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Tight Bounds for Schrödinger Potential Estimation in Unpaired Data Translation

Nikita Puchkin, Denis Suchkov, Alexey Naumov, Denis Belomestny

TL;DR

This work studies learning Schrödinger potentials for unpaired data translation under an Ornstein–Uhlenbeck reference process. It defines population and empirical risks using only iid samples from the initial and target marginals and proves a non-asymptotic, high-probability bound on the KL divergence between the true optimal coupling π^* and its empirical estimator, capturing fast rates when the approximation error Δ is small. The key technical contribution is a nearly parametric rate for the KL error, driven by OU mixing properties and a Bernstein-type concentration framework, with explicit dependence on data geometry and the time horizon T. The approach is validated through numerical experiments on Gaussian mixtures, single-cell data, and unpaired image-to-image translation, showing improvements over baseline LightSB and illustrating practical applicability to high-dimensional generative tasks.

Abstract

Modern methods of generative modelling and unpaired data translation based on Schrödinger bridges and stochastic optimal control theory aim to transform an initial density to a target one in an optimal way. In the present paper, we assume that we only have access to i.i.d. samples from initial and final distributions. This makes our setup suitable for both generative modelling and unpaired data translation. Relying on the stochastic optimal control approach, we choose an Ornstein-Uhlenbeck process as the reference one and estimate the corresponding Schrödinger potential. Introducing a risk function as the Kullback-Leibler divergence between couplings, we derive tight bounds on generalization ability of an empirical risk minimizer in a class of Schrödinger potentials including Gaussian mixtures. Thanks to the mixing properties of the Ornstein-Uhlenbeck process, we almost achieve fast rates of convergence up to some logarithmic factors in favourable scenarios. We also illustrate performance of the suggested approach with numerical experiments.

Tight Bounds for Schrödinger Potential Estimation in Unpaired Data Translation

TL;DR

This work studies learning Schrödinger potentials for unpaired data translation under an Ornstein–Uhlenbeck reference process. It defines population and empirical risks using only iid samples from the initial and target marginals and proves a non-asymptotic, high-probability bound on the KL divergence between the true optimal coupling π^* and its empirical estimator, capturing fast rates when the approximation error Δ is small. The key technical contribution is a nearly parametric rate for the KL error, driven by OU mixing properties and a Bernstein-type concentration framework, with explicit dependence on data geometry and the time horizon T. The approach is validated through numerical experiments on Gaussian mixtures, single-cell data, and unpaired image-to-image translation, showing improvements over baseline LightSB and illustrating practical applicability to high-dimensional generative tasks.

Abstract

Modern methods of generative modelling and unpaired data translation based on Schrödinger bridges and stochastic optimal control theory aim to transform an initial density to a target one in an optimal way. In the present paper, we assume that we only have access to i.i.d. samples from initial and final distributions. This makes our setup suitable for both generative modelling and unpaired data translation. Relying on the stochastic optimal control approach, we choose an Ornstein-Uhlenbeck process as the reference one and estimate the corresponding Schrödinger potential. Introducing a risk function as the Kullback-Leibler divergence between couplings, we derive tight bounds on generalization ability of an empirical risk minimizer in a class of Schrödinger potentials including Gaussian mixtures. Thanks to the mixing properties of the Ornstein-Uhlenbeck process, we almost achieve fast rates of convergence up to some logarithmic factors in favourable scenarios. We also illustrate performance of the suggested approach with numerical experiments.

Paper Structure

This paper contains 38 sections, 22 theorems, 371 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Related work.
  4. Notations.
  5. Main Result
  6. Numerical Experiments
  7. Evaluation on the Gaussian Mixture
  8. Evaluation on the Single Cell Data
  9. Evaluation on Unpaired Image-to-Image Translation
  10. Proof Insights
  11. Proof of Theorem \ref{['th:main']}
  12. Proof of Lemma \ref{['lem:cgf_lower_bound']}
  13. Proof of Lemma \ref{['lem:log-potential_bernstein_condition']}
  14. Log-potential variance and conditional variance bounds
  15. Proof of Lemma \ref{['lem:conditional_variance_upper_bound']}
  16. ...and 23 more sections

Key Result

Theorem 1

Grant Assumptions as:base-process, as:bounded_support, as:sub_gaussian_density, as:log-potential_quadratic_growth, and as:lipschitz_parameterization. For any $n \in \mathbb N$ and $\delta \in (0, 1)$, introduce Then there exists such that for any $\delta \in (0, 1)$ and any $T \geqslant T_0$, with probability at least $(1 - \delta)$, the coupling $\widehat{\pi}$ defined in eq:erm satisfies the i

Figures (4)

  • Figure 1: Translation results from the latent space of ALAE using LightSB (left) and LightSB - OU(right). Top: Adult to Child — skin color and face shape preserved. Middle: Male to Female — glasses and face shape preserved. Bottom: Male to Female — skin color and face shape preserved.
  • Figure 2: Translation results from standard normal distribution using LightSB. Top: uniform grid and standard covariance matrices. Middle: random location on the grid and standard covariance matrices. Bottom: uniform grid and anisotropic random covariance matrix.
  • Figure 3: Translation results from standard normal distribution using LightSB and our modified approach. Top: uniform grid and standard covariance matrices. Middle: random location on the grid and standard covariance matrices. Bottom: uniform grid and anisotropic random covariance matrix.
  • Figure 4: Translation results from standard normal distribution using LightSB and our modified approach. Top: uniform grid and standard covariance matrices. Middle: random location on the grid and standard covariance matrices. Bottom: uniform grid and anisotropic random covariance matrix.

Theorems & Definitions (22)

  • Theorem 1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma A.1: restatement of Lemma \ref{['lem:cgf_lower_bound']}
  • Lemma A.2: restatement of Lemma \ref{['lem:log-potential_bernstein_condition']}
  • Lemma D.1
  • Lemma D.2
  • Lemma D.3
  • Lemma D.4
  • Lemma E.1
  • ...and 12 more