[Paper Review] Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem

Gonzalo E. Mena, Jonathan Niles‐Weed|arXiv (Cornell University)|May 28, 2019

Geometric Analysis and Curvature FlowsMathematics68 citations

TL;DR

The paper derives new sample complexity bounds and a central limit theorem for entropic optimal transport with subgaussian measures in any dimension, and applies these results to entropy estimation under Gaussian noise.

ABSTRACT

We prove several fundamental statistical bounds for entropic OT with the squared Euclidean cost between subgaussian probability measures in arbitrary dimension. First, through a new sample complexity result we establish the rate of convergence of entropic OT for empirical measures. Our analysis improves exponentially on the bound of Genevay et al. (2019) and extends their work to unbounded measures. Second, we establish a central limit theorem for entropic OT, based on techniques developed by Del Barrio and Loubes (2019). Previously, such a result was only known for finite metric spaces. As an application of our results, we develop and analyze a new technique for estimating the entropy of a random variable corrupted by gaussian noise.

Motivation & Objective

• Motivate understanding of the statistical behavior of entropic OT beyond bounded supports.
• Provide nonasymptotic bounds on the difference between population and empirical entropic costs for unbounded measures.
• Establish a central limit theorem for entropic OT in general subgaussian settings.
• Apply entropic OT to entropy estimation for variables corrupted by Gaussian noise.

Proposed method

• Use dual formulation of entropic OT and optimal potentials (f,g) to relate S(P,Q) and S(Pn,Qn).
• Develop exponential‑free bounds by controlling Hölder norms of optimal potentials on compact sets.
• Employ empirical process theory and covering numbers for function classes to bound E|S(P,Q)-S(Pn,Qn)|.
• Prove a central limit theorem for S(Pn,Qn) when P and Q are subgaussian, following the Del Barrio–Loubès approach.
• Derive and analyze a plug‑in estimator for the entropy of X+Gaussian noise using entropic OT, with variance characterizations.
• Provide simulations validating the theoretical claims and illustrating the entropy estimation application.

Experimental results

Research questions

• RQ1 What are the rates at which the empirical entropic OT cost S(Pn,Qn) converges to the population cost S(P,Q) for subgaussian measures in R^d?
• RQ2 Do entropic OT costs admit a central limit theorem in the subgaussian setting, and what is the asymptotic variance?
• RQ3 How can entropic OT be used to estimate the differential entropy of a random variable convolved with Gaussian noise, and what are the statistical guarantees?
• RQ4 How do unbounded supports affect the statistical behavior of entropic OT compared to bounded or finite settings?

Key findings

• For σ^2-subgaussian P,Q, the expected error E|S(P,Q)-S(Pn,Qn)| ≤ C_d(1+σ^{⟨5d/2⟩+6})/√n, improving previous bounds and removing dependence on large diameter.
• A central limit theorem is established: √n(S(Pn,Q)-E[S(Pn,Q)]) converges to N(0,Var_P(f(X))).
• In the two-sample setting with independent samples from P and Q, √(mn/(m+n))(S(Pn,Qm)-E[S(Pn,Qm)]) converges to a normal with variance (1−λ)Var_P(f(X)) + λ Var_Q(g(Y)).
• The paper links entropic OT to entropy of convolved measures, enabling a plug‑in estimator with √n convergence for the entropy of X+Gaussian noise.
• Simulations corroborate the asymptotic normality and the improved sample complexity bounds, and illustrate entropy estimation via OT.
• An application estimator for differential entropy based on entropic OT achieves the stated CLT-based guarantees.