Exponential Convergence Guarantees for Iterative Markovian Fitting
Authors: Marta Gentiloni Silveri
Giovanni Conforti
Alain Oliviero Durmus
Venue: NeurIPS 2025 poster ()
Keywords: Iterative Markovian Fitting
Convergence guarantees
Schrödinger Bridge
Generative Models
Primary area: theory

Abstract: The Schrödinger Bridge (SB) problem has become a fundamental tool in computational optimal transport and generative modeling.
To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed—alongside practical approximations like Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, a quantitative, non-asymptotic understanding remains unknown. In this paper, we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a sufficiently large time horizon. Our results encompass two key regimes: one where the marginals are log-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.

Exponential Convergence Guarantees for Iterative Markovian Fitting
Marta Gentiloni Silveri
École polytechnique
Route de Saclay, 91 12 Palaiseau, France
marta gentiloni silleri@@olyytechiigg edu
Giovanni Conforti
Alain Durmus
Università degli Studi di Padova
École polytechnique
Via Trieste, 63, 35131 Padova, Italia giovanni. conf @math. unipd.
Route de Saclay, 91 120 Palaiseau, France
alain. durmus@@olyyecchniiuee edu
Abstract
The Schrödinger Bridge (SB) problem has become fundamental tool in com- putational optimal transport and generative modeling. To address this problem, ideal methods such as Iterative Proportional Fitting and Iterative Markovian Fitting (IMF) have been proposed- proposed-alongside practical approximations like Diffusion Schrödinger Bridge and its Matching (DSBM) variant. While previous work have established asymptotic convergence guarantees for IMF, quantitative, asymptotic understanding remains unknown. In this paper. we provide the first non-asymptotic exponential convergence guarantees for IMF under mild structural assumptions on the reference measure and marginal distributions, assuming a suffi- ciently large time horizon. Our results encompass two key regimes: where the marginals are cog-concave, and another where they are weakly log-concave. The analysis relies on new contraction results for the Markovian projection operator and paves the way to theoretical guarantees for DSBM.
Introduction
Generative models play foundational role in modern machine learning, with widespread applications ranging from image synthesis IRPG+211 221] and natural language generation [BMR 20] to molecular structure design [XLH 22). These models aim to learn the underlying probability distribution of a given dataset and can generate new, high-fidelity samples that resemble the original data. Among the various generative frameworks, diffusion and flow-based models have emerged as particularly powerful, thanks to both their empirical success and theoretical grounding. These models typically rely stochastic differential equations (SDEs) to gradually transform simple prior (often Gaussian noise) into samples from complex target distribution, often through learned score function [SE19, HJA20. SSDK+211 However, despite their effectiveness, traditional score- based generative models (SGMs) often suffer from long sampling times, due to the need for finely discretized time grids and numerically stable SDE integration over long time horizons.
An increasingly popular and theoretically grounded alternative is to generative modeling Schrödinger Bridge (SB) problem. Originally introduced by Erwin Schrödinger in the 1930s [Sch32], the SB problem arises from statistical physics and seeks the most likely evolution of cloud of inde- pendent particles (typically Brownian) that interpolates between two observed empirical distributions μ, ddd prescribed initial and final times. Formally, the SB problem consists in finding a path
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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measure that minimizes the Bullback-Leibles (KL) divergence with respect to reference diffusion process (usually stationary Brownian motion), subject to fixed marginal constraints. Although originally motivated by principles from statistical mechanics, the SB problem is now known to be equivalent to a regularized version of classical optimal transport (OT), where an entropie term added to the transport cost [Nut]. In this formulation, the KI. divergence acts as a soft transport penalty, leading to the interpretation of the SB as entropy-regularizedd OT in path space.
In addition to the path space (dynamical) formulation, it admits a static reformulation, where the optimization is performed over couplings between the given marginals. The optimal coupling x². often referred to as the Schrödinger bridge, can be used to sample joint initial and terminal states. Conditional trajectories are then obtained by interpolating these endpoints with standard Brownian bridges. When the initial distribution is taken as the standard Gaussian y", solving the SB problem identifies the most likely diffusion process in terms of KI. divergence from the reference - whose marginals are 7d and v. As result, this formulation enables sample generation from v in finite time, and provides compelling alternative to traditional score-based generative approaches.
Several algorithms have been developed to compute or approximate solutions to the generalized SB problem, especially when the reference process is not necessarily Brownian motion. Among these, two families of algorithms stand out: the Iterative Proportional Fitting (IPF) algorithm (also known as the Sinkhorn algorithm) [Cut13, Nut], and the Iterative Markovian Fitting (IMF) algorithm (also known as Iterated Diffusion Bridge Mixture) [Pel23, SDBCD24J. IPF, on one hand, can be interpreted as an alternative minimization scheme where the marginals constraints are alternatively relaxed, which leads to a sequence of forward and backward diffusion processes with the target measures 1l and v as initial distributions. On the other hand, IMF defines sequence of stochastic interpolations between μ and v/ processes and their Markov projections, building on the Diffusion Flow Matching framework [Pel21, AVE22] and extending it recursively. In addition, practical and approximate implementations of IPF and IMF have been proposed. In particular, the Diffusion Schrödinger Bridge algorithm [DBTHD21a] 1a] along with its Matching version [SDBCD24, Pel23] or the Iterative Proportional Maximum Likelihood (IPML) algorithm [VTLL21a]
Although empirical performance of IMF is promising, its theoretical analysis remains limited. In particular, existing works only provide asymptotic convergence results [SDBCD24, Pe123], and no non-asymptotic rates or guarantees are currently known. In this paper, we fill this gap by providing the first non-asym con-syymtotiic convergence guarantees for IMF. Under mild assumptions on the reference diffusion process and for sufficiently large time horizon T, we derive explicit convergence bounds in KL divergence. Our analysis distinguishes two important regimes: strongly log-concave marginals and weakly log-concave marginals the latter being more general class that includes multimodal and non-convex distributions [Con24]. A central technical contribution of our work is the derivation of new contraction estimate for the Markovian projection operator (see Theorem 4), which plays fundamental role in controlling the error propagation over iterations. Our main results - Theorem 5 and Theorem 6 establish the first quantitative convergence rates for sehrödinger-brrdge--aaeed generative models.
Outline of the Paper. Section 2 introduces the necessary background on the Schrödinger Bridge problem and details the Iterative Markovian Fitting (IMF) algorithm. Section 3 presents our main theo- retical results under both the strongly log log-concave (Section 3.1) and weakly log log-concave (Section 3.2) regimes. Section 3.3 discusses connections to prior work. Full technical details and supporting lemmas are deferred to the supplement.
Notation. For metric space (E,d), d), denote by P(E) the set of probability measures on E and by B(E) its Borel ofild.. Given two probability measures μ. v E P(E), the relative entropy (or KL-divergence) of μ with respect continuous with respect to v, and KL(p))) = too otherwise. Note that we also consider an extension of this definition to the case where is only a-finite measure on E. Similarly, the Fisher information of i with respect to P is defined by (H(p)) = J V log(dn/do))lddp if is absolutely continuous with respect to v. and I (ulp) = too otherwise. When, E = gd. denote by P) the set of couplings between j and P, i.e., E I1(,,)) v) if and only if & is probability measure d x Rd and Ç(A " ((AA and Ş(R" A) = =(A) for all measurable A C Rdd If μ and have finite second moment, the 2 Wasserstein distance is defined by
is defined by KL(μ/p) : l' / dv du if μ is absolutely

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(4, == infgem)ue) [| - v/2 dg(x, Denote by 7 the density of the standard Gaussian distribution on R4. With abuse of notation, identify the standard Gaussian measure with its density. For d E N, denote by Leb" the Lebesgue measure on IR Denote by Cr C( T)) the set of continuous functions defined on the time-interval [0. T]. Equipped with the L -nomm || |l refer to it as Wiener space. Given Pla. E P(Cr) and Ple.t] E P(C((ss (])) respectively the marginal time distribution at time t, the joint law at times 5 and t, the conditional distribution with respect to the marginals at times 8 and and the the restriction to the time sub-interval [s, of P. Also, we denote by PR P(Cr) the reverse-time measure of P. i.e., the path-measure defined, for any A B(Cr), as (P)((A = P(A") where A := (t + WT t) w E A}. Given two matrices A, B.B. E Rddd we write A 2 B A B is positive semi-definite. Given two vectors z, y E R. we denote by (r. y) and ||r|| the standard scalar product between and y and the Euclidean norm of æ. Last, denote by ( the set of Lipschitz functions _2 IR with Lipschitz constant smaller than 1.
P(CT) and 8, E [o. T) denote by P. E P(TR)) P..t. E P(
2 Schrödinger Bridge Problem and Iterative Markovian Fitting
In this section, we present the SB problem in details and its resolution through IMF.
Schrödinger Bridge problem. The first key tool for defining the SB problem is path measure R" on Cr. referred to as the reference measure, We consider in this paper for R the distribution of the process (X.))e(o.T) solution of the Langevin dynamics
dx; VU.VU(X + V2d, e [0,T], X. m(dx) x exp(-U(z))dr,
(1)
where U R + R is potential. We also consider the special case U = 0 which corresponds to taking as R the measure on Cr associated with the Brownian motion initialized at the volume measure 3,, Annexe A]. We make the following assumption on (1)
H1. Kither U = 0 or the Langevin system (1) has a unique stationary solution.
Under mild assumptions on U, Hi holds; see e.g., [Ken78].
Equipped with R given the two marginal distributions J4, E ((RT)) the SB problem can be expressed as the following minimization problem:
minimize (T.(P)R)). under the constraint P(CT), p Py
This formulation is known as the Dynamical Schrödinger Bridge Problem, where the optimization is performed over measures defined on the path space. Interestingly, this dynamic problem has an equivalent static formulation [Léo13, Proposition 2.3]:
minimize (L(T)R(T)) under the constraint æ E I),,v)
(3)
where R denotes the joint law between the marginal distributions at times f and T' of R This formulation is known as the Static Schrödinger Bridge problem. To ensure existence of a unique solution for (2)-(3, we consider the following assumption.
H2. There exists (at least) a coupling I( v) such that KL(+|R(,)) +00.
Under H2, [Nut, Theorem 2.1.] shows that problem (3) admits unique solution, called the Schrödinger Bridge, which can be expressed as
" (dr, dy) = exp (-p(x) - <b())) (dx, dy)
(4)
where sp, 1/ Rd (-o0, too).
Consider the bridge o associated with R i.e. the Markov kernel on IR24 XCr corresponding the conditional distribution of (X) (X))eeo given (Xo, Xr) and therefore satisfying for any A E B(Cr), R (A) I dRR, (æo, bRR 4 ((ro, IT), A). Equipped with bR can easily construct an optimal solution for (2) based on (4). Indeed, denoting by IP' the path distribution of (Xf) [e((.))
3

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such that (G) ~ n" and (X; (E(O)) ~ bR ((Xg, -), and using that for any P E P(Cr)
holds [Léo13, Annexe A]
(L(P)IR (L(F R + [P 0.7 (-1(Xo, (0. Xr)),,
with (Xo, X ~ Po,r, we immediately get that P a solution for (2). For further details on SB, refer to the two surveys [Léo13] and [Nut].
One of the most famous scheme to solve (2) (or equivalently (3)) is the Iterative Proportional Fitting (IPF) algorithm, also known as the Sinkhorn algorithm. This scheme roughly consists in solving this problem, relaxing the constraints on one of the marginals alternatively. However, these steps typically intractable but, leveraging learning and statistical techniques, [DBTHD2Ib, VTLL21a] propose practical implementation for IPF.
Besides IPF, a recent class of algorithms have recently emerged as alternative for solving (2). They can be interpreted as an extension of Diffusion Flow Matching models that we now present.
Diffusion Flow Matching Models. Given the two marginal distributions jr, v E P(RR), Diffusion Flow Matching (DFM) models learn transform samples from μ into samples from by following continuous stochastic flow. The key idea is to construct an interpolating process, or bridge, between ju and v. To formalize this, again, we consider the bridge bR associated to (1) and a coupling No.r (),, We then define, the resulting stochastic interpolant (Yı)ce(o,T) as
(Yo: Yr) ~ To, (Y))ee0or))) Yr) ~ bR ((%, )r).
(5)
The process (Ye) provides a smooth, random evolution from μ to v, but not Markov process in general and does not satisfy simple Stochastic Differential Equation (SDE), making direct sampling difficult. However, we can consider the non- oon-homogeneous Markov diffusion process (x("))eeooo)) solution of the SDE
ax!") = (x( V2dB, LE (0, X p.
(6)
where, for any & E [0, T), the mimicking drift is given, for any Mo Rd by
It (wv) E क (Y,, YT) ve VU(y.)
(7)
Here, for any t E [0, 7), the vector field $; is defined, for any Vc, yr Rd by
T he( vr) = 2V v. log (urlu))
(8)
where pf denotes the conditional density with respect to the Lebesgue measure of XT given X Indeed, under the technical conditions Tl and T2 given in the supplement, it can be shown (see Theorem 3 in Appendix B) that the Markov process (x(")ce(ozz)) preserves the same marginal distributions as (Y1) i.e.
(1) dist Yr, E [0,27.
In particular, x( y and x:()) that is (X!") Drejo.ry interpolates between μ and This process known as the Markovian Projection of the stochastic interpolant and originates from [Gy886] and [Kry84].
In its most conventional implementation, DFM models typically consider either U or a quadratic potential U. corresponding to Ornstein-Uhlenbeck process. In such cases, the conditional log- density (yt, 3r) + log , (yrly)) (ye) is available in closed form. The drift function f(()) is then the solution to regression problem, which can be learned from samples of the process (Y:)re [o.]]- often using neural network approximation. In practice, DFMs generate approximate samples from the target distribution v by applying the Euler-Maruyama scheme to discretize the SDE in (6), substituting the true drift _ with its learned approximation. This approach offers an efficient and trainable method to transform samples from μ into samples from & via a continuous stochastic flow.
It turns out that DFM models can be extended to solve the SB problem (2) as describe.

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Iterative Markovian Fitting Iterative Markovian Fitting (IMF) (also known as Iterated Diffu- sion Bridge Mixture) [SDBCD24, Pel23] is an algorithm designed iteratively approximate the Schrödinger Bridge. It builds upon the Diffusion Flow Matching (DFM) framework and can be interpreted as a recursive extension of it. At a high level, a DFM model takes a given coupling No.r [T, v) as initial input, constructs the corresponding stochastic interpolant (Y)) (5), JeEoo.T (6), and outputs path measure IP(1) and new computes its Markovian projection (xE"O)t
(00 coupling O = PO. E I[(,,)) 1/) corresponding to the distribution of (x(")<eto and the pair (x("), x respectively. By iterating this procedure multiple times, we obtain the IMF algorithm whose pseudo-code is provided in Algorithm 1.
Algorithm 1: Iterative Markovian Fitting
Input:
Initial coupling (0)
"0. II,,,) and Langevin bridge bR associated with (1)
For each iteration & = O......N 1: Step 1. Stochastic Interpolant Update.
Define (Y, (y((++1) )eE]T as:
(y ,y y(t+1) Ta. Y/ (y (ott), bR ((ygues) ((vttt))yatt))
Step 2. Markovian Projection Update.
Consider p(++1) as the path math-distribution of the solution (X )EJO,TT to the SDE:
dx((+1) = (((+1) (x( )dl + v2dB,, x ~ M
(10)
where:
f1 [k+ (y)) =B क (y, yfa (x(() vi - VV(()))
(11)
and 4 is given in (8).
Set (k++1) pph+")
End For Output: The path distributionn P(N) and the coupling Tor E I(p,v)
(N)
has been shown [Léo13, LRZ14] that the SB coupling n², defined in (4), the unique fixed point from the two Steps defining Algorithm 1, which can therefore be interpreted as a fixed point algorithm. Therefore, the sequences (IP()) JoeN and (=f0)) defined in Algorithm converge to the SB solutions P and n² In particular, asymptotic convergence has been established in [SDBCD24, Pe123] under appropriate conditions. Further discussions on IMF, and its connection with Sinkhorn and Diffusion Schrödinger Bridge, are postponed in Appendix C.
expected
During computational implementation, at each step k, the mimicking drift Le (1+1) defined in (11) learned by solving a regression problem using neural networks, following the same approach as in DFMs. This leads the first version of the Diffusion Schrödinger Bridge Matching (DSBM) algorithm. However, from a practical standpoint, has been observed that to avoid bias accumulation, is beneficial to leverage not only the fact that [p(++1) is the law of a diffusion process but also its time- reversal. Denote by Ip(k+1) the distribution of the time reversal (x(*+ (k+1) )re(o.T) of (xx( (x+(k))) ")
defined for any E [o, T] by X (1+1) = x(++1) It can be shown that (X, (k+1) )cE(O.2) is solution of the SDE (see Proposition in Appendix D)
(xx(A++)_ = 9i
(k+1) (X(! (k++)) )t + V2dB,, tE (O,T))
X ((+1)) ~N,
(12)
where, for any E [0, T), the drift function is given, for any yr E Rd, by
(k+1) (vr-,) = E [5, (vo (y+++) y1++ y/h) = vr- + VU(rT-),
91
(13)
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and, for any & [0, R,, the vector field Φ; is defined, for any yo, yr E R by
T i vT- = vv 1 Jogp?--o((VT--ll))
Since up time inversion, ((++1) and p[++1) are equivalent, an alternative to Step 2 in Algorithm 1
to consider [(++1) instead of Tp(++1, and the resulting coupling (X,, (X:(++1) X x E IT v).
While this substitution is equivalent for the idealized IMF scheme, it becomes non non-trivial when approximating the mimicking drifts. Indeed, comparing (10)-(1) with (12)-13), we observe that the two processes are initialized at different distributions: the forward diffusion starts from μ4, while time-reversed process starts from v. Alternating between Step 2 and its time-reversed variant thus mitigates the bias introduced by using approximate rather than exact mimicking drifts. This strategy leads to the final version the DSBM algorithm proposed [SDBCD24, Pe123]. Further details are provided in Appendix D.
3 Main Results
In this section, we establish exponential convergence guarantees in Kullback-Leible (KL) divergence for the IMF Algorithm I, under two different settings: the strongly log-concave and the weakly log-concave regimes. Although [SDBCD24, Theorem 8] and [Pel23, Theorem 2] demonstrate that, under strong assumptions, the IMF sequence [To_T) ), converges asympiotically in KL divergence to (n) x²) neither work provides quantitative bounds
the Schrödinger Bridge, i..., lim (L on the convergence rate. This section aims to fill that gap.
Before proceeding further, we introduce an assumption on the reference measure (1), which will be considered in both the strong and weak log-concavity settings. Recall that for 0 < < t, we denote by Pffe the conditional distribution of X, given X. where (Xt):00 is the unique stationary solution of (1) under HI.
T and any x,y R², the maps 2 + V log log (2ly) and z + V, log (1l()) are Lu Lippsshhtzz with yo
H3. There exists LU > such that, for any 0 S s <
Lu
Lv(8, S 2(1 x)
Remark 1. First note the case where for any 0 S < <tsT, (2,x) + log bogpl,.uul (vlz) is C², then H3 is equivalent the fact that (z,y) v) V. log log Pi. (ulz) is bounded by L(s
Remark 2. We observe that in the common setting where = 0, the above condition satisfied. Indeed, in that case, it holds
[1y
x=|33 -
U. ((l(()lz)= (Am(t d/2 exp
t, [0,7] sct, 2. 2. Wd. (14)
4(t - s)
Therefore, for any 0 S < and any x, Rdd, they hold
Va log p (=|x)
V (ogg)((())z) 2(1 - s)
y-2
2(1-8)
Similar conclusions hold for Ornstein- this is the case, then VU (x) 2Ax and the SDE (1) becomes linear process:
Ornstein-Ulhenbeed Ulhenbeck processes, i.e., as Uxx) (Ax, x). Indeed, if
(OU)
dx, di + V2aB., LE [, Xo m(dx).
This process admits a Gaussian transition density, More specifically,
exp (- (x) En Tx ma(o(zt))
18 P(jx(2)x)
/ (2 dot Egjo
det
where
(z) = - -2 Egjs =2 I' 22A(t- -2A -2 (c-

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Therefore. it holds
Va log U/ (o)a(=) Met Egjs (2- majo(z)) = c- 2/1 (2 -A((-)))
It follows that the map 2 + v log p (z|z) ) is affine, hence Lipschitz. Being
V V log Po (2|x)] & ||e-2A(t-o)]| |
with
(e (e-2A(t- 2A (t-)) ld,
- 1 2(t 8)
then, holds
Lu
V loggl & - 2A 111111171111111 < 2(1-8) 8)
with Ly depending only on A. This and Remark I complete the proof. We expect that Assumption H 3 holds true also if is infinitely differentiable with bounded derivatives, but this problem is out of the scope of the paper and left for future work.
In the sequel, for any measure Ç on (R", B(RR)) absolutely continuous with respect to the Lebesgue measure, we denote by Pc its density. Furthermore, denote by mo.e7 T and Pb. the probability distribution and the density with respect the Lebesgue measure respectively of (Xo: X,, Xx) with (Xt)re)o,r; as in (1), for t E (O.
3.1 Strongly Log-Concave Setting
We consider in this section the following additional assumptions.
H4. The two distributions μ and v are absolutely continuous with respect to the Lebesgue measure and have positive density functions Рμ and Pu. In addition, + log (x() and log (x)) are twice continuously differentiable. In addition, (ro, + llog logro,t, (To, Tt. (T) is twice continuously differentiable.
HS. (i) The two distributions u and are strongly log-concave, i.e., there exist ap ay (0, +00)
such that for any x. E Rdd,
V2( log logg))()) t22 an Id, (2 2 a ld.
(ii) There exists a E [0, +oc) such that for any No, Tt, æt E Rd, E (0,T),
v2 Vg. (-logrö,,.1) (Fo, 3t, uT) 2 ald.
where V2 (- log
2T) denotes the Hessian with respect to the variables (Io,
Remark 3. We remark that in the common setting U = 0 in (1), the part of H4 and H5 related to -log Po.rr holds. Indeed, when U = 0, it follows from (14) that
xT
[Tx - 1) -1)-0-o--Lx LXT
Po (50, 23, IT)
exp
4T
4TT
which makes it clear that - log log Pb.,. satisfies H4 and H5 since for any Fo. Ft, FT E Rd and E (0, T), 22 I- log log (xo, TT, 0. The conditions hold true when U is quadratic, i.e., when Uxx) = (Ax, x)) for a positive defimite matrix A. In this setting, the process (Xt)uefo,1) defined process. Consequently, for any O < l < T, the joint distribution of the triple (Xg, X, Xx) IR& a multivariate Gaussian, (Xo, X, x, for some mean vector μ 3R3d and for positive definite covariance matrix Σ 3dxx3d This immediately implies that H4 and H5 are satisfied. Indeed, let [Σ " denote the principal submatrix of E-1 corresponding to the variables in and IT. Then,
in (1) is multivariate Ornstein-Uhlenben
N(J, E),
V2 20.27 (-log Ph Dz, RT, = =E 2 alld,
where the final inequality follows from the positive definiteness of E,, which implies that all its principal submatrices and thus those of S.-1-ars also positive definite. Therefore, there exists such that the above inequality holds.
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This setting ensures robust stability properties, enabling strong convergence guarantees for IMF. The following theorem formalizes our result in this regime.
Theorem 1. Assume HI to H5. Let (orTh22 be the IMF sequence defined in Algorithm If T max ,.y") then, for any E N, it holds
Lu
(n) TT) S T(ap ag + a)) KL
(L(TOTTTT')
where Oipp 00p - T and aep av -T-1
Remark 4. Theorem establishes that IMF converges to the Schrödinger Bridge nt at an exponential rate in KL-divergence. The convergence is explicitly given by the factor
LU
T(ap + Ha t --------1 2T < <1,
provided that T max(a, a Ly + +ag + +a) + 2}. This shows that larger values of the convexity parameters Og and (vv, as well as stronger convexity of the reference (via larger (.), lead to faster convergence. Conversely, as I approaches max az Lu Lu + 0tp at) 2 from above, the rate p approaches 1, and the convergence becomes arbitrarily slow. This highlights the trade-off between the time horizon T and the strength of nog-concavity in determining the practical efficiency of the IMF algorithm.
Remark 5. We remark that the convergence bound provided in Theorem is dimension-free in the same sense as in the literature on Langevin Monte Carlo: the bound does not depend explicitly on the ambient dimension, but rather on parameters of the marginal distributions, specifically, their log- conccvivv constants. While these parameters are independent of the dimension in some specific settings, they may degrade with dimension in more general scenarios.
Remark 6. We highlight that Theorem can equivalently be reformulated terms of the reference process's 's variance rather than time horizon, as done in [GSK 24). This alternative presentation direct consequence of the Brownian scaling property (Bg o (B.jaz) and does not alter the asymptotic scaling behavior of the convergence rule.
3.2 Weakly Log-Concave Setting
To extend results beyond the strongly log-concave case, consider weaker notion of convexity. which is particularly relevant for machine learning applications involving multimodal distributions or heavy-tailed ttiled priors.
For given differentiable vector field B IRd + Rd, we define its weak convexity profile as
(r(r) = inf (3(x) xx) Byy) y) : || - yll =r) r r>0.
I y/
This function quantifies non-uniform convexity lower bounds, capturing long-range dependencies in nigh-dimensional distributions. This definition is widely used in the study of Fokker-Plande equations via coupling methods (see [Ebe16]).
Define for any L > 0 and r > 0, (r(rr) = 2 tanh(rvLL/2)
In addition, for a distribution absolutely continuous with respect to the Lebesgue measure with positive and continuously differential density function Pç. we write Kg := K- loge:-
H6. The two distributions y and are absolutely continuous with respect to the Lebesgue measure and have positive density functions Pμ and pu. In addition, I + log (x()) and x i (a)) are continuously differentiable. In addition, (Io, RT) + -log poggh.rr. Tt, xT) continuously differentiable.
H7. (i) The two distributions P and v are weakly log-concave, iee, there exist Ceps Ove (0, +00), L,p Lu > O such that for any r O,
kg(r) 2 0pp
r (r), (r(r) 2 ar ar
(ii) There exists a E [0, +00), L 2 0 such that for any t E (0,T) and De E Rdd, denoting by Pt.z, : (I0, + R log pi (Ira It, IT, and Ko,tT == Kp it holds for any r > 0
(r, 2 a - 0z(r).
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Remark 7. Note that H5 (ii) implies H7 (ii). Therefore, in light of Remark 3, we note that either in the classical and in the quadratic case H7 (ii) holds true.
Remark 8. Weakly log-concave distributions constitute a broad and expressive class of probability measures that can be seen as perlurbations of log-concave distributions. For instance, this includes cases where the negative log-density, - 'logg p. has the form of double-well potential
- (() = |==1| - M||2]| + C
for some M > 0,C R. More generally. if -ogp
W with V strongly convex and Lipschitz with Lipschitz derivative, then p weakly log-concave. The class of weakly log-concave distributions also includes Gaussian Mixtures, as shown in i [GSO25, Appendix A In fact, we remark that weak tog-concavity property is also referred as strong convexity at infinity in the literature related to the convergence of the Langevin diffusion. Indeed, such a hypothesis on the potential of the target distribution has been considered various works including [Ebel6, CCAY 18, BRE21, CZS22, MCJ* 19, CM25].
By leveraging weak convexity properties, we derive exponential convergence results for IMF also in this more general setting.
Theorem Assume HI to H4 and H6 and Let { the IMF sequence defined in Algorithm 1. IfT >> max(o7] ,ort), then, for any TL E N, it holds
Lu (TCp
KL KL 31 O
(l(morl(x*)
where
9 (9max ETyss Lu LugLi] Oip + ap + a
Ce = 4(a + av +a)
exp
and Otip = Ckp. T-11 0 = C2y T
Remark 9. Theorem 2 extends the exponential convergence guarantee of Theorem I to the setting where and the reference bridge are only weakly log-concave. The rate
Lu
max (Typs T.u, L.) ag + +a a 00-22-1 2T <1,
exp
(ap +0 + -2T-11
provided that
4LU
18 maxx Lun L}) }
T max a -1 a,
exp
Og + au +
O2 + a2
remains exponential, but is dampened by the factor C.p.u.U. which worsens as the deviations from strong convexity (quantified by Lus Lyg L) increase. This highlights that IMF still converges under weaker assumptions, but the speed is sensitive to the regularity and smoothness of the imput and reference measures.
Remark 10. Theorem 2, as the previous one, can also be reformulated terms of the reference process's 's variance rather than time horizon. The reasons are the same as those discussed in Remark 6. Moreover, the same considerations regarding the dependence on the dimension of the space, discussed in Remark for the previous theorem, also apply here.
3.3 Related Literature and Our Contribution
The study Schrödinger Bridges and their applications has seen surge of interest in recent years. Among the classical methods for solving the SB problem. prominent role is played by the Sinkhorn algorithm, also known as the Iterative Proportional Fitting (IPF) procedure [For40b, Kul68a, Cut13]. More recently, advances in generative modeling have extended the reach of these techniques to continuous-tine settings [Rü95, DBTHD21a). Notably, the Diffusion Schrödinger Bridge (DSB) algorithm (DBTHD21b) lb] and the Iterative Proportional Maximum Likelihood (IPML) method [VTLLL21b] represent two pioneering efforts to approximate Schrödinger Bridge solutions via iterative procedures grounded in continuous-time IPF and time-reversal arguments. While

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IPML leverages Gaussian processes and maximum likelihood estimation, DSB relies on score- matching techniques and can be viewed recursive extension of Score-Based Generative Models (SGMs) [SE19, HJA20, SSDK+21J. SGMs focus on learning the score function of stochastic differential equation (SDE), which then used to simulate the time-reversed diffusion process for sample generation. In recent years, Flow Matching methods [Pel21, LAVE 22, AVE23b, AVE23a, SCD24] have become an alternative to SGMs. These methods construct transport maps using ordinary or stochastic differential equations, bypassing the need for forward noising process that converges to a reference distribution, as required in SGMs [SSDK SSSDK++1J. 21J. Unlike SGMs, Flow Matching methods interpolate between arbitrary probability distributions in finite time, offering greater flexibility in the transport mechanism and improved numerical stability. Building on the principles of Flow Matching, two recent iterative schemes have been proposed for approximating the Schrödinger Bridge: the Iterative Markovian Fitting (IMF) algorithm, also referred to as the Iterated Diffusion Bridge Mixture, and the Diffusion Schrödinger Bridge Matching (DSBM) algorithm (SDBCD23, Pe123). In contrast to classical IPF-hased methods [DBTHD21b], IMF and DSBM incorporate Markovian projections that preserve either Markovian and marginals' consistency across iterations. This refinement leads to improved empirical performance in generative modeling tasks and provides a more principled approach entropy-regularizzed optimal transport. A key distinction between the two algorithms lies in how they handle estimation errors in the drift terms. While IMF does not account for such errors, DSBM explicitly corrects for them via neural networks and L'-estimates and mitigates the resulting bias on the terminal marginal y by alternating between forward and backward Markovian projections. This bidirectional strategy enhances the robustness of the generated samples and leads to more reliable approximation: of the underlying SB solution. Although prior work has extensively analyzed the theoretical properties of score-based generative models [CDS25, GSO25), IPF-type Schrödinger Bridge approximations [CDG23, CCGT24), and flow matching techniques [SCD24, BDD23, AVE22], the Iterative Markovian Fitting (IMF) algorithm has so far lacked rigorous, non- asymptotic guarantees. Existing analyses provide only asymptotic convergence results [SDBCD24, Pe123], leaving open the question of its quantitative behavior in finite iterations. In this work, we address this gap by establishing the first non-asymptotic exponential convergence guarantees for the IMF algorithm. Under mild regularity assumptions on the reference process R and the marginals μ and v, and in the regime of large time horizon T,, we derive explicit convergence rates in terms of KL divergence. These results provide a theoretical foundation for the use of IMF in practical generative modeling applications.
4 Conclusion
We provided the first quantitative theoretical guarantees for the Iterative Markovian Fitting algorithm, under mild assumptions on the marginals and the reference measure. Our analysis covers both strongly log-concave and weakly log-concave distributions the latter being broad class that includes, for example, double-well potentials and yields explicit convergence rates, the first in the literature for this setting. A key technical ingredient is the proof of novel contraction properties of Markovian projections, which are not only essential to our convergence results but also of independent mathematical interest. Our results hold in the regime where the time horizon T is sufficiently large, while in the Schrödinger Bridge problem T is typically fixed. Moreover, our analysis is purely theoretical and does not account for the mimicking drift estimation error or the discretization error arising in practical implementations. These limitations point to natural directions for future work, particularly the extension to finite-time settings and the inclusion of approximation errors in the theoretical analysis.
Acknowledgments and Disclosure of Funding
The work of Marta Gentiloni- Centiloni-Silveri has been supported by the Paris Ile-de-France Région in the framework of DIM A14IDF. The work by Alain Durmus is partially funded by the European Union SAADD--SGG-OCAN--OOUUI
Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them.
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21

---

A Technical Appendices and Supplementary Material
Additional Notation
Denote by 11 E P(Cr) the law on the Wiener space of the stochastic interpolant (Yo) (5). For given & [0, 7" and given Rd, denote by II, the marginal time distribution time E (0, T) of Yt, i..,
II. == L(Y) E P(RR).
follows from the very definition (5) of stochastic interpolant that II, is absolutely continuous with respect the Lebesgue measure with density given, for any aı Rd, by
P (v:)
U Pil(o.T) (Weluo. UT) (dyo, dyr)
(15)
JR
with P) denoting the conditional density of X, given (Xo, Xr). Also, denote by K₁ the regular kernel associated with the conditional distribution of (Yo, Yr) given Y,, i.e., the map K, Rd x (d + [0, 1] such that
(i) + (v, A) is measurable for any A E BR (ii) A K, A) probability measure for any y Rd;
(iii) almost surely it holds
Ky (V, == E (1,6%, V+) A E B(R22))
It follows again from (5) that K, admits transition density with respect to Leb?d given by ky yrly)) PilIo.T) (Melyo. (p) (1x))= T, yT)
(16)
Furthermore, denote by P()) P(Cr) the law on the Wiener space of the (first update of the) forward
Markovian Projection (x," DeEOT (6) of the stochastic interpolant (5) and by Po() E P((r)) the law on the Wiener space of its time-reversal (x(1) (s=(0.z] (x("))<e(or)) (21).
We observe that, with the introduced notation, for any t E T)) and yr, vr- E R&,, we have that
(w)) 2 / V. log log ( (Ur ( (dyo, dyrlu)) VU(V)),
(ye)
(17)
of (vT = / VVrr 1o (dhe
+ VU((Y-
where f()) and (L)) defined in (7) and (13) are drifts of the Markovian projection and time-reversal.
B Technical conditions
We introduce the set of assumptions under which the Markovian projection is well-defined. Ti. For s, (s, v) + p (y1x)) continuously differentiable in the and variables and twice continuously differentiable in the x and variables. Furthermore, O.pH,(ule (ulx)) (.p,,, (ulz), (ylx), (uPH) (ulp), VepHi (ulz) are bounded.
Let b Rd x (0,T) Fdd be a given vector field. We make the following assumption:
T2 (b). b Rd x T) - Rd is locally bounded and such that, for at least one probability solution Na the Pokker-Plances equation
0.u. + div(b,ue) Aus =,, LE (0,TT), no p
it holds f [x)[ (dx)dt < t00.
22

---

Theorem 3. Assume TI and (2( with (()) in (7). Then, the solution (X: DuEOoT of (6) is
such that X dizt for any E [0, .
Remark 11. We remark that TI and T2(J()), with (() in (7) are satisfied in the standard case, i.e, when U = and refer to ISCD24, Remark 9] for more details.
Proof. The thesis is a direct consequence of [SCD24, Theorem 1] for t < T. The case = T, follows essentially from T2((()),, with f()) as in (7). Indeed, T2(f(1) implies that for any {tn In converging from below
J." 1 (x, = L," lim / +
(x jde,
well-defined and that any two almost-surely convergent subsequences must have the same limit almost surely (see e.g. [Bal17, Proposition 1.5)). Therefore,
{-"
(x." dt == a.s lim + I f (x
where (fn denotes any almost surely convergent subsequence of {tn)n. This means that (eEpor) can be extended to the closed time interval [0, I] by defining
{%,-0 Ao f (x. ddt + V2B... }
x (1) =as lim
X
=as
lim 1-++00
(18)
40+000
where {fne denotes any almost surely convergent subsequence of (tn)n. Fix F E C² (10, xR*) with bounded derivatives and let (C(.) uE(T,T denoting the generator of (x(())uE(oTT). i...,
C Fx( (1 (z), (x)) (F)
[0, T).
Using (2))(1) and dominated convergence theorem, we get that for any {tn)n converging from below to T
[( (0.+ CQ Fx := I.? lim / (0, =
well-defined and that any two almost-surely convergent subsequences must have the same limit almost surely (see e.g. [Bal17, Proposition 1.5]). This means that
J." (0. F
a.s lim k-++00
(0, FL
(19)
where ftny L denotes any almost surely convergent subsequence of {tn In is well-defined. As consequence, for any F E (L, T] x Rd) with bounded derivatives, we have that
(
('(3. + C())F.
(20)
<E(O,T]
a martingale. Indeed, for T. (20) follows from [SCD24, Theorem 1]. Whereas, for T, by
continuity of F, (18), (19) and Ito formula, given any almost surely convergent subsequence {mme }k: we have that
f(0, -
Fo(x -FoX)))
Fing (x(()) F6(x /
a.s
lim
(0.+ C
Pank VFF(X lim v2 /2/("
= a.s
too
= 2 l VF
where, in the very last step, we used dominated convergence theorem. Condition (20) makes a diffusion on the closed time interval [0, Moreover, we have that (!)) ~ v: by
continuity and (9), T-, we have that x Y, Yr and Yr
23

---

C Geometric Perspective on Iterative Markovian Fitting
For any measure E P(Cr), denote by bP the bridge associated with it, i.e,, the Markov kernel on I2dd CT satisfying for any A B(CT), P(A)
(T), xT), From geometric perspective, the Iterative Markovian Fitting (IMF) algorithm alternates between two different projections: constructing the stochastic interpolant corresponds projection onto the reciprocal class R" of the Langevin diffusion (1) defined as
Rd : {IP P(CT) bP = bR
while computing the Markovian projection corresponds to projection onto the space M of diffusion Markovian processes. To make this structure explicit, we denote, for any given path measure Q E P(Cr), these projections respectively by projæ (Q) and (Q)) They hold the followings:
projge arg min (QP) PER"), projm (Q) arg min (KT(())P) P M)
We refer ISDBCD24, Propositions 2,4] for the detailed proofs. This means that at each iteration, the IMF algorithm moves back and forth between these two constraint sets:
Algorithm 2: Iterative Markovian Fitting (Geometric Interpretation)
Input: Initial coupling (0)
"0,7 I1(, 2) and Langevin bridge bR associated with (1).
For each iteration k == 0, O,...,
, NN 1:
Step Stochastic Interpolant Update
Set
(k+1)) projæe" (mm²t (zo, xT) bT) ( ((Z0)
Step 2. Markovian Projection Update Set
p(k+1) + proj (II(+t))
(k+1) = 1)) To
End For Output: The path-distribution P(N) and the coupling 7o,r E IT(,, v).
(()
This formulation highlights the alternating nature of the method: each iteration refines the coupling by first enforcing Markovian consistency (via the Markovian projection) and then reintroducing the structure of Langevin bridges (via the stochastic interpolant). This iterative refinement process enables IMF to approximate the SB efficiently. Indeed, the Schrödinger Bridge Pa defined in Section 2 is the unique fixed point of the above iterative scheme, i.e. the unique path-measure Cr such that
P E R nM.
For rigorous proof, we refer the reader [Léo13] and [LRZ14].
C.1 Comparison with Sinkhorn and Diffusion Schrödinger Bridge
The geometric formulation (2) highlights that IMF naturally integrates principles from the Iterative Proportional Fitting (IPF) method [For40a, Ku168b], also known Sinkhorn algorithm, classi- cal technique for approximating Schrödinger bridges. Indeed, both algorithms alternate between projecting onto two different sets of constraints: for IMF, they correspond to Markovian and re- ciprocal processes, whereas for IPF these sets correspond to measures with prescribed first and second marginals. As a result, one key difference is that while IMF provides, at each iteration, coupling between the two distributions, Sinkhorn does not. A true coupling is only recovered at convergence. Moreover, when least one of the distributions continuous, Sinkhorn algorithm can not be implemented exactly: one must rely on approximations, for instance, parameterizing the potentials via kernel expansions [GCPB16 16] or neural networks [SDF- 17]. This is precisely why Diffusion Schrödinger Bridge (DSB) [DBTHD21 algorithm has been adopted. Such algorithm can be seen as dynamical formulation of Sinkhorn and requires similar approximations as those used in IMF. However, from practical perspective, [FVEC22] highlighted the "forgetting" issue of DSB,

---

where drift errors accumulate over iterations, causing intermediate models to lose alignment with the true Schrödinger bridge. In contrast, as observed by [SDBCD24, Appendix FJ, IMF mitigates this issue by explicitly projecting onto the reciprocal class of the reference measure, thereby maintaining the correct bridge structure each iteration and reducing bias accumulation. Moreover, IMF requires only samples at the initial and terminal times, reconstructing intermediate trajectories through the reference bridge, which reduces memory and computational overhead compared to Sinkhorn-basee methods that cache all intermediate trajectory samples.
D Diffusion Schrödinger Bridge Matching
As seen in Section 2, Diffusion Schrödinger Bridge Matching (DSBM) leverages time-reversals of Markovian projections to mitigate the bias accumulation on the second marginal v during practical implementation of Iterative Markovian Fitting (IMF). As shown in [SCD24, Theorem 1J and recalled in Section 2, the mimicking drift f()) (z) can be rewritten conditional expectation:
(w)) = (Y,, Y. = 2i] VU()),
+ (36. vr) = 2V ye log PTy (yr (1))
As consequence, it can be approximated via an L projection [Kle13, Corollary 8.17], by minimiz- ing:
E 110000 (t, (v)) (r>l dt,
0
over a rich enough family of parametrized functions {(t,z) + fo² (t, x) )eEe. Remarkably, the same holds for the drift of the time-reversal of the Markovian projection (Xx")) )eeo..). Indeed, it holds the following result.
)oE(o.T] of the Markovian projection (x(")) )EE(,,T] (6) evolves Proposition 1. The rime-reversal (Xx("))=eoon)
accordingly to
dx =t -x" + 22d,, EE (0,T), X ~ v
(21)
where (Be))E)O,T) is a Brownian motion as in [HP86, Remark 2.5,, for any & E [0, T), the drift function 9t is given, for any yr-t E RR,, by
9: ① (yr (
= [2 Yr Yr- = vr-1 VU(U--))
and, for any E T), the vector field Tb is defined, for any yuu -r- E I2,, by
t
e DT = 2V yr log (UT (
Remark 12. While the Brownian motion in (21) differs from that in (6), since our subsequent analysis is conducted purely in terms of distributions, it can be identified for simplicity.
Proof of Proposition 1: It follows from [And82] and Theorem that the time-reversal of (x(") )tEjo,y (6) evolves accordingly to an SDE with drift function given by
(v:) + 2V log YY (()))
and volatility term equals to 22. Therefore, remains to show that
(22, = $ (Yo. Yo) = yı + vU()) = - -s( (v)) + 2V logp! (vr)
Observe that, as consequence of (15), we have that
V log p. (vx) = JRaxR= Vyr
log p (urlu:))
P ( (v+) plyj (vr (oo)
To. (dyo- dyr)
ad (v:)
25

---

On the other hand, by using (17) and (16), we get that
(v)) JRaxRa V. log Prie (UT w)) Paia (an (or lum)
To, dyr)
(vro()))
(U))
pt (vx)
Therefore, we have that
- ff (vr) + 2V log pi (yt)
Pijo (oo) (err lus) Prr vo) To,T (dyo, dyr)
SRAXRd Vy. log Pl (vo (80)
+ VU(Y))
P (va)
It follows from (16) that
11 (v) 2V log p (w:) E (Yo. = vi] (U()))
which concludes the proof.
Using again [Kle13, Corollary 8. 17], we can therefore approximate also the drift function g2) in (21) by minimizing a standard L²=functional:
0
[/ J0
E
10
(1) (t,
Yr
" (Yr (Yr-))]
rich enough family of parametrized functions {(t,z)) 90 (t, The resulting DSBM algorithm looks as follows.
E Proof of the main results
E.1 Preliminaries on Functional Inequalities.
In this section, recall some well-known definitions and results functional inequalities that will be useful later on.
Definition 1. We say that measure E ((RT)) satisfies Talagrand inequality with parameter T2(E), if. for any p E P(R"), it holds
(p, S (LL())))
This inequality was originally introduced by Talagrand [Ta196] for Gaussian measures. Blower in [Blo03] gave refinement and proved that g-strong log-concavity, i.e., (=(--og gTa leads to T2(E/2).
Definition 2. We say that measure P E P(R") satisfies Log-Sobolev inequality with parameter g <> 0, LST(E), if. for any p E P(RA), holds
KL(P)D) (
It is worth to mention that (CLP23, Theorem 5.7] shows that asymptotic positivity of the integrated convexity profile of the log-density of a probability measure is enough to establish that the measure is Lipschitz image of Gaussian and satisfies LST. More precisely, it shows that if
k log d (r) 2 r r- (r)))
for some à i O, then و satisfies LSI(E) with
{ = 2(a + 1) exp
I 1+à
Moreover, it well known that if satisfies IST(E), then satisfies T2(E). This result was conjectured by Bobkov and Gotze in [BG99] and first proved by Otto and Villani in [OV00].
26

---

Algorithm 3: Diffusion Schrödinger Bridge Matching
Input:
Initial coupling "o.r (H, and Langevin bridge bR associated with (1)
(9)
For each iteration k O,...,.. -:
Step 1. Stochastic Interpolant Update.
Define (Y₂ (2(2k+1)) )eE|o,Ty as:
(ybatt)) v xfak+t) To.T [0,7]
1 (yg (2taht)) r)aat1) ~ bR ((xoatt)) y yfaett)). ::)
Step 2. Forward Markovian Projection Update.
Consider p(24+11 as the path-distribution of the solution (x/2k+1 1E to the SDE:
(2k+1)
dx. (2++1) f/z 1) (1, X, (22 ydl v2dB., x x(2k+1) M
where l the minimizer of
5 122+1) (1, (.y((ktt))) - ((2b+1)) (y,(att))
+ E
with
f 112 (y) = -B o (r (2(2++1) ,, (2tkt)i y( = = vU()),
and te is given in (8).
Set (2k+1) "0,7
(22++1)
0,7"
Step 3. Stochastic Interpolant Update.
Define (y(22++)) ) as:
(y
y22k+2)) .Y
(2++1) ToT
M0ot1331 (yo (2(2)+2) (eaatt),,eattt) ylakt2)) bR ((y / yfact2)
Step 4. Backward Markovian Projection Update.
Consider (1(2 as the path-distribution of the solution (x
)ee(o,T] to the
SDE:
dx (26+2) (2k+2) (t, x (26+2) )tt + v2dB,
x x(2k+2)
gø-
~ v.
where 6* the minimizer of
(t, (, (2k+2)
(2k+2)
(2(2h221)
θ -
E
90
Di T-
with
St
(2k+2)
= FF.
Tb (y (2(2)+2) y(2at2)
| v) (U()).
and de is given in (8).
(2k+2)
(2k++) For
Set 70
End For
Output: The path-distributions P(2N-1) p((N) and the couplings
(2N-1) (2N)
To.T "0,TT
27

---

E.2 Contractivity Properties of the Markovian Projection
In the sequel, we derive contractivity properties for Markovian Projections. Specifically, we show that if two couplings To,T, #o E [[, v) between μ and are close in terms of KL-divergence, then their Markovian projections are even closer KT.-divergence.
In this section, for different objects considered in Algorithm starting from uo. E I[ v), c.g.. the stochastic interpolant, the corresponding Markovian projection, and their associated mimicking drifts, laws, marginals, and kernels, we adopt the same notations as those introduced for No,r, with the addition of a hat symbol. Specifically, if an object was denoted by in relation to Torr, will
(1) now be denoted by [] for øo In particular, we denote by #0,7 the first iteration of Algorithm starting from Norr-
Theorem 4. Assume HI to H3. Let Tu,7, torr E I[ v). Assume that for any E (O, and any y E RRd, the Markov kernel -) -) P(R2d) associated the stochastic interpolant (5) built for and the bridge bR associated to (1) satisfies (E),), for some > O. Then, it holds
KT.
(1) 0,T O,
S
KT. (To,T////
2CT
Proof of Theorem 4: From the data processing incquality [Nut, Lemma 1.6], we get
KL
T
(1)
To.
(1)
KL
(P/)) | (0.T))
From the standard decomposition of the KL divergence based on Girsanov theorem [CLL23, CCL CDS25, SCD24], we obtain
KL (p(1)
pp(1) [0.21]
[0,2]
T/2
(xx²)) dt
1
E -| ( dt.
(L((l))
E
T/2
By the additive property [Léo13, Appendix A] of the KL divergence, we have that
()))
((1) (T/2.T]
KL
E KT. C
(x()
11/2.17 x172) |c
(1)
xx122) + KL. (1/4) (2))
[1/2,11
1 TT
((()) fu0l| (x(2)) de + KI. (D(1) 2/2 2/2)
T/2
The non-negativity of the KL divergence yields
1
|1 - 1/0 (x dt
E
4
T/2
KL (p(1) (1/7)2.7]
(21)) (11/2.m)
KI (1 (B0ry/2)
1 (1
T/2
(00h) - 1/ (x(11) di
1
T/2
2:00 1 11(xx1)) |= de
= KTT(u/)) + J
E
E
By fixing 0 8 < T/2 and putting these inequalities together, we get using Fatou's lemma,
KL
(1)
[11)
[0.27
[0.21
1
T/2
((0) - -.a01 (x((1)] + |1020 -
1601| -
(8(0)) de
J.
I/2
1 -(00
|.00| (xx+0)
+ E 1 الاد (1lt1) at.
lim inf
E
4
8-
28

---

Fix t E [6, T/2) and y Rd. Using the dual representation of the 2 distance
Wi p) sup { |M=>o p)())) Lipei(12
in the expressions (17) we have for the mimicking drifts and H3, we get that
120 - ft f()l s Lo Lo 2 ((((.))).(((..)))
and that
- -x) (w)) I Lu - 2 ( K, ((y-
Since, by assumption, Re( -) and ( satisfy T2(E) when t E [ō, T2), then we have that
()) f w|)l L 1) KL (ke(v.)) -) Re(v.-))
and that
(0) S LoE (T KT. (kr- RT (K)-
By putting all these bounds together and using (9), we get
(1) (1) KL
& lim inf
Lu
[s"
T/2
[1-
t)2 (Vee KL (Ke(v.)) all((e)
o.f Fo
80
4
KL (Kr- (Kr dlly dltr-s(0))
dt
Rd
Now, observe that, for any (0, T), we have that
KL
[prr (om)) 2 KL (C(Y,, Ys, Yr) (x.
(r)o C(Po. Yo, (r.
KL (.. + / KL (Ks(V.-) Ro(v. (I)
/ KL (R.c..))RR(v.)) dll.(y).
This fact follows once again from tha data processing inequality [Nut, Lemma 1.6], the additive property [Léo13, Appendix A] and the non-ncgativity of the KL divergence. then follows that
Lu KL |
f:
T/2
KL
(1) T
O.
< lim inf 6-00
(T- 1)2 df
2
From the additive property of the KL divergence [Léo13, Appendix A] it follows that
KL (Mozz (..21)
KT. ( f + f KT. (bTT RT) - (0 (xo,
+ f KI. (DR" ((ra, (oRa(( oT), BR (To, dzo,r(Zo, KL (xo.r) Ao.r)
where we used that, for any A E ((CT), they hold II (A) = d IT bbI (2) A) and (A) J d (æa, or )bT (o.r) ((), A) together with the fact that bll [oT]
or),
būo,, = 5R Therefore, we get
Lu KT. (TO.7((0.7) f,v2
2212
KL. (1)
*(1) 70,2
S lim inf 8-20
(T z)2 dt
2g
Lu KL (TO.r(
2ºT
The proof is completed.
29

---

E.3 Proof of the Main Theorems
In this section we establish exponential convergence for the Iterative Markovian Fitting (IMF) algorithm 1, Before proceeding further, let us point out that the results of this section hold true under more general assumptions on the log-densities of the probability distributions p.v/ than the ones considered in the main body.
Specifically, consider, in place of H5 (i) and H7 (i) respectively, the following generalized conditions.
H5. (i)' There exist Ou O2 E (0,+ E (0, +o0 such that for any E R
ag ld (-1 (a) = B Id, au Id = v log = B, ld
H7. (i)" There exist On3 E (0, +o0) 8p: 3. E (0, to0 Tup Tu M,, Me > 0 such that for any 0
r
(r(r) 2 a₂ - 0 (r), -k((r) S Bv + r M, (r) kr(r) 02 - 0 (r) -ku(r) S Bu r 0
Remark 13. Note that when Bps By +00, H5 (iY' and H7 (il reduce H5 (i) and H7 (i) respectively.
Theorem 5. Assume H1 to 4 and H5 (iy-H5 (ii). Let (n) ( be the sequence defined in Algorithm IfT max(a, ,07" then, for any n it holds
(n) (T)) S (T KL(TOT
KL(TOOT(T*)
TT(ap + ap + a
where
(ap Ohpp 0pp (an+vaa
4ap
1
400
+
a +
(22)
T2
T
T2
T
Remark 14. Note that when Bu, Bu = +00, Okp, Ogg defined in (22) reduce to Olip = Opp T and
Og = a, TT respectively. Therefore, Theorem is the limit case when Bys By too of Theorem 5.
Theorem 6. Assume HI to 3, H6 and H7 -17 (i). Let (n) be the sequence defined in Algorithm 1. IfT max(a, aay" }, then. for any N. it holds
(L (n) (L(To,T(TT)
(I(TOT((T*))
TC
where
9 19mxx Lug Lug
CpA.U = 4(0 + axp + a
exp
(23)
Cip + as + a
with 0gp Opp --1 (resp. Orp au TT-1) smallest fixed point of
Gv.((a, (a,
1 T
G (a,((,2)
= O2 T +
rexp.
(24)
272
272
and
V5OL, (V8)
(a =nff(f 20: Fr. (a, 2 ut.
(a, BBs
T(1+Ta)
(1+
(resp.
VSUL. (v8)
Gapla, u) inf 20 Faz 2222 u) Fu = Bus T(1 + Ta)
(I+TaY
Remark 15. Note that when Bun 3r = to0, the fixed points Cups Ogg of (24) are exactly a = ap T-1 and ag. = Orr - --1 respectively. Therefore, Theorem is the limit case when Bu, By +00 of Theorem 6.
30

---

Proof of Theorem 5: From [Léo13, Theorem 2.12b] and [LRZ14, Theorem 2.14]. the Schrödinger Bridge πT" is a fixed point of the IMF algorithm 1. We thus wish iteratively apply Theorem 4 with fo.r - this aim, we simply need to show that, under the assumptions of Theorem 5, the kernel (y associated with the stochastic interpolant built on ão, - =x and the bridge bR" associated and for any fixed & (0, T') and y Rd. Fix E (0, 7) and y E Rd Note that, as a consequence of (16) and (4), the kernel K2 (z, admits density with respect to the Lebesgue measure given by
to (1) satisfies (g))), for some
(y.
dLeb (yo. VT)
(25)
x (o(T)) (1lo.r)yyll00. (Mo, yT)
(o..) (ylvo: yrd exp (-p(w0) - (v(r)) Ro Vrr
= Po y. MT) exp (-v(10) -b(T))
exp (- {1(6)) ((rr) log ro.e v,
When T > max o ), the byproduct of conditions H4, H5(i)' and [Gre24, Theorem 6.6.1)
leads to
V2y = ag Id,
V = av Id.
(26)
with aep Oup as in (22). follows from (25), (26) and H5 (ii) the 2g-strong log cog-cccavity of Ky. +), with { = (ap toz +a)/2. As seen in Section E, the 2g-strong log log-concavity of K.(y,) combined with [Blo03] yield that (w:-)) satisfies T2(3). The arbitrariness of L (0, T) and y Rd together with the iterative application of Theorem 4 for मे, x- and & = (app + ap + a)/2 allow to conclude.
Proof of Theorem 6: Once we prove that, under the assumptions of Theorem 6, the kernel R2 (M, )) associated with the stochastic interpolant built on the Schrödinger Bridge uu. = π² and the bridge bR" associated to (1) still satisfies T2(E), for some & > 0 and for any fixed t E (0,T) and y E Rd. we can proceed as in the proof of Theorem 5 and conclude. Fix (T, and y E TRd. Denote by
hiyy = log( (dRR(y, 2)). It follows from (25) that
( yr) x ((86) (vrr log logpd...r V. vr) log (yo. yr)
follows from H6 and H7 (i)' combined with [Con24, Theorem 1.2] and H7 (ii) that for any > 0 kin, (r) 2 ap r - (r) + as r 101 -ar -1
Now, observe that, for any r >> O, the maps L + tanh(rv and L (r) increasing:
0n (tanh(rv[//2)) = LoL (rvosh 0,
r
1 tanh(rr
0r,n((r) =
>0
VL
2 coshi coshh2(rV (rVL/2)
Hence,
+ 0z(r) < 30 max f (..
maxx Lyss Lus 1}//2
2 max Lyp Lu, L} tanh
(r/9max(( (Tpps Jows I.) T.)/2
< 2 max( Tops Tw L.} tanh
2 1a (T)
Therefore, we have that kh., (r) 2 (ap + av + - +a
r
-1 0g (...
31

---

As seen in Section E.I, this combined with [CLP23, Theorem 5.7] yields that ke(y.)) satis- fies L..(C....12) with Ca.., in (23) and, in turn, [OV00] yields that (y, satisfies T2(Cp,,,12).
32


Reviews:
[{"review_id": "ATluf57hzH", "reviewer": "Reviewer_NJSy", "summary": "This work studies the convergence of the IMF method for solving the Schr\\”odinger bridge problem. This method consists in alternating Markovian projection steps with stochastic interpolation steps. This method can sometimes outperform Sinkhorn’s algorithm in practice. The paper shows that under log-concavity assumptions on both end marginals as well as the reference path measure, one achieves exponential convergence with rate depending roughly on $1/(\\alpha_\\mu + \\alpha_\\nu)$. The result is also extended to the case of ``weak convexity’’.", "questions": "**Major**\n\n- I am not particularly familiar with the definition of “weak convexity” used here, so some greater explanation around Theorem 2 would be nice. In particular, is this the expected rate? Can we hope to do better in any meaningful sense?\n- Is the large $T$ condition fundamental, or a limitation of the analysis?\n\n**Minor**\n\nL. 177: Steps $\\to$ steps", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n\n- The result (in the strongly convex case) is extremely elegant and stated under interpretable but generic conditions. In particular, the dependence on the convexity parameters $\\alpha_\\mu, \\alpha_\\nu$ makes sense. \n- As far as I can tell, the proof is also very clean and readable, and may inspire further analysis in this direction. The paper in general is quite well-written.\n- I think this is a strong contribution to the literature. As IMF is a well-known algorithm, this will be of interest both to theoreticians and practitioners.\n\n**Weaknesses**\n\n- The algorithms are idealized, and most discretization errors (SDE discretization, estimating the conditional drift) are ignored.\n- There is an additional requirement that $T> \\alpha^{-1}$ in e.g., Theorem 1, which seems strange. Why would this be necessary?", "was_revised": true, "final_justification": "I believe this paper makes a strong contribution as it obtains a relatively sharp (if unsurprising) convergence bound for a canonical algorithm. I believe the lack of experimental validation is NOT a hindrance to this paper's acceptance, and that it should be assessed on the merits of its theoretical contributions only; I am willing to champion its acceptance.", "rebuttal": "We sincerely thank the reviewer for the very positive evaluation. We are glad that the reviewer finds our convergence result elegant, interpretable, and potentially inspiring for future work. We also appreciate the kind comments on the clarity of the paper and the interest this may generate among both theoreticians and practitioners.\n\n*There is an additional requirement that T>α^{-1}, in e.g., Theorem 1, which seems strange. Why would this be necessary?*\n\nThe condition $T>\\max\\{\\alpha_\\mu^{-1}, \\alpha_\\nu^{-1}}$ is a technical assumption required by our proof technique to ensure that the convexity parameters (either strong or weak log-convexity) of the Sinkhorn potentials are well-defined and strictly positive. This is formalized in [2], which we refer to in our analysis. \n\n*I am not particularly familiar with the definition of “weak convexity” used here, so some greater explanation around Theorem 2 would be nice. In particular, is this the expected rate? Can we hope to do better in any meaningful sense?*\n\nWeak log-concavity generalizes log-concavity and includes a broad class of distributions, such as those arising from double-well potentials (Remark 5) and mixture of Gaussian distributions. In fact, weak log-concavity property is also referred to as strong convexity at infinity in the literature related to the convergence of the Langevin diffusion. Indeed, such a hypothesis on the potential of the target distribution has been considered in various works including [10], [16],[17],[18],[19], and [20]. Regarding the rate,  exponential convergence of the Sinkhorn algorithm for weakly log-concave distributions, convergence rate that decays polynomially in $T$, has only been established recently [4]. For IMF, we currently do not know whether our rate or the condition on $T$ can be improved. This remains an open question that we are not able to address.\n\n*Is the large T condition fundamental, or a limitation of the analysis?*\n\nWe thank the reviewer for the question. At this stage, we do not know whether the assumption of a large time horizon $T$ is genuinely necessary for convergence, or if it reflects a limitation of our current analysis . In our proof, a large $T$ plays two roles: (i) it guarantees (weak) convexity of the Sinkhorn potentials (as detailed in the previous answers), and (ii) it controls the mimicking drift in the Markovian projection (see the proof of Theorem 4). The sharper condition required for (ii) is discussed in Remarks 3 and 6. Extending the analysis to handle arbitrary or small values of $T$ is an important open problem. We have already spent time investigating this direction, but we currently do not have a viable approach.\n\n\t\n[2] Conforti, G. (2024). Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges. Probability Theory and Related Fields, 189(3), 1045-1071.\n\n[4] Chiarini, A., Conforti, G., Greco, G., & Tamanini, L. (2024). A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm. arXiv preprint arXiv:2412.09235.\n\n[10] Eberle, A. (2016). Reflection couplings and contraction rates for diffusions. Probability theory and related fields, 166(3), 851-886.\n\n[16] Cheng, X., Chatterji, N. S., Abbasi-Yadkori, Y., Bartlett, P. L., & Jordan, M. I. (2018). Sharp convergence rates for Langevin dynamics in the nonconvex setting. arXiv preprint arXiv:1805.01648.\n\n[17] Bou-Rabee, N., & Eberle, A. (2023). Mixing time guarantees for unadjusted Hamiltonian Monte Carlo. Bernoulli, 29(1), 75-104.\n\n[18] Cheng, X., Zhang, J., & Sra, S. (2022). Efficient sampling on Riemannian manifolds via Langevin MCMC. Advances in Neural Information Processing Systems, 35, 5995-6006.\n\n[19] Ma, Y. A., Chen, Y., Jin, C., Flammarion, N., & Jordan, M. I. (2019). Sampling can be faster than optimization. Proceedings of the National Academy of Sciences, 116(42), 20881-20885.\n\n[20] Chak, M., & Monmarché, P. (2025). Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case. IMA Journal of Numerical Analysis, draf045."}, {"review_id": "H9bViJoLcC", "reviewer": "Reviewer_KTkT", "summary": "The paper analyses the convergence of the (theoretical) IMF scheme to recover the solution of a Schrödinger Bridge (SB) problem for which no quantitative, non-asymptotical results were yet derived. The key ingredient of the proof is to show that the updates are contractive under the assumptions made in the paper. More specifically, in Theorem 4, it is shown that if the Markov kernel associated with the stochastic interpolant of the optimal coupling (which is in fact the solution to the dynamic problem (2) right?) satisfies a Talagrand inequality, then one step of the IMF is contractive. Then, the paper leverages the recent results of Conforti 2024 to recover weak semi-convexity estimates on the optimal coupling. To handle the non-strongly log concave case, the paper uses the fact that weak semi-convexity implies log-Sobolev which in turn implies Talagrand.", "questions": "See above + why use IMF rather than Sinkhorn?", "limitations": "I feel that the guarantees offered by IMF are overall much weaker than Sinkhorn (especially with $\\varepsilon = 1$ as it seems to be the case here) for several reasons:\n- Sinkhorn can be implemented \\emph{exactly} (I do not agree with l. 129-133) while IMF cannot because the function (11) needs to be approximated and no guarantees are provided with this approximation.\n- Sinkhorn converges fast in much larger settings.\nIn my opinion, this comparison should have been made in the paper.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The strength of the paper is to bring the first quantitative, non-asymptotic guarantees under fairly general assumptions ; remarkably, no uniform log-smoothness assumptions is made. \nWeaknesses:\n- from a technical point of view, the main contribution seems to be Theorem 4 since the weak semi-convexity estimates were proven in Conforti 2024. Since I am quite far from the SB community, I cannot tell how hard it was to derive (I acknowledge theoretical results always seem obvious once they are derived though).\n- because of this proof scheme, that relies on weak-semi-convexity estimates, the quantitative guarantees are in fact quite weak; indeed, weak-log-concavity implies exponentially bad log-Sobolev constants. In particular, how does the exponential bound in Theorem 2 compares to practical settings: does IMF fail when the distributions are not strongly log-concave?", "was_revised": true, "final_justification": "I maintain my score. I believe the analysis is most-likely ad-hoc and suboptimal but it has the merit of being the first. Therefore, I am not opposed to acceptance.", "rebuttal": "We thank the reviewer for their thoughtful summary and detailed appreciation of our contributions. We are particularly grateful for the recognition of our work as the first to provide quantitative, non-asymptotic guarantees for IMF, under general assumptions that avoid uniform log-smoothness. \n\n*From a technical point of view, the main contribution seems to be Theorem 4 since the weak semi-convexity estimates were proven in Conforti 2024. Since I am quite far from the SB community, I cannot tell how hard it was to derive (I acknowledge theoretical results always seem obvious once they are derived though).*\n\nWe appreciate the reviewer’s perspective. While the (weak) semi-convexity estimates build upon the prior work [2], our contribution goes significantly beyond this. A central technical novelty of our work lies in deriving a new contraction estimate for the Markovian projection operator (Theorem 4), which is nontrivial and, to the best of our knowledge, has not appeared previously in the literature. Integrating this estimate into a recursive convergence proof under (weak) log-concavity, required careful analysis and constitutes an essential step in establishing non-asymptotic guarantees for IMF. \n\n*Because of this proof scheme, that relies on weak-semi-convexity estimates, the quantitative guarantees are in fact quite weak; indeed, weak-log-concavity implies exponentially bad log-Sobolev constants. In particular, how does the exponential bound in Theorem 2 compares to practical settings: does IMF fail when the distributions are not strongly log-concave?*\n\nWe thank the reviewer for their comment. We would like to emphasize that up to our knowledge, estimating a log-Sobolev constant is highly nontrivial in general and only tractable in specific settings. In particular, outside the log-concave case, such constants typically depend exponentially on the parameters of the distribution. Finally, in the literature on Langevin diffusions and gradient-based MCMC methods (see e.g.,[16],[17],[18],[19], and [20]) , both types of assumptions (e.g., convexity vs. functional inequalities) are commonly treated as complementary. Therefore, we do not view our assumptions as a weakness. Furthermore, empirical studies (e.g., [1], [9]) suggest that IMF performs well even when the marginals are not strongly log-concave. This is analogous to MCMC algorithms, where theoretical convergence guarantees can be pessimistic in non-log-concave settings, yet the algorithms remain relatively effective in practice.\n\n*Why use IMF rather than Sinkhorn?*\n\nWe thank the reviewer for the question. One key difference is that IMF provides, at each iteration, a coupling between the two distributions, whereas Sinkhorn does not. More specifically, from a primal perspective, Sinkhorn alternates between projections onto sets of joint distributions that match either the first marginal $\\mu$ or the second marginal $\\nu$, but not both simultaneously. As a result, the intermediate distributions produced by Sinkhorn typically have only one correct marginal at a time. A true coupling is only recovered at convergence. From a practical perspective, [15] highlighted the “forgetting” issue of DSB, where drift errors accumulate over iterations, causing intermediate models to lose alignment with the true Schrödinger bridge. In contrast, as observed by [1] in Appendix F, IMF mitigates this issue by explicitly projecting onto the reciprocal class of the reference measure, thereby maintaining the correct bridge structure at each iteration and reducing bias accumulation. Moreover, IMF requires only samples at the initial and terminal times, reconstructing intermediate trajectories through the reference bridge, which reduces memory and computational overhead compared to Sinkhorn-based methods that cache all intermediate trajectory samples.\n\n*Sinkhorn can be implemented \\emph{exactly} (I do not agree with l. 129-133) while IMF cannot because the function (11) needs to be approximated and no guarantees are provided with this approximation.*\n\nWe are not aware of any setting in which Sinkhorn can be implemented exactly when at least one of the distributions is continuous. In practice, one must rely on approximations, for instance, parameterizing the potentials via kernel expansions [12] or neural networks [13], which introduces non-trivial computational and analytical challenges (see e.g., [1], [11], [14]). This is precisely why the dynamical formulation of Sinkhorn has been adopted in [1],[11], which also requires similar approximations as those used in IMF. However, to our knowledge, no theoretical results are currently available that account for the approximation errors arising in these practical implementations of the dynamical Sinkhorn approach. In contrast, quantitative convergence results do exist for the ideal (exact) Sinkhorn algorithm. For IMF, such quantitative results were not previously available. Therefore, a necessary first step toward understanding the effect of approximation errors in practical implementations is to establish convergence guarantees at the ideal level. This is exactly what we do in this work. \n\n*Sinkhorn converges fast in much larger settings. In my opinion, this comparison should have been made in the paper.*\n\nWe thank the reviewer for this useful remark. We agree that an explicit comparison with Sinkhorn would strengthen the presentation and plan to include it in the revised version of the manuscript. In particular, we will discuss recent results establishing exponential convergence of the Sinkhorn algorithm for weakly log-concave distributions and all values of $T$, with a rate that decays polynomially in $T$ (see [4]).\n\n\n[1] Shi, Y., De Bortoli, V., Campbell, A., & Doucet, A. (2023). Diffusion schrödinger bridge matching. Advances in Neural Information Processing Systems, 36, 62183-62223.\n\n[2] Conforti, G. (2024). Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges. Probability Theory and Related Fields, 189(3), 1045-1071.\n\n[4] Chiarini, A., Conforti, G., Greco, G., & Tamanini, L. (2024). A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm. arXiv preprint arXiv:2412.09235.\n\n[9] Peluchetti, S. (2023). Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling. Journal of Machine Learning Research, 24(374), 1-51.\t\n\n[11] Vargas, F., Thodoroff, P., Lamacraft, A., & Lawrence, N. (2021). Solving schrödinger bridges via maximum likelihood. Entropy, 23(9), 1134.\n\n[12] Aude Genevay, Marco Cuturi, Gabriel Peyré, and Francis R. Bach.Stochastic optimization for large-scale optimal transport.In Neural Information Processing Systems, 2016.\n\n[13] Vivien Seguy, Bharath Bhushan Damodaran, Remi Flamary, Nicolas Courty, Antoine Rolet, and Mathieu Blondel.Large-Scale Optimal Transport and Mapping Estimation.In International Conference on Learning Representations, 2018.\n\n[14] Peyré, G., & Cuturi, M. (2019). Computational optimal transport: With applications to data science. Foundations and Trends® in Machine Learning, 11(5-6), 355-607.\n\n[15] Fernandes, D. L., Vargas, F., Ek, C. H., and Campbell, N. D. (2021). Shooting Schrödinger’s cat. In Fourth Symposium on Advances in Approximate Bayesian Inference. \n\n[16] Cheng, X., Chatterji, N. S., Abbasi-Yadkori, Y., Bartlett, P. L., & Jordan, M. I. (2018). Sharp convergence rates for Langevin dynamics in the nonconvex setting. arXiv preprint arXiv:1805.01648.\n\n[17] Bou-Rabee, N., & Eberle, A. (2023). Mixing time guarantees for unadjusted Hamiltonian Monte Carlo. Bernoulli, 29(1), 75-104.\n\n[18] Cheng, X., Zhang, J., & Sra, S. (2022). Efficient sampling on Riemannian manifolds via Langevin MCMC. Advances in Neural Information Processing Systems, 35, 5995-6006.\n\n[19] Ma, Y. A., Chen, Y., Jin, C., Flammarion, N., & Jordan, M. I. (2019). Sampling can be faster than optimization. Proceedings of the National Academy of Sciences, 116(42), 20881-20885.\n\n[20] Chak, M., & Monmarché, P. (2025). Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case. IMA Journal of Numerical Analysis, draf045."}, {"review_id": "ZaW26AweQm", "reviewer": "Reviewer_CFWL", "summary": "In this paper, the authors propose a non-ordinary method for proving the exponential convergence rate of the IMF algorithm under specific log-concave assumptions on marginal distributions and reference dynamics. This makes possible to use the duality of the Kantorovich metric and the Talagran inequality to prove contraction of the Markovian projection in IMF by the Kullback-Leibler divergence. The authros derive two convergence rate results, for strongly and weakly log-concave settings. The constant in the resulting inequality requires the time horizon to be large enough.", "questions": "- Would the authors consider reformulating the theorem in terms of the reference process's variance rather than time horizon? This alternative presentation might enhance interpretability and better align with established physical interpretations of the convergence behavior.\n- What evidence suggests that large time horizons are fundamentally necessary? Could you provide examples where compression fails for smaller horizons to clarify this requirement?\n - Might the authors' methodology be, in principle, extensible to more general classes of marginal distributions? One wonders if there exist implicit conditions under which such generalizations could hold?\n - Could numerical experiments be added at least to: (1) verify convergence behavior at various time horizons, including sorter ones? (2) Test the practical implications of the marginal distribution, including possibly violating log-concavity constraints?", "limitations": "The authors have addressed the limitations of their approach in the conclusion.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\nThe paper is clearly written and all the required notions are properly introduced. The authors prove a rather nontrivial result for an important class of marginals and a reasonable family of reference measures. The proof of the main technical result (Theorem 4) seemed to be quite an interesting technique.\n\n**Weaknesses**\n- First, the dependence of the Markov projection's compression constant on the time horizon appears somewhat unclear. The existing analysis might alternatively suggest that the IMF convergence rate depends primarily on noise variance, as compellingly demonstrated in prior work [1] (see Figure 6, \"Dependence on the variance of the prior process\"). It would be particularly insightful if the authors could reformulate their main theorem to express the compression constant in terms of the reference process's noise variation while keeping the time horizon fixed. Such refinement could provide a more intuitive interpretation of the convergence behavior.\n- Second, the requirement for a sufficiently large time horizon needs additional justification. The manuscript would be significantly strengthened by including explicit examples demonstrating the absence of compression properties for shorter time horizons. The absence of such examples might suggest that this condition reflects limitations of the current proof technique rather than fundamental properties of the iterative procedure. \n- Third, the assumptions regarding marginal distributions appear somewhat restrictive. By focusing exclusively on unimodal distributions (log-concave and weakly log-concave families), the current theoretical guarantees exclude potentially important cases of multimodal distributions that frequently arise in practical applications. Addressing this limitation through theoretical extension would enhance the paper's applicability.\n- Finally, the absence of numerical validation represents a notable gap in the current research. Empirical results would be particularly valuable for addressing: (1) the behavior of convergence rates for different time horizons, including shorter ones, and (2) the depended of convergence for the different marginal distributions,  possibly including the ones that do not fall ini described by authors assumptions. Including such experimental validation would substantially strengthen confidence in the theoretical findings.\n\nThese suggested improvements, while not diminishing the paper's theoretical contributions, could help establish a more comprehensive understanding of the method's properties and limitations. The authors' attention to these aspects would likely increase both the depth and practical relevance of their work.\n\n\n**References.**\n\n[1] N. Gushchin, D. Selikhanovych, S. Kholkin, E. Burnaev, A. Korotin. Adversarial Schrödinger Bridge Matching. The Thirty-eighth Annual Conference on Neural Information Processing Systems. 2024", "was_revised": true, "final_justification": "The authors have addressed my concerns partially. *In my opinion*, the lack of numerical experiments that validate the presented convergence rates is a serious shortcoming of the paper. However, this paper still advances an interesting topic and despite having some gray areas in their results, e.g., small values of $T$, I'd rather see this paper accepted than rejected.", "rebuttal": "We thank the reviewer for their positive feedback. We appreciate the recognition of the novelty in our approach to analyzing the convergence of IMF under (weak) log-concave assumptions. We are glad the reviewer found our use of Kantorovich duality and Talagrand-type arguments compelling, and that Theorem 4 was seen as a technically interesting contribution.\n\n*On the first weakness: Would the authors consider reformulating the theorem in terms of the reference process's variance rather than time horizon? This alternative presentation might enhance interpretability and better align with established physical interpretations of the convergence behavior.*\n\nWe thank the reviewer for this insightful question. There is, in fact, no contradiction between our analysis and the observation in [7] that the convergence behavior depends on the noise variance. This relationship is a direct consequence of the Brownian scaling property: $B_t \\sim \\sigma B_{t/\\sigma^2}$. As a result, our bounds can equivalently be reformulated in terms of the noise variance $\\sqrt{2T}$ while fixing the time horizon to $1$, as the reviewer suggests. However, such a reformulation does not alter the asymptotic scaling behavior of the compression constant, which remains of order $O(T^{-1})$. More precisely, if one fixes the time horizon to $1$ and sets the volatility parameter to $\\sigma = \\sqrt{2T}$, then the mimicking drift appearing in the Markovian projection includes a prefactor of order $1/(1 - t)$. The standard Girsanov decomposition of the KL divergence [8] introduces a multiplicative factor of $\\frac{1}{2\\sigma^2} = \\frac{1}{4T}$, and this ultimately leads to a compression constant of order $O(T^{-1})$, even when the analysis is expressed in terms of the noise variance. We will clarify this equivalence and its implications in the revised manuscript.\n\n*On the second weakness: What evidence suggests that large time horizons are fundamentally necessary? Could you provide examples where compression fails for smaller horizons to clarify this requirement?*\n\nWe thank the reviewer for raising this important point. We fully acknowledge that, at present, we do not know whether convergence still holds for small values of $T$. It remains an open and interesting question and we do not currently know if this is a limitation of our current analysis. Nevertheless, we believe our contribution is still significant: to the best of our knowledge, this is the first result establishing geometric convergence for IMF under relatively broad assumptions. While we aimed to provide a comprehensive analysis, scientific progress often advances incrementally. As a point of comparison, geometric convergence for Sinkhorn in compact settings has been known for over a decade, but results in non-compact spaces or with unbounded cost functions have only appeared recently [4]. \n\n*On the third weakness: Might the authors' methodology be, in principle, extensible to more general classes of marginal distributions? One wonders if there exist implicit conditions under which such generalizations could hold?*\n\nWe thank the reviewer for this important observation. While we acknowledge that extending the theory to any multimodal marginals would be valuable, the assumptions we consider (i.e. log-concavity and weak log-concavity) are, to our knowledge, standard in the literature, particularly when aiming for dimension-free convergence rates. This situation is closely analogous to that of Langevin-based algorithms, where similar assumptions are commonly adopted to obtain geometric convergence bounds that are uniform in dimension. Beyond this setting, convergence guarantees, if they exist, typically depend on the dimension or are no longer geometric. That said, empirically, as also seen with Langevin dynamics, IMF method performs well even outside the log-concave setting (see e.g., [1], [9]).\n\n*On the last weakness: Could numerical experiments be added at least to: (1) verify convergence behavior at various time horizons, including sorter ones? (2) Test the practical implications of the marginal distribution, including possibly violating log-concavity constraints?*\n\nWe thank the reviewer for the constructive suggestion. Our work builds upon [1] and [9], who propose practical implementations of IMF, and already include empirical evaluations that address many of the questions raised. Indeed, [1] reports ablation studies on the time horizon, showing improved convergence with larger $T$ in 2D Gaussian settings (Figure 3). Moreover, either [1] and [9] consider non–log-concave marginals, including mixtures, uniform distributions, and real datasets (e.g., MNIST or StyleGAN-generated images), where IMF still performs well despite the lack of theoretical guarantees. That said, in non-Gaussian scenarios, IMF implementations rely on numerical approximations (e.g., of the mimicking drift), which prevent direct verification of convergence rates. In particular, the optimal Schrödinger bridge is not available in closed form. This challenge is not unique to IMF: to the best of our knowledge, there are no empirical studies verifying known convergence rates for Sinkhorn-type methods in continuous, non-Gaussian settings.\n\n\n[1] Shi, Y., De Bortoli, V., Campbell, A., & Doucet, A. (2023). Diffusion schrödinger bridge matching. Advances in Neural Information Processing Systems, 36, 62183-62223.\n\n[4] Chiarini, A., Conforti, G., Greco, G., & Tamanini, L. (2024). A semiconcavity approach to stability of entropic plans and exponential convergence of Sinkhorn's algorithm. arXiv preprint arXiv:2412.09235.\n\n[7] Gushchin, N., Selikhanovych, D., Kholkin, S., Burnaev, E., & Korotin, A. (2024). Adversarial schrödinger bridge matching. Advances in Neural Information Processing Systems, 37, 89612-89651.\n\n[8] Chen, H., Lee, H., & Lu, J. (2023, July). Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learning (pp. 4735-4763). PMLR.\n\n[9] Peluchetti, S. (2023). Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling. Journal of Machine Learning Research, 24(374), 1-51."}, {"review_id": "FxJE3RCaBk", "reviewer": "Reviewer_THDL", "summary": "The authors analyze the convergence of the Iterative Markovian Fitting (IMF) algorithm for estimating Schrödinger Bridges. In particular, they develop the first non-asymptotic convergence rates for Iterative Markovian Fitting. An analysis is provided for two important regimes:  strongly and weakly log-concave marginal distributions, where the latter case even captures certain non-convex marginals.", "questions": "- Can you please elaborate on the dimension-dependence of your analysis (at least for some relevant examples / instances)?\n- Can you please elaborate why assumption H2 is reasonable? When is it satisfied?", "limitations": "yes -- the discussion could be strengthened though by highlighting where the key challenges in the current analysis lie that would need to be overcome to analyze more practical variants of Iterative Markovian Fitting.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The authors provide the first non-asymptotic convergence result for an important algorithm for generative modeling.\n- The new contraction results for Markovian projections could be of independent interest.\n- Mostly clear structure and exposition, with assumptions are clearly spelled out. Most of them (with one exception, see below) are discussed in detail and settings provided where they hold.\n\nWeaknesses:\n- The main weakness and acknowledged limitation is that the analysis applies to an idealized (impractical) version of the Iterative Markovian Fitting algorithm, which assumes exact projections, and ignores effects of discretization. Nevertheless, I think the contribution is still a meaningful first step given the importance of the problem and interest of the machine learning community in the area.", "was_revised": true, "final_justification": "The authors have satisfactorily addressed my concerns, and I still think the paper makes a valuable contribution.", "rebuttal": "We thank the reviewer for their encouraging and constructive remarks. We are pleased that the reviewer highlights the novelty of our non-asymptotic analysis of IMF in both the strongly and weakly log-concave regimes. We are also grateful for the appreciation of the clarity of our assumptions and exposition, and the potential independent interest of the new contraction results.\n\n*Can you please elaborate on the dimension-dependence of your analysis (at least for some relevant examples / instances)?*\n\nWe clarify that the convergence bounds we obtain are dimension-free in the same sense as in the literature on Langevin Monte Carlo: the bounds do not depend explicitly on the ambient dimension, but rather on parameters of the marginal distributions, specifically, their (weak or strong) log-convexity constants. While these parameters are independent of the dimension in some specific settings, they may degrade with dimension in more general scenarios.\n\n*Can you please elaborate why assumption H2 is reasonable? When is it satisfied?*\n\nRegarding assumption H2, when the reference potential $U$ in either null or quadratic, a simple sufficient condition for its validity is that the product measure $\\mu \\otimes \\nu$ is absolutely continuous with respect to the reference measure $R^U$ and that the logarithm of its Radon–Nikodym derivative is integrable, i.e., $\\log⁡ \\frac{d(\\mu\\otimes\\nu)}{d R^U}\\in L^1(\\mu\\otimes\\nu)\\$. This condition is standard in optimal transport and Schrödinger bridge literature, see e.g. [5]. Moreover, when $U$ is sufficiently smooth, we expect Assumption H2 to hold whenever $\\mu$ and $\\nu$ are absolutely continuous with respect to the Lebesgue measure.\n\n[5] Nutz, M. (2021). Introduction to entropic optimal transport. Lecture notes, Columbia University."}, {"review_id": "N0mr3ywlsh", "reviewer": "Reviewer_gX7U", "summary": "The paper studies non-asymptotic convergence guarantees for IMF procedure. The exponential-speed convergence is established under suitable assumptions, of which the key ones are strong or weak log-concavity of the reference distributions. The manuscript comes with remarks which connect theoretical layouts with practical use-cases.", "questions": "1. Assumptions H4, H6 - need to be explicitly stated that $(x\\_0, x_T) \\mapsto - \\log p_{0,t,T}^U(x_0, x_t, x_T)$ is [properties from these assumptions] hold for each $x_t$\n\n2. Theorem 1 and Theorem 2: It seems we need $T > \\max{(\\alpha_\\mu^{-1}, \\alpha_\\nu^{-1}, \\alpha^{-1})}$, e.g., for the contraction in Theorem 1 holds true. \n3. A lot of repeated bibtex records: \n  - [AVE22] , [AVE23a] ([AVE23a] does not exists, note that [AVE22] is ICLR-23)\n  - [AVE23b] - missed author Nicholas M. Boffi\n  - [DBTHD21a], [DBTHD21b]\n  - [For40a], [For40b]\n  - [Kul68a], [Kul68b]\n  - [SDBCD23], [SDBCD24] (One from NeuriPS 23, the other from NeurIPS 24 ???)\n  - [VTLL21a], [VTLL21b]\n    \n   Double check bibtex files if you generate bibtex records with llms. \n\n4. Theorem 4: From the proof it seems we do not need the bridge $\\text{bR}^U$ to satisfy $\\text{T2}(\\xi)$\n\n5. Line 645: “the byproduct of conditions [...] leads to” - I think for clarity it is better to be more specific and detailed here, so does in line 656 in the proof of Theorem 6.\n\n6. Formula after line 656: $k\\_{\\overline{h}\\_{t, y}}$ -> $k_{\\nabla \\overline{h}_{t, y}}$.", "limitations": "Discussed above", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "I found the paper to be more-or-less well-written and the considered problem (convergence of IMF) to be interesting. At the same time, from a practical point of view, paper has no practical implications; from a theoretical point of view - it is interesting to study whether the assumptions needed for the convergence hold true could be relaxed - current assumptions (on reference distributions) seems to be rather restrictive. Overall, I have a good impression of the paper.\nMore specific points:\n- The paper has no experiments validating the theoretical conclusions. In particular, it is reasonable to run DSBM on, e.g., gaussian measures/perturbed gaussian measures to see the convergence rates\n- There is a preprint studying the convergence of IMF [Kholkin]. Their analysis seems to be rather restrictive and hold only for gaussians, but anyway they are highly related.\n\n[Kholkin] Kholkin et. al., Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting, 2024", "was_revised": true, "final_justification": "The paper considers an important and interesting area of theory in the context of modern diffusion bridge models, providing novel results. However, the experimental validation of the obtained bounds is lacking. While I generally appreciate the authors' contribution (and therefore vote for acceptance), I am inclined to borderline due to this concern.", "rebuttal": "We thank the reviewer for the overall positive impression and for finding the topic of convergence of IMF both interesting and relevant. We appreciate the comments on the theoretical contributions and the effort to relate them to practical use-cases.\n\n*The paper has no experiments validating the theoretical conclusions. In particular, it is reasonable to run DSBM on, e.g., Gaussian measures/perturbed Gaussian measures to see the convergence rates.*\n\nWe thank the reviewer for the suggestion. While numerical validation of the bounds could certainly provide additional insights, such an empirical investigation lies outside the scope of this paper and is left for future work.\n\n*There is a preprint studying the convergence of IMF [Kholkin]. Their analysis seems to be rather restrictive and hold only for gaussians, but anyway they are highly related.*\n\nWe thank the reviewer for bringing the preprint [6]. While related in spirit, their work analyzes a different algorithm, the Iterative Proportional Markovian Fitting (IPMF), which blends ideas from both the Iterative Markovian Fitting (IMF) and Iterative Proportional Fitting (IPF) algorithms. Their theoretical guarantees are established under highly restrictive conditions, specifically for one-dimensional Gaussian marginals. In contrast, our analysis focuses on the original IMF algorithm and provides non-asymptotic convergence guarantees under significantly more general assumptions. Nonetheless, we agree that their contribution is related, and we will cite it appropriately in the revised version.\n\n*Questions*\n\nWe thank the reviewer for the careful and detailed check. \n\n*Clarity of assumptions H4, H6:*\n\nRegarding assumptions H4 and H6, we will clarify the statements and explicitly specify all relevant quantifiers to ensure full precision.\n\n*Time horizon’s condition in Theorem 1 and Theorem 2:*\n\nConcerning the bounds in Theorems 1 and 2, these hold under the explicit assumption $T > \\max \\{ \\alpha_{\\mu}^{-1} , \\alpha_{\\nu}^{-1} \\}$, which is stated in the theorems and guarantees that the convexity parameters (whether strong or weak log-convexity) of the Sinkhorn potentials are well-defined and strictly positive [2]. To achieve a contraction rate strictly less than $1$, stronger conditions are indeed necessary. In the strongly convex case, this requires $T > \\max\\{ \\alpha_\\mu^{-1}, \\alpha_\\nu^{-1}, \\alpha^{-1}\\}$, and we will correct this in Remark 3 following Theorem 1. We thank the reviewer for identifying this typo. In the weakly convex case, as correctly noted in Remark 6, the condition becomes $T > \\max\\{ \\alpha_\\mu^{-1}, \\alpha_\\nu^{-1}, C^{-1}\\}$. \n\n*Repeated or incorrect BibTeX entries:*\n\nRegarding the bibliography, we will carefully review and fix all the entries to address the issues pointed out.\n\n*Possible redundant assumption on the bridge in Theorem 4:*\n\nWe thank the reviewer for the comment. We believe there may be a misunderstanding. In Theorem 4, the Talagrand-type inequality is required for the Markov kernel associated with the interpolating process, not for the bridge itself. This distinction is essential and follows directly from the structure of the proof. More precisely, after applying the standard KL decomposition [8], one is led to control an $L^2$-norm of the difference between the mimicking drifts of two projections. These mimicking drifts are expressed as expectations with respect to the corresponding Markov kernels (see Equation (17)), not the bridges. This is why the Talagrand inequality is imposed on the Markov kernels. We then use the dual formulation of the 1-Wasserstein distance, followed by a Talagrand-type inequality, to control the $L^2$-distance between the drifts in terms of a KL divergence. This step is crucial to establishing a contraction estimate in KL, and it entirely bypasses the need to assume any transport inequality for the bridge itself. \n\n*Lines 645 and 656:*\n\nFinally, we will revise the indicated formula and provide a clearer, more detailed explanation of line 645 in the updated version.\n\n[2] Conforti, G. (2024). Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges. Probability Theory and Related Fields, 189(3), 1045-1071.\n\n[6] Kholkin, S., Ksenofontov, G., Li, D., Kornilov, N., Gushchin, N., Suvorikova, A., ... & Korotin, A. (2024). Diffusion & adversarial schr\\\" odinger bridges via iterative proportional markovian fitting. arXiv preprint arXiv:2410.02601.\n\n[8] Chen, H., Lee, H., & Lu, J. (2023, July). Improved analysis of score-based generative modeling: User-friendly bounds under minimal smoothness assumptions. In International Conference on Machine Learning (pp. 4735-4763). PMLR."}]

Author rebuttal:

Meta-review: The authors proved the convergence of the IMF approach for solving the Schrodinger bridge problem. The paper demonstrates that under the log-concavity assumptions the IMF achieves an exponential convergence rate which depends only on the convexity parameters. Also the authors extended the results to the case of weak convexity.

As a strengths of the paper I would highlight 
- conditions of the theorem are sufficiently generic
- the text is well written
- the results is an important step in justification of the IMF algorithm under fairly general assumptions

As a weaknesses I would highlight
- the authors did not take into account discretization or approximation errors
- the dependence of the Markov projection's compression constant on the time horizon and  the requirement for a sufficiently large time horizon
- unimodality of the marginal distributions
- numerical simulations to check the behaviour of convergence rates for different time horizons and marginal distributions
- some additional clarifications should be added to the final text, see comments below

The main reasons to accept are
- the paper provides non-asymptotic guarantees under fairly general assumption for the important algorithm being the first results of this kind. The new contraction results for Markovian projections could be of independent interest
- the paper is mostly well written

During the discussion with the reviewers several issues were raised that should be addressed in the final version of the paper. Here is the list of the main issues
- add discussion around weak convexity (reviewer NJSy)
- discuss benefits of using IMF instead of Sinkhorn (reviewer KTkT)
- discuss reformulation of the theorem in terms of the reference process's variance rather than time horizon; possibility to generalize the results to more general classes of marginal distributions (reviewer CFWL)
- discuss dimension-dependence of the proposed analysis (reviewer THDL)
Decision: Accept (poster)