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A Sieve M-Estimator for Entropic Optimal Transport

Rami V. Tabri

TL;DR

The paper tackles statistical estimation for entropically regularized OT with compactly supported marginals under minimal regularity. It develops a sieve M-estimation framework that recasts the EOT problem as an information-projection with moment-inequality constraints, enabling tractable finite-dimensional dual programs estimated via sample average approximation. The author proves almost-sure consistency of the sieve-based estimators, derives finite-sample mean-error bounds with a nonparametric bias–variance trade-off, and provides asymptotic Gaussian-process-based bounds that support confidence intervals for the EOT value. Unlike approaches reliant on smooth Schrödinger potentials or empirical marginals, this method leverages population-level moment structures and VC properties of the moment class, extending statistical OT to a broader class of costs and constraints. The framework also points to extensions to martingale constraints and other constrained transport problems, offering a versatile tool for inference in entropic transport settings.

Abstract

Entropically regularized optimal transport between probability measures supported on compact subsets of Euclidean space admits a representation as an information projection under moment inequality constraints. Exploiting this structure, I develop a sieve-based approximation of the Fenchel dual, yielding a sequence of finite-dimensional convex programs whose sample analogues provide tractable estimators of the regularized optimal value and associated dual optimizers. Under minimal assumptions--compact support and continuity of the cost function--I establish almost sure consistency of these estimators. I further derive finite-sample bounds for the estimation error of the optimal value, featuring only logarithmic dependence on sieve complexity, and obtain asymptotic stochastic bounds characterized by suprema of centered Gaussian processes. The results furnish general statistical guarantees for sieve-based estimation of entropic optimal transport and apply to settings not covered by existing theory for the empirical Sinkhorn divergence and other sieve-based methods.

A Sieve M-Estimator for Entropic Optimal Transport

TL;DR

The paper tackles statistical estimation for entropically regularized OT with compactly supported marginals under minimal regularity. It develops a sieve M-estimation framework that recasts the EOT problem as an information-projection with moment-inequality constraints, enabling tractable finite-dimensional dual programs estimated via sample average approximation. The author proves almost-sure consistency of the sieve-based estimators, derives finite-sample mean-error bounds with a nonparametric bias–variance trade-off, and provides asymptotic Gaussian-process-based bounds that support confidence intervals for the EOT value. Unlike approaches reliant on smooth Schrödinger potentials or empirical marginals, this method leverages population-level moment structures and VC properties of the moment class, extending statistical OT to a broader class of costs and constraints. The framework also points to extensions to martingale constraints and other constrained transport problems, offering a versatile tool for inference in entropic transport settings.

Abstract

Entropically regularized optimal transport between probability measures supported on compact subsets of Euclidean space admits a representation as an information projection under moment inequality constraints. Exploiting this structure, I develop a sieve-based approximation of the Fenchel dual, yielding a sequence of finite-dimensional convex programs whose sample analogues provide tractable estimators of the regularized optimal value and associated dual optimizers. Under minimal assumptions--compact support and continuity of the cost function--I establish almost sure consistency of these estimators. I further derive finite-sample bounds for the estimation error of the optimal value, featuring only logarithmic dependence on sieve complexity, and obtain asymptotic stochastic bounds characterized by suprema of centered Gaussian processes. The results furnish general statistical guarantees for sieve-based estimation of entropic optimal transport and apply to settings not covered by existing theory for the empirical Sinkhorn divergence and other sieve-based methods.
Paper Structure (28 sections, 17 theorems, 138 equations, 1 figure, 1 table)

This paper contains 28 sections, 17 theorems, 138 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Summary of Contributions
  3. Organization of the paper
  4. Moment Inequality Information Projection Problem and Approximation
  5. Sieve M-Estimation
  6. Discussion
  7. Other Approaches
  8. Part 1 of Theorem \ref{['Thm Stochastic Opt Sieve']}
  9. Part 2 of Theorem \ref{['Thm Stochastic Opt Sieve']}
  10. Comparison to Theorem 3 in GenevayCuturiPeyreBach2019
  11. Bias--variance trade-off in the rate
  12. Part 3 of Theorem \ref{['Thm Stochastic Opt Sieve']}
  13. Comparison with Empirical Sinkhorn Divergence
  14. Confidence Interval for EOT value
  15. An Extension: Entropic Optimal Transport with Martingale Constraints
  16. ...and 13 more sections

Key Result

Proposition 2.1

Let the constraint set $\mathcal{M}$ be given by (eq - constraint set M), and suppose that Assumption Assump -Primities OTP holds.

Figures (1)

  • Figure 1: Boxplots of simulated estimates

Theorems & Definitions (41)

  • Proposition 2.1
  • proof
  • Definition 1: Gelfand--Pettis integral in $L_1(R_\gamma)$
  • Corollary 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 3.1: Measurability
  • ...and 31 more