Title: Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr\"{o}dinger Bridge Estimation
Venue: NeurIPS 2025
Accepted: yes
Keywords: Schrödinger bridge

Sinkhorn algorithm

statistical analysis
Abstract: The Schrödinger 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 Schrödinger 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: $\mathcal{O}(1/m+1/n + r^{2k})~(r \in (0,1))$, 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 Schrödinger 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.

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

Abstract

The Schrödinger 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 Schrödinger 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 4 k ) ( r ∈ (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 Schrödinger 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.

1 Introduction

In recent years, there has been increasing interest in Schrödinger bridge problems, with applications in mathematical biology (Chizat et al., 2022; Lavenant et al., 2024; Yao et al., 2025), Bayesian posterior sampling (Heng et al., 2024), generative modeling (De Bortoli et al., 2021; Wang et al., 2021), among others. The Schrödinger bridge problem aims to minimize the Kullback-Leibler (KL) divergence with respect to the Wiener measure, the distribution of Brownian motion over the path space, given two fixed marginal distributions µ, ν as source and target distributions.

The Schrödinger Bridge problem is also strongly connected to the entropic optimal transport (EOT) problem, which offers an efficient and regularized alternative to classical optimal transport, and is solvable via the Sinkhorn algorithm (Cuturi, 2013). The Sinkhorn algorithm operates in the dual space, optimizing the Schrödinger potentials between empirical distributions, and enjoys wellestablished convergence guarantees (Franklin and Lorenz, 1989; Peyré and Cuturi, 2018). Recently, Pooladian and Niles-Weed (2024) introduced the Sinkhorn bridge method, which plugs in the optimal Schrödinger potentials into the time-dependent drifts of stochastic differential equations to estimate

the Schrödinger bridge between the source and target distributions µ and ν . Their analysis established statistical bounds, adaptive to the intrinsic dimension of the target distribution ν , in the one-sample estimation setting, where the true source distribution µ is fully known.

1.1 Contributions

In this work, to better understand the algorithmic and statistical convergence rates of the Sinkhorn bridge method, we study the statistical performance of intermediate estimators obtained by Sinkhorn iterations in the two-sample estimation setting, where only empirical distributions µ m and ν n of m -and n -finite samples from µ and ν are available.

Statistical guarantee for Sinkhorn bridge iterations. First, we establish the following statistical bound (Theorem 1) for the Sinkhorn bridge estimator by leveraging analytical tools for the Schrödinger bridge in the one-sample setting (Pooladian and Niles-Weed, 2024) and for entropic optimal transport (Stromme, 2023b):

where P ∗ ∞ , ∞ and P ∗ m,n denote the true Schrödinger bridge and its Sinkhorn bridge estimator based on the optimal EOT solution, and P ∗ , [0 ,τ ] ∞ , ∞ and P ∗ , [0 ,τ ] m,n are restrictions of them on the time-interval [0 , τ ] . The result improves upon the bound established in (Pooladian and Niles-Weed, 2024) by extending the analysis to the two-sample setting and sharpening the rate in n from n -1 / 2 to n -1 at the cost of a deterioration in the dependence on ε and R , where ε is the strength of entropic regularization and R is the radius of the data supports.

Next, we analyze the intermediate iterations of the Sinkhorn algorithm and prove a convergence rate (Theorem 2) for the corresponding path measures:

where P k, [0 ,τ ] m,n denotes the Sinkhorn bridge estimator at the k -th Sinkhorn iteration for µ m and ν n . Together, these results imply the convergence rate E [ TV 2 ( P ∗ , [0 ,τ ] ∞ , ∞ , P k, [0 ,τ ] m,n )] = O (1 /m + 1 /n + r 4 k ) ( r ∈ (0 , 1)) regarding the number of samples m and n , and the number of Sinkhorn iterations k .

Relationship with the other Schrödinger Bridge estimators. Remarkably, our theoretical results on the Sinkhorn bridge can be directly applied to the representative Schrödinger Bridge solvers: [SF] 2 M(Tong et al., 2023), DSBM-IMF (Shi et al., 2023; Peluchetti, 2023), BM2 (Peluchetti, 2024), and lightSB(-M) (Korotin et al., 2023; Gushchin et al., 2024). First, the optimal estimator produced by these methods coincide with the Sinkhorn bridge estimator (Proposition 1). Moreover, when DSBM-IMF and BM2 are initialized with the reference process, the estimator after k iterations matches the Sinkhorn bridge obtained after k iterations of the Sinkhorn algorithm (Proposition 2). Furthermore, Algorithm 1, which learns the Sinkhorn bridge drift via a neural network, is a special case of the procedure proposed in [SF] 2 M. Consequently, our generalization-error analysis applies to all of these methods, endowing the algorithms with theoretical guarantees.

1.2 Related Work

Schrödinger bridge problem. The connection between entropic optimal transport and the Schrödinger bridge (SB) problem has been extensively studied; for a comprehensive overview, see the survey by Léonard (2013). More recently, there has been growing interest in computational approaches to the SB problem, particularly those leveraging deep learning techniques (De Bortoli et al., 2023; Shi et al., 2022; Bunne et al., 2023; Tong et al., 2023). In parallel, several studies have explored classical statistical methods for estimating the SB (Bernton et al., 2019; Pavon et al., 2021; Vargas et al., 2021). For example, Bernton et al. (2019) proposed a sampling framework based on trajectory refinement via approximate dynamic programming. Pavon et al. (2021) and Vargas et al. (2021) introduced methods that directly estimate intermediate densities using maximum likelihood principles. Specifically, Pavon et al. (2021) presented a scheme that explicitly models the target

density and updates weights accordingly, while Vargas et al. (2021) estimated forward and backward drifts using Gaussian processes, optimized via a likelihood-based objective. Assuming full access to the source distribution, the statistical performance of the Schrödinger bridge estimator was analyzed in the one-sample setting by Pooladian and Niles-Weed (2024). Other estimators, such as those based on neural approximations or efficient algorithmic variants, have been evaluated in recent works (Korotin et al., 2023; Stromme, 2023a).

Entropic optimal transport. Entropic Optimal Transport (EOT) introduces an entropy term to the standard Optimal Transport (OT) problem (Cuturi, 2013), primarily proposed to improve computational efficiency and smooth the transport plan. This regularization makes the OT problem strictly convex, enabling the application of the Sinkhorn algorithm, an efficient iterative method for the dual problem (Cuturi, 2013). The introduction of the Sinkhorn algorithm has allowed the OT problem, previously computationally very expensive, to obtain approximate solutions in realistic time even for large-scale datasets, leading to a dramatic expansion of its applications in the field of machine learning. For example, there are diverse applications such as measuring distances between distributions in generative modeling (Genevay et al., 2018), and shape matching or color transfer in computer graphics and computer vision (Solomon et al., 2015). The theoretical aspects of EOT have also been deeply studied. Analyses regarding the convergence rate of the Sinkhorn algorithm (Franklin and Lorenz, 1989) and the statistical properties (such as sample complexity) when estimating EOT from finite samples (Genevay et al., 2019) have been established. Furthermore, EOT is known to have a close relationship with the Schrödinger Bridge (SB) problem (Léonard, 2013). EOT in a static setting is equivalent to the problem of matching the marginal distributions of the SB problem, which deals with dynamic stochastic processes, and both share common mathematical structures in duality and optimization algorithms. This connection suggests that theoretical and computational advances in one field can contribute to the other.

1.3 Notations

For a metric space X , let P ( X ) be the space of probability distributions over X , and P 2 ( X ) be the subset of P ( X ) with finite second-order moment. Let R d be the d -dimensional Euclidean space and B ( a, R ) = { x | ∥ x -a ∥ 2 ≤ R } ⊂ R d be the Euclidean ball of radius R > 0 centered at a ∈ R d . For real numbers a and b , we set a ∨ b = max( a, b ) and a ∧ b = min( a, b ) . We write a ≲ b to denote that there exists a constant C > 0 , independent to a and b , such that a ≤ C b . We denote by δ x the Dirac measure concentrated at the point x ∈ R d . We denote by ✶ d ∈ R d the d -dimensional vector whose components are all equal to one. For a measure µ and a function f , we write µ ( f ) = ∫ f ( x ) µ ( dx ) . Let TV( µ, ν ) be the total variation distance between two probability distributions µ and ν . For ε > 0 and any real number or real-valued function h , write exp ε ( h ) := exp( h/ε ) . We denote the d -dimensional probability simplex by ∆ d := { α ∈ R d ≥ 0 | ∑ d i =1 α i = 1 } .

2 Preliminaries

2.1 Entropic Optimal Transport

Let µ, ν ∈ P 2 ( R d ) and ε > 0 be fixed. The entropic optimal transport (EOT) cost between µ and ν is

with the cost function c ( x, y ) := 1 2 ∥ x -y ∥ 2 2 . Here, Π( µ, ν ) ⊂ P 2 ( R d × R d ) denotes the set of couplings with marginals µ and ν , and

is the Kullback-Leibler divergence between π and µ ⊗ ν . If π is not absolutely continuous with respect to µ ⊗ ν , we set H ( π ∥ µ ⊗ ν ) = + ∞ .

The optimization problem in Eq. (3) is strictly convex with respect to π , and thus it admits a unique solution π ∗ ∞ , ∞ ∈ Π( µ, ν ) . Furthermore, the problem has a dual formulation given by

The dual problem (4) admits the Schrödinger potentials f ∗ ∞ , ∞ and g ∗ ∞ , ∞ as optimal solutions, and the first-order optimality condition implies

which is known as Schrödinger system. Note that the solution is constant-shift invariant, i.e. for any a ∈ R , ( f ∗ ∞ , ∞ + a, g ∗ ∞ , ∞ -a ) is also a solution to the Schrödinger system (5). It can be shown that with a constraint ν ( g ∗ ∞ , ∞ ) = 0 , the solution to (5) is unique. The optimal solution π ∗ ∞ , ∞ of the primal problem (3) can be expressed using the dual solution via

We refer to (Nutz, 2021) for more details of entropic optimal transport problems.

2.2 Schrödinger Bridge

Let Ω = C ([0 , 1]; R d ) be the path space consisting of all continuous maps from the time interval [0 , 1] to R d . Let W ε be the law of a reversible Brownian motion on R d with volatility ε . The Schrödinger Bridge problem is then defined through the following entropy minimization problem:

where P t ∈ P ( R d ) is the marginal distribution at time t ∈ [0 , 1] of the path-space distribution P , i.e. X t ∼ P t if X is a stochastic process with distribution P . Though W ε is an unbounded positive measure over Ω , the above optimization problem is still well defined as illustrated by Léonard (2013).

The Schrödinger Bridge problem is closely connected with the EOT problem. Precisely, the optimal solution P ∗ ∞ , ∞ ∈ P (Ω) of Eq. (7) can be derived by first solving the EOT problem (3) and then connect the optimal coupling with the Brownian bridge W ε | x 0 ,x 1 = W ε ( · | X 0 = x 0 , X 1 = x 1 ) , i.e.,

The optimal solution P ∗ ∞ , ∞ can also be characterized as the path measure of the SDE defined by

where B t is the standard Brownian motion, and the drift coefficient is given by

with γ ∗ ,t ∞ , ∞ ( y, z ) = exp ε ( g ∗ ∞ , ∞ ( y ) -1 2(1 -t ) ∥ y -z ∥ 2 2 ) . We refer to Chen et al. (2021); Léonard (2012, 2013) for more details of the Schrödinger Bridge.

2.3 Sinkhorn Algorithm

Given the empirical distributions µ m = 1 m ∑ m i =1 δ X i and ν n = 1 n ∑ n j =1 δ Y j , we consider the empirical EOT problem OT ε ( µ m , ν n ) . Let f ∗ m,n and g ∗ m,n be the optimal Schrödinger potentials. The Sinkhorn algorithm is an iterative method for approximating these optimal potentials by alternately updating the dual potentials f ( k ) m,n and g ( k ) m,n as follows.

We note that f ( k ) m,n , g ( k ) m,n can be interpreted as m - and n -dimensional vectors, respectively. For efficient computation, it is common to rewrite these updates in an alternative form. Let the kernel matrix K ∈ R m × n be defined as K ij = exp ε ( -1 2 ∥ X i -Y j ∥ 2 2 ) , and denote the vectors u ( k ) ∈ R m and v ( k ) ∈ R n by u ( k ) i = exp ε ( f ( k ) ( X i )) and v ( k ) ( Y j ) = exp ε ( g ( k ) ( Y j )) . Then the update rule (10) becomes

where ⊘ denotes element-wise division 1 . This reformulation greatly improves computational efficiency and enables large-scale applications.

Let d Hilb be the Hilbert (pseudo-)metric on R d + defined by

It is obvious that d Hilb ( x, y ) = 0 if and only if there exists λ ∈ R + such that x = λy . Let the initial values be ( u (0) , v (0) ) = ( ✶ m , ✶ n ) . Denote by ( u ( k ) , v ( k ) ) the solution obtained after k iterations of the Sinkhorn algorithm (10), and by ( u ∗ , v ∗ ) = (exp( f ∗ m,n /ε ) , exp( g ∗ m,n /ε )) the optimal solution. Then, it is shown that (Franklin and Lorenz, 1989; Peyré and Cuturi, 2018),

Here, λ ( K ) < 1 is defined as λ ( K ) := √ γ ( K ) -1 √ γ ( K )+1 where γ ( K ) := max i,j,k,l K ik K jl K jk K il . This result implies the exponential convergence of the Sinkhorn iterates u ( k ) and v ( k ) .

2.4 Convergence Rate of One-Sample Estimation

Pooladian and Niles-Weed (2024) analyzed the one-sample estimation task for the Schrödinger Bridge. In this setting, full access to the source distribution µ is assumed, while the target distribution ν is observed only through the empirical distribution ν n = 1 n ∑ n j =1 δ Y j based on i.i.d. samples Y 1 , . . . , Y n . Let ( f ∗ ∞ ,n , g ∗ ∞ ,n ) denote the optimal dual potentials solving OT ε ( µ, ν n ) , and let P ∗ ∞ ,n ∈ P (Ω) be the corresponding Schrödinger Bridge. Under Assumptions 1, 2 and 3 stated below, Pooladian and Niles-Weed (2024, Theorem 4.1) proved that there exists a constant C ν , depending only on ν and geometric constants of its support supp( ν ) , such that for any τ ∈ [0 , 1) the following holds:

where d ν is the intrinsic dimensionality of supp( ν ) .

3 Main Results

In this section, we provide statistical and algorithmic convergence rates of Sinkhorn bridge (Pooladian and Niles-Weed, 2024) in two-sample estimation setting.

3.1 Sinkhorn Bridge in Two-sample Case

Given empirical distributions µ m and ν n , the Sinkhorn bridge constructs the estimator P k m,n ∈ P (Ω) for the Schrödinger bridge between true distributions µ and ν as follows. First, the Sinkhorn algorithm runs for k iterations to solve OT ε ( µ m , ν n ) , yielding the dual potentials f ( k ) m,n and g ( k ) m,n . Then, the estimator P k m,n is defined as the path measure of the following SDE starting from µ :

where the drift function b ( k ) m,n is

3.2 Convergence Rates of Sinkhorn Bridge

First, we analyze the statistical convergence rate of the path measure P ∗ m,n ∈ P (Ω) using the optimal Schrödinger potentials f ∗ m,n and g ∗ m,n . Specifically, P ∗ m,n is defined analogously to P k m,n by replacing the potential g ( k ) m,n with its optimal counterpart g ∗ m,n . For details of the definition, see Appendix A.1. The analysis will be carried out under the following assumptions.

Assumption 1. The potentials g ∗ ∞ , ∞ , g ∗ ∞ ,n , g ∗ m,n satisfy

Assumption 2. The supports of µ and ν are subsets of a ball with radius R centered at the origin.

Assumption 3. The measure ν is supported on a compact, smooth, connected Riemannian manifold ( N,h ) without boundary, embedded in R d and equipped with the submanifold geometry induced by its inclusion into R d ; furthermore, dim N = d ν ≥ 3 . Moreover, ν admits a Lipschitz continuous and strictly positive density with respect to the Riemannian volume measure on N .

The manifold hypothesis suggests that data distributions often lie on lower-dimensional manifolds, and statistical learning methods can exploit this structure. In the EOT literature, Stromme (2023b) established the Minimum Intrinsic Dimension (MID) scaling, where the minimum of the intrinsic dimensions governs the convergence rate. For the Sinkhorn bridge under one-sample setting, Pooladian and Niles-Weed (2024) demonstrated that the convergence rate is independent of the ambient dimension d and depends only on the intrinsic dimension of the target distribution ν . Following these works, we also impose data-manifold assumptions solely on the target distribution ν .

The following theorem provides the squared total variation error between P ∗ ∞ , ∞ and P ∗ m,n .

Theorem 1 (Statistical convergence rate) . Suppose Assumptions 1, 2 and 3 hold. Let P ∗ ∞ , ∞ be the path measure of the true Schrödinger bridge for marginals µ, ν . Then, it follows that for any τ ∈ [0 , 1) ,

where P ∗ , [0 ,τ ] ∞ , ∞ and P ∗ , [0 ,τ ] m,n are restrictions of P ∗ ∞ , ∞ and P ∗ m,n on the time-interval [0 , τ ] .

Remark 1. In this section and related proofs, we suppress non-dominant terms such as polynomial factors of d ν , geometric characteristics of supp( ν ) , and exponential terms of order O ( d ν ) in uniform constants, in the notation ≲ .

We make the following observations: (1) Regarding the data dimensions, the intrinsic dimension d ν governs the growth of the convergence rate, as in Stromme (2023b); Pooladian and Niles-Weed (2024). Although our bound, unlike theirs, involves the ambient dimension d , the dependence is merely linear. Moreover, we note that the term d arises from the covering number of the data space

and can also be replaced by the intrinsic dimension of the source distribution µ by additionally imposing a manifold assumption on µ . (2) The degeneration of the estimated SDE toward a specific sample point Y j arises as τ → 1 because the estimated drift b ( k ) m,n points toward a particular sample Y j near t = 1 , causing the deviation from the target distribution ν at the terminal time t = 1 . Therefore, it is common practice to restrict the time interval to [0 , τ ] ( τ < 1) to ensure generalization. (3) Our result extends and sharpens the result of Pooladian and Niles-Weed (2024) to the two-sample setting, improving the convergence rate in n from n -1 / 2 to n -1 at the cost of a slight deterioration regarding, ε , R , and d .

In the next theorem, we give the algorithmic convergence rate of P k m,n → P ∗ m,n attained by running the Sinkhorn iterations k →∞ .

Theorem 2 (Algorithmic convergence rate) . Under Assumptions 1 and 2, we get for any τ ∈ [0 , 1) ,

where P k, [0 ,τ ] m,n are restrictions of P k m,n on the time-interval [0 , τ ] .

These results immediately imply the convergence rate E [ TV 2 ( P k, [0 ,τ ] m,n , P ∗ , [0 ,τ ] ∞ , ∞ )] = O (1 /m +1 /n + r 4 k ) for some r ∈ (0 , 1) regarding the sample size m and n , and the number of Sinkhorn iterations k .

3.3 Drift Estimation using Neural Networks in the Schrödinger Bridge Problem

The drift estimator (13) of the Sinkhorn bridge between µ m and ν n is defined as an empirical average regarding ν n . As a result, simulating the SDE (12) using a time discretization scheme (e.g., Euler-Maruyama approximation) incurs an O ( n ) -computational cost at every discretization step. To improve computational efficiency, we consider a neural network-based drift approximation.

Recall that W ε | x 0 ,x 1 is the Brownian bridge connecting x 0 and x 1 , so its marginal distribution at time t ∈ [0 , 1] is a Gaussian distribution W ε t | x 0 ,x 1 = N ( (1 -t ) x 0 + tx 1 , εt (1 -t ) I d ) . For dual potentials f ( k ) m,n ∈ L 1 ( µ m ) and g ( k ) m,n ∈ L 1 ( ν n ) obtained by the Sinkhorn algorithm, we define the joint distribution π ( k ) m,n analogously to Eq. (6) by replacing the optimal potentials and µ, ν with f ( k ) m,n , g ( k ) m,n , and µ m , ν n . Using π ( k ) m,n , we define the mixture of Brownian bridge (a.k.a. reciprocal process ): Π ( k ) = ∫ W ε | x 0 ,x 1 d π ( k ) m,n ( x 0 , x 1 ) ∈ P (Ω) . The drift function b ( k ) m,n in Eq. (13) is known to have the following expression ( Markovian projection (Shi et al., 2023)) (see Appendix B.2):

Therefore, we see that the function b ( k ) m,n minimizes the following functional L :

where Π ( k ) t 1 is the marginal distribution of Π ( k ) on time points t and 1 . This suggests training a neural network b θ with parameter θ to minimize the above functional. We note that samples from Π ( k ) t 1 can be obtained by sampling ( x 0 , x 1 ) ∼ π ( k ) m,n , t ∼ U [0 , τ ] , and x t ∼ W ε t | x 0 ,x 1 and discarding x 0 . Therefore, with this sampling scheme, SGD can be efficiently applied to minimize L ( b θ ) . The detail of the procedure is described in Algorithm 1. This procedure yields a neural network approximation of the drift function defined by the Sinkhorn bridge.

3.4 Other Estimators of Schrödinger Bridge

Our theoretical results on Sinkhorn bridge (Pooladian and Niles-Weed, 2024) can be directly applied to the representative Schrödinger Bridge solvers such as [SF] 2 M (Tong et al., 2023), lightSB(M) (Korotin et al., 2023; Gushchin et al., 2024), DSBM-IMF (Shi et al., 2023; Peluchetti, 2023), and BM2 (Peluchetti, 2024). First, we show that the optimal estimator produced by each of these methods

Algorithm 1 Drift Approximation via Neural Network

input Joint distribution π m,n defined by potentials f ∈ L 1 ( µ m ) , g ∈ L 1 ( ν n ) ; neural network b θ output Trained neural network b θ that approximates the drift function repeat

Sample batch of pairs { x i 0 , x i 1 } N i =1 ∼ π m,n

Sample batch t i ∼ U [0 , τ ]

0 1 t t i | x 0 ,x 1

For each triplet ( x i , x i , t i ) , sample x i ∼ W ε i i (Brownian bridge)

Compute the loss:

Update θ using ∇ L ( θ ) until converged;

coincides with P ∗ m,n attained by Sinkhorn bridge with k → ∞ (Proposition 1). Moreover, when DSBM-IMF and BM2 are initialized based on the reference process, the estimator after k iterations matches P k m,n obtained after k iterations of the Sinkhorn bridge (Proposition 2). Furthermore, it follows from the proof of Proposition 1 that Algorithm 1, which learns the Sinkhorn bridge drift via a neural network, is a special case of the algorithm proposed in [SF] 2 M. The proofs of these results are postponed to the Appendix. Consequently, our statistical and algorithmic convergence analysis applies to these methods, endowing the theoretical guarantees.

Proposition 1. The optimal estimators produced by [SF] 2 M, lightSB(-M), DSBM-IMF and BM2 coincide with P ∗ m,n attained by Sinkhorn bridge with k →∞ .

Proposition 2. With initialization based on the reference process, both DSBM-IMF and BM2 produce the same estimator as P k m,n of the Sinkhorn bridge for all iterations k .

4 Experiments

In this section, we verify our theoretical findings through numerical experiments. In Section 4.1, we verify the dependence of the statistical error on the sample size presented in Theorem 1, and the algorithmic convergence rate presented in Theorem 2. Next, in Section 4.2, we evaluate the neural network approximation of drift coefficient as illustrated in Section 3.3. All experimental implementations are conducted with PyTorch 2.6.0 (Paszke et al., 2019).

4.1 Experimental Verification of Theorems 1 and 2

We use 3 -dimensional normal distributions µ = N (0 , I 3 ) and ν = N (0 , B ) as source and target distributions, where B ∈ R 3 × 3 is a random positive definite matrix. Under this setting, Bunne et al. (2023, Eq. (25)-(29)) provided explicit expressions for the drift coefficient b ∗ ∞ , ∞ and the marginal distribution P ∗ ,t ∞ , ∞ . We use these expressions to evaluate the errors of the drift b ∗ m,n and the corresponding Schrödinger Bridge estimators P ∗ m,n in the finite-sample settings. These estimators are approximated by running the Sinkhorn bridge with a sufficiently large number of iterations ( k →∞ ) .

Statistical convergence. To verify that the estimator b ∗ m,n converges to the true drift b ∗ ∞ , ∞ when the sample sizes m and n increase, we draw m samples from µ and n samples from ν , respectively, and compute the estimator b ∗ m,n ( · , t ) for an arbitrary time t ∈ [0 , 1) . Subsequently, we evaluate the mean squared error (MSE) against the true drift b ∗ ∞ , ∞ ( · , t ) .

To compute this, the norm ∥ · ∥ L 2 ( P ∗ ,t ∞ , ∞ ) is approximated using Monte Carlo with 10,000 samples drawn from P ∗ ,t ∞ , ∞ . The expectation over the samples is computed by averaging over 10 independent sampling trials. With a fixed parameter ε = 0 . 1 , we generated heatmaps for several t ∈ [0 , 1) while varying the sample sizes m,n used in the estimator definition (Figure 1).

MSEsample(m, n, 0.1)

MSEsample(m, n, 0.3)

MSEsample(m,n, 0.7)

MSEsample(m, n, 0.9)

0.9

MSEsample(m,n,t)t

104

101

103

100

MSE

m

102

10-1

101

10-2

101

102103104

101102103104

101102103104

10

102103104

101102103104

n

n

n

n

n

As evident from the figure, the mean squared error deteriorates as t → 1 , but the overall convergence rate remains unchanged. For all t shown in the figure, the convergence rate is observed to be approximately proportional to ( m -1 + n -1 ) . This aligns with the result predicted by Theorem 1.

Algorithmic convergence. To verify that the estimator P ( k ) , [0 ,τ ] m,n exponentially approaches P ∗ , [0 ,τ ] m,n as the iteration count k increases, we obtain m and n independent samples from distributions µ and ν , respectively. We then consider the integral of the difference between estimators b ( k ) m,n ( · , t ) and b ∗ m,n ( · , t ) over the interval [0 , τ ] . Specifically, we evaluate the mean squared error integrated over t ∈ [0 , τ ] :

Time integrals over the interval [0 , τ ] are approximated via Monte Carlo integration by uniformly sampling 1 , 000 time-points. The norm ∥ · ∥ L 2 ( P ∗ ,t m,n ) is also estimated via Monte Carlo integration, using 1 , 000 samples drawn from P ∗ ,t m,n . We set m = n = 1 , 000 and compute the expectation by averaging over 10 independent samplings. With the regularization parameter fixed at ε = 0 . 005 , we generate graphs of the integral values over [0 , τ ] for varying k and multiple τ values (Figure 2). For all τ values shown in the figure, the convergence rate exhibits exponential decay with respect to k . This observation corroborates the theoretical prediction in Theorem 2.

100

D

10-1

10-2

二

=0.1

0

二

=0.5

t=0.9

10-3

100

101

102

Moreover, under the experimental setup of Section 4.2, we illustrate in Figure 3 the evolution of the drift when the number of Sinkhorn iterations is set to 1, 5, and 10. In the figure, it is observed that as the number of iterations k

k

increases, the drift b ( k ) m,n rapidly converges to the optimal drift b ∗ m,n .

4.2 Experimental Verification of Neural Network-Based Drift Estimation

Next, we evaluate the effectiveness of the drift approximation by a neural network using Algorithm 1. In this experiment, we set ε = 0 . 1 , defined µ as the eight-Gaussians distribution and ν as the moons distribution, and drew 1 , 000 independent samples from each. Using these samples, the Sinkhorn algorithm is employed to approximate the optimal EOT coupling π ∗ m,n between µ m and ν n , and, via Algorithm 1, a neural network approximation b θ of the drift b ∗ m,n is obtained. We employ an 4 -layer neural network with 512 -512 -512 hidden neurons for b θ , and train it using the AdamW optimizer with a learning rate of 1 × 10 -3 , weight decay of 1 × 10 -5 , and a mini-batch size of 4 , 096 . Finally, starting from each sample of µ m , we simulate trajectories using either b ( k ) m,n or the neural network drift b θ , and present the results in Figure 3. Trajectory simulations are performed using the Euler-Maruyama approximation with 1 , 000 discretization steps.

b(1)

m,n

b(5)

m,n

b(10)

m,n

b*

m,n

bo

5 Conclusion and Discussion

In this study, we provide a comprehensive analysis of statistical guarantees for the Schrödinger Bridge problem. Specifically, we establish theoretical guarantees in two key settings: (i) the two-sample estimation task and (ii) intermediate estimators during the learning process.

Our main contributions are as follows. First, we establish a statistical convergence analysis in the two-sample estimation setting for the Schrödinger bridge estimator with k →∞ , which demonstrates a statistical convergence rate of O ( 1 n + 1 m ) . Second, we establish new convergence guarantees for the dual potentials obtained during intermediate iterations of the Sinkhorn algorithm, proving exponential convergence in the finite-sample setting. These results allow for a clear estimation of the number of samples m,n and Sinkhorn iterations k required to achieve a desired precision. Experimental results align with our theoretical analysis, confirming that the error decreases at a rate of approximately ( m -1 + n -1 ) with respect to the sample size, and that the error decreases exponentially with respect to the number of Sinkhorn iterations k . These findings strengthen the statistical guarantees for existing methods proposed in works such as Korotin et al. (2023); Peluchetti (2023); Gushchin et al. (2024); Shi et al. (2023); Peluchetti (2024) through the connections among these methods. The theoretical framework of this study deepens the understanding of generalization error in the Schrödinger Bridge problem and offers new insights into the convergence properties and stability of the Sinkhorn algorithm during intermediate iterations.

Future research directions include extending the analysis to more complex distributional settings and more general reference processes, and removing linear dependence on the ambient dimension d . Another interesting research direction is to explore alternative optimization methods for EOT problems and analyze their performance using our theoretical framework beyond the Sinkhorn algorithm.

Acknowledgment

This research is supported by the National Research Foundation, Singapore, Infocomm Media Development Authority under its Trust Tech Funding Initiative, and the Ministry of Digital Development and Information under the AI Visiting Professorship Programme (award number AIVP-2024-004). Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not reflect the views of National Research Foundation, Singapore, Infocomm Media Development Authority, and the Ministry of Digital Development and Information.

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Appendix

A Omitted Proofs

A.1 Proof of Theorem 1

We here provide the complete definitions of path measures P k m,n , P ∗ m,n ∈ P (Ω) . Using the Sinkhorn iterations ( f ( k ) m,n , g ( k ) m,n ) with respect to µ m and ν n , we define the drift function b ( k ) m,n as follows:

where

Then, P ( k ) m,n is defined as the path measure of the following SDE:

P ∗ m,n ∈ P (Ω) is also defined in the similar way by replacing the Sinkhorn iterations to the corresponding optimal Schrödinger potentials. That is, using the optimal Schrödinger potentials ( f ∗ m,n , g ∗ m,n ) with respect to µ m and ν n , we define the drift function b ∗ m,n as follows:

where

Then, P ∗ m,n is defined as the path measure of the following SDE:

Theorem 1 (Statistical convergence rate) . Suppose Assumptions 1, 2 and 3 hold. Let P ∗ ∞ , ∞ be the path measure of the true Schrödinger bridge for marginals µ, ν . Then, it follows that for any τ ∈ [0 , 1) ,

where P ∗ , [0 ,τ ] ∞ , ∞ and P ∗ , [0 ,τ ] m,n are restrictions of P ∗ ∞ , ∞ and P ∗ m,n on the time-interval [0 , τ ] .

Proof of Theorem 1. The proof of Theorem 1 is based on the main idea presented in Pooladian and Niles-Weed (2024); Stromme (2023b): we introduce the following entropic plan.

Here, the function ¯ f ∞ ,n : R d → R is defined as follows:

Denoting by ¯ P ∞ ,n the measure obtained by substituting the coupling (17) into (8), Proposition 3.1 in Pooladian and Niles-Weed (2024) implies that the drift ¯ b ∞ ,n with law ¯ P ∞ ,n is expressed as follows:

By incorporating the path measure ¯ P ∞ ,n into the bound via the triangle inequality and subsequently applying Pinsker's inequality, we obtain:

We analyze these two terms separately. For the first term, we apply Proposition 4.3 of Pooladian and Niles-Weed (2024)

For the second term, we apply Girsanov's theorem to derive the difference between the drifts.

where for the second inequality, we used Lemma 1.

Lemma 1. Under assumptions of Theorem 1, we have

Proof of Lemma 1. We perform the expansion while keeping Y j and z fixed

From the case ( p, q ) = (1 , ∞ ) of Lemma 12, we have

By applying Lemma 11, we obtain

A.2 Proof of Theorem 2

Theorem 2 (Algorithmic convergence rate) . Under Assumptions 1 and 2, we get for any τ ∈ [0 , 1) ,

where P k, [0 ,τ ] m,n are restrictions of P k m,n on the time-interval [0 , τ ] .

Proof of Theorem 2. We start by applying Girsanov's theorem to obtain a difference in the drifts

where we applied Lemma 2.

Lemma 2. Under assumptions of Theorem 1, we have

Proof of Lemma 2. We perform the expansion while keeping Y j and z fixed

We define w ∗ , w k , v ∗ , v k ∈ R n as follows:

where both w ∗ and w k satisfy the conditions of a probability vector, i.e., w ∗ j , w k j > 0 , ∑ n j =1 w ∗ j = ∑ n j =1 w k j = 1 .

Since

the quantity λ ( K ) can be upper bounded as follows:

Since the upper bound of λ ( K ) is independent of Y j and z , the proof is complete.

A.3 Proof of Proposition 1

Proposition 1. The optimal estimators produced by [SF] 2 M, lightSB(-M), DSBM-IMF and BM2 coincide with P ∗ m,n attained by Sinkhorn bridge with k →∞ .

Proof of Proposition 1. It suffices to show that the drift estimated by each method coincides with b ∗ m,n .

Proof of optimal estimator for [SF] 2 M

The SDE representation of the Brownian bridge W ε | x 0 ,x 1 with endpoints ( x 0 , x 1 ) is given by

The marginal density of the Brownian bridge at time t is p t ( x t | x 0 , x 1 ) = N ( x t | (1 -t ) x 0 + tx 1 , εt (1 -t ) I d ) . There exists an ordinary differential equation (ODE) that preserves the same marginal distributions p t ( x t | x 0 , x 1 ) as the SDE (18). This ODE, referred to as the probability flow ODE , takes the form

Consequently, the drift term of the Brownian bridge SDE can be decomposed into the sum of the probability flow ODE drift u ◦ t ( x t | x 0 , x 1 ) and the score function ∇ log p t ( x t | x 0 , x 1 ) :

In the [SF] 2 Mframework, let π ∗ m,n denote the optimal coupling for the entropic optimal transport (EOT) between µ m and ν n , and consider the mixture distribution Π ∗ = π ∗ m,n W ε | x 0 ,x 1 . Under Π ∗ , two neural networks v θ and s φ are trained to minimize the following objectives:

The estimator for the Schrödinger bridge drift in [SF] 2 Mis then defined as

When both models are sufficiently expressive, the optimal solutions admit the following conditional expectation representations:

Therefore, the drift estimator in [SF] 2 Mcan be written as

Proof of optimal estimator for LightSB(-M)

Let S ( µ ) denote the set of Schrödinger bridges whose source distribution is µ . In lightSB(-M), the drift estimator g v is defined with a function v : R d → R as follows:

and let S v be the law of the SDE

For any coupling π m,n ∈ Π( µ m , ν n ) , consider its Brownian-bridge mixture Π = π m,n W ε | x 0 ,x 1 . Then, by (Gushchin et al., 2024, Theorem 3.1),

Restricting S to S v , one obtains (Gushchin et al., 2024, Theorem 3.2)

where Π t 1 denotes the marginal law of Π at times t and 1 , and C ( π m,n ) is a constant independent of S v . lightSB(-M) trains a parameterised model v θ that minimises

The drift g v θ ∗ obtained from the learned v θ ∗ serves as the drift estimator produced by lightSB(-M). When the model capacity is sufficiently large, g v θ ∗ satisfies

Proof of optimal estimator for DSBM-IMF and BM2

Let F ( v f ) and B ( v b ) denote the laws induced by the following SDEs:

Here, in the second SDE, dt represents an infinitesimal negative time step, and ¯ B t denotes the time-reversed Brownian motion.

Given drifts v f ′ and v b ′ , define the following loss functions:

In DSBM-IMF, starting from initial drifts v (0) f and v (0) b , we perform the following iterative updates:

In BM2, given initial drifts v (0) f and v (0) b , the iterative procedure is:

For both methods, F ( v ( k ) f ) and B ( v ( k ) b ) converge to Π ∗ as k → ∞ (Shi et al., 2023, Theorem 8), (Peluchetti, 2024, Lemma 1).

Therefore, letting v ∗ f and v ∗ b denote the limits of v ( k ) f and v ( k ) b as k → ∞ , we have that v ∗ f equals b ∗ m,n :

A.4 Proof of Proposition 2

Proposition 2. With initialization based on the reference process, both DSBM-IMF and BM2 produce the same estimator as P k m,n of the Sinkhorn bridge for all iterations k .

Proof of Proposition 2. We begin by introducing the Iterative Proportional Fitting (IPF) method (Fortet, 1940; Kullback, 1968; Rüschendorf, 1995). IPF provides one means of solving Eq. (7) and generates a sequence of path measures ( ˜ P ( k ) ) k ∈ N according to

We initialize with ˜ P (0) = W ε . It is known (Léonard, 2013; Nutz, 2021) that the sequence ( ˜ P ( k ) ) k ∈ N satisfies

Here the coupling π f,g is defined by

Hence it suffices to show that the drifts estimated by each algorithm coincide with b ( k ) m,n .

When the initial drift for DSBM is set to v (0) b = 0 , the sequence B ( v (0) b ) , F ( v (1) f ) , B ( v (1) b ) , F ( v (2) f ) , B ( v (3) b ) , . . . coincides with the IPF sequence ( ˜ P ( k ) ) k ∈ N (Shi et al., 2023, Proposition 10). Consequently, v ( k +1) f coincides with b ( k ) m,n .

When the initial drift for BM2 is initialized as v (0) f = v (0) b = 0 , the sequence B ( v (0) b ) , F ( v (1) f ) , B ( v (2) b ) , . . . coincides with the IPF sequence ( ˜ P ( k ) ) k ∈ N (Peluchetti, 2024, Theorem 1). Thus, v (2 k +1) f is given by

Similarly, the sequence F ( v (0) f ) , B ( v (1) b ) , F ( v (2) f ) , . . . corresponds to IPF with the update order reversed, and the same conclusion holds for v (2 k ) f .

B Technical Lemmas

B.1 Lemmas used for Theorems 1 and 2

Lemma 3 (Stromme, 2023b, Proposition 14) . Under assumptions of Theorem 1, we have

.

In the following, p ∗ ∞ , ∞ denotes the Radon-Nikodym derivative regarding µ ⊗ ν , defined in Eq. (6), and p ∗ m,n is also defined in a similar manner replacing f ∗ ∞ , ∞ , g ∗ ∞ , ∞ with f ∗ m,n , g ∗ m,n .

Lemma 4 (Stromme, 2023b, Proposition 15) . The population dual potentials f ∗ ∞ , ∞ and g ∗ ∞ , ∞ are 2 R -Lipschitz over supp( µ ) and supp( ν ) , respectively. The extended empirical dual potentials f ∗ m,n and g ∗ m,n are also 2 R -Lipschitz over supp( µ ) and supp( ν ) , respectively. In particular, the population density p ∗ ∞ , ∞ and the extended empirical density p ∗ m,n are each 4 R ε -log-Lipschitz in each of their variables over supp( µ ⊗ ν ) .

Lemma 5 (Stromme, 2023b, Lemma 25) . Under assumptions of Theorem 1, with probability at least 1 -1 n e -20 R 2 /ε

Lemma 6 (Stromme, 2023b, Lemma 26) . Under assumptions of Theorem 1, we have

And, with probability at least 1 -1 n e -20 R 2 /ε

Lemma 7 (Stromme, 2023b, Lemma 27 adapted) . Under the assumptions of Theorem 1, if ε/R is sufficiently small, then

Lemma 7 are the extensions of Stromme (2023b, Lemma 27) that only treats the case of m = n . This extension can be readily verified by a slight modification of the proof in Stromme (2023b).

Lemma 8. Let U m ( y ) = ( µ m -µ ) ( p ∗ ∞ , ∞ ( · , y ) ) . Then the following holds:

.

Proof. We first note that µ ( p ∗ ∞ , ∞ ( · , y )) = 1 since the marginal of π ∗ ∞ , ∞ on y is ν , and hence U m ( y ) = µ m ( p ∗ ∞ , ∞ ( · , y )) -1 . Hoeffding's lemma, with 0 ≤ p ∗ ∞ , ∞ ≲ ( R ε ) d ν , implies that there exists a constant C > 0 such that

Consequently, U m ( y ) is subgaussian with variance proxy C 4 m ( R ε ) 2 d ν . Orlicz ψ 2 -norm is defined for a random variable ξ by

Chen et al. (2023, Theorem 3.7 and Theorem 4.1) with φ ( x ) := x 2 / 2 , p = 2 and d ( y, y ′ ) := ∥ U m ( y ) -U m ( y ′ ) ∥ ψ 2 imply the following estimates:

Let Z i := p ∗ ∞ , ∞ ( X i , y ) -p ∗ ∞ , ∞ ( X i , y ′ ) . Since each Z i is an independent, zero-mean random variable, by Vershynin (2018, Proposition 2.6.1) d ( y, y ′ ) is bounded above as follows:

For each Z i we have ∥ · ∥ ψ 2 ≲ ∥ · ∥ L ∞ ( µ ) , hence

∥ Z i ∥ ψ 2 = ∥ ∥ p ∗ ∞ , ∞ ( X i , y ) -p ∗ ∞ , ∞ ( X i , y ′ ) ∥ ∥ ψ 2 ≲ ∥ ∥ p ∗ ∞ , ∞ ( · , y ) -p ∗ ∞ , ∞ ( · , y ′ ) ∥ ∥ L ∞ ( µ ) .

By | e a -e b | ≤ e a ∨ b | a -b | and Lemma 4, 6, for any x ,

Therefore,

This concludes N (supp( ν ) , d, r ) ≤ N ( supp( ν ) , ∥ · ∥ 2 , r A ) ( A = C √ m ( R ε ) d ν +1 where C is a constant hidden in the above inequality).

We evaluate the first term in (23) by using Proposition 43 in Stromme (2023b) that shows there exists 0 < c ν ≤ R such that, for any 0 < δ ≤ c ν , the covering number satisfies δ -d ν ≲ N (supp( ν ) , ∥ · ∥ 2 , δ ) ≲ δ -d ν .

where ≲ also hides the intrinsic dimensionality d ν .

From (24), the second term of (23) is bounded as

Combining the results of (26) and (25) completes the proof.

Lemma 9. Let V n ( x ) = ( ν n -ν ) ( p ∗ ∞ , ∞ ( x, · ) ) . Then the following holds:

Proof. Lemma 9 can be established by a minor modification of the proof of Lemma 8. The only difference is the evaluation of the covering number: the bound over supp( µ ) is replaced by that over B (0 , R ) . This substitution introduces a dependence on the ambient dimension d .

Lemma 10. It follows that

Proof. Let

Using ¯ f ∞ ,n , we evaluate ∥ f ∗ ∞ , ∞ -f ∗ m,n ∥ 2 L ∞ ( µ ) as follows.

For the first term, the Schrödinger system (5) implies

where V n ( x ) := ( ν n -ν )( p ∗ ∞ , ∞ ( x, · )) and we used ν ( p ∗ ∞ , ∞ ( x, · )) = 1 .

For any η ∈ (0 , 1)

, the following holds:

For the first term in (28), the mean value theorem implies that for | u | < 1 we have | log(1 + u ) | ≤ | u | 1 -| u | . Hence,

For the second term in (28), since exp ε ( -10 R 2 ) ≤ p ∗ ∞ , ∞ ≤ exp ε (10 R 2 ) , an application of Markov's inequality yields

Setting η = 1 / 2 and applying Lemma 9 yields

For the second term in (27), Eq. (6.4) in Stromme (2023b) implies

Let E denote the event on which Lemma 5 holds. From Lemma 3 and P [ E C ] ≤ 1 n , it follows that

Combining this with Lemma 7 and Lemma 5 yields

Lemma 11. It follows that

Proof. Let

Using ¯ g m, ∞ , we evaluate ∥ g ∗ ∞ , ∞ -g ∗ m,n ∥ 2 L ∞ ( ν ) as follows.

For the first term, the Schrödinger system (5) implies

where U m ( y ) := ( µ m -µ )( p ∗ ∞ , ∞ ( · , y )) and used µ ( p ∗ ∞ , ∞ ( · , y )) = 1 .

For any η ∈ (0 , 1) , the following holds:

For the first term in (31), the mean value theorem implies that for | u | < 1 we have | log(1 + u ) | ≤ | u | 1 -| u | . Hence,

For the second term in (31), since exp ε ( -10 R 2 ) ≤ p ∗ ∞ , ∞ ≤ exp ε (10 R 2 ) , an application of Markov's inequality yields

Setting η = 1 / 2 and applying Lemma 8 yields

For the second term of (30), since the gradient of the log-sum-exp function is the softmax and its ℓ 1 -norm equals 1 , we obtain

Therefore, by Lemma 10, we have

Combining the above results, we obtain

Lemma 12 (Lipschitzness of softmax) . For λ > 0 and d ≥ 2 , define σ λ : R d → ∆ d by

Then for any 1 ≤ p, q ≤ ∞ ,

Proof. Let s = σ λ ( z ) . The Jacobian of σ λ at z is

By the fundamental theorem of calculus along the segment t ↦→ y + t ( x -y ) , hence, for any 1 ≤ p, q ≤ ∞ ,

It suffices to bound sup z ∥ J ( z ) ∥ q → p .

Step 1: Decomposition. With the standard basis { e i } d i =1 ,

and by the triangle inequality and ∥ e i -e j ∥ p = 2 1 /p (with the convention 2 1 / ∞ = 1 ),

Step 2: Maximization over s ∈ ∆ d . Let i min = arg min i u i , i max = arg max i u i . Since | u i -u j | ≤ | u i max -u i min | and ∑ i<j s i s j = 1 2 ( 1 -∑ d i =1 s 2 i ) ,

The last inequality is tight when s i max = s i min = 1 2 and all other s i = 0 . Therefore,

Because σ λ ( R d ) is the interior of ∆ d and the right-hand side of (38) is continuous in s , the supremum over z equals the supremum over s ∈ ∆ d .

Thus for any u ∈ R d ,

Step 3: Maximization over ∥ u ∥ q = 1 . For any a, b ∈ R and q ≥ 1 , | a -b | q ≤ 2 q -1 ( | a | q + | b | q ) . With a = u i max , b = u i min and ∥ u ∥ q = 1 ,

with equality when u has exactly two nonzero entries u i max = 2 -1 /q and u i min = -2 -1 /q .

Step 4: Combine. From (38)-(40),

Plug this into (36) to obtain the claim. The bound is tight by the equality cases noted in (39) and (40).

Lemma 13. For any probability vectors u, v ∈ ∆ d ,

Proof. By Cohen and Fausti (2024)[Theorem 5.1],

Since tanh( x ) ≤ x for x ≥ 0 and d Hilb ( u, v ) ≥ 0 , letting x = d Hilb ( u,v ) 4 gives

This proves (41).

Lemma 14. For any u , v ∈ R d + , the following holds:

Here, ⊙ denotes the element-wise product.

Proof. The first two properties are established in (Lemmens and Nussbaum, 2013, Lemma 2.1). For the third property, we have

B.2 Lemma used for the drift estimation using neural networks

We provide an alternative expression of the Sinkhorn bridge drift estimator as a Markovian projection.

Recall that W ε | x 0 ,x 1 is the Brownian bridge connecting x 0 and x 1 , so its marginal distribution at time t ∈ [0 , 1] is a Gaussian distribution W ε t | x 0 ,x 1 = N ( (1 -t ) x 0 + tx 1 , εt (1 -t ) I d ) . For any distributions µ ′ , ν ′ ∈ P ( R d ) , and dual potentials f ∈ L 1 ( µ ′ ) , g ∈ L 1 ( ν ′ ) , we define

We here only consider the case where π becomes a joint distribution π ∈ P ( R d × R d ) , that is, ∫ dπ ( x 0 , x 1 ) = 1 . Using π ( k ) m,n , we define the mixture of Brownian bridge (reciprocal process): Π = ∫ W ε | x 0 ,x 1 dπ ( x 0 , x 1 ) ∈ P (Ω) .

Lemma 15. Under the above setting, it follows that

Proof. Given x t ( t < 1) , we denote by Π( x 1 | X t = x t ) the conditional density of Π with respect to Lebesgue measure dx 1 . Note that the marginal density Π t 1 of Π on time points t and 1 with respect to dx t dx 1 satisfies that for a measurable A ⊂ R d

Therefore, we get

This means

Therefore,

C Omitted Experiments

C.1 Stopping and guidance

Theorems 1 and 2 suggest that beyond a certain point additional Sinkhorn iterations are not beneficial. The total error decomposes into a 'sampling error' and an 'optimization error'. The latter decreases

only through iterations, whereas the former does not. By the triangle inequality for the total variation distance,

Once the second term on the right-hand side (optimization error) becomes no larger than the first term (estimation error), further iterations no longer yield a meaningful reduction of the total error. Hence it is reasonable to choose the stopping point as the smallest k such that

From Theorems 1 and 2 , a sufficient condition for this is

Since log ( tanh ( R 2 ε )) < 0 , iterations are beneficial only when B m + A + B n < 1 . Equivalently, unless the sample size is large enough to make the numerator negative-roughly ( m ∧ n ) ≈ ( A ∨ B ) -the reduction achievable by iterations is smaller than the estimation error, leading to wasted computational resources.

In Figure 3, the fact that b (1) m,n already captures the shape of ν n to a certain extent is consistent with this reasoning. Increasing k becomes meaningful only when µ m and ν n are sufficiently close to the true distributions µ and ν , namely when m and n are sufficiently large.

C.2 Intrinsic dimension

Theorem 1 shows that the orders of the orders of R and ε depend on the intrinsic dimension d ν rather than the ambient dimension d . In this section, we verify that the error varies with d ν under the following setup.

We fix the ambient dimension at d = 50 , the regularization strength at ε = 0 . 5 , the evaluation-interval endpoint at τ = 0 . 9 , and the sample sizes at m = n = 10000 . The distribution µ is sampled uniformly from the unit hypercube [0 , 1] d , while ν is sampled uniformly from the d ν -dimensional manifold embedded in the unit sphere of radius 1 ,

By varying d ν , we examine whether the convergence behavior of the estimation error with respect to the sample size depends on the manifold dimension.

Specifically, for each d ν ∈ { 5 , 10 , 20 , 30 } we compare

The results are reported in Table 1. An increasing trend of the error with larger manifold dimension is observed.

C.3 The role of epsilon

We examine the numerical stability of the regularization parameter ε in the Sinkhorn algorithm and provide selection guidelines in relation to the sample sizes ( m,n ) . Using the same setup as

Experiments 4.1, for each given m,n we search for the smallest ε that achieves the target error δ = 1 . 0 .

As m ∧ n increases, the ε required to meet the target error exhibits a two-phase behavior: a steep initial decrease followed by an asymptotically mild decay. Representative values are reported in Table 2.

By Theorem 1, the ε required to satisfy the target error δ obeys

This indicates a steep improvement in the smallm ∧ n regime followed by logarithmically slow decay, which aligns with the empirical results. Since excessively small ε can cause numerical instability, using the above expression as a lower-bound guideline for selecting ε is effective.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper analyzes the statistical and algorithmic behavior of Sinkhorn-based Schrödinger bridge estimators in the two-sample setting,\nwhere both source and target distributions are accessible only via finite samples. The authors establish a total variation error bound,\nquantifying how the estimator improves with more samples and Sinkhorn iterations. They further show that several popular Schrödinger\nbridge solvers (e.g., [SF]²M, DSBM-IMF, BM2, lightSB) are equivalent to the Sinkhorn bridge and thus inherit the same guarantees. Finally,\nthey propose a neural network approximation for the drift to reduce simulation cost. Experiments validate both theoretical rates and the\nefficiency of the learned drift.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n1. Provides a sharp finite-sample error bound $\\mathcal{O}(1/m + 1/n + r^{2k})$ for Sinkhorn-based Schrödinger bridge estimation,\ncovering both sample and iteration complexity.\n2. Extends prior work from the one-sample setting to the more general and practical two-sample setting.\n3. Unifies several existing SB methods (\\[SF]²M, DSBM-IMF, BM2, lightSB) under the Sinkhorn bridge framework, giving them new\ntheoretical guarantees.\n4. Introduces a neural network–based drift approximation that significantly reduces simulation cost from $\\mathcal{O}(n)$ to $\\mathcal{O}\n(1)$ per step.\n5. Empirical results validate both the statistical convergence and the efficiency of the neural approximation.\n\n**Weaknesses:**\n1. Experimental results are primarily demonstrated in low-to-moderate dimensions; scalability to high-dimensional applications remains to be tested.\n\nQUESTIONS:\n1. How does the proposed Sinkhorn bridge estimator perform compared to standard reverse diffusion interpolations or Schrödinger bridge\nsolvers that use learned forward-backward dynamics, both in accuracy and computational cost?\n2. The neural network drift approximation reduces per-step cost from $\\mathcal{O}(n)$ to $\\mathcal{O}(1)$, but does this come at the cost of\naccuracy?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper extends the analysis of the plug-in Schrödinger bridge estimator proposed in [1] in several ways:\n* The analysis is generalized to the two-sample setting.\n* The convergence rate with respect to the sample size $n$ is improved compared to [1] (Theorem 1).\n* The analysis incorporates the number of Sinkhorn iterations as a parameter, which was absent in [1] (Theorem 2).\n\nIn addition, the paper provides an alternative computational implementation using neural networks, which avoids the computation at every discretization step (Algorithm 1).\n\n\n[1] Aram-Alexandre Pooladian, Jonathan Niles-Weed. Plug-in estimation of Schrödinger bridges. Arxiv preprint https://arxiv.org/abs/2408.11686, 2024.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n* As mentioned in the summary, the paper extends the plug-in Schrödinger bridge estimator proposed in [1] in several ways. In particular, improving the convergence rate in $n$ and including the Sinkhorn iteration parameter in the bound are new contributions that were absent in the previous literature.\n\n* The paper shows that the estimator proposed in [1] can be interpreted as the minimizer of a certain loss functional. Consequently, the authors propose an alternative implementation of the estimator using a neural network. This implementation is favorable, as one can use a single neural network that takes the time-step as input, rather than computing the drift term at each time-step.\n\n### Weaknesses\n* It seems that in order to improve the rate from $O(n^{-1/2})$ to $O(n^{-1})$ (assuming $m \\to \\infty$), one obtains a substantially worse rate with respect to $R$. This restricts the implications of the analysis, as the bound becomes significantly worse unless $R$ is very small.\n\n* While I find the estimator proposed in Algorithm 1 interesting, I suspect that the estimator obtained from this algorithm may have different statistical guarantees compared to the plug-in estimator analyzed in the main body. The statistical error between the population loss minimizer (Line 210) and the empirical loss minimizer (Algorithm 1) should involve some dependence on $N$. This statistical error should be carefully analyzed, as the loss is non-convex and thus convergence is not straightforward.\n\nQUESTIONS:\n* Minor typo: Line 196 contains a typo. It probably should be $\\mathbb{E}\\left[TV^2(P_{m,n}^{k,[0,\\tau]}, P_{\\infty,\\infty}^{*,[0,\\tau]})\\right]$?\n\n* As mentioned in the weaknesses section, the result from this paper seems to indicate a trade-off between $n$ and $R$. Do you think this trade-off is unavoidable? It seems that the cost is quite large.\n\n* In Remark 1, the authors mention that the dimension $d$ can be replaced by the intrinsic dimension $d_{\\nu}$, but Line 308 states that the theorem relies on the ambient dimension rather than the intrinsic dimension. Could you elaborate on this?\n\nLIMITATIONS:\nThe authors expressed limitations of the work. I have stated additional limitations that I found in the weaknesses section.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-25"}

comments: SUMMARY:
This works investigates the statistical guarantees of the Sinkhorn bridge algorithm in the two-sample estimation setting.

STRENGTHS AND WEAKNESSES:
I enjoyed reading through the results. The results extend the results of Pooladian and Niles-Weed (2024) to the two-sample estimation setting. The theoretical analysis is built on the techniques of Pooladian and Niles-Weed (2024) and seems sound.

The empirical validation is also clearly presented for synthetic Gaussian settings.

Overall, I think this paper is a solid theoretical contribution to the community. I do have to say that I don't work in this field, so my overall evaluation of this paper is limited to the presentation of this paper. I do find the problem and the results great.

That said, I'm not very sure about the relevance of this paper to the NeurIPS community. I'd appreciate if the authors could provide the relevance of this result for more general audience and in their final version.

QUESTIONS:
See above.

LIMITATIONS:
See above.
I think some clarifications on the relevance to the greater AI community would be nice for general audience to appreciate the main contributions. But I should say I'm not quite familiar with the literature.
TITLE: Official Review
CLARITY: 4
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-01

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper provides a theoretical study of a numerical scheme for approximating Schrodinger bridges between pairs of probability distributions.   More specifically,   it considers the Sinkhorn bridge approach of \"Plug-in Estimation of Schrödinger Bridges\" (Pooladian and Niles-Weed, 2024), and extends the theoretical guarantees.    In the original paper the authors assumed that one had access to the full source distribution, and only samples of the target.    In this paper, the novel contribution is the extension of this to the setting where only finitely many samples are available of both the source and the target (m and n respectively),   and provide an error bound on the Sinkhorn bridge iterates in TV distance.   \n\nAdditionally, the authors note connections with other Schrodinger bridge approximations, and demonstrate that they can be recovered for various parameter choices and initialisations of the Sinkhorn Bridge algorithm.      The authors additionally perform a number of numerical experiments as empirical validation of the theory.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n* The paper fills an important gap in (Pooladian and Niles-Weed, 2024), extending the bounds in that paper to the more realistic setting where only samples are available for both the source and the target.\n* The non-asymptotic bounds are provided for both the full estimator and the intermediate iterates.  This should, in practice, allow for practical guidance on sample size and number of iterations.\n* The neat observation that many recent algorithms ([SF]^2 M, BM2, DSBM-IMF, LightSB, etc) are special cases of the Sinkhorn estimator is really neat - and means that the theoretical guarantees in this paper carry over.\n\nWeaknesses:\n* The limitation on the ambient dimension rather than the intrinsic dimension is a potential limitation, but one which the authors readily address.   \n* The numerical experiments - while highly elucidating - do not really provide any insight into how well the method scales in general - as they are simple examples, really focused on confirming the theory.\n* The \"coup de grace\" that is missing from this paper, really whether the non-asymptotic results could be used to provide recommendations or guidance on how to choose the parameters, or even better provide some form of adaptive stopping (this might be asking too much given that the bounds are TV).\n* As reductive as it sounds, the paper isn't a huge step, neither conceptually nor theoretically from (Pooladian and Niles-Weed, 2024) - but still very interesting and useful.\n\nQUESTIONS:\nI appreciate some of these questions are a bit open ended, I think they would add to the quality of the paper, but I already think this is a nice contribution as-is.\n\n1. (The role of epsilon):   In practice, for Sinkhorn and related algorithms one needs to select the epsilon regularisation parameter wisely.   Depending on the setting, this is either selected through some cross-validation process, or sometimes annealed, or somestimes chosen as small as possible while maintaining numerical stability.    I can imagine that the sample size and epsilon are inter-related, particular due to the 'step-down' of the Holder exponent of the map (marginals -> coupling) in sinkhorn.    Can the authors comment on the numerical stability of the algorithm as epsilon->0.    Clearly, the bounds blow up as epsilon->0 in (14), etc, and this is expected, but in addition to this:  (i) is there some domain of validity, that can be elucidated through nuumerical experiment and (ii) do the bounds offer some recommendation on how to chose epsilon for given n, m?\n\n2. (Experiments and d):  The authros  mention one limitation of the convergence bounds, namely the dependence on ambient dimension - something that is different from the original (Pooladian and Niles-Weed, 2024) bounds.    Would the authors be able to provide numerical experiments to demonstrate that the tightness or lack of tightness of the bounds in the case to provide support for their claim that \"when the data are supported on a k -dimensional submanifold, the convergence behavior is expected to depend on the submanifold dimension k\"  as they state in the conclusion?\n\n3 (Stopping and guidance):  This might be asking too much, but can these bounds be applied to some sort of stopping rule for the iterates?   The bounds clearly suggest that there's a point at which there's no point in doing further sinkhorn iterations.   Can this be made explicit?  This would be  genuinely useful for practitioners.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}