``shuffled privacy model'' with Gaussian noise and improved generic consistency guarantees for empirical Bayes procedures in compound decision problems.

math.ST↗

arXiv2024

Statistical optimal transport

We present an introduction to the field of statistical optimal transport, based on lectures given at \'Ecole d'\'Et\'e de Probabilit\'es de Saint-Flour XLIX.

math.ST↗

arXiv2024

Plugin Estimation of Smooth Optimal Transport Maps

We analyze a number of natural estimators for the optimal transport map between two distributions and show that they are minimax optimal. We adopt the plugin approach: our estimators are simply optimal couplings between measures derived from our observations, appropriately extended so that they define functions on $\mathbb{R}^d$. When the underlying map is assumed to be Lipschitz, we show that computing the optimal coupling between the empirical measures, and extending it using linear smoothers, already gives a minimax optimal estimator. When the underlying map enjoys higher regularity, we show that the optimal coupling between appropriate nonparametric density estimates yields faster rates. Our work also provides new bounds on the risk of corresponding plugin estimators for the quadratic Wasserstein distance, and we show how this problem relates to that of estimating optimal transport maps using stability arguments for smooth and strongly convex Brenier potentials. As an application of our results, we derive central limit theorems for plugin estimators of the squared Wasserstein distance, which are centered at their population counterpart when the underlying distributions have sufficiently smooth densities. In contrast to known central limit theorems for empirical estimators, this result easily lends itself to statistical inference for the quadratic Wasserstein distance.

math.ST↗