Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schrödinger Bridge Estimation

• URL: http://arxiv.org/abs/2510.22560v1
• Date: Sun, 26 Oct 2025 07:39:36 GMT
• Title: Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schrödinger Bridge Estimation
• Abstract summary: We study the statistical performance of intermediate Sinkhorn iterations in the two-sample estimation setting.<n>We establish a statistical bound on the squared total variation error of Sinkhorn bridge iterations.<n>This result provides a theoretical guarantee for the finite-sample performance of the Schr"odinger bridge estimator.
• Score: 11.478463820527745
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The Schr\"odinger bridge problem seeks the optimal stochastic process that connects two given probability distributions with minimal energy modification. While the Sinkhorn algorithm is widely used to solve the static optimal transport problem, a recent work (Pooladian and Niles-Weed, 2024) proposed the Sinkhorn bridge, which estimates Schr\"odinger bridges by plugging optimal transport into the time-dependent drifts of SDEs, with statistical guarantees in the one-sample estimation setting where the true source distribution is fully accessible. In this work, to further justify this method, we study the statistical performance of intermediate Sinkhorn iterations in the two-sample estimation setting, where only finite samples from both source and target distributions are available. Specifically, we establish a statistical bound on the squared total variation error of Sinkhorn bridge iterations: $O(1/m+1/n + r^{4k})~(r \in (0,1))$, where $m$ and $n$ are the sample sizes from the source and target distributions, respectively, and $k$ is the number of Sinkhorn iterations. This result provides a theoretical guarantee for the finite-sample performance of the Schr\"odinger bridge estimator and offers practical guidance for selecting sample sizes and the number of Sinkhorn iterations. Notably, our theoretical results apply to several representative methods such as [SF]$^2$M, DSBM-IMF, BM2, and LightSB(-M) under specific settings, through the previously unnoticed connection between these estimators.

Related papers

• Kernel-based potential mean-field games with unbiased random Fourier $U$-statistics [0.0]
  We study the subclass of potential mean-field games in which the running interaction cost and the terminal target cost are both expressed through reproducing- kernel maximum mean discrepancy penalties.<n>Both costs are estimated from finite-sample empirical distributions using a random Fourier U-statistic representation that is unbiased and has linear cost in the batch size.
  arXiv  Detail & Related papers  (2026-05-28T05:15:25Z)
• Entropy Across the Bridge: Conditional-Marginal Discretization for Flow and Schrödinger Samplers [68.9946633641494]
  Flow matching and Schrdinger bridges define probability, yet their inference grids are usually or inherited from one-endpoint diffusion.<n>We derive a conditional-marginal entropy-rate objective for bridge-aware discretization, separating endpoint-conditioned bridge geometry from marginal flow evolution.<n>On EDM/CIFAR-10, the entropic time-discretization gives the best tested five-step FID (186.3 pm 4.0 versus 200.5 pm 2.9 for linear and 238.0 pm 5.3 for cosine)
  arXiv  Detail & Related papers  (2026-05-15T16:11:10Z)
• Discrete Flow Matching: Convergence Guarantees Under Minimal Assumptions [13.856153548136826]
  Flow Matching has recently emerged as a popular class of generative models for simulating a target distribution.<n>We study two DFM models on $mathbbZ_md = 0,ldots,m-1d$, sampled through time discretization.<n>We derive non-asymptotic bounds in Kullback--Leibler divergence for the early-stopped version of the target distribution.
  arXiv  Detail & Related papers  (2026-05-09T11:06:33Z)
• Direct Estimation of Schrödinger Bridge Time-Series Drifts: Finite-Sample, Asymptotic, and Adaptive Guarantees [0.0]
  We study nonparametric estimation of Schrdinger bridge drifts from i.i.d. data.<n>Our approach works directly at the drift level and isolates emphstatistical error from optimization, approximation, and discretization error.
  arXiv  Detail & Related papers  (2026-05-06T20:44:44Z)
• Iterative Sampling Methods for Sinkhorn Distributionally Robust Optimization [2.345146665577353]
  Distributionally robust optimization (DRO) has emerged as a powerful paradigm for reliable decision-making under uncertainty.<n>This paper focuses on DRO with ambiguity sets defined via the Sinkhorn discrepancy: an entropy-regularized Wasserstein distance.<n>We study Sinkhorn DRO from the primal perspective, by reformulating it as a bilevel program with several infinite-dimensional lower-level subproblems over probability space.
  arXiv  Detail & Related papers  (2025-12-14T04:42:51Z)
• Beyond likelihood ratio bias: Nested multi-time-scale stochastic approximation for likelihood-free parameter estimation [49.78792404811239]
  We study inference in simulation-based models where the analytical form of the likelihood is unknown.<n>We use a ratio-free nested multi-time-scale approximation (SA) method that simultaneously tracks the score and drives the parameter update.<n>We show that our algorithm can eliminate the original bias $Obig(sqrtfrac1Nbig)$ and accelerate the convergence rate from $Obig(beta_k+sqrtfracalpha_kNbig)$.
  arXiv  Detail & Related papers  (2024-11-20T02:46:15Z)
• Relative-Translation Invariant Wasserstein Distance [82.6068808353647]
  We introduce a new family of distances, relative-translation invariant Wasserstein distances ($RW_p$) We show that $RW_p distances are also real distance metrics defined on the quotient set $mathcalP_p(mathbbRn)/sim$ invariant to distribution translations.
  arXiv  Detail & Related papers  (2024-09-04T03:41:44Z)
• Plug-in estimation of Schrödinger bridges [15.685006881635209]
  We propose a procedure for estimating the Schr"odinger bridge between two probability distributions. We show that our proposal, which we call the emphSinkhorn bridge, provably estimates the Schr"odinger bridge with a rate of convergence that depends on the intrinsic dimensionality of the target measure.
  arXiv  Detail & Related papers  (2024-08-21T15:07:25Z)
• Soft-constrained Schrodinger Bridge: a Stochastic Control Approach [4.922305511803267]
  Schr"odinger bridge can be viewed as a continuous-time control problem where the goal is to find an optimally controlled diffusion process. We propose to generalize this problem by allowing the terminal distribution to differ from the target but penalizing the Kullback-Leibler divergence between the two distributions. One application is the development of robust generative diffusion models.
  arXiv  Detail & Related papers  (2024-03-04T04:10:24Z)
• A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport [92.96250725599958]
  Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples. We show that our SSN method achieves a global convergence rate of $O (1/sqrtk)$, and a local quadratic convergence rate under standard regularity conditions.
  arXiv  Detail & Related papers  (2023-10-21T18:48:45Z)
• Minimax estimation of discontinuous optimal transport maps: The semi-discrete case [14.333765302506658]
  We consider the problem of estimating the optimal transport map between two probability distributions, $P$ and $Q$ in $mathbb Rd$. We show that an estimator based on entropic optimal transport converges at the minimax-optimal rate $n-1/2$, independent of dimension.
  arXiv  Detail & Related papers  (2023-01-26T18:41:38Z)
• Robust computation of optimal transport by $\beta$-potential regularization [79.24513412588745]
  Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. We propose regularizing OT with the beta-potential term associated with the so-called $beta$-divergence. We experimentally demonstrate that the transport matrix computed with our algorithm helps estimate a probability distribution robustly even in the presence of outliers.
  arXiv  Detail & Related papers  (2022-12-26T18:37:28Z)
• Faster Wasserstein Distance Estimation with the Sinkhorn Divergence [0.0]
  The squared Wasserstein distance is a quantity to compare probability distributions in a non-parametric setting. In this work, we propose instead to estimate it with the Sinkhorn divergence. We show that, for smooth densities, this estimator has a comparable sample complexity but allows higher regularization levels.
  arXiv  Detail & Related papers  (2020-06-15T06:58:16Z)
• Breaking the Sample Size Barrier in Model-Based Reinforcement Learning with a Generative Model [50.38446482252857]
  This paper is concerned with the sample efficiency of reinforcement learning, assuming access to a generative model (or simulator) We first consider $gamma$-discounted infinite-horizon Markov decision processes (MDPs) with state space $mathcalS$ and action space $mathcalA$. We prove that a plain model-based planning algorithm suffices to achieve minimax-optimal sample complexity given any target accuracy level.
  arXiv  Detail & Related papers  (2020-05-26T17:53:18Z)
• Distributionally Robust Bayesian Quadrature Optimization [60.383252534861136]
  We study BQO under distributional uncertainty in which the underlying probability distribution is unknown except for a limited set of its i.i.d. samples. A standard BQO approach maximizes the Monte Carlo estimate of the true expected objective given the fixed sample set. We propose a novel posterior sampling based algorithm, namely distributionally robust BQO (DRBQO) for this purpose.
  arXiv  Detail & Related papers  (2020-01-19T12:00:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.