Statistical Analysis of the Sinkhorn Iterations for Two-Sample Schr\"{o}dinger Bridge Estimation
Authors: Ibuki Maeda
Rentian Yao
Atsushi Nitanda
Venue: NeurIPS 2025 poster ()
Keywords: Schrödinger bridge
Sinkhorn algorithm
statistical analysis
Primary area: theory
TL;DR: We provide the statistical guarantee for Sinkhorn bridge method to estimate Schrödinger bridge

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
Ibuki Maeda'
Rentian Yao?
maeda. 1buki15540mall htt.a/aaaa..y.th.....aaa kyutech.
rentian20math, ubc.
Atsushi Nitanda
atsushi Stsushi_nitandaaee www.iniiinnaaaaaeeeeee staeeaasstrr..o edu.. sg
Department of Artificial Intelligence, Kyushu Institute of Technology, Japan
Department of Mathematics, University of British Columbia, Canada
3 Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore Centre for Frontier Al Research, Agency for Science, Technology and Research (A*STAR), Singapore College of Computing and Data Science, Nanyang Technological University, Singapore
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 solve the static optimal transport problem, recent work (Pooladian and Niles Weed 2024) proposed the Sinkhorn bridge, which estimates Schrödinger bridges by plugging optimal transport into the depeeeeeddnnt 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, 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, the squared total variation error of Sinkhomn bridge iterations: O(1/m + 1/n + ,4h) ) (r E (0, 1)). where m and n are the sample sizes from the source and target distributions, respectively, and & is the number of Sinkhorn iterations. This result provides a theoretical guarantce for the finite-saple performance of the Schrödinger bridge estimator and offers practical guidance for selecting sample sizes and the number Sinkhorn iterations. Notably, our theoretical results apply to several representative methods such as [SF] M, DSBM-IME, BM2, and lightSB(-M) under specific settings, through the previously unnoticed connection between these estimators.
establish statistical bound on
Introduction
In recent years, there has been increasing interest in Schrödinger bridge problems, with applications in mathematical biology (Chizat et 2022 Lavenant et al., 2024 Yao al.] 2023). Bayesian
posterior sampling (Heng et al.) 2024). generative modeling (De Bortoli et al. 2021 : Wang et al.
202I), among others. The Schrödinger bridge problem aims minimize the Beulback-Leibbee (KL) divergence with respect to the Wiener measure, the distribution of Brownian motion over the path space, given two fixed marginal distributions (u, v as source and target distributions.
The Schrödinger Bridge problem is also strongly connected to the entropic optimal transport (EOT) problem, which offers efficient and regularized alternative 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 well- established 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 dime-dependent drifts of stochastic differential equations to estimate
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

the Schrödinger bridge between the source and target distributions and v. Their analysis established statistical bounds, adaptive the intrinsic dimension of the target distribution v, in the one-sample estimation setting, where the true source distribution ja 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 ymm and in of m- and re-finite samples from and // are available.
Statistical guarantee for Sinkhorn bridge iterations. First, we establish the following statis- tical bound (Theorem D for the Sinkhorn bridge estimator by leveraging analytical tools for the Schrödinger bridge in the one-sample setting (Pooladian and Wiles-Weed 2024) and for entropic optimal transport (Stromme 2023b):
("1 m :)
E TV ,, Pr, =0
(1)
where Poo, and P mmn denote the true Schrödinger bridge and its Sinkhorn bridge estimator based on the optimal EOT solution, and p00 and p= are restrictions of them on the time-interval [0, 7]. 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 172 to at the cost of deterioration in the dependence on e and R, where & is the strength of entropic regularization and R the radius of the data supports.
Next, we analyze the intermediate iterations of the Sinkhorn algorithm and prove a convergence rate (Theorem for the corresponding path measures:
T RA T-T
R2
) 4k
(P=.10.r) pk. (ETV" m,
Lanh
(2)
where m.n denotes the Sinkhorn bridge estimator at the k-th Sinkhorn iteration for ymm and Un- Together, these results imply the convergence rate E[TV² (P=,..) Phijer) y p4h) (r E (0,1) 1)) regarding the number of samples m and n, and the number of Sinkhorn iterations k.
O(1/m + 1/n +
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] (Tong 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 [ Moreover, when DSBM-IMF and BM2 are initialized with the reference process, the estimator after k iterations matches the Sinkhorn bridge obtained after l iterations of the Sinkhorn algorithm (Proposition 2 Furthermore, Algorithm|] [ which learns the Sinkhorn bridge drift via neural network, is a special case of the procedure proposed in [SF] Consequently, our generalization 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 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.) [021] Vargas et al.) 2021). For example, [Bertton et al. (2019) proposed a sampling framework based on trajectory refinement via approximate dynamic programming. Pavon et al. (2021) and Vargas et al. (202T) introduced methods that directly estimate intermediate densities using maximum likelihood principles. Specifically, Pavon et al. (202I) presented 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 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 dramatic expansion of its applications in the field of machine learning. For example, there are diverse applications such as measuring distances between distributions 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 al. 2019) have been established. Furthermore, EOT is known to have a close relationship with the Schrödinger Bridge (SB) problem [[éónnrd] 2013). EOT in 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 metric space X, let P(X)) be the space of probability distributions over X, and P2(X') be the subset of P(X) with finite second-order moment. Let Rd be the d-dimensional Euclidean space and B(a, Rt) = {x1 ||æ--a|2z S Rt C R be the Euclidean ball of radius R > 0 centered at E IRd. For real numbers and b, we a V max b) and a AAD min(a, b). We write a S b denote that there exists constant C > O, independent to a and b, such that a < C.b. We denote by 85 the Dirac measure concentrated at the point x E Rd. We denote by 1 R the d-dimensional vector whose components are all equal to one. For measure and function f. we write n(J) J [(x) p(dz). Let TV(4,, (j4, v) be the total variation distance between two probability distributions μu and v. For & > 0 and any real number or real-valued function h, write exp, (h) exp(h/e). We denote the
d-dimensional probability simplex by Aa {a Rd E Oi 1}.
Preliminaries
2.1 Entropic Optimal Transport
Let [u, E (2(R)) and >> 0 be fixed. The entropic optimal transport (EOT) cost between, y and is
U c(x, v) dr(x, v) + t ( +
(T, == inf nell(ur))
(3)
with the cost function clz, y) = || vil Here, M(,, c PTRT d x RR)) denotes the set of couplings with marginals y and v, and
/ JadxRd
da (I, dr (z(
H(T 11)
log
d(u 8 v)
the Kullback-Leible divergence between π and μ 8 v. If not absolutely continuous with respect to μ @ v, we set ( μ X v) t00.
3

---

The optimization problem in Eq. 3 is strictly convex with respect to and thus it admits a unique solution Toc [, v). Furthermore, the problem has a dual formulation given by
OT(()))=
sup SEL (EL((W))EL(())
(F)))
where
@ (f.g) = / f fdu+ / gdv ((xp. (F())+
2 =11 2112 -1) du(x) dv(y).
(4)
The dual problem (4) admits the Schrödinger potentials fo0,0 and g00,00 as optimal solutions, and the first-order optimality condition implies
1 go0, (v)
500,00 (2) = -E =log
exp=
--21z --11--vi vll de(y),
(5a)
3d
00 (V) - log
expe
S (x)
(1)
2 || - vl dp(x),
(5b)
nd
which is known as Schrödinger system. Note that the solution constant-shift invariant, i.e. for any E R, (f + +a, D6c, a) is also solution to the Schrödinger system (5). It can be shown that with constraint ( O, the solution to (5) is unique. The optimal solution & of the primal problem 3 can expressed using the dual solution via
d d(u @v)
Po0.00 (2,
(5.y) = exx= (Feccolo)) g6..00 21100 u1B
(6)
We refer to Nutz 2021 for more details of entropic optimal transport problems.
2.2 Schrödinger Bridge
Let Ω = C([O, 1]; R be the path space consisting of all continuous maps from the time interval [0, 1] to IRd Let We be the law of a reversible Brownian motion on IR with volatility e. The Schrüdinger Bridge problem is then defined through the following entropy minimization problem:
inf eH(P W*) subject to PEP())
Po = 14, P =v
where P. E P(R") is the marginal distribution at time t E [0. 1] of the path-space distribution P. ie. X, P; X is a stochastic process with distribution P. Though W an unbounded positive measure over Ω the above optimization problem is still well defined as illustrated by Léonard (2013).
The Schrödinger Bridge problem closely connected with the EOT problem. Precisely, the optimal solution P Pe, E P(?) of Eq. (7) can be derived by first solving the EOT problem (3) and then connect the optimal coupling with the Brownian bridge We
(= Xo = xo, X = x1), i.e.,
|20.01
= = =f
p 20,00
Wi iioo do (IO.
(8)
The optimal solution P
00,00 can also be characterized as the path measure of the SDE defined by
dX. b (Xe, 1) di + VedB., Xo ~
where B, is the standard Brownian motion, and the drift coefficient is given by
Ey By (Y,
6x (z,t)
1-
Bo (Y, 2)]
with oc (y. 2) = exp (v)
1 201 7()+)) 1ly 2113). We refer to Chen et al. 200 1; Léonard
(2012 2013) for more details of the Schrödinger Bridge.

---

2.3 Sinkhorn Algorithm
Given the empirical distributions Pm 1 Σ 0x, and U, = - oy,, we consider the empirical EOT problem OT, (pm, Vn). Let fm.. and Im,n n be the optimal Schrödinger potentials. The Sinkhorn algorithm an iterative method for approximating these optimal potentials by alternately
updating the dual potentials p!A)) and gm,n as follows.
(R)
p(k+1) argmax THmmpn (J. (k) m. (e())
(++1) argmax P. ((k+1) mm .9).
(10)
(ET)(M.)
(k) We note that Jm.n, gm.. 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 E IR m be defined as Kij = expx (- 22,, - Y,112), and denote the vectors u(*) E m
and (4) E In by ii exps (f()) (X:)) and e (Y;) = (x (x) (Y))). Then the update rule I0 becomes
2 (k+1) m1m (K v ) (b++)) n1, @ (K" u(++1)).
where @ denotes element-wise division ! This reformulation greatly improves computational efficiency and enables large-scale applications.
Let dHib be the Hilbert (pseudo-)mettie E4. defined by
dHib(u, u') = log
max U U
max
Vu, e R..
It is obvious that (I, y) = 0 if and only if there exists À E R4 such that x = Ay. Let the initial values be (n("), ((())) (1m, 1... Denote by (u(*), r((k) the solution obtained after k iterations of the Sinkhorn algorithm (TO), and by (u" v²) (exp(fmnn// / exp 9 n /=)) the optimal solution.
Then, it is shown that (Franklin and Lorenz 1989 Peyré and Cuturi 2018).
drub (R) u² s X(K)2 2h-1 dHuD (v) v²). ( v²) S A(K)2* 2 drun (1)) v²).
V/(K)-1
where 7(K) jaxijid KuKu, This result KyeKs"
Here, X(K) defined X(K)
V>(K)+1
implies the exponential convergence of the Sinkhorn iterates (k) and v(())
2.4 Convergence Rate of One-Sample Estimation
Pooladian and Niles-Weed 2024) analyzed the oam-sa estimation task for the Schrödinger Bridge. In this setting, full access to the source distribution μ is assumed, while the target distribution
observed only through the empirical distribution v,, = 1/2 =j=1 8y, based on i.i.d. samples Y1 Y..
Let (f=c... 9 denote the optimal dual potentials solving OT, (u, 1,), and let P P E P(2) be the corresponding Schrödinger Bridge. Under Assumptions 2 and stated below, Pooladian and
Niles- Weed (2024] Theorem 4.1) proved that there exists constant Cy, depending only v and
geometric constants of its support supp(v), such that for any E [0, 1) the following holds:
Tv
(TTV((" (pn(00r), pr. s Co
E
d1
+
R² d,
(11)
Vn
(1
d. ))d 12
where d, is the intrinsic dimensionality of supp(u).
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.
'The standard Sinkhorn algorithm updates as (k+1) 1m, @ (*)) ((++ 1,, 2 (nk"T((+1) (k+1))
This update is derived from the formulation of the primal problem 3 that incorporates the regularization term H (IT), and there is substantive difference.
5

---

3.1 Sinkhorn Bridge in Two-sample Case
Given empirical distributions Jm and Uns the Sinkhorn bridge constructs the estimator Pk P(2)) for the Schrödinger bridge between true distributions şt and v as follows. First, the Sinkhorn algorithm runs for & iterations to solve (fmm Vn), yielding the dual potentials fm. and gm, Then, the estimator Pk is defined as the path measure of the following SDE starting from p m.n
(k)
dx, (()) (X dl VedBe,
Xo ~ M,
(12)
where the drift function bro.n is
t j= Yj exps (1) () 2[1-(] MY =113)
(0+) mnn (2,
(13)
I-1
j== exps
Fmmm (Y₁) 2(-1)
| - =112)
3.2 Convergence Rates of Sinkhorn Bridge
First, we analyze the statistical convergence rate of the path measure Pm.... P(22) using the optimal Schrödinger potentials mm, and Im.n- Specifically, Pm.. defined analogously to m.n by replacing the potential gin.n with its optimal counterpart Pn... For details the definition, see Appendix A.1 The analysis will be carried out under the following assumptions.
Assumption 1. The potentials Fs0,00 I Pm,n satisfy
( subsets of ball with radius R centered at the origin. Assumption 3. The measure u is supported on a compact, smooth, connected Riemannian manifold (N, fi) without boundary, embedded in Rd and equipped with the submanifold geometry induced by its inclusion into Ra; furthermore, dim N = d, 3. Moreover, admits Lipschitz continuous and strictly positive density with respect to the Riemannian volume measure on N.
f ( (y)u(dy) = 1o ( - ff n = 0.
Assumption 2. The supports of and
The manifold hypothesis suggests that data distributions often lie lower-dimensional manifolds, and statistical learning methods can exploit this structure. In the EOT literature, Strommee| (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, Poola dian and Niles-Weed (2024) demonstrated that the convergence dimension d and depends only on the intrinsic dimension of the target distribution v. Following these works, we also impose data-manifold assumptions solely on the target distribution v.
is independent of the ambient
The following theorem provides the squared total variation error between P Poo.o and P....
Theorem 1 (Statistical convergence rate). Suppose Assumptions the path measure of the true Schrödinger bridge for marginals Then, it follows that for any [0,1).
2 and hold. Let P> be
(Pa. 10.r p+.10.=])]
00,00
R2
dRt e²
R
9d, +1
(14)
(1 22
[0,r] where Poc,00 and mm,. are restrictions of oo and P P... on the time-interval [0,
[0,])
Remark 1. In this section and related proofs, we suppress non-dominant terms such as polynomial factors of de, geometric charucteristics of supp(y), and exponential terms of order (ddy) uniform constants, in the notation E
make the following observations: (1) Regarding the data dimensions, the intrinsic dimension d, governs the growth of the convergence rate, in Stromme (2023b); Pooladian and Niles Weed (2024). Although our bound, unlike theirs, involves the ambient dimension d, the dependence merely linear. Moreover, 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 by additionally imposing a manifold assumption on μr. (2) The degeneration of the estimated SDE toward a specific sample point Y arises as - 1 because the estimated drift bin.. points toward a particular sample Y; near causing the deviation from the target distribution at the terminal time = Therefore, it is common practice to restrict the time interval to [0. (T < 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 from 172 to at the cost of slight deterioration regarding. R. and d.
In the next theorem, we give the algorithmic convergence rate of pk m.n
P m,n
attained by running
the Sinkhorn iterations + oc.
Theorem (Algorithmic convergence rate). Under Assumptions 7
and
we get for any T E [0, 1).
E (P".
(P"...,,,, m.n
T
R
tanh
R² {
1-
(0,) where m. are restrictions of Pk the time-interval [0,
These results immediately imply the convergence rate (P p=. pak) for some r E (0, regarding the sample size m. and n, and the number Sinkhorn iterations k.
3.3 Drift Estimation using Neural Networks in the Schrödinger Bridge Problem
The drift estimator (33 of the Sinkhorn bridge between Hmn and Vn is defined as empirical average regarding V/-- As result, simulating the SDE (12) using a time discretization scheme (e.g.. Euler-Maruyame approximation) incurs an (n) -computational cost every discretization step. To improve computational efficiency, we consider a neural setwork-basee drift approximation.
Recall that W [20.21 is the Brownian bridge connecting and 21, so its marginal distribution at a Gaussian distribution Wi = NN1(1 - t)xo + tæ1,Et(( t)la). For dual
time t E [0, 1] potentials fm.m L' (Hm) and gmn E L' (p,) obtained by the Sinkhorn algorithm, we define the joint distribution (k) Im,n Im..,, and Mm, Vn- Using Tm,,, we define the mixture of Brownian bridge (a.k.a. reciprocal process): (k Wi ( (zo, (F0, x1) P(t). The drift function (/)) in Eq. (33 known to
an, analogously Eq. (6) by replacing the optimal potentials and μ.v with
(k)
Iso.ay have the following expression (Markovian projection (Shi et al. 2023)) (see Appendix B.2
X1
min (2,, Ex-I(K)
X; 2z
1
Therefore, see that the function (k) minimizes the following functional L:
C(b) = [(f
AR XR bxr,
21 zt 1 1 an (nz, 21) dt,
where 11(4) is the marginal distribution of (Aa) on time points t and 1. This suggests training a neural network by with parameter 0 to minimize the above functional. We note that samples from IT()) can be obtained by sampling (zo, .1) ~ Tii,n t ~~ U(0, T]. and x, We t| and discarding zo- Therefore, with this sampling scheme, SGD can be efficiently applied to minimize L(by). The detail of the procedure is described Algorithm 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]*M TTog et al. (023)) IightSB(
M) (Korotin et al. 2023, Gushchin al. 2024), DSBM-IMF (Shi et al. 2023; Peluchetti (023). and
BM2 (Peluchetti] 2024). First, we show that the optimal estimator produced by each of these methods
7

---

Algorithm Drift Approximation via Neural Network
input Joint distribution *m,, defined by potentials f E (pm),, g (Mn); neural network by output Trained neural network by that approximates the drift function
repeat
Sample batch pairs {ag, =i>== Tm,
Sample batch & 0/10, T] For each triplet (2), = (i), sample xt ~ W falggaa (Brownian bridge)
Compute the loss:
L(0) t N Σ
bo(0),
21
Update / using VI((0) until converged;
coincides with P m.n. attained by Sinkhorn bridge with k - oo (Proposition Morcover, when DSBM-IMF and BM2 are initialized based the reference process, the estimator after k iterations matches Ph. obtained after & iterations the Sinkhorn bridge (Proposition 2) Furthermore, follows from the proof of Proposition T that Algorithm [] which learns the Sinkhorn bridge drift via neural network, is a special case of the algorithm proposed in M. The proofs of these results statistical and algorithmic convergence analysis
postponed to the Appendix. Consequently, applies to these methods, endowing the theoretical guarantees.
Proposition 1. The optimal estimators produced by [SF] lightSB(-M), DSBM-IMF and BM2 coincide with P alluined by Sinkhorn bridge with k co.
Proposition 2. With initialization based the reference process, both DSBM-IMF and BM2 produce the same estimator as Ph.,. of the Sinkhorn bridge for all iterations k.
Experiments
In this section, we verify our theoretical findings through numerical experiments. In Section 4. verify the dependence of the statistical error the sample size presented in Theorem I and
the algorithmic convergence rate presented in Theorem 2 Next, in Section 4.2
we evaluate the
neural network approximation of drift coefficient 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 and
We use 3-dimensional normal distributions μ (N 1z) and v (O, B) source and target distributions, where B E R3x3 is a random positive definite matrix. Under this setting, Bunne et al. (2023 Eq. (25)-(29)) provided explicit expressions for the drift coefficient &e, and the marginal distribution p=s We use these expressions evaluate the errors of the drift 6".,. and the corresponding Schrödinger Bridge estimators P in the finite-sample settings. These estimators are approximated by running the Sinkhorn bridge with a sufficiently large number of iterations (A: - oo).
Statistical convergence. To verify that the estimator bin... converges to the true drift 6i when the sample sizes m and n increase, we draw m samples from μ and n samples from v. respectively, and compute the estimator bū, b (, for an arbitrary time E [o, 1). Subsequently, evaluate the mean squared error (MSE) against the true drift b oo (,)
MSEsamplc (m, t) = Ex Yi..., Yare
(2, t) 1
bo. (2,t)))) (PPS)t
To compute this, the norm (1 is approximated using Monte Carlo with 10,000 samples drawn from p The expectation over the samples is computed by averaging over 10 independent sampling trials. With a fixed parameter 0.1, we generated heatmaps for several [0, 1) while varying the sample sizes used in the estimator definition (Figure

---

Sinkhorn MSE mo..e.
10
101
10"
10
10"7
10
10
101 10'
10*
10a
10
10a
10*
101
10"
10a
Figure 1: Heatmaps illustrating saEsa (m, n, t) function of sample sizes and n for various time points 1. The error decreases roughly proportionally to (m- n n=1).
As evident from the figure, the mean squared error deteriorates as f + 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-)). This aligns with the result predicted by Theorem
Algorithmic convergence. To verify that the estimator Pm. (0), the iteration count & increases, we obtain m and independent samples from distributions j and v, respectively. We then consider the integral of the difference between estimators b t) and mm (,,t) over the interval r]. Specifically, we evaluate the mean squared error integrated over e [0,r]:
exponentially approaches p=
MSEsinkhom (k, f) dt L'' Ex Y. Viu...Yaa Ya
((a)) (mmm(,,)) (2,)) (P(Pm)h)
dt.
Time integrals the interval [0, 7] are approximated via Monte Carlo integration by uniformly sampling 1, 000 time-points. The norm [ is also estimated via Monte Carlo integration, using 1, 000 samples drawn from pe.t We set m = = =,,000 000 and compute the expectation by averaging With the regularization parameter fixed at we generate graphs of the integral values over [0, for varying k and multiple values (Figure 2 For all т values shown in the figure, the convergence rate exhibits exponential decay with respect to k. This observation corroborates the theoretical prediction Theorem
('MS dr
10 independent samplings.
0.005,
10:2
0.1
0.9
10-1
Figure 2: Integrated mean squared error
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 set to 1, 5, and 10. In
as function Sinkhorn iterations & for multiple integration intervals [0, r].
the figure, it is observed that as the number of iterations k increases, the drift (() rapidly converges to the optimal drift bm...
4.2 Experimental Verification of Neural Network- Based Drift Estimation
Next, we evaluate the effectiveness of the drift approximation by neural network using Algorithm In this experiment, we set & = 0.1, defined μ as the eight-Gaussians distribution and v as the moons distribution, and drew 000 independent samples from each. Using these samples, algorithm is employed to approximate the optimal EOT coupling T between jm and and, via Algorithm] ncural network approximation by of the drift br,,, is obtained. We employ an 4-layer neural network with 512-512-512 hidden neurons for by, and train it using the AdamW optimizer with a learning rate x weight decay of 1x10- and a mini-batch size of 4, 096. Finally, starting from each sample of pm.m we simulate trajectories using either (/k) or the neural network drift be. and present the results in Figure Trajectory simulations are performed using the Fuler-Maruyama approximation with 1, 000 discretization steps.

---

Figure From left to right, the simulation results of the Schrödinger bridge using the estimated drifts bm.ns (6) and (1))) obtained by terminating the Sinkhorn iteration after 1, 5, and 10 iterations, respectively, the optimal drift m and the neural network apprrxxmaae drift by.
Conclusion and Discussion
In this study, we provide comprehensive analysis of statistical guarantees for the Schrödinger Bridge problem. Specifically, establish thcoretical 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 statistical convergence analysis in the two-sample estimation setting for the Schrödinger bridge estimator with oc, which demonstrates a statistical convergence rate of O (1 + ) 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 clear estimation of the number of samples m, and Sinkhorn iterations required to achieve a desired precision. Experimental results align with our theoretical analysis, confirming that the error decreases at rate of approximately (m -1 + - with respect to the sample size, and that the error decreases exponentially with respect 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. ) 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 thcoretical framework beyond the Sinkhorn algorithm.
Acknowledgment
This research is supported by the National Research Foundation, Singapore, Infocomm Media Devel- opment 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 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.
References
Bernton, E., Heng, J.. Doucet, A., and Jacob, P. (2019). Schrödinger Bridge Samplers.
Bunne, C., Hsich, Y.PP., Cuturi, M., and Krause, A. (2023). The Schrödinger Bridge between Gaussian Measures has Closed Form.
Chen, Y., Georgiou, T., and Pavon, M. (2021). Stochastic control liaisons: Richard sinkhorn meets gaspard monge schrodinger bridge. Siam Review, 63(2):249-313.
10

---

Chen, Y., Li, P., Liu, D., and Wang, H. (2023). On talagrand's 's functional and generic chaining.
Chizat, L., Zhang, S., Heitz, M., and Schiebinger, G. (2022). Trajectory inference via mean-field langevin in path space. Advances in Neural Information Processing Systems, 35:16731-16742.
Cohen, S. N. and Fausti, E. (2024). Hyperbolic contractivity and the hilbert metric probability measures.
Cuturi, M. (2013). Sinkhorn distances: Lightspeed computation of optimal transport. Advances in neural information processing systems, 26.
De Bortoli, V., Thortoo, J., Heng, J., and Doucet, A. (2021). Diffusion schrödinger bridge with applications to score-based generative modeling. Advances in Neural Information Processing Systems, 34:17695-
De Bortoli, V., Thorton,, J., Heng, J., and Doucet, A. (2023). Diffusion Schrödinger Bridge with Applications to Score-Based Generative Modeling.
Fortet, R. (1940). Résolution d'un système d' l'équations de m. schrödinger. Journal de mathématiques pures et appliquées, (911-))83-100
Franklin, and Lorenz, (1989). On the scaling multidimensional matrices. Linear Algebra and its Applications, 114-115:71777338
Genevay, A., Chizat, L., Bach, F., Cuturi, M., and Peyré, G. (2019). Sample Complexity of Sinkhorn Divergences. In Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics, pages 1574-1583. PMLR.
Genevay, A., Peyre, G., and Cuturi, M. (2018). Learning Generative Models with Sinkhorn Diver- gences. Proceedings the Twenty-First International Conference on Artificial Intelligence and Statistics, pages 1608-1617. PMLR.
Gushchin, N., Kholkin, S., Burnaev, E., and Korotin, A. (2024). Light and Optimal Schrödinger Bridge Matching.
Heng, J., De Bortoli, V., and Doucet, A. (2024). Diffusion schrödinger bridges for bayesian computation. Statistical Science, 39(1)900-99
Korotin, A., Gushchin, N., and Burnaev, E. (2023). Light Schrödinger Bridge. hutps:/arrx..oggaas/331000111.1
Kullback, S. (1968). Probability densities with given marginals. The Annals of Mathematical Statistics, 1236-1133.
Lavenant, H., Zhang, S., Kim, Y.-H., Schiebinger, G., et al. (2024). Toward mathematical theory of trajectory inference. The Annals of Applied Probability, 34(1A))428-500
Lemmens, B. and Nussbaum, R. (2013). Birkhoff's version of hilbert's metric and its applications in analysis.
Léonard, C. (2012). From the schrödinger problem to the monge-kantorovich problem. Journal of Functional Analysis, 1899199.89.
Léonard, C. (2013). A survey of the Schrödinger problem and some of its connections with optimal transport.
Nutz, M. (2021). Introduction to entropic optimal transport. Lecture notes, Columbia University.
Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z. Gimelshein, N., Antiga, L.,, Desmaison, A., Kopf, A., Yang, E., Vitoo Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., and Chintala, S. (2019). Pytorch: An imperative style, high- performance deep learning library. In Advances Neural Information Processing Systems 32, pages 8024-8035.

---

Pavon, M., Trigila, G., and Tabak, E. G. (2021). The Data-Driven Schrödinger Bridge. Communica- tions on Pure and Applied Mathematics, 7407):155--153.
Peluchetti, S. (2023). Diffusion Bridge Mixture Transports, Schrödinger Bridge Problems and Generative Modeling. http:://aaxiioorgaaa/222000000
Peluchetti, S. (2024). BM2: Coupled Schrödinger Bridge Matching.
Peyré, G. and Cuturi, M. (2018).
Computational
Optimal
Transport.
https://arivv.rrgaa///88883....
Pooladian, A.-. and Weed, J. (2024). Plug-in estimation of Schrödinger bridges.
https://ardiv.or//aa//2448 11101//....0...1...11.11 11686v1.
Rüschendorf, L. (1995). Convergence of the iterative proportional fitting procedure. The Annals of Statistics, pages 1 1160-1174
Shi, Y., Bortoli, V. D., Deligiannidis, G., and Doucet, A. (2022). Conditional simulation using diffusion Schrödinger bridges. In Proceedings of the "hirty-Eighht Conference on Uncertainty in Artificial Intelligence, pages 1792- 1802. PMLR.
Shi, Y. De Bortoli, V., Camphell, A., and Doucet, A. (2023). Diffusion Schrödinger Bridge Matching.
Solomon, J., de Goes, F., Peyré, G., Cuturi, M., Butscher, A., Nguyen, A., Du, T., and Guibas, (2015). Convolutional wasserstein distances. ACM Transactions on Graphics, 34(4):66:1-66:1
Stromme, A. (2023a). Sampling From Schrödinger Bridge. In Proceedings of The 26th International Conference on Artificial Intelligence and Statistics, pages 4058-4067. PMLR.
Stromme, A. J. (2023b). Minimum intrinsic dimension scaling for entropic optimal transport.
Tong, A., Malkin, N., Fatras, K., Atanackovic, I.., Zhang, Y., Huguet, G., Wolf, G., and Bengio, Y. (2023). Simulation-free Schrödinger bridges via score and flow matching. hutps//arrv..og//a//2/07...3777
Vargas, F., Thodoroff, P., Lawrence, N. D., and Lamacraft, A. (2021). Solving Schrödinger Bridges Maximum Likelihood. Entropy, 23(9):1134.
Vershynin, R. (2018). High-Dimensional Probability: An Introduction with Applications Data Science. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, edition.
Wang, G., Jiao, Y., Xu. Q., Wang, Y., and Yang, C. (2021). Deep generative learning via schrödinger bridge. In International conference machine learning, pages 10794- 10804. PMLR.
Yao, R., Nitanda, A., Chen, X., and Yang, Y. (2025). Learning density evolution from snapshot data. arXiv preprint 173333
12

---

NeurIPS Paper Checklist
1. Claims
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Question:
Answer: [Yes]
Justification: The claims in the abstract and introduction accurately reflect the theoretical claims in the paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long it is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
Justification:
We discuss the limitations of our work the section "Conclusion and Discus-
sion".
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested few datasets with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or speech-to-texx system might not be used reliably provide closed captions for online lectures because it fails handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairess.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
13

---

Answer: [Yes]
We list all assumptions in the main text and include complete proofs in the Appendix.
Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proof' in the paper should be numbered and cross referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors encouraged provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided not)?
Question:
Answer: [Yes]
Justification: To support reproducibility, we provide information on the experimental setup and hyperparameter configurations in Section 4.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reprodueibility may be tricky in some cases, in which authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
14

---

Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Question:
Answer: [Yes]
Justification:
We submit the source code and data through OpenReview.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurIPS code and data submission guidelines (https://nips.co cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (c.g., for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurIPS code and data submission guidelines (https
//nips //iiss.cc/pill wttpcccpppppppppeeeeeeeeeeeeeeee httpa///ggiieee/eeeSebiiisssssss /Cc////oddiiisssseeeeeee ssiiiiiiccyyyyyyy for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Question:
Answer: [Yes]
Justification: We provide information on the experimental setup and hyperparameter config- urations in Section 4.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification:
We report the average accuracies over 10 independent runs.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi-
dence intervals, statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for
example, train/test split, initialization, random drawing of some parameter, or overall with given experimental conditions).
15

---

The method for calculating the error bars should be explained (closed form formula, call a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative enror rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification: We use a single A100 GPU card for our experiments. All examples are reproducible within one hour.
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't Itdn't make it into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics ptttts://neurips. httprrpp..../pbblcc/cchcccccaudd
Answer: [Yes]
Justification: Our research is theoretical study on the Schrödinger bridge and does not raise any ethical concerns. Therefore, the paper does not violate the NeurlPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is special consid- eration due to laws or regulations their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative socictal impacts of the work performed?
Answer: [NA]
Justification: Our research conducts statistical analysis of existing methods for the Schrödinger bridge problem. As such, the paper does not present any direct positive negative socictal impacts.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
16

---

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Our research focuses on theoretical studies of the Schrödinger bridge problem, and all experiments are designed validate the theoretical findings. As such, the paper does not pose any risk of misuse.
Justification:
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification:
We cite the original papers for the assets used in our experiments.
Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
17

---

For scraped data from a particular source (e.g., website). the copyright and terms of service of that souree should be provided.
If assets are released, the license, copyright information, and terms use in the package should be provided. For popular datasets, paperswi thcode. com/ com/dataseta has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: Our research focuses on theoretical studies of the Schrödinger bridge problem, and all experiments designed validate the theoretical findings. As such, the paper does not release any new assets.
Guidelines:
The answer NA means that the paper does release new assets.
Researchers should communicate the details the dataset/codeemmodll as part their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset used.
At submission time, remember anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsoureing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and sereenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
This paper does not involve crowdsourcing or research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then much detail as possible should be included in the main paper.
According to the NeurlPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [NA]
Justification: This paper does not involve research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
18

---

Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if' applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: Our research does not rely on LLMSs for any important, original, or non- standard components of the core methods.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to our LLM policy (https://neuri)) attt::e//eearrrg..c/cooooffreeer /ctp:///e/eeeeeecc//22/////./. for what should or should not be described.
19

---

Appendix
A Omitted Proofs
A.1 Proof of Theorem
We here provide the complete definitions of path measures P mnn Pr m.n E P(2). Using the Sinkhorn
iterations (m,, gm,n) with respect to and Vn, we define the drift function b as follows:
Σ 5== Y, expe (Com.n (8) (Ym) 20-1) || =113)
600 (2 (2,
2
Σ == exP: (9min (Y₁) - 2(1-1) +) ||8, - =1(2)
1
Σ ブー= =1 Ym, (Y,, , Y
-x
j=1 Mm. (Y,,
(k),e
where
(k),.
(Y,, (0,)aoY)333
exp:
Im. (Y2)
(k)
2(1 ||5 - =112
Then, p(k) defined as the path measure of the following SDE:
dX, ((#) (X.) dt + ded Xo μ.
Pa P()) is also defined in the similar way by replacing the Sinkhorn iterations the correspond- ing optimal Schrödinger potentials. That is, using the optimal Schrödinger potentials (fmm. Fm, with respect to pm and Un. we define the drift function b as follows:
Σ Yy exp. (om (Y3) 2(1-t) +) IY - 21
1
b (z)
1-1
Ej= expe (9tn (Y₁) 2(1-1) |Y, 2 211
1
Σ =1 Win (Y,, 2)
-2
(15)
1
> 3 mm (M
n mit
where
ret. Mm, (Yj, 2) = exp, 9in, (Y₁) 2(1 t) ||, -
Then, P is defined as the path measure of the following SDE:
dX2 by n (X,) dt + V Xo μ.
(16)
Theorem 1 (Statistical convergence rate), Suppose Assumptions the path measure of the true Schrödinger bridge for marginals p. Then, follows that for any E [0, 1),
2 and 3 hold. Let P oo be
(0.. p+ 00,00
min
R2
dR 2²
R
9d, +4
1 (1+1)
(14)
11 du
1-T
where P and [0.])
are restrictions of P and P on the time-interval [0,
The proof of Theorem is based on the main idea presented in Pooladian and Niles- Weed (004 Stromme 2023b): we introduce the following entropic plan.
Proof of Theorem
d (z. v) = (F (z) + goc (v) 2 | =lli) du(x)) du, (v).
(17)
20

---

Here, the function formn RR - IR is defined as follows:
n
fx) = -E lolog Σ
exp2 (N
(Y))
- || - Y
j=1
Denoting by P Poo.. the measure obtained by substituting the coupling (17) into ( Proposition 3.1 in
Pooladian and Niles-Weed (2024) implies that the drift boen with law Poo.n expressed as follows:
Ej=1 Yoo,00 (Y,, 2)y; Σ 160,00 (Y,,2) 2)
so (2)
By incorporating the path measure Poo.n into the bound via the triangle inequality and subsequently applying Pinsker's inequality, we obtain:
[[TV'(P",10.) 0c,00
[=*(0.=]]]
& ETV poo.)) + E (Pl0. p=] 20,00 s ( P + ( 0 P [0.1l)]
oc,
We analyze these terms separately. For the first term, we apply Proposition 4.3 of Pooladian and Niles Weed (2024)
R²
B|H(PP:000))10001) X
a(1
72
For the second term, we apply Girsanov's theorem to derive the difference between the drifts.
E EiH (P0.7] [0.]]
00.7t
S
E
bocnn
by
(2 dt
1
Σ== 700,00 Y,,2)y,
ΣE=1 f== ndt m 2) dt
E
t)²
j== 10,,00 (Y),2)
Σド= Tmi (Y,
LP(PP..))
1
2 +
dR
R
9d,
(1 1 d
(1
t)2
E²
0
T
2
dR
9d +4
+
72
where for the second inequality, we used Lemma
Lemma 1. Under assumptions of Theorem 7 we have
Ej= Toe. (Y Σj= == (Y,,))
E net (Y,,
E
Σj= Poi (Yz,,)
dR
+4
x
2²
n
Proof of Lemma 7
We perform the expansion while keeping Y, and = fixed
E vo. (Y,,))y,
Ej
2x. immn
(Y,, 2),,
= 100,00 (Yj,z)
+ = mm
(Y,, 12
(Y, (Y,, 100,20
Neih (Y,, 2) Ej=1 x, (Y,, 2)
Y,
Σj=1 1= 000 (Yj,2)
12
Noc 2)
2ith (Y, 2)
Σ 700,00 (Y,,2)
プラ 1= Vin (Y),
1
21

---

From the case (p.g) (1, 00) of Lemma 12 we have
プz=1 00,
2);
2j=1 min (Y3, 2
プソ= 700,
20,
Σラ= Yin. ({)²
R
(x,)
1
|| =112
(Smmn(())) 2(1
| 21
max J
2(1 - 1)
Wn 120,00 - (mm
By applying Lemma
we obtain
j=1 700,00 (x,,
プド= til (Y,, 2)} 2
E
==1 000 (Vz=z)
== mim (V (Vz:z)
R²
|9,,
000,000l1..00
g²
dR
R
9d
1 + -
6²
62
e
m
11
A.2 Proof of Theorem
Theorem (Algorithmic convergence rate). Under Assumptions and 2
we get for any T E [0. 1).
5 R
R
4k
(P%.or) m.n
taanh
I-T E²
where (0.)
are restrictions of Pk Pm,, on the time-interval [0, r]
Proof of Theorem
We start by applying Girsanov's 's theorem to obtain difference in the drifts
E p*. p
% E 11/0
b mn L2(Pm.t)) di
2
//¹ E
E 1=1 mii (Y,, -Y,,
Eド= f=1 Mm,n
(Y,.
dt
1
E)=1 Poin (o, (Y),,)
= Ym, (Y,,:)
hut
La(P)())
R
R22
4k
tanh
dt
J0
(1
t)2
c²
E
T
R6
R²2
4k
tanh
where we applied Lemma
2
Lemma 2. Under assumptions of Theorem 7 we have
Σ - Mmn (Y,,
Σ -ー mm, (Yj, 2)y
h.t
S R
R22
4k
tanh
E m m (Y,,
Σ
k.t n
(Y,
c²
E
(P(()))))
Proof of Lemma
2
We perform the expansion while keeping Y; and = fixed
Σn Imm (Xg,2)X,
ΣØ= Aet (Yg, 2),
Σ/= Mmm (Y),z)
kil ΣE Yon (Y₁,
22

---

Mm (Xj12) 2)
mm (X,, K,t
j=1 Tim (1)
Σラ Tm. (23x 2)
Mmn (Yj.
Mn (Y,, 2) k
||X/|12
a.t 1=1 Mm (Yj,
7m, (Y 2) Tmi, (Y 2)
Imin (2),2)
R
プー= mi m (8,,2)
Σラ=1 7m (1)
We define w', wh, v², v E R" as follows:
mi (Yj, 2)
Mmn
R.t (Y₁, z)
w
W
ブド= j=1 Ymin (Y,, 2)
EØ=1
7mm (Yz, z)
Nit
Im, (Y₁)
(k) (omi((y))
Vj = exp
= exp
where both w' and w satisfy the conditions of probability vector, i.e., wj. wj.w. > 0, E wj. Σj=1 w 1.
Σ Tmm Soan((y) (Y,;))) 2)
Σ j=1 mm (oannyyy 2),;
Σ 11 Min (V,,
j=1 Yman (Yj, 2)
ax (Y,,
Nit (Y,,
R
Σj= Vinn ()
j= 7m, (Y,,z)
R² I|w"
I R² dHun (wn, whi
(using Lemma 13
R² dhubl (v", vlia
(using Lemma [4]
s Rd ( AK dHilOV' vV²
R ((K) 1K dHmb(v", 1m)²
-RAA(K) (1O8 ( (maxxxxp exp= (Jm, (Yi)) (mm))()))) (maxexxp ( (- (Xi)
s 182 (())k (log (exp, (4R²2) exp, (41?²)))
(using Lemma 3
R2 S क ((K)at
Since
Ki Kzt Kat
(xxx:-X,) y) s exp, (4R²).
((K) max 1ikl
max exp
sikl
the quantity A(K) can be upper bounded as follows:
ví(K-1
A(K)
Vx(K) +
exps (2R²)
( 2
(x > 0) is increasing function
exp. (2R2) +
I +
tanh R²
Since the upper bound of X(K) is independent of Y, and the proof is complete.
23

---

A.3 Proof of Proposition
1
Proposition 1. The optimal estimators produced by [SF] lightSB(-M) DSBM-IMF and BM2 coincide with m,, attained by Sinkhorn bridge with k oo.
Proof of Proposition It suffices to show that the drift estimated by each method coincides with bm
Proof of optimal estimator for [SF]
The SDE representation the Brownian bridge W 200.21
with endpoints (za, 21) is given by
X dt + VedBi, Xo Fo.
dx,
(18)
The marginal density of the Brownian bridge at time is (2 .n) = lan, et(1 (a). There exists ordinary differential equation (ODE) that preserves the same marginal distribu- tions pe (ar/2 v1) as the SDE (8). This ODE, referred to as the probability flow ODE, takes the form
Φ1 Xt
dx,
V log (X dt
1-t
1 2l
1
1
X, +
28 5o
dl
21(1 - t)
2(1 - 1)
"n(((x (Xxx|ao 21) dl.
Consequently, the drift term of the Brownian bridge SDE can be decomposed into the sum of the probability flow ODE drift un (2r x1) and the score function V log pe (
T1 Zt t ug (10, #1) + v (1))
In the M framework, let T denote the optimal coupling for the entropic optimal transport (EOT) between jm and Pn, and consider the mixture distribution II" um W Wf Under II", two neural networks uo and Ri are trained to minimize the following objectives:
|=0.=]"
V [lvo(X.,,)) - (X, X x,)11 dt
= arg min EX-II
To (X.,)) V log p. (X,|Xo X )1l dt
8p arg min E Ex-I-
The estimator for the Schrödinger bridge drift in [SF] is then defined as
Ug- (xt, t) 2 Sp- (xr,t).
When both models are sufficiently expressive, the optimal solutions admit the following conditional expectation representations:
vg- (Ie, t) Ex-m- (x X1) | X. xz]]
8 (rx, t) Ex-n- (V po X1) |
X, = F..
Therefore, the drift estimator in M can written as
eg. t) + 2 S. (x,,
Ex-m- [u] (2ra Xo, X1) |X, 24] EX-I- (V log (xa)Xo, X1) m.)
-
ug No. X1)
Vlogpt (2el X1) XX, = 2z
Xi Zt 1-t
-Ex-II
X, = 2t
24

---

6 (zt, t).
Proof of optimal estimator for LightSB(-M)
Let S(1)) denote the set of Schrödinger bridges whose source distribution is jr. In lightSB(-M), the drift estimator 9u is defined with function I + I. follows:
go 1) = >V log ( ar. e - ()]a) exp
|| 112
v(x') de''
and let S. be the law of the SDE
dx,- = 9r (Xt, dt VedB,
Xo pm.
For any coupling Tm.n E [M P,), consider its Brownian- bridge mixture II = Tm.n We by (Gushchin et al. 2024 Theorem 3.1),
(m0.11 Then,
argmin HII 8) IT. SES(um)
Restricting S to Sv, one obtains (Gushchin et al.
2024
Theorem 3.2)
(x,,)) X1 X HIII S.) Cm + 28 / /
1
dII, XI)
1-1
where I1 denotes the marginal law II at times i and 1, and C(Tm,,) is constant independent of S.. lightSB(-M) trains parameterised model vo that minimises
Iro (X,,)) X1 X1 0/- argmin HII S.,,) = argmin [
2
au (X,, dt.
The drift Geg- obtained from the learned ey- serves as the drift estimator produced by lightSB(-M).
When the model capacity is sufficiently large, Mrg- satisfies
X1 Zt
Jug- (Ie, t) IX-II-
Xx
21 = b',nn (2t, t).
Proof of optimal estimator for DSBM-IMM and BM2
Let F(Uy) and ((vs)) denote the laws induced by the following SDEs:
dX, = 8y (X,, + V Xo ~ Pm:
dx, = 0o (X,, t)dt + Ved X1 ~m
Here, in the second SDE, represents an infinitesimal negative time step, and B, denotes the time-reversed Brownian motion.
Given drifts vp and uh, define the following loss functions:
[%o' 10 (0)(X
X1 X,
Toooward (1y; (v) = Ex~B(By) los We
dt
1
Xx - Xo Linckward Ub3 Up) = Do lra L''l (xa(Xe)t)
dt
t
In DSBM-IMF, starting from initial drifts of and u/ we perform the following iterative updates:
(k+1) argmin Leorward (Vy: (t)), (k+1) argmin Laackwand (Ub: of (k+1))
vy
%
In BM2, given initial drifts of ( of (k+1) argmin Lowward (1y; 26 (2)))
and Do (0) the iterative procedure is:
(k-1) argmin Loackward (bb; 2/ (1)))
25

---

For both methods, F(ug)) (1)) and B(rg") converge to II as h: oo (Shi et al. 2023
Theorem
8), (Peluchetti 2024 Lemma 1).
Therefore, letting vj and vy denote the limits of of (e) b
and vbb
(6)
as k - we have that vy equals
X1 2t
(1),) Ex-B(ug)an W 1=0:=1
X₂ =
X1 2t
-x-IT
X₁ =
1
Um, (Fis, 1).
A.4 Proof of Proposition 2
Proposition 2. With initialization based the reference process, both DSBM-IMF and BM2 produce the same estimator as Pk of the Sinkhorn bridge for all iterations
m.n
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. ( and
generates a sequence path measures (P()) MEN according
p(2k+1) argmin P
H(P p(2k))
Po p4
p(28+2) argmin
H(P p(2k+1))
P Un
We initialize with P(0) W. is known Léonard 2013 satisfies
Nutz 2021 that the sequence (P(K)) KEN
p(2k+1) = n (k++1) (k)
We
120,211
p(2k+2) Mitt (lle1))
We W1o.e:
(k+1)
Here the coupling zf is defined by
á y) exp, (f(z) + g(y) 111= 111--vli v112) dpm(x) dvn(y).
Hence suffices to show that the drifts estimated by each algorithm coincide with b
When the initial drift for DSBM is set to Ub B(x))) (0)) ), F() ()), B(ug ), F(o B(u,)) )... coincides with the IPF sequence (P(K) keN
(0)
0. the sequence
Shi et al. 2023 Proposition 10). Consequently, vf
(k+1) coincides with (k)
vf (k+1)
(xt, 1) E Bx~B(u Dou Woi (sgwe]
X1 Ft.
X, 2t
1 -
X1 It
E
X₂ =
X mia om We
t
X1
Ex
X, 24
1
B(k) (2t, t).
(0)
(0)
When the initial drift for BM2 is initialized vf B(u (o), F(og)), ((u(")). )... coincides with the IPF sequence (i(k)
vb
0, the sequence
kEN Peluchetti
2024 Theo-
rem 1). Thus, of (2k+1) given by
(2k 1) (II, t) = =E xx ) We 130
X1 Ft
X₁ = Zt
26

---

X1 It
Nup (B) (k)
X, It
W 20.1
t
X1 Di
X = It
t
Similarly, the sequence F( B( F(uf))) )
corresponds to IPF with the update order
reversed, and the same conclusion holds for ff
(2R)
Technical Lemmas
B.1 Lemmas used for Theorems and 2
Lemma 3 Stromme, 2023b
Proposition 14). Under assumptions of Theorem 7 we have
f loo, ol loo S 2R², alfm.alls il lloo I 4R²
In the following, Po denotes the Radon-Nikodym derivative regarding μ @ v, defined in Eq. (6). and Pm,, is also defined similar manner replacing for, Ixc.c with f Im,
Lemma 4 (Stromme 2023b Proposition 15). The population dual potentials fx and Ixo.on. are 2R-Lipschitz over supp(u) and supp(v), respectively. The extended empirical dual potentials fm and Fm,n also 2R-Lipschitz population density Po and the extended empirical density pm,, are euch AR Ee--oo--iipchie each of their variahles over supp(u $ v)-
supp(u) and supp(v), respectively. In particular, the
Lemma (Stromme
2023b Lemma 25). Under assumptions of Theorem 7 with probability at least
20
inf =Esupp(y)
Vn (B
E
2
(19)
2R
R
Lemma 6 (Stromme, 2023b Lemma 26). Under assumptions of Theorem 7 we have
R d,
(20)
And, with probability at least
Le -2002/ /e
R dp
ll Il (NSV)
(21)
Lemma Stromme
2023b Lemma 27 adapted). Under the assumptions Theorem 7 ife/ is
sufficiently small, then
R
+4
1 1
Em 00 (I S
+
(22)
E²
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 slight modification of the proof Stromme (2023b).
Lemma 8. Let (ml = (Am - (1) (p (-, v)). Then the following holds:
R²
2d.+2
sup Un( S
yεsupp(y)
Proof. We first note that ((p (v))) = since the marginal of =oo,00 on is and hence
Um(y) (y) pm (o (c) - 1. Hoeffding's lemma, with Poo, $ R de implies that there
exists a constant C > 0 such that
m
E Exxp {sU, (x)) - exp -
11)
m
27

---

C
R
2d.
exp
8m
C
R
2d
82
exp
4m
E
Consequently, Um (v) is subgaussian with variance proxy random variable g by
R
2ds
Orlicz 4/2mnorm is defined for
Im
| = inf { > O: Ee ///e <2
Chen et al. (002 [0
Theorem 3.7 and Theorem 4.1) with p(x) = =²/2, = 2 and d(y. =
(y)
Um (v')
ez imply the following estimates:
1/2
E sup
Um(
0 Vlog.N(supp(u))dd....
sup
d(y.y'..
(23)
vEsupp(e)
v.wy Esupp(v)
Let Zi ;= Dio. (Xi, y)
Poc do (X,, s/). Since each Z, is an independent, zero-mean random variable, by Vershynin (2018 Proposition 1. d(y, v²) is bounded above as follows:
d(y, 3¹) = Pm oo (- v) -o (1, (:. B3))
2
m
i=1
1/2
Σ
2Z
For each Z, we have || 1l S -1. - ( (et) hence
1Zillez =II25o,00 (X,, 8) P (X,, v') 1/2 2 1o 1.0.
By leah - s S caavbl a - bl and Lemma for any x, (x. - oc (x. s/) S 1o L (W@V)
(2, v) log (z.
d2 4R
1
v'112
8
R
3/ 112.
Therefore,
2d
1/2
R
d(y.v') x
Σ
11y - v l1
m
du+1
R
mm [y 1/ ll2
1
R
dut1
|1
1/1/2-
(24)
Vm
This concludes ((supp(p), d,, s N (supp(y), || 112, A)) (A constant hidden in the above inequality).
C du+t /
(1)dxt1 where C is
We evaluate the first term in (23) by using Proposition 43 in Stromme (2023b) that shows there exists 0 < < R such that, for any 0 < 8 S Cy. the covering number satisfies 5-ds S (ssupp(v), || 112. 20- ddd.
Joop((supp(v),eo dr
28

---

log.N (supp()):l ll 112. A
r
dr
A
C
R
dut1
mm
2n
A
VlogW (supp(v): 22, 5) d6
0 =
r A
0
2R
A
(suppps))
22. dð
Vlog N (supp(r), |l 112. 8) d6
Vlog8 4, dō A(2R cv) Voggo c
2 A
/-/ 11120" du MARVA ARVd,
1) au:= log
A
2 AR
R
d++1
(25)
m
where S also hides the intrinsic dimensionality de.
From 24 the second term of (23 is bounded as
d.+1
R
d.+1
sup
d( S
sup
11 - 1/112
(26)
3.9/ Esppp()))))
9.y Esupp(e)
vm
Combining the results of 26 and
25
completes the proof.
d,+1
d
R2
R
2d, +2
sup yEsupp(v)
( z
m
m
m
Lemma 9. Let (x) (V/ v) (o ().
Then the following holds:
dR²
R
2d,+2
E
sup #Esupp(μ)
VT((1)
7
Proof. Lemma can be established by minor modification of the proof of Lemma The only difference is the evaluation of the covering number: the bound over supp(μ) is replaced by that over ((0, R). This substitution introduces a dependence on the ambient dimension d.
Lemma 10. It follows that
dRa
74
E - 1 S
62
c²
.
+ m
Proof. Let
1 Σ exp, (9 c(2, Yj))
fo),n(x) := - lelog
Using Joo.ns we evaluate
|| fth lili.-x (w) as follows.
Bl 1 Ellf S Bllfoo,n -m. -m (27)
For the first term, the Schrödinger system implies
(x) - (x) = elog
(fx, Σ
(x))
exp. (sc - clz, YT))
1=1
29

---

n Poo,oe (ce, Yj)
E log Σ
==1
=loog (1+ (z()))
where Vn (x)) (Un- 10 (x, .)) and we used vI (z,
1.
For any 17 (0, 1), the following holds:
f (z) (z))( 2º sup =Esupp(a)
|log(1 + (x(22)
1 sup #Esupp(H)
VV < m/ sup =Esupp(H)
|log(1 + (x))
+ =
1 #Eaupp(u)
sup
[V 2 7)
sup zEsup((a)
[log(1 + Vn (x)))²
(28)
For the first term in 28 the mean value theorem implies that for |u| < we have llog(1 + 2) -
|u| 1-|u|"
Hence,
Vn(x))
E
1
sup =Esupp(p)
VV,
m7]
sup лεsupр(и)
|log(1 + [=()))
E sup
(1 (1-17) 7)
zEsup((a)
For the second term in inequality yields
(28) since exp. (-10R²) S 0o S (10R) an application Markov's
R4
E
sup
|V 2 n]
sup zEsup(())
|lg[[1 + ([())|
S
sup #Esup(())
[V,(1)] 2 m)
#εSupp(H)
R SuS pnesupp)))))
(x))
Setting 77 =1/2 and applying Lemma yields
Rdd 2²
||| oo (x) ( S 1
E
sup
[x)(2))
xESUpP(()
2 + d.R.
2d,
(29)
For the second term in (27) Eq. (6.4) Stromme 2023b) implies
-1/2
[z)) Smn (x) S 1190e.00 In (22
sup =Esupp(v)
Un
2, 2R
Let denote the event on which Lemma 5 holds. From Lemma and P
< n
it follows that
R²
Bl 1 (w) S [[E11 1m Combining this with Lemma 7 and Lemma yields
m
El f (m)))))))) x E 18 Sin limml (u))
n
R22
1[
(1I - -Joonlli + |1Joomn - (immlliz(w))
22
dR
R
2d
R d,
R2
24
Bllgoo,0 - sm. +
n
E
72
dRs
R
+4
2
2
2
2
m
30

---

Lemma 11. It follows that
dR
R
7d
1
1
Bl - P S
6²
x²
+
m
n2.
Proof. Let
1 Σ
exp, (f (Xi) - c(Xi: v))
Jm. = -5 log
Using co, we evaluate 10x0,00 Ii.. li (v) as follows.
Ellgbc, Im.ullim 2 El 00 (1: + Bllgm,,o - Im. 11000
(30)
For the first term, the Schrödinger system
implies
100,00 (v) Do (y) = log
exps (6o, (v)) Σ
exps (S (o))) -o( c(Xx.
e log
0o0, (Xir
c log
(m)
where Um (v) (mmm p oo For any 77 (0, 1), the following holds:
(:: v))
and used oo
E0 (2) o0 (y)l (v)l s2B sup yEsupp(1)
log(1 + Um ()))²
E
sup yεsupp(w)
Ilml
< n] sup yEsupp(r)
1log(1 + Um (1))
e
1
sup yEsupp(p)
U
n
sup yEsupp(>)
llog(1 + (1)))
(31)
For the first term in 31 the mean value theorem implies that for |u] we have [log(1 + u)) s
Hence,
2y)
E
[[ sup yEsupp(v)
[0ml
< y
sup yεsupp(v)
log(11 + Un (2)))
<
E sup yEsupp(v)
(1 (1-7)²
For the second term in 31). since exp. (-10())) S Poc,o0 (10R application of Markov's inequality yields
R4 P( sup
E
1 sup
1mml 2 nl
sup yEsupp(v)
|log(1 + Um()
(m)( 2 n)
μEsupp(y)
yεsupp(y)
R E Buu Eesppp(p))
Um (y))²
g2
Setting 77 1/2 and applying Lemma 8 yields
Rd
El (v) Im, (m) (v)
$
14
E
sup
22m(y))2
yEsupp(y)
82 +
R
R
2d.
(32)
31

---

For the second term of (30) since the gradient of the log-sum-exp function is the softmax and its -nomm equals 1, we obtain
Om (1)
Fm,n S 11 Smm.lle (mm))
(33)
Therefore, by Lemma 10 we have
E 00
Im,n (1)==() S BBI - -fmm (1
dR
R
7d,
1
-
$
82 +
-
62
Combining the above results, we obtain
dR
R
7du+4
1
1
00 gmmll
2
(34)
x²
m
n2
Lemma 12 (Lipschitzness of softmax). For A 0 and d 2 2, define σ% Fdd - Da by
cxp(Ax;) (oa(z)), = E/h exp(Axx))
(i = 1,.1....
l.).....)). d).
Then for any p,9 00, [ox(x)) ox(iil>> s Lpg |1 - vll,
Lpng 22 22-141/p-1/1
Proof. Lets= The Jacobian of σ% at z is
J(z) Vo (z) X(diag(s) TST).
(35)
By the fundamental theorem of calculus along the segment + y + U(x - y).
xx(x) -xx)) [ JV + 1(x w)) (or y) dl,
hence, for any 1 p.q S 0c.
(lxx())ox>())) - ox1il S (sup [|(())la=>>) ||x - 811
(36)
suffices to bound sup: [1)
Step 1: Decomposition. With the standard basis Jei diag(s) - TS = Σ
8,88 (e: - e;)(e; - e,)
(37)
icg
Thus for any E Rdd
(()u A Σ
8:8, (u: - (d:-uy)(e; - Ej),
and by the triangle inequality and ||e - -eilo = 2 (with the convention 2//²0 1),
[[(#)ullp S 1/1 s:3y Ius uzl.
(38)
Step 2: Maximization over E Ad Let imin = arg min; Uis imax arg max; Uj.
Since
[u4 Uz S [Wimax Mimino and Eies 8:85 =
Σd s7
Σ 8,85 [1u - uzl S Uiiaa Uiminl
Uimav Uinstn
2
The last inequality is tight when Stmax Stmin
and all other 8; 0. Therefore.
sup Σ SEA,
8:8j |1:
ugl Uinmax
Wimial
(39)
4
Because ax (RRd) is the interior of A and the right-hand side of 38 is continuous 8, the supremum over equals the supremum over Ad.
32

---

Step 3: Maximization over |lullg 1.
For any a, b E IR and 2 1, -b0 2 2 + }
With a Wimax mimin and Jull,
1/9
S 2 1//1//
[iimax| 1 + M
I 2 21-1// 1/9
(40)
with equality when u has exactly two nonzero entries Wimax
2-1/9 and Usmin = -22
Step 4: Combine.
From 38
(40)
|I(z)ullp I X21/0.
22-1/4 111111/12--11:1 -22-141/p-1/g
sup sup
4
Plug this into (36) obtain the claim. The bound is tight by the equality cases noted in (39 and
(40)
Lemma 13. For any probability vectors v E Ad.
e|1 S 2 duanb v)
(41)
Proof. By Cohen and Fausti 22024 [hhoremm 5.1].
||u w|h tanh dhin v)
(42)
Since tanh(x) & for 2 and duab (u, v) 2 O, letting x dssuuu,)) gives
||u - 111 S tanh dhub (u, (u,
42
duuubu,, v)
dhill (u, v).
(43)
This proves
Lemma 14. For any v. E Rd, the following holds:
duib(u, i, v) drab (v, u),
duab (u, (v) = dumn(u, v) Ve > 0, Vc E Rd.
duab(c ④ u, C v) = dumm(u, v) Here, @ denotes the element-wise product.
Proof. The first two properties are established in Lemmenn and Nussbaum, 2013 Lemma 2.1). For the third property, we have
max C.T C2
C.ts C.U,;
d ② u, ② v) log
max
log
max 2/
U4
ddiib (1iib(u,V v..
B.2 Lemma used for the drift estimation using neural networks
We provide an alternative expression of the Sinkhorn bridge drift estimator Markovian projection. Recall that W (o. is the Brownian bridge connecting zo and Ø1, so its marginal distribution at f|20,.11 =N((11 l)ar tai, e((1 1) For any distributions μ', v' E P(R))), and dual potentials E L (1'), E L' (v'), we define
time / E [0, is Gaussian distribution We
d x1) = exp [(woo) + g(m)) 100 æ1 113
d(pl @ v/ (2o, æ1).
33

---

We here only consider the case where becomes a joint distribution π E P(R x RR", that is, (k)
dr (Iu, 21) 1. Using Tm.a. we define the mixture of Brownian bridge (reciprocal process):
II = Wee (00 dr(wo, 21) E P(2).
Lemma 15. Under the above setting, it follows that
X1 - Dtt
b(2,, t) == Ex-1l
X1 2t
1-
Ex Ex, 1x
-t +
--t
Ex xx))
where (4), = exp ((0(0)) - शा- || - xxel3) /8).
Proof. Given 24 (1 < 1). we denote by [[(r]Xx xz) the conditional density of with respect to Lebesgue measure 1 Note that the marginal density 11.1 of I1 on time points and I with respect to dæ,džı satisfies that for a measurable
Rdd
f. [x 21) dx =
/ W 2120.20 (x))T((Zo A).
/
Therefore, we get
[T Xx = 12) dei xx { f exp
2et(1 t) Il,re (1-
(1-1)zg 1)200 Lxn 1x1113
(((x))
1 b1100 =11l2 du (z(( de1()1)
exp
(2||01) ||=11122- -2tx 21 + +t(11 - t)æe x11)
exp
2=t(1
t)
1
(1|=|] 112 - 2xx x1) del du' (xo) j -d((xr))
eep
-
25
g(x1)
1
2e(1 (1|=0 1|2 200 x1) de' (21)
exp
[((1)) - 2(1 1) ||:1 - xz 12 du du
x exp
This means
2(11, (x1)
11(x1|X, = dr
E Ex 1> (X1,
Therefore,
b(x,,)) 1
Ft. Ex-Il [X1 Xx, mr))
{51,, Iit)
-Bt + 11 Ex, (x xz1] (12)
1
-Pt + Ex (1,,,,
Ex (x(,
C Omitted Experiments
C.1 Stopping and guidance
Theorems and suggest that beyond a certain point additional Sinkhorn iterations are not beneficial. The total error decomposes into "sampling error" and an "optimization error". The latter decreases
34

---

Table 1: Comparison of L
sample (m, n,,,,,,0 t)dt across intrinsic dimensions du-
dv
[0.9 MSF sampiee (m,, n,, 1)dt
0.031
10
0.138
20
0.562
30
1.178
only through iterations, whereas the former does not. By the triangle incquality for the total variation distance,
E ( (P= PM...) S TV2 (P Pm..) E (TV2 (P Pk p
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 meaningful reduction of the total error. Hence it is reasonable to choose the stopping point as the smallest k such that
E TV2 (P P pm,n)]
> E [TV2 (P (mm.r pk m,n)]
From Theorems] and Q. sufficient condition for this is
log
1 + A-B
R
9d,+4
1:
A
B =
+ dR²
log tanh 11a
T(1 - 711-7)du+1
2 gd.
176
Since log taan
Ra
< O, iterations are beneficial only when B + A+B < 1. Equivalently, unless
the sample size is large enough to make the numerator negative- routhlyy (m A n) x (A v B reduction achievable by iterations is smaller than the estimation error, leading to wasted computational
resources.
In Figure the fact that b already captures the shape of to a certain extent is consistent with this reasoning. Increasing becomes meaningful only when fmm and vn. are sufficiently close to true distributions μ and namely when m and are sufficiently large.
C.2 Intrinsic dimension
Theorem D shows that the orders of the orders of and ε depend on the intrinsic dimension d, rather than the ambient dimension d. In this section, we verify that the error varies with d, under the following setup.
We fix the ambient dimension at 50, the regularization strength at = 0.5, the evaluation- endpoint 0.9, and the sample sizes at uniformly from the unit hypercube [0, 1]4, while sampled uniformly from the dy-dimensional manifold embedded in the unit sphere of radius 1,
10000. The distribution μ is sampled
{ xx 2
||=||| 1, Vi d, 0)
By varying du, 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 E {5, 10, 20, 30} we compare
0.9
MSEsample (m,, dl.
The results are reported in Table I An increasing trend of the error with larger manifold dimension observed.
C.3 The role of epsilon
We examine the numerical stability of the regularization parameter e in the Sinkhorn algorithm and provide selection guidelines in relation to the sample sizes (m, n).. Using the same setup as
35

---

Table 2: Representative E achieving the target error Experiments 4.1) )
1.0 for varying m (same setup as
m A 71
10
2.684 x 10
20
7.979 X 10c
30
3.200 x 10a
40
6.797 x 10
50
3.291 X 10
60
1.299 X 10-1
70
2.010 x 10-1
80
9.766 X 10
Experiments 4.1 for each given m, we search for the smallest that achieves the target error 1.0.
As m A n increases, the required to meet the target error exhibits a two-phase behavior: a steep initial decrease followed by asymptotically mild decay. Representative values are reported in Table
By Theorem 1 the required satisfy the target enror 8 obeys
2 max
(0 A m)
R
1/ds
R
Rduu+2
Sdy
+1+1 8(m An)
1/(9d.+6)
This indicates steep improvement in the small-mA regime followed by logarithmically slow decay. which aligns with the empirical results. Since excessively small can cause numerical instability, using the above expression as a lower-bound guideline for selecting E is effective.
36


Reviews:
[{"review_id": "y734eR1shU", "reviewer": "Reviewer_g9fY", "summary": "This 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.", "questions": "I 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.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The authors have provided additional details which substantially fill in some gaps of the paper, and make it more well-rounded.   Specifically, the authors addressed my queries about (i) the role of $\\epsilon$, (ii) the dependence on dimension, and even provided a good stopping rule condition derived from their bounds.\n\nGenerally this paper is a generally strong theory paper, which I believe should be accepted.", "rebuttal": "We thank the reviewer for carefully reading our paper.\n\n**The role of epsilon**\n\nRevisiting the proof, we improved the exponential factor in equation (14) to $\\exp\\bigl(R^2(60+\\varepsilon^{-1})\\bigr),$\nand the following is written with that improvement in mind (to be reflected in the final version).\n\nThe numerically unstable parts are the Sinkhorn Algorithm and the drift $b\\_{m,n}^{(k)}$ with the instability being especially pronounced in the drift. This stems from the term appearing in equation (13)\n$$\n\\exp\\left(-\\frac{\\|Y_j - z\\|_2^2}{2(1-t)\\varepsilon}\\right),\n$$\nwhere the denominator contains $(1-t)\\varepsilon$; as $t$ approaches $1$, this induces sharp scaling. However, since the expression is in a softmax form, standard stabilization by subtracting the maximum of the exponent’s argument can be applied to ensure numerical stability.\n\n(i) To verify the empirical behavior, we ran the experiment for finding $\\varepsilon$ that achieves target error $\\delta=1.0$ under the same setup as in “Experiments (Statistical convergence)”, for given $m,n$. As predicted, the error drops sharply as $\\varepsilon$ increases, then transitions to a regime of slower, asymptotic improvement—i.e., a steep improvement in $m\\wedge n$ followed by a gradual change. The observed values were:\n\n| $m\\wedge n$ | $\\varepsilon$ |\n|--------------|----------------|\n| 10           | $2.684\\times10^{8}$ |\n| 20           | $7.979\\times10^{6}$ |\n| 30           | $3.200\\times10^{1}$ |\n| 40           | $6.797\\times10^{-1}$ |\n| 50           | $3.291\\times10^{-1}$ |\n| 60           | $1.299\\times10^{-1}$ |\n| 70           | $2.010\\times10^{-1}$ |\n| 80           | $9.766\\times10^{-4}$ |\n\n(ii) This phenomenon is supported theoretically: from equation (14), the $\\varepsilon$ required to achieve target error $\\delta$ satisfies\n$$\n\\varepsilon \\gtrsim \\frac{R^2}{\\log(\\delta \\cdot (m\\wedge n))}.\n$$\nThis indicates that when $m\\wedge n$ is moderately large, error reduction proceeds rapidly, and thereafter improves only logarithmically.\n\n\n**Experiments and d**\n\nWe conducted experiments to verify the dependence of the Sinkhorn bridge estimator on the intrinsic dimensionality. The experimental setup is as follows: the ambient dimension $d = 50$, the regularization strength $\\varepsilon = 0.5$, and endpoint $\\tau = 0.9$. We sampled the source distribution $\\mu$ uniformly from the unit cube $[0,1]^d$, and the target distribution $\\nu$ uniformly from the $d_\\nu$‑dimensional unit sphere embedded in the ambient space $\\mathbb{R}^d$: $\\\\{ x \\in \\mathbb{R}^{d_\\nu} | \\||x\\||_2 = 1\\\\} \\subset \\mathbb{R}^d$.\n\nBy varying $d_\\nu$, we examined whether the convergence behavior of the estimation error with respect to sample sizes depends on the manifold dimension. For $d_\\nu \\in \\\\{5,\\,10,\\,20,\\,30\\\\}$, we compared the time‑integrated mean squared error $\\int_{0}^{0.9} \\mathrm{MSE}_{\\mathrm{sample}}(m,n,t)\\,\\mathrm{d}t,\n$ and observed that the error tends to increase as the manifold dimension grows, suggesting room for improvement.\n\n| $d_\\nu$ | Error |\n| -------: | --------------------------------------------------------: |\n| $5$     | $0.031$                                                  |\n| $10$    | $0.138$                                                  |\n| $20$    | $0.562$                                                  |\n| $30$    | $1.178$                                                  |\n\n**Stopping and guidance**\n\nConsidering the decomposition of TV-distance, the number of iterations $k$ satisfying the following inequality is sufficient: $\\mathbb{E}[ TV^2(P\\_{\\infty,\\infty}^{\\*,[0,\\tau]}, P\\_{m,n}^{\\*,[0,\\tau]})] \\geq \\mathbb{E}\\bigl[TV^2(P\\_{m,n}^{\\*,[0,\\tau]}, P\\_{m,n}^{k,[0,\\tau]})]$. This implies $k \\geq C\\frac{\\log\\bigl(A/n + B/(m\\wedge n)\\bigr)}\n     {\\log\\bigl(\\tanh(R^2/\\varepsilon)\\bigr)}$ where $A = \\frac{1}{(1-\\tau)^{d+1} \\varepsilon^{d-1}R^2}$ and $B = \\frac{C_{d}R^{5(d+1)}}{(1-\\tau)\\tau\\varepsilon} e^{R^2\\bigl(60 + \\varepsilon^{-1}\\bigr)}$. Since $\\log(\\tanh(R^2/\\varepsilon))<0$, the numerator $\\log\\bigl(A/n + B/(m\\wedge n)\\bigr)$ must also be negative for the right-hand side to be meaningful. This in turn implies that\n$m \\wedge n \\approx \\max\\{A, B\\}\n$ samples are needed.\n\nThis prediction is also confirmed by Figure 3. In particular, $b_{m,n}^{(1)}$ obtained after a single Sinkhorn iteration, already transports points to their nearest neighbors and captures the bulk of $\\nu_n$. \n\nThe only remaining discrepancy with the optimal drift $b_{m,n}^*$ lies in the mismatch of the marginal distributions, similar to the case of entropic optimal transport. As a result, a sufficiently large $k$ becomes necessary only when the empirical distributions $\\mu_m$ and $\\nu_n$ are already close to the true distributions $\\mu$ and $\\nu$ i.e., when $m$ and $n$ are sufficiently large."}, {"review_id": "9KZVOaRcFD", "reviewer": "Reviewer_EHkZ", "summary": "This works investigates the statistical guarantees of the Sinkhorn bridge algorithm in the two-sample estimation setting.", "questions": "See above.", "limitations": "See above.\nI 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.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "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.\n\nThe empirical validation is also clearly presented for synthetic Gaussian settings.\n\nOverall, 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.\n\nThat 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.", "was_revised": true, "final_justification": "The authors clarified the relevance of the paper. I'd like to maintain my positive assessment.", "rebuttal": "We thank the reviewer for reading our paper.\n\n**Relevance to NeurIPS community**\n\nSchrödinger bridges (SB) have attracted significant attention at NeurIPS as a key research theme at the intersection of diffusion models and optimal transport. Notably, several influential papers on SB have been published at NeurIPS, including Diffusion Schrödinger Bridge (NeurIPS 2021) [1] and Diffusion Schrödinger Bridge Matching (NeurIPS 2023) [2]. Our work reinforces the theoretical foundations that underpin this growing line of research.\n\nOn the applied side, SB techniques have already been utilized in a wide range of real-world tasks. These include image restoration and transformation using I$^2$SB [3] (e.g., inpainting and super-resolution), unpaired image translation via UNSB [4] and SB Flow [5], domain adaptation through distribution alignment using maximum-likelihood SB solvers [6], and trajectory inference for single-cell RNA-seq data in the life sciences [6, 7].\n\nTaken together, the theoretical contributions of this work strengthen the foundation of SB-based methods and enhance our understanding of their practical applicability, making the results broadly relevant to the NeurIPS community.\nWe will include the above related works in the final verstion.\n\n\n[1] De Bortoli, Valentin, et al. \"Diffusion Schrödinger bridge with applications to score-based generative modeling.\" NeurIPS, 2021.\n\n[2] Shi, Yuyang, et al. \"Diffusion Schrödinger bridge matching.\" NeurIPS, 2023.\n\n[3] Liu, Guan-Horng, et al. \"I$^ 2$SB: Image-to-Image Schrödinger Bridge.\" arXiv:2302.05872, 2023.\n\n[4] Kim, Beomsu, et al. \"Unpaired image-to-image translation via neural Schrödinger bridge.\" arXiv:2305.15086, 2023.\n\n[5] De Bortoli, V., Korshunova, I., Mnih, A., & Doucet, A. Schrödinger bridge flow for unpaired data translation. NeurIPS, 2024\n\n[6] Vargas, Francisco, et al. \"Solving Schrödinger bridges via maximum likelihood.\" Entropy, 2021.\n\n[7] Chizat, Lénaïc, et al. \"Trajectory inference via mean-field Langevin in path space.\" NeurIPS, 2022."}, {"review_id": "1gYWY4FiWt", "reviewer": "Reviewer_Lkjn", "summary": "The 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.", "questions": "* 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?", "limitations": "The authors expressed limitations of the work. I have stated additional limitations that I found in the weaknesses section.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### 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.", "was_revised": true, "final_justification": "As elaborated, I found the Section 3.2 as an incremental improvement compared to [1]; while there is an improvement in the rate, I find the improvement relatively restrictive. On the other hand, I find Section 3.3 favorable, as it offers the computational advantage which [1] lacked. My initial concern on Section 3.3 was its algorithmic convergence, and summing up these points I rated the borderline accept at the first review. However, I found out that I had some misunderstanding regarding its algorithmic convergence, and as a result my concern was resolved during the discussion period.\n\nIn conclusion, I would like to rate accept to this paper.", "rebuttal": "We thank the reviewer for carefully reading our paper.\n\n**Tradeoff between $n$ and $R$**\n\nSeveral recent works that established the faster convergence rate of $O(1/n)$ for EOT assumed either the boundedness of $R$ or Lipschitz continuous bounded cost (i.e, the bounded $R$ with Euclidean cost.) These results suggest a trade-off between $n$ and $R$. However, the optimal dependence on these constants remains unclear. \nOur bound exhibits exponential dependence on $R$, but we  note that such dependence is common in the literature of the entropic transport [1,2]. In fact, the term $\\exp(CR^2)$ in our bound is inherited from the existing bound on the Schrödinger potential functions [2]. More recently, [3] established the polynomial bound of the form $R^d$ for $L^2$-distance between potential functions. However, translating their result for EOT into statistical bounds for the Schrödinger bridge is nontrivial and is a future research direction.\n\n\n[1] Rigollet, Philippe, and Austin J. Stromme. \"On the sample complexity of entropic optimal transport.\" The Annals of Statistics, 2025.\n\n[2] del Barrio, Eustasio, et al. \"An improved central limit theorem and fast convergence rates for entropic transportation costs.\" SIAM Journal on Mathematics of Data Science, 2023.\n\n[3] Stromme, A. J. \"Minimum intrinsic dimension scaling for entropic optimal transport.\" In International Conference on Soft Methods in Probability and Statistics, 2024.\n\n**Dependence on (intrinsic) dimension**\n\nOur bound relies on the following decomposition:\n$\\mathbb{E}[TV^2(P\\_{\\infty,\\infty}^{\\*,[0,\\tau]},P\\_{m,n}^{\\*,[0,\\tau]} )] \\lesssim \\mathbb{E}[KL(P\\_{\\infty,\\infty}^{\\*,[0,\\tau]} | \\bar{P}\\_{\\infty,n}^{[0,\\tau]})] + \\mathbb{E}[KL(\\bar{P}\\_{\\infty,n}^{[0,\\tau]} | P\\_{m,n}^{\\*, [0,\\tau]}) ]$. \n\nThe first term admits an upper bound of the form: $\\frac{R^2}{n (1-\\tau)^{d_{\\nu}+2} \\varepsilon^{d_{\\nu}}}$ (Proposition 4.3 in [4]) when the support of $\\nu$ is a $d_{\\nu}$-dimensional submanifold. However, for the second term, we have not obtained a bound that depends on the intrinsic dimension $d_{\\nu}$. As a result, the overall bound depends on the ambient dimension $d$ rather than $d_\\nu$. \n\n[4] Pooladian, A.-A. and Niles-Weed, J. Plug-in estimation of Schrödinger bridges. arXiv:2408.11686, 2024.\n\n**Weakness 2: relationship between Algorithm 1 and plug-in estimator**\n\nWe note that Algorithm 1 performs online SGD to minimize $\\mathcal{L}(b)$ (line 210), and this is feasible since $\\mathcal{L}(b)$ is tractable. Specifically, the expectation in $\\mathcal{L}(b)$ taken with respect to a mixture of Brownian bridges defined by the transportation plan $\\pi$, from which we can sample to perform SGD. As a result, Algorithm 1 converges to the plug-in estimator, and no additional statistical error is incurred if the neural network is sufficiently large. It is also important to note that $N$ denotes the mini-batch size of SGD and is unrelated to the definition of the objective $\\mathcal{L}(b)$.\n\n**Minor typo**\n\nThank you for noticing the typo. It has been corrected accordingly."}, {"review_id": "Jm781Zsy70", "reviewer": "Reviewer_AupN", "summary": "This 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.", "questions": "1. 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?", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "The authors have addressed my concerns thoroughly, and I have no further issues. This paper presents strong theoretical results and has been well-received by all reviewers. However, due to my unfamiliarity with the relevant theory, I will keep my score and confidence level unchanged. I appreciate the efforts of both the authors and the AC—apologies for not being able to fully engage in the rebuttal and adjust my rating accordingly.", "rebuttal": "We thank the reviewer for reading our paper.\n\n**Accuracy and computational cost of Sinkhorn bridge**\n \n- Accuracy: The Sinkhorn bridge estimator performs well in terms of accuracy. In fact, [1] observed that it outperforms Schrödinger bridge solvers based on learned forward–backward dynamics under the experimental setting of [2]. For details, see Table 1 in [1].\n\n\n- Computational cost: The Sinkhorn Bridge requires the coupling $\\pi_{m,n}$ which can be obtained using Sinkhorn iterations. These iterations converge exponentially fast and have a per-iteration cost of $mn$. In the sampling phase, the Sinkhorn bridge applies Euler–Maruyama discretization, which incurs a cost of $O(n)$ per discretization step. \nTherefore, generating a new sample from the estimated target distribution costs $O(nK)$, where $K$ is the number of discretization steps. To reduce the inference cost, we also proposed a neural network-based approximation of the drift.\n\n\n[1] Pooladian, A.-A. and Niles-Weed, J. \"Plug-in estimation of Schrödinger bridges.\" arXiv:2408.11686, 2024.\n\n[2] Gushchin, N., et al. Building the bridge of Schr ̈odinger: A continuous entropic optimal transport benchmark. NeurIPS, 2023.\n\n**Neural network based drift approximation**\n\nWe note that Algorithm 1 performs online SGD to minimize $\\mathcal{L}(b)$ (line 210), and this is feasible since $\\mathcal{L}(b)$ is tractable. Specifically, the expectation in $\\mathcal{L}(b)$ is taken with respect to a mixture of Brownian bridges defined by the transportation plan $\\pi$, from which we can sample to perform SGD. As a result, there are approximation errors (due to model expressivity) and optimization errors (due to finite training iterations), but with sufficient network capacity and training time, the NN-based estimator will converge to the Sinkhorn bridge drift, which is the minimizer of $\\mathcal{L}(b)$."}]

Author rebuttal:

Meta-review: This work establishes finite-sample error bounds for Sinkhorn bridge iterations in the two-sample Schrödinger bridge setting, providing theoretical guarantees, practical guidance on sample sizes and iteration numbers, and unifying several existing methods under a common framework.

Strength: This paper extends the results of Pooladian and Niles-Weed (2024) to the more practical/realistic two-sample estimation setting.

Weakness: Some limited empirical results. 

The paper makes an important contribution to the recent literature on analysis of Sinkhorn algorithms. All reviewers unanimously recommend acceptance, which the AC also concurs with in their own reading.
Decision: Accept (poster)