Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting
Authors: Sergei Kholkin
Grigoriy Ksenofontov
David Li
Nikita Maksimovich Kornilov
Nikita Gushchin
Alexandra Suvorikova
Alexey Kroshnin
Evgeny Burnaev
Alexander Korotin
Venue: ICLR 2026 Poster (2026)
Keywords: Schrödinger Bridge
Optimal Transport
Entropic Optimal Transport
Unpaired Learning
Primary area: generative models

Abstract: The Iterative Markovian Fitting (IMF) procedure, which iteratively projects onto the space of Markov processes and the reciprocal class, successfully solves the Schrödinger Bridge (SB) problem. However, an efficient practical implementation requires a heuristic modification-alternating between fitting forward and backward time diffusion at each iteration. This modification is crucial for stabilizing training and achieving reliable results in applications such as unpaired domain translation. Our work reveals a close connection between the modified version of IMF and the Iterative Proportional Fitting (IPF) procedure-a foundational method for the SB problem, also known as Sinkhorn’s algorithm. Specifically, we demonstrate that the heuristic modification of the IMF effectively integrates both IMF and IPF procedures. We refer to this combined approach as the Iterative Proportional Markovian Fitting (IPMF) procedure. Through theoretical and empirical analysis, we establish the convergence of the IPMF procedure under various settings, contributing to developing a unified framework for solving SB problems. Moreover, from a practical standpoint, the IPMF procedure enables a flexible trade-off between image similarity and generation quality, offering a new mechanism for tailoring models to specific tasks.

Published as a conference paper at ICLR 2026
DIFFUSION & ADVERSARIAL SCHRÖDINGER BRIDGES VIA ITERATIVE PROPORTIONAL MARKOVIAN FITTING
Sergei Kholkin*
Grigoriy Ksenofontoy
David Li
Nikita Kornilov*1.77
Nikita Gushchin 12
Alexandra Suvorikova
Alexey Kroshnin
Exgeny Burnaev
Alexander Korotin
AApllied AI Institute PAXX*
MMoscow Independent Research Institute of Artificial Intelligence!,
* "Mohamed bin Zayed University of Artificial Intelligence,
WWeierstrass Institute for Applied Analysis and Stochastics,
'Mathematical Foundations of Machine Learning Institute
Basicc Research of Artificial Intelligence Laboratory
ABSTRACT
The Iterative Markovian Fitting (IMF) procedure, which iteratively projects onto the space of Markov processes and the reciprocal class, successfully solves the Schrödinger Bridge (SB) problem. However, an efficient praticall implementation requires a heuristic modificationnaatermaann between fitting forward and back- ward time diffusion at each iteration. This modification is crucial for stabilizing training and achieving reliable results in applications such as unpaired domain translation. Our work reveals a close connection between the modified version of IMF and the Iterative Proportional Fitting (IPF) procedure- a foundational method for the SB problem, also known as Sinkhorn's 's algorithm. Specifically, we demonstrate that the heuristic modification of the IMF effectively integrates both IMF and IPF procedures. We refer to this combined approach the Itera- tive Proportional Markovian Fitting (IPMF) procedure. Through theoretical and empirical analysis, establish the convergence of the IPMF procedure under various settings, contributing to developing a unified framework for solving SB problems. Moreover, from practical standpoint, the IPMF procedure enables a flexible trade-off between image similarity and generation quality, offering a new mechanism for tailoring models to specific tasks. The code for our method can be
found at: https: httptgggtttt/tt//// com/ gregkseno/ /
harrttwe Proparionnll 2iting (iP))
Markosien
Ps
Fating (1M7)
Fi
mangrad
proj:
proje
SB
5B
9B
projo
projs
Lanks property
M
R
M
R
M
M
2
2
2
the reciproch classs (R2 Barkociam
Propartional Markowian Fittin (IPMF)
P
Ps
Ps
Fi
proj: (eroj.)
SB
proje
SB
(pros))
proje
M
R
M
R
M
R
M
projs
Figure 1: Diagrams of IPF, IMF, and unified IPMF procedure. All procedures aim to converge to the Schrödinger Bridge, i.e., Markovian process in the reciprocal class, with marginals po and p1-
"Equal contribution.
1 Coonesonn author: www.kkkissseeeoc gaaanndddmmmmii. com?
Moscow, Russia.

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Published as a conference paper at ICLR 2026
INTRODUCTION
Diffusion Bridge models inspired by the Schrödinger Bridge (SB) theory, which connects stochastic processes with optimal transport, have recently become powerful approaches in biology (Tong et al.
2024 Bunne al. 2023). chemistry (Somnath et al. 2023 Igashov et al.; Kim et al. 2024). im-
age processing (Liu et al. 2023a Shi et al. 2023] Korotin et al. Gushchin et al. (024a) and speech
processing (Chen al. 2023b). Most of these applications deal with either supervised domain trans-
lation, c.g., image super-resolution and inpainting (Liu et al.) 2023a) or with unpaired translation, e.g., image style-transfer (Shi al. 2023) or single-cell data analysis (Tong et al. 2024).
This work specifically focuses on unpaired domain translation (Zuu et al., 2017 Fig. 2). In this setting, given two domains represented solely by unpaired samples, the goal is to transform a sample from the input domain into sample related to it in the target domain. In this context, researchers usually use SB-based algorithms because they enforce two key properties: the optimality property, ensuring similarity between the input and the translated object, and the marginal matching property, ensuring the translation of the input domain to the target domain. The motivation for relying on such specialized methods, rather than general text-to-data models, is further discussed in Appx. B
Early works (De Bortoli et al. 2021 Vargas et al. 202 Chen et al. 2021 Pavon et al. 2021 on using the SB for unpaired domain trans lation employed the well-celebrated lterative Proportional Fitting (IPF) procedure (Kullback 1968), also known as the Sinkhorn algorithm (Cutur & Doucet, 2014). The IPF procedure is initialized with simple prior process satisfying the optimality prop- erty. It then refines this process iteratively through marginal matching property is achieved. In each iteration, IPF decreases the forward KL-ddivereence KL(q" (Lgg between the current approximation q and the ground-truth Schrödinger Bridge q", How- ever, in practice, approximation errors may cause IPF to suffer from the "prior forgetting" where the marginal matching property is achieved but optimality is lost VVrrgas et al. 2024 2021)
transformations until the
The Iterative Markovian Fitting (IMF) procedure (Shi al., 2023 Peluchetti 2023a Gushehin
et al. 2024b Ksenofontov & Korotin 2025) emerged as promising competitor to IPF. Contrary to
IPF, IMF starts from a stochastic process satisfying the marginal matching property and iteratively achieving optimality. Each iteration of IMF decreases the reverse KL-ddveegeece KIl(gll")) between the current approximation g and the ground-trutt SB q" (cf. with IPF). The approach generalizes rectified flows (Liu et al., (022)) to stochastic processes, which are employed (Liu et al. 2022bb Yan
jet al. 2024) in modern foundational models such as Stable Diffusion 3 (Esser et al. (0224). Like
IPF, IMF may also accumulate errors. Specifically, it may fail to approximate data distributions due to an imperfect fit at each iteration, causing the marginal matching property to be lost.
In practice, to stabilize IMF training, prevent eπror accumulation and loss of marginal matching property, practitioners use a hcuristic modification of IMF This is a bidirectional procedure alter- nating between learning forward and backward processes, either by diffusion- Diffusion Schrödinger Bridge Matching (DSBM) algorithm (Shi et al., 2023) or GANs in Adver- sarial Schrödinger Bridge Matching (ASBM) algorithm (Gushchin al. (024b). In this work, we investigate the properties of the heuristic modification of the IMF. Our contributions:
models in the
1. Theory. We show that the heuristic bidirectional IMF procedure used in practice is closely related IPF in fact, it secretly uses IPF iterations. Therefore, we propose calling the bidi- rectional IMF procedure Iterative Proportional Markovian Fitting (IPMF, IB.I). We prove that the IPMF procedure exponentially converges for Gaussians under various settings. We also guarantee that IPMF converges to q², if po and p1 have bounded supports and conjecture that IPMF converges under very general settings, offering a promising way of developing a unified framework for solving the SB problem (E3.2).
2. Practice I. We empirically validate our conjecture through series of experiments, including the Gaussian setup (E44), toy 2D setups ( (44.)) the Schrödinger Bridge benchmark (34.3) setup with real-world colored MNIST
c and CelebA image data (34.4).
3. Practice II. Thanks to the proposed IPMF framework, we introduce novel way to trade-off be- tween generation quality and ouput- similarity of Schrödinger Bridge solvers by designing the starting coupling. Empirically. demonstrate real-world image data that the proposed initializations outperform classical ones ($4.4).
These contributions demonstrate that the IPMF procedure has significant potential to unify a range of previously introduced SB cechhds--nncuuii IPF and -bF-d ones in both discrete (Gushchin
2

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Published as a conference paper at ICLR 2026
et al. 2024b De Bortoli et al. 2021) and continuous time (Shi el. 2023 Peluchetti 2023a Vargas et al., 2021) settings, as well as their online versions (De Bortol et al. 2024; Peluchetti 2025 Karimi et al., 2024p. Furthermore, the forward-backwant IPMF framework could enable rectified flows to avoid error accumulation, making them even more powerful in generative modeling.
Notations. P2 is set of absolutely continuous distributions R.D with finite second moment and finite entropy. We fix N 2 intermediate time moments and set 0 = to < t < ... IN < tN+1 = 1. Let E P2.ac(PPx(N+2)) be an associated discrete stochastic process on this grid. For any such q. we denote the density at ( Fty Ii ) E DxxxN++) as g( Pin v1), and initial distribution
with Fin Fexx W is Wiener process with volatility
N+1
Po- Let p n be its discretization, i.e., p0 Fin 21) po
W Ive,. 1: 1.:e(n-nn--i IL)))
where (V---) is conditional Gaussian distribution. ((g) is the differential entropy of q.
2 BACKGROUND
This section details the study's 's key concepts; 22. Introduces the Schrödinger Bridge (SB) problem, 2.2 presents the Iterative Proportional Fitting (IPF), desccibbs the Iterative Markovian Fitting (IMF), discusses the heuristic modification of the IMF (Bidirectional IMF).
Recall that the SB problem (Schrödinger (931), IPF, and IMF admit both discrete- and continuous- time setups leading the same problem solution. Moreover, the explicit formulas for IPF and IMF in the discrete setting are expressed terms of probability densities, which helps to convey the main idea of our paper. Thus, for the sake of presentation flow, the main text focuses exclusively the presents the continuous setup.
discrete setup, while Appendix C
2.1 SCHRÖDINGER BRIDGE (SB) PROBLEM
The SB problem with a Wiener prior in the discrete-time setting (De Bortoli et al. 2021), given the initial distribution po (20) and the final distribution pı (æ1), is stated as
min
KL(g(10, Tin; xi )lpp" W" ( Tin, 21)),
where II, (po. P1) C. P2 (R Dx (2+ (ND)()+)) is the subset of discrete stochastic processes with marginals
q(20)) = (o()) ((x1) = (x1). ). The objective function in ① admits decomposition
KI. (I Zin, m1)lp"" (vo, Tin, a1)) (I #1)lpp" (.o, x1)
/ KL (g(2m)/ x1) 1p W' (Iink 1)) q 11 )dæodz1.
All ((TTulzo 21) can be chosen independently of q(20, 21). Thus, consider q(Zin 21) w (Iin (oo Ti )and get KL(g(0in No, æ1 "1pl (zinlzo. x1)) 0.
This leads to the Static SB problem:
min q (p)).))))
KL <1(20, (x0,=))))
where II(po, PI) C (I DxDD is the subset of joint distributions g(20, 21 s.t. g(rn) po (=(o))
((x1) = (1) Finally, the static SB objective can be expanded (Gushchin et al. 2023a Eq. 7)
- xoll2
(L(g(xo, x Ill (1),) =
dg x1)
(q(x0, x1)) + C,
(3)
2c
that is equivalent to the objective of the entropic optimal transport (EOT) problem with the quadratic
cost up to additive constant (Cuturi 2013 Peyré al. 2019
Léonard 2013
Genevay 2019).
2.2 ITERATIVE PROPORTIONAL FITTING (IPF)
Early works SB Vargas et al. 202 De Bortoli al. tional methods based on the IPF procedure (Kullback] 168). The IPF-based algorithm is started by
2021 Tang al. 2024 propose computa-
setting the process q (Iu, Iin Pu(zu)p" (Tin I1 zg). Then, the algorithm alternates between
two types of IPF projections, proj, and projo- given by
(De Bortoli et al. 2021 Prop. 2):
N
N
q 2k+1 = proj ( 2k (x1) II
a (# (et
def pı (x1)
II n=0
g 2k (Ir,, (
(4)
n=0
g2k (ong (2o
q2h (1g)
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Published as a conference paper at ICLR 2026
N+1
N++11
22h+2 projo( ( (
II
q 2k (I (x))-)))
der
Po (m(o
II 1=
q 26+
(I (em-)))
(5)
2 ( (10))
2 +1 (110)
Thus, proji and projo replace marginal distributions ((21) ) and ((x)) in q(20, Tiny 21 by pı (x1) and po(2o) respectively. The constructed sequence {4") converges to the solution of the SB problem qº and causes the forward KL-divergence (l( to decrease monotonically at each iteration.
In practice, since the prior process n is used only for initialization, the imperfect fit may lead to a deviation from the SB solution at some iteration. This problem is called "prior forgetting" and was discussed in |Vargas al., 2024. Appx. E.3). The authors of Vargas et al. (022 consider continuous analog of the IPF procedure using inversions of diffusion processes (see Appx. C.2
2.3
ITERATIVE MARKOVIAN FITTING (IMF)
Gushchin al. 2024b) emerged as a strong competitor the IPF procedure. contrast to IPF. IMF does not suffer from the "prior forgetting". The procedure is initialized with any g" E II (po( (Po. Then it alternates between reciprocal projection projR and Markovian projection proj
The Iterative Markovian Fitting (IMF) procedure (Peluchetti 2023a Shi et al., 2023
2k+ projo (q
(22t) def (x0, x1 W (Iin zo. x1),
(6)
N+1
N
2k+2 = (1 2 def 2h+ (zo)
II n=1
q 2
(Ire Ir 1) =
2h (x1)
II n==
q
2h+ (2e, (2m111)
(7)
forward representation
backward representation
The reciprocal projection projæ creates a new (in general, non-Markovian) process combin-
ing the distribution (( and y" (Jrim (rm, I1). The latter is called the discrete Brownian Bridge. The Markovian projection proj.M uses the set of transitional densities 1 or {q( (x..,,)) to create a new Markovian process starting from ((ro) or g(z1) respectively. Markovian projection keeps the marginal distributions each timestep, but, in general, changes the joint distributions between them. The sequence {g*} converges to the SB q" and causes the reverse KL-divereence (L(g"\lg") to decrease monotonically each iteration (ef. with IPF). The authors of Shi al. (2023 Peluchetti (2023a) consider continuous-time version the IMF.
2.4 HEURISTIC BIDIRECTIONAL MODIFICATION OF IMF
The result of the Markovian projection ( admits both forward and backward representation. To learn the corresponding transitional densities, one uses neural networks {ge (xt,_,)) (forward parametrization) or (4o (xt,, ( [ (backward parametrization). The starting distributions are
as follows: qo (3ro = Po( for the forward parametrization and do ( Pr (rx1) for the backward
parametrization. In practice, the alternation between representations of Markovian processes used
in both implementations of continuous-time IMF by DSBM algorithm (Shi et al. 2023 Alg. 1)
based on diffusion models and discrete-time IMF by ASBM algorithm (Gushchin et al. 2024b Alg. 1) based the GANs. This bidirectional procedure be described as follows:
q Ak+1 - (10, W
q 1k+2 p(x1) II
Ak+1 1
( (ot.)))
projz (Qth
backward parametrization
N+1
9 4k43 4 (Tu Iz )p W' (zmn 20, x1),
q Ah+4 (x)))
II n=1
(Iz, (2)..))
(9)
projze (g+)2)
forward parametrization
Thus, only one marginal is fitted perfectly, e.g., qo(xu) Po( in the case of forward representation, while the other marginal is only learned, e.g. qo (x)) J (o(o)) N+1 go(x,, (e,,, ddzgdx ..- Rs (x().
3 ITERATIVE PROPORTIONAL MARKOVIAN FITTING
This section demonstrates that the heuristic bidirectional IMF ($2.4) is, in fact, the alternating imple- mentation of IPF and IMF projections. T...e that this heuristic defines the unified Iterative
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Proportional Markovian Fitting (IPMF) procedure. $3.2 provides the analysis of the convergence of the IPMF procedure under various settings, with the proofs provided in Appendix D
3.1
BIDIRECTIONAL IMF IPMF
For given Markovian process y, we recall that its IPF projections (projo (g) 5 and proju (g) (4)
replace the starting distribution (zo) with (xx) and g(z1) with p1 (21), respectively. Further, the process g (8) is a result of a combination of the Markovian projection proj.M in forward
parametrization and of the IPF projection projı
N
N II ==
q 4k+2 = p(x1)
II
4k++ (It,,, (2t,. proji (1 (Qukt)(() (x1)
1
th (are, (sPentt))
n=0
peojy (prej ( (1)
Next, the process y 1 9 results from combination of the Markovian projection proj. in backward parametrization and of the IPF projection projo (5):
N+1
N+1
g2++3 p(20)
II
1k+3 (It, (tt_1) = projo (g 16+3 (0))
II
q 1k+3 (tt, ((tn-)))
projo (proj (g
Thus, we can represent the heuristic bidirectional IMF given by and as follows: Iterative Proportional Markovian Fitting (Discrete time)
ak+1 (I0, (I 21 x1)p )p" (Iin 20: 22),
g p(x1) II n=0
ak+ (It (ree);
proiz (0(k)
proj, (proj in ((*)+))
N+
kk43 gehe 2 (To, ( I )p W' (z
gak+4= = p(x))
II n=1
4 q
(It, (mn-.))-
projye (k4)
prujy (projj (g (k-)
The heuristic bidirectional IMF alternates between two IMF projections (proj M (projæ (-))) during which the process "became more optimal" (step towards optimality property) and two IPF projec- tions (projo and proji) during which the marginal fitting improves (step towards marginal matching property). We refer to this procedure as Iterative Proportional Markovian Fitting (IPMF), An IPMF step consists of two IMF projections and two IPF projections. We hypothesize that IPMF converges from any initial process (Zo, Fin Iz unlike IPF and IMF, which require a specific form of the starting process. We emphasize that IPMF reduces to IMF when the initial coupling has the correct marginals Pa and p1 and has Brownian Bridge between the marginals. Similarly, the initial coupling is Markovian, in the reciprocal class, and has the correct initial marginal Po P1, then IPMF reduces to IPF. Fig. [ visualizes these cases, clarifying the role of the initial coupling and the iterative steps. A similar analysis for continuous-time IPMF provided in Appx. CK and the summarization of the conceptual differences between prior bidirectional SB heuristics and the proposed IPMF framework is presented Table
3.2 THEORETICAL CONVERGENCE ANALYSIS VARIOUS CASES
Our first result introduces novel approach quantify the optimality property for Gaussian plan. We show that any 2D Gaussian distribution (D > 1) is an entropic OT plan between its marginals for certain transport cost. Let Q, S DDD be positive definite matrices (Q, S > and P R DDD
be s.t. Q P(S) PT > O. Define
E(P, Q,
-
(10)
Theorem 3.1. Let g(00, x1) ) be Gaussian with marginals p Nn, Q) and 5- N(v,
g(zo. 21) = -N (e) (
Q
5)
P) S
PT
Let A A= (PP,, Then q is the unique minimizer of
{ { [-x] Axg) d' xn dd)dd -H())) d'Ell(p.p)
min
(1I)
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Problem [IT)
is the OT problem with the transport cost CA(20. (. 21):= -oi Aro and entropy regu- larization (with weight 1) (Cuturi 2013 Genevay 2019). In other words. for any 2D Gaussian dis-
tribution q, there exists matrix (9) R"** that defines the cost function for which q solves the
EOT problem. We name A(g) the optimality matrix. If is such that A(q) = - - Dp. then the cor- responding transport cost is ca (300 211 -c-1, (211 which is equivalent -| 3o 1//2
Consequently, is the static SB 22) between its (o())) and @ (21) for the prior W recall 3
Main result. We prove the exponential convergence of IPMF (w.r.. the parameters) to the solution q" of the static SB problem (2) between Po and p1 under certain settings.
Theorem 3.2 (Convergence of IPMF for Gaussians). Let po Nuo, Eo) and pı =N(un, 21) be D D-dimensional Gaussians. Assume that we run IPMF with c > O, starting from some 2D Gaussiar
(((()) )())
(<0, x1) -
(
Eo Po
Po So
E P2. x xRR).
We denote the distribution obtained after ls IPMF steps by
q (FO,
def N
(
pio
Li
Ps Sk
P2, (2, RD)
Pk
and Ag E(Pr. Eg. Sx). Then in the following settings
D = 1, discrete- or continuous-time IMF (N = any e <>>0 0;
D >>> 1, discrete-time IMF, > 0 (see Appendix D.4:
the following exponential convergence bounds hold:
||S| || S - Ipll S a 2h 1|S 2 E1 3-12l 112l22,
||Σ (ux - -1)ll2 S a |= (vo- -1)1l,, - S 322 - c Iblla.
(12)
with a, 3 < and || 1l2 being the spectral norm; a, depend on IPMF type (discrete or continuous), initial parameters Su, vo, Po. marginal distributions po: PL, and Consequently, KL (1th KL (I")l4"") k=300 0.
Proof idea. An IPF step does not change the "copula", i.e., the information about the joint distribu- tion that is invariant w.r.t. changes in marginals po and p1. In the Gaussian case can be represented by the optimality matrix Ag (see Lemma In contrast, an IMF iteration changes the copula but preserves the marginals. Next, we analyze closed formulas for the IMF step in the Gaussian case (Peluchetti] 2023a Gushchin et al. 2024b) and show that the IMF step makes As closer to E - Specifically, ly, we verify the contractivity of each step w.r.t. Ag.
Our next result shows that the convergence of IPMF holds far beyond the Gaussian setting.
Theorem 3.3 (Convergence of IPMF under boundness assumption). Assume Po and pr have bounded supports. Then for both discrete-time and continuous-time IPMF holds 9th (zo, x1) " (Z0, where " denotes weak convergence.
General conjecture. Given our results, we believe that IPMF converges under very general settings (beyond the Gaussian and bounded cases). Moreover, in the Gaussian case, we expect exponential convergence for all 0, all D, and IMF rypes. We verify these claims experimentally ($4)
Related works and our novelty. Our work provides the first theoretical analysis of bidirectional IMF, whereas prior studies analyzed only vanilla IMF, (Shi al. 2023] Theorem 8) and (Gushchin et al. 2024b Theorem 3.6) proved sublinear convergence of IMF in reverse KL divergence for continuous and discrete cases. For IPF, sublinear convergence in forward KL divergence under mild assumptions, as well as geometric convergence for Gaussians, are shown in (De Bortoli et al., 202T Propositions 4 and 43). Previous results cannot be directly generalized to IPMF. First, IPH and IMR converge in different divergence measures, and a decrease in one does not imply reduction in the other. Second, IPF updates marginals at each step, so IMF must optimize toward a moving target, whereas pure IMF has fixed optimum. Unlike Shi et al. 2023). which proves convergence only from the IPF starting coupling, our analysis applies to arbitrary starting couplings. We also view the starting coupling
tunable hyperparameter and examine its effect in the next section.
We assume that (*a) po(zo), i.c., the initial process starts at Po at time = 0. This is reasonable, as after the first IPMF round the process will satisfy this property thanks to the TPF projections involved.

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Forward KL vs IPMF step (D 128, c=0.3)
Reverse KL vs IPMF step (D 128, E=0.3)
(Xlg"(Xxxxx)lll x))ll (Dx, x())
10a
kL.gq (x), (oo.xilllq "No, X)))
10"
IIT starting process
IIT starting processs
10'
10
(ASBM] process
10"
Independent. As pa starting a process
30
Independent By starting process
10
10"
10.9
10n4
10-4
10a9
20
40
60
80
100
0
20
40
60
80
100
IPMF steps (k)
IPMF steps
(a) Forward -i--dieegencce during IPMF steps.
(b) Reverse KL-divergence during IPMF steps.
Convergence of optimality matrix H.
Convergence of v.
Convvegence of S.
1P starting proceess D)
Jull
I+ starting p acesss
fall2
IiF starting processs
INF 165810 start
199 (osgmn star darting precess ( U1
199 (ASEN) stariing process I- Irdependden Io -5a slarting process
Intependent
Independient Py -po start ng process
INF stops (0)
IPMF stops [k]
IPMF stops (k)
(c) Convergence of matrix A.
(d) Convergence of v.
(e) Convergence of S.
Figure 2: Convergence of IPMF procedure with different starting process qº.
EXPERIMENTAL ILLUSTRATIONS
This section provides empirical evidence that IPMF converges under more general setting- specifically, from any starting process-unlike IPF and IMF. The first goal is to achieve the same or similar results across all used starting couplings and for both discrete- time (ASBM) and continuous- time (DSBM) solvers on illustrative setups ($4.1 4422 $4.3) (4.4). The second one is to highlight that, while all initializations converge to qualitatively similar outcomes, in practice, some offer bet- ter generation quality and others better input- similarity on real-world data, due to different starting points. This allows one to choose initializations based on specific task requirements ($4.4).
In $3.1 and Appx. C we show that the bidirectional IMF and the proposed IPMF differ only in the initial starting process. Since both practical implementations of continuous-tine IMF (Shi et al.) 2023 Alg. 1) and discrete-time IMF Gushhchnn et al. 2024b) Alg. 1) use the considered bidirectional version, we use practical algorithms introduced in these works, i.c., Diffusion Schrödinger Bridge Matching (DSBM) and Adversarial Schrödinger Bridge Matching (ASBM) respectively.
Experimental setups. We consider multivariate Gaussian distributions for which we have closed- form IPMF update formulas, illustrative 2D example, the Schrödinger Bridges Benchmark (Gushchin et al., 2023b) and real-life image data distributions, i.e., the colored MNIST dataset and the CelebA dataset (Liu et al., 20156). All technical details can be found in the Appx.
Starting processes. We focus on running the IPMF procedure from various initializations, re- ferred as starting processes. The starting processes are constructed by selecting different cou- plings gº (zo. R1) and incorporating the Brownian Bridge process We [z0.01 (i.e., W conditioned on T1). In the discrete-time setup, for each selected coupling qº (1) we construct the starting process as (00, -in,
(xo, x1 21)p"" (Fin|x0, (1) and 7 = Wi= dg((0= x1) for the continuous-time case (see Appx. C. We consider three "starting" scenarios: IMF-like starting process of the form q" qz x11 = Po (2)))) (II). IPF-like starting process of the form (Io, æ1) = Po po("))p"' (z1 (z)), and various starting processes which cannot be used to initial- ize IMF or IPF. The latter demonstrates that IPMF converges under more general setting.
The results of DSBM and ASBM with different starting processes are denoted as (D/A)SBM- *coupling*, e.g., DSBM-IMF for DSBM with IMF as the starting process.
Remark. Notably, in practice, both the IPF and IMF procedures can be recovered through different implementations. For example, IPF can be realized through (D/A)SBM with the IPF starting cou- pling, or alternatively via DSB (De Bortoli et al.] (021). IMF, in turn, can be implemented using (D/A)SBM with a one-directional parametrization. However, in practice, matching-based meth- ods exhibit superior performance (Shi et al.) 2023). Furthermore, the authors of (Shi et al.) 2023 Peluchetti] 2023a] Gushchin et al.) (024b) observed that bidirectional IMF does not accumulate approximation errors, whereas relying solely
one direction parametrization leads to error accu- mulation and eventual divergence (De Bortoli et al.) 22224, Appx. 1). Therefore, we argue that a direct comparison between the IPMF procedure and previous practical implementations is unnecessary.
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0.1
Algorithm Type D-2
72- 16
D-64
D-128
7--
D-16
77-11
D1-126
D-2
D-10 10
27 04
D7- 128
Bos algunihim henchmark'
Varies
1.91
13.67
11.74
11.4
1.04
9.05
15.05
15.25
1.40
1:27
2.36
1.31
1.21
4,61
9.81
19.5
0.68
0.63
1.5
29.5
11.31
5.45
68.9
362
DSBM IFF
2.05
17.4
15.85
17.45
0.29
0.76
4.05
20.59
11.35
3.58
55.2
210
DSRM-ldrutyy
ASUM EMT
IPMF
0.89
1.25
18.56
9.2
21.71
13.5
21.39
53.7
0.10
0.36
0.69
7.46
5.5
28.5
0.13
0.13
5.50
0.4
88.2
917
4.7
AS6M-IPF
3.06
16.37
64.35
52.5
0.25
20.47
0.13
0.36
2.28
4.99
ASBM Alentiy
0.58
80.2
0.35
M-SM-
Bridge Manching
0.54
10.9
0.51
4.9
Table I: Comparisons of -BV3-UVP J. (%) between the static SB solution o (20 2t1) and the learned solution on the SB benchmark.
The best metric is bolded. Results marked with are taken from (Gushehin et al. [024bb and (Guahchin et al (0223)) The results of DSBM and ASBM algorithms starting from different starting processes denoted us (DIA)SBM "Mamme of starting process*
8
2 2 2
2
2222
2
2
2 2 2222
3
22
2
2
2 2
2
2
222
2
2
3
2 2222
2 2 2 2 2
2
2222
2 2 2
2
3
2 2 2
2
2 2
2 2 2
2
2 2
2
2
(1) Po
(b) DSBM IMF
(c) DSBM Inverted 7
(d) ASBM-IFF
(e) ASBM Inverted 7
Figure 3: Samples from DSBM and ASBM Iearned with IPMF using IMF and jo77
starting processes on
Colored MNIST 3-2 (32 x 32) translation for = 10.
4.1 HIGH DIMENSIONAL GAUSSIANS
This section experimentally validates the convergence of IPMF for the multivariate Gaussians (see our General conjecture, 8..2). We use explicit formulas for the discrete IPF and IMF (Gushchin et al. 2024b Thm. 3.8) and follow the setup from (Gushchin et al. 2023a Sec. 5.2). Specifically, we consider the Schrödinger Bridge (SB) problem with D = 128 and 0.3, where the marginal distributions are Gaussian: Po (0, Eg) and P1 (3, E,), with 0 RRD denoting the vector of all zeros and 3 E D the vector of all threes. The eigenvectors of En and Σ₁ are sampled from the uniform distribution on the unit sphere. Their eigenvalues are sampled from the loguniform distribution on | log 2, log 2). We choose N = 3 intermediate time points uniformly between / = 0 and 1 and run 100 steps of the IPMF procedure, each consisting of two IPF projections and two Markovian-Reciproeee Reciprocal projections (see $3.1). Denote as 9th = 9 (Iu, x11 the IPMF output at the k-th step and let q* = q" (2 21) ) be the solution of the static SB. Fig. 2 sshows that both the forward KL (9 ( and reverse (L(g">llg") divergences converge. The quantities from (12 converge to zero exponentially, as expected. Note that particular starting processes can cause an immediate match of the parameters: an IMF starting process that already has the required marginals converges only in optimality, while IPF converges to the true marginals with the unchanged optimality matrix.
4.2 I.LUSTRATIVE 2D EXAMPLE
We consider the SB problem with e = 0.1, po being a Gaussian distribution on R² and P1 being the Swiss roll. We train DSBM and ASBM algorithms using IMF and IPF starting processes. Ad- ditionally, we consider Identity starting processes induced by ( un1) po(2o) Jzz (21). ), i.e. we set x1 = To after sampling To ~ p( The purpose of testing the Identity coupling is to verify that IPMF converges even when initialized with a naive coupling. Furthermore, we hypothesize that this coupling is the best in terms of the optimality property. We present the starting processes and the results in Fig. in Appendix E. In all the cases, we observe similar results.
4.3 EVALUATION ON THE SB BENCHMARK
We use the SB mixtures benchmark Guushhhn et al. 2023b) with the ground truth solution to the SB problem to test ASBM and DSBM with IMF, IPF, and ldentity (#4.2) the starting processes. The benchmark provides continuous distribution pairs po. P1 for dimensions D E {2, 16, 64, 128) that have known SB solutions for volatility E (0.1. 10}. To evaluate the quality of the recovered SB solutions, we use the -BW2-UVP metric (Gushchin et al. (023b). Tab. [ provides the results.
We also include the results of the standard baseline for Bridge Matching tasks called SF²M-Sink (uushchin et al., 2003b)). We provide training details and additional results in Appx. B All starting processes yield similar results within each solver type (DSBM or ASBM).
4.4 UNPAIRED IMAGE-TO- TO-MAGE TRANSLATION
To test IPMF real data, consider two unpaired image-to too--tt---ni to-image translation setups: colorized 3 + colorized 2 digits from the MNIST dataset with 32x32 resolution size and male -femmae faces from the CelebA dataset with 64x 64 resolution size.
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Initialisation (coupling)
DSBM
ASBM
IMF
DDPM
SDEdit
SDEdit
SD
Identity
IMF
DDPM
SDEdit
SD SDEdit
Identity
IMF
DDPM
SDEdit
SD SDEdit
Identity
FID4
0.0
35.23
28.77
61.56
13.65
14.84
22.65
33.11
19.32
21.84
20.64
19.58
MSE(20, Φ
0.16
0.02
0.02
0.0
0.16
0.09
0.04
0.03
0.17
0.07
0.08
0.07
Table 2: Qualitative results on CelebA (64 x 64) for male- translation with ASBM and DSBM across different starting processes. Generative quality (FID.) and similarity (MSE( ži )J) are reported on the test set. Best and second-best values for solvers are marked in bold and underline, respectively.
Colored MNIST. We construct train and test sets by RGB colorization of MNIST digits from corre-
sponding train and test sets of classes "2" and "3". We train ASBM and DSBM algorithms starting from the IMF process. Additionally, we test a starting process induced by the independent cou- pling of the distribution of colored digits of class "3" (po) and the distribution of colored digits of class "7" with inverted RGB channels (p 1y (x1)
We refer to this process as Inverted 7, i.e., (20, x1) po(oo)) (21) (see Fig. 10 Appx. E contains further technical details. We learn
DSBM and ASBM on the train set of digits and visualize the translated test images (Fig.) 3
Both DSBM and ASBM algorithms start- ing from both IMF and Inveried 7 starting process fit the target distribution of colored MNIST digits of class "2" and preserve the color of the input image during translation. This supports that the limiting behavior of
IPMF resembles the solution of SB.
CelebA. We consider the IMF-OT varia- tion of the IMF starting process. It is in- duced by mini-batch optimal transport cou-
(a) Io
(b) DSBM-IME-OT
(e) ASBM-IMI-OT
(d)
pling gdt (Io, _1) (Tong et al. 2024 Poola- dian et al. 2023). We also test Identity (:4.2) starting process. Additionally, we test starting processes induced by DDPM
SDEdit and SD SDEdit couplings, which
is the SDEdit method (Meng al.] 2022)
(e)zo potff DSBM-DDPM SDEdit
(g) ASBM DDPM (DEdit
used for male Sfemale translation with (1)
DDPM (Ho et al.] 2020)) model trained
on the female part of CelebA and (2)
Stable Diffusion v1.5 (Rombach et al. 2022
with designed text prompt. see Appx.
E.3
The aim of introducing such a coupling is to test the hypothesis that well-designed SDEdit couplings can improve the metries of both properties. We approximately the same number of parameters for the DSBM and the ASBM generators and 10% of images for evaluation (see Appx. E other details).
i mo no
DSBM- SD SDEdit
(k) ASBM-SD SDE
We provide qualitative results in Fig. 4 Ad- ditionally, we report the final FID score (gen- eration quality) and the Mean Squared Error (MSE) between the input Ty and the trans- lated image T1 (input-output similarity) in Table 2 Figure
(nn) 3ig ~P Po (n) DSBM -deuiyy
(o) ASBM Ideniry
(p) r"g"
Figure 4: Results of CelebA at 64x64 size for male-sfemale translation learned with ASBM and DSBM using various starting processes for e 1. Samples zo ~ po are samples from the source marginal. Samples x1 ~ q are samples from the initialization coupling gº (41 for a given zo from po-
g illustrates that the mod- els (1) converge to the target distribution and (2) preserve semantic alignment between in- put and output (c.g., hair color, background). Despite this, their outputs differ due to the influence of initialization on optimization trajectories. For DSBM, our couplings (SD SDEdit, DDPM SDEdit, Identity) maintain generation quality while greatly improving similarity. For ASBM. they boost similarity but slightly reduce quality. Results with Identity couplings support our hypothesis ( (64.) whereas experiments with SDEdit offer only partial validation and yield moderate FID.

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Published as a conference paper at ICLR 2026
BN DOOM STONE
JANH
610
6163
030
(a) MSE(Fo. &1) for (D/AJSBM
(b) FID for DSBM
(c) FID for ASBM
Figure 5: Test metrics in CelebA male femalee (64 64) function of IPMF iteration for various starting couplings.
AFHQ images
DSBM IMF OT translation results.
DSBM-ldentity trassattonn results.
Figure Results AFHQ at 512x512 size cat->wild translation learned with DSBM using various starting processes for e
Furthermore, in Figure we present quantitative study IPMF convergence, reporting FID and the Mean Squared Error (MSE) between the inputs and the translated outputs functions of the IPMF iteration. Both metrics are computed on the CelebA male-tfemale (64 x 64) test set. We observe consistent pattern: the higher the similarity or generation quality of the coupling, the better the model performs on the corresponding metric. For additional quantitative results on CelebA 64 x 64, we refer the reader to Appendix E.4
AFHQ. For AFHQ (Choi et al. (020), we con- sider classes cat and wild with 512 512 reso- lution images. Each class contains approximately 5000 samples. We run DSBM with IMF-OT and Identity couplings and present the results in Fig- ure and Table E.10 We observe similar quality- similarity tradeoff Celeba setup. We provide technical details for this setup in Appendix [..10
Coupling
FID
MSEJ
CMMDJ
DSBM-IMF-OT DSBM-Identity
53.42
0.085
0.591
65.19
0.054
0.731
Table 3: Results of AFHQ 512x512 size for cat-swild translation learned with DSBM using various starting processes for e
For broader discussion of the potential implications and limitations of this work, Appendix A.
BROADER IMPACT
This paper presents work whose goal is to advance the field of Artificial Intelligence, Machine Learning and Generative Modeling. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
Acknowledgements. We thank Kirill Sokolov for the careful feedback and support refining the proofs. The work supported by the grant for research centers in the field of AI provided by the Ministry of Economic Development of the Russian Federation in accordance with the agreement 00000C313923PP4F002 and the agreement No 139-10-2025-03..
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6 LLM USAGE
Large Language Models (LLMs) were used only assist with rephrasing sentences and improving the clarity the text. All scientific content, results, and interpretations in this paper were developed solely by the authors.
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APPENDIX CONTENTS
Discussion
15
B Motivation for SB over Foundational Models
16
Continuous-time Schrödinger Bridge Setup
16
Schrödinger Bridge (SB) Problem in contimuous-tim
16
Iterative Proportional Fitting (IPF) for continuous-time
Iterative Markovian Fitting (IMF) for continuous-time
18
Iterative Proportional Markovian Fitting (IPMF) for continuous-time
18
D
Theoretical Analysis for Gaussians
19
D.I Gaussian plans as entropic optimal transport plans
19
D.2 IPF step analysis
20
D.3 Discrete IMF step analysis: multidmmensionl case for large
23
D.4 Proof of D-IPMF Convergence Theorem 3.2 D
D.5 IMF step analysis in 17)
31
D.6 Proof of IPMF Convergence Theorem 3.2
35
D.7 Proof of IPMF General Convergence Theorem 3.3
35
Experimental supplementary
38
Illustrative 2D example visualization.
38
E.2
SB benchmark -VP
39
CclebA SDEdit starting processes description
39
E,4
CelebA experiment additional quantitative study
39
General experimental details
40
E,6
Illustrative 2D examples details
41
SB benchmark details
42
E,8
CMNIST details
42
CelebA details
43
E.10
AFHQ details
43
Computational resources
44
A DISCUSSION
Potential impact. The IPMF procedure demonstrates a potential to overcome the error accumulation
problem observed distillation methods-such as rectified flows (Liu et al. 2022 2023b 2023b) which used to accelerate foundational image generation models like StableDifiission 3 in (Esser et al.) 2024). These distillation methods are the limit of one-directional IMF procedure with - 0. The one- directioaio version accumulates errors, which may lead to the divergence (De Bortoli et al. 2024 Appx. D). The use of the bidirectional version (with e > 0) should correct the marginals and make diffusion trajectories straighter to accelerate the inference of diffusion models. We believe
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Method
View of bidirectional procedure
Convergence guarantees
Starting coupling
DSBM
A houristic approech for mitigat-
Only for osc-directionll continuous-tine pro- cedure with
Shi al. 2025
ing error accumulation
starting couplings
Only IMF and IPF
ASBM
heuristic approach for mitigat- ing error accumulation
Only for one-directional discrete-time proce- dure with IMF starting couplings
Gushchin el al. 2024b
Only IMF
A thooretically grounded ap- proach for mirigating ervor accumulation and managing the trade-oft off between inpui cuiput similarity generative quality
For Gaussian marginals in discrete and contin- uous time (Theorem [. and convergence for bounded suppori distributions discrete continuous time (Theorem 3.3
IPMF (our work)
Atbitrary
Table Positioning of our IPMF framework relative prior bidirectional SB heuristics.
that considering such distillation techniques from the IPMF perspective may help to overcome the current limitations of these techniques.
Another potential impact of our contribution is the advancement of multi-marginal SB methods.
This direction has been explored only rarely in the literature (Chen et al. [2019] 2023a Shen et al.)
2025 Howard et al., 2025; Lavenant et al. 2021] Theodoropoulos et al. (025)) mainly because the
mulii-marginal case is inherently difficult: it requires solving multiple
marginal (classical) SB instances. A notable examples are (Howard et al. 2025) Theodoropoulos et al. 2025), which extends the IMF procedure to the
setting. Within this context, our framework provides way to select a suitable starting coupling for initialization, thereby offering a potential route to reducing the training burden. In this sense, our contribution may encourage more systematic and deeper analysis of multi-marginal SB.
Limitations. While show the proof of exponential convergence of the IPMF procedure in the Gaussian case in various settings, and present a wide set of experiments supporting this procedure, the proof of its convergence in the general case still remains promising avenue for future work.
B MOTIVATION FOR SB OVER FOUNDATIONAL MODELS
At first glance, one might consider foundational models as potential baselines, since translation via large text-to-image models trained on extensive image corpora may work adequately on the bench- mark datasets we consider (CelcbA, MNIST). However, they do not constitute a relevant baseline for the unpaired translation task, because their training data may lack the domain-specific examples required. In contrast, methods for solving the unpaired translation (including the SB) are designed to address domain- specific tasks across various scientific fields (Schneider et al.) 2022 Singh al.) 2024 Shi et al. (023). Moreover, these methods successfully address non-image-relan stream tasks such single-cell data analysis (Tong et al. 2024 Section 6), where large text-to-image models are just irrelevant.
down-
C CONTINUOUS- SCHRÖDINGER BRIDGE SETUP
For considering the continuous version of Schrödinger Bridge we denote by P(C((O, 1)), P) the set of continuous stochastic processes with time t E [0, 1], i.e., the set of distributions on continuous trajectories [0, 1] > RR" We use dW, to denote the differential of the standard Wiener process. We denote by pT E P(R (X the discrete process which is the fiite-dimensions projection of T to time moments 0 = to < f1 < ... < Iy < 1N+1 =
C.1 SCHRÖDINGER BRIDGE (SB) PROBLEM IN CONTINUOUS--TIM
This section covers the continuous- introduce several notations to better align the continuous version with the discrete-time version considered in the main text. Consider the Markovian process defined by the corresponding forward backward (time-reversed) SDEs:
formulation of SB as its IPF and IMF procedures. First,
T : dx. + (xr, f)dt + VedW., so po(
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T: dz 00 (xi, + VedW.. 21 (P(()))
where we additionally denote by Wt+ and W; the Wiener process in forward backward time. We say Tim and Tix denotes the conditional process of T fixing the marginals using delta functions, ie., setting po(zo) = 6zo (z) and p1 (1(z)) = =z, (z)):
T To dæ = x (xr, + VedW,, so ~ (zo(2))
Tia : dx, = v (xz: t)dt + VedWT, In ~ -,, (x).
Moreover, we use I= denote the stochastic process which starts by sampling 2o ~ p(zo) and then moving this 20 according the SDE given by Ti i..., p(zo)T]zz short for the process ((w)ddd Finally, we use the shortened notation of the process (2), conditioned on its values at times and I, saying p" (zg, Z1 jo, (Io, x1) = To (zo, I1 x1))p" (Io. x Jzzdd11. This links the following equations with the discrete-time formulation.
Schrödinger Bridge problem. Considering the continuous case, the Schrödinger Bridge problem is stated using continuous stochastic processes instead of one with predefined timesteps. Thus, the Schrödinger Bridge problem finds the most likely in the sense of Kullback-Leible divergence stochastic process T with respect to prior Wiener process W i.e.:
min TET(Po.P))
KL(T))
(13)
where F(po.P) Pi) C P(C 1)), RD) is the set of all stochastic processes pinned by marginal dis- tributions po and P1 at times 0 and 1, respectively. The minimization problem (13) has unique solution 7 which can be represented as forward or backward diffusion (Léonard 2013):
T dr, =0" (æt,
(x t)dt VedWit, Fo po(2)),
T dr, v²- (x= t) + VedW,, an ~ (1()),
where x++ and w* the corresponding drift functions.
Static Schrödinger Bridge problem. As in discrete-time, Kullback-Leibler divergence 13
could be decomposed follows:
KL(T = (L (2o, x1)lpp W (zo, x1)) + KL [ mo ddp (zo, x1).
(14)
It has been proved [éonnadd 2013) that for the solution T it's conditional process is given by T 22 We | Thus, can set Tixp W [=0.2, zeroing the second term in (14) and
minimize over processes with T T10.21 = We [oo.1 This leads the equivalent Static formulation of
the Schrödinger Bridge problem:
min @e (ET)())P1))
KL(g(zo, x1 x1)lle" (wo, x))),
(15)
where 11(po. Pi) is the of all joint distributions with marginals po and pl. Whether time discrete or continuous, the decomposition of SB leads to the same static formulation, which is closely related Entropic OT as shown
C.2 ITERATIVE PROPORTIONAL FITTING (IPF) FOR CONTINUOUS- -IMME
Following the main text, we describe the IPF procedure for continuous-time setup using stochastic processes. Likewise, IPF starts with setting TDD = po(xo) (mo and then it alternates betwethe followinging projections:
26+ = =proj (p (21)IT2) def
pi PI(11)
(16)
22k+2 = projo
p T (NO)T Imo
del
po (2o))
(17)
As in the discrete-time case, these projections replace marginal distributions (T(1) and p (ar)) in the processes D (I Ti,, and p" ( im
by pi (x1) and po(zo) respectively. Similarly to discrete- time formulation, the sequence of t converges to the solution of the Schrödinger Bridge problem
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T implicitly decreasing the reverse Kullback-Leible divergence IL(TT*IT*) between the current process T and the solution to the SB problem T'x Additionally, it should be mentioned that existing methods perform projections via numerical approximation of forward and time-reversed conditional processes, Taa and 7 Te by learning their drifts via one of the methods: score matching (De Bortoli
et al. 202T maximum likelihood estimation (Vargas al. (02T).
C.3
ITERATIVE MARKOVIAN FITTING (IMF) FOR CONTINUOUS
IMF introduces new projections that alternate between reciprocal and Markovian processes starting from any process 7o pinned by po and p1 at times and 1, i.e., To F(po.
T2k+1 projR (I def pTdh (Io, (1)
(20
210221
(18)
7 proj.m (T2k+1) def p Tauti (5o) Miz0
T
(T))
(x
M|n
(19)
forward representation
backward representation
where we denote by TMr the Markovian projections of the processes T. which can be represented as the forward or backward time diffusion as follows (Gushchin et al.) 2024b) Section 2.1):
/
1 x,t p² (2) 1a dr11
Tas dx.t oxt (at t) + VedW, zo p" (mu),
2 (at t)
-
/
Tas doy = - (5) t)dt + VedWE, 21 ~ pf (a1),
(T
I0 Ft x² ( are )dao- 1
This procedure converges to a unique solution, which is the Schrödinger bridge T (éconard 2013 While reciprocal projection can be easily done by combining the joint distribution p (Iu, #1 of the process 7 and Brownian bridge W |m the Markovian projection is much more challenging and
must be fitted via Bridge matching (Shi et al.
2023 Liu et al. Peluchetti (023bb).
Since the result of the Markovian projection ward representation, in practice, neural networks Vg (forward parametrization) or Us (backward parametrization) are used to learn the corresponding drifts of the Markovian projections. In turn, starting distributions are set be (x()) for forward parametrization and p1 (x1) for the backward parametrization. So, this bidirectional procedure can be described follows:
19 can be represented both by forward and back-
4
Trth (wo, 21 W W (RO.
4h+
pi (zi (Z1) 1 ME
(20)
prjj (T
backward parametrization
+
(wo x1 We
4
Po Po (1oo))T T
(21)
projz (T44+3)
forward parametrization
C.4 ITERATIVE PROPORTIONAL MARKOVIAN FITTING (IPMF) FOR CONTINUOUS-TIME
Here, we analyze the continuous version of the heuristical bidirectional IMF, First, we recall, that the IPF projections (T) and proji (T)) given by (66) and (17) of the Markovian process T is just change the starting distribution from pd (zo)
) to po (20) and p (x1) to (z1).
Now we note that the process 4-4+22 in (20) is obtained by using combination of Markovian
projection proj. given by (19) in backward parametrization and IPF projection proji given by (17)
T pi (x) (Z (r proj, Th (II)T
proj, (proj A ( (T))+1))
In turn, the process 4 in 21 is obtained by using combination of Markovian projection proj.M given by (19) in forward parametrization and IPF projection projo given by (16):
T+++3 p Mi
projo qoa--
( )
proju (projas 13))+
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Combining these facts rewrite bidirectional IMF in the following manner:
Iterative Proportional Markovian Fitting (Conitnious time setting)
pm (Io, 21 (1o..r
1
( Mas
(22)
projie (T48)
proj, (proj M (Tck+1))
T
7 +2
(oo: 11 W
T
Po( T
(23)
pmip (74412)
Thus, obtain the analog of the discrete-time IPMF procedure, which concludes description of the continuous setups.
D THEORETICAL ANALYSIS FOR GAUSSIANS
Here, we study behavior of IPMF with volatility c between D-dimensional Gaussians Pu Nuu: Σo) and P1 E1), For various settings, prove that the parameters of qa with each step geometrically converge to desired values po, M1, Eo, Ei,e. The steps are as follows:
1) In Appendix D.1 we reveal the connection between 2 2Dddimensional Gaussian distribution and solution of entropic OT problem with specific transport cost, i.e., we prove our Theorem 3.1
2) In Appendix D.2 we study the effect of IPF steps the current process. We show that during these steps, the marginals become close to and pt, and the optimality matrix does not change. We also prove that the spectral norms of the marginal matrices are bounded during the whole IPMF procedure.
3) In Appendix D.3 that after discrete IMF step, the distance between current optimality matrix and desired one can be bounded by the scaled previous distance.
study the effect of IMF step the current process when D >>>
1. We show
4) In Appendix D.5 we study the effect of IMF step in particular case D = 1. We show that after IMF step (continuous or discrete with N 1), marginals remain the same, and the intermediate distribution becomes closer to the intermediate distribution of the static c-EOT solution between the marginals.
5) Finally, in Appendices D.4 and D.6 we prove our main Theorem 3.2 for the case D > 1 and
D respectively.
D.1 GAUSSIAN PLANS AS ENTROPIC OPTIMAL TRANSPORT PLANS
Proof of Theorem
3.1
The conditional distribution of ((_p]=/]))))))) has closed form:
((0o(=1)
N -
Zmo Zn, exp (zg (Q - (QQ-
Za, Zz, exp (o] Arr),
(24)
where factors Zz, and 2n, depend only on op and æ1, respectively, and the matrix A is
A PS) (Q PP))
(25)
Theorem 3.2 from Gushchin et al. (023b) states that the conditional distribution g(r) (mo) be expressed as:
(x0 mm) x exp(-c(2n, mn) fe((ro))
(26)
where o(Ho, is lower bounded cost function, and the function fc depends only on zo. then
q solves 1-entropic OT with the cost function c( 21). Equating the terms in (24) and (26) which depend on both Zy and Øz, we derive the formula for the cost function is c(#o, 31 = -x0 Az We denote it as (Zo, 2
= Ar show the dependency the optimality matrix A.
We only need to note that we can add any functions f(xp) and g depending only on Zn or Z1, respectively, the cost function ca (I0, -x: Axo. and the OT solution will not change. This
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is because the integrals of such functions any transport plan will be constants, they will depend only on the marginals (which are given) but not on the plan itself. Thus, for any A E R can rearrange the cost term c (To. 21) so that it becomes Lower-bounder
(Too, 21) = | 12 - = Aro + || 1/2/2 2 = || - x 2211//220 0,
where č (20, is lower bounded function.
D.2 IPF STEP ANALYSIS
We run IPMF with the desired volatility parameter between the desired distributions po
N (No Mo) and pı = -N(m1, (1m1,>ii,, E1), starting with the process N the correlation matrix P.
Mo
En
5)) which has
P) 5)
v
One IPMF step can be decomposed into the following consecutive steps:
1. IMF step: projections (proj))) refining the current optimality matrix,
2. IPF step: projection proj changing final prior at time t = 1 to p1 = [N1,, E1).
3. IMF step: projections proj (projzz), refining the current optimality matrix.
4. IPF step: projection projp changing starting prior at time / 0 Po NNoo Eg).
We use the following notations for the covariance matrices changes during IPMF step:
En
IMF
Σ
p
IPF
Q (P')
pr
P
S
PT
s
T
IMF
Q
IPF
Σg (P")T T
P"
PT
Σ1
S
and for the means, the changes are:
IMF
N4
IPF
IMF
IPF
1
11
Lemma D.1 (Improvement after IPF steps). Consider an initial 2D-dimensional Gaussian joint
distribution N
((W)
(5
Eg
P) 5))= E (R xR.).. We run IPMF step between distributions
PT
NHo, Eo) and N N En) and obtain new joint distribution N (H)) Ea P") )) Then,
((PI))
the distance between ground truth p11 En and the new joint distribution parameters decreases as:
(I(5")- (S') Ipll2 s P 2 || - -pll2 (27)
|Σ (z' 11)12 < | lll |P
||T 2 -1)1l
(28)
where Pn = Σn 1/2 1/2 = (Q)- P. == 12) and PM =
E 2-112 1/2 12'"(((()= -51) are normalized matrices whose spectral norms greater than 1.
Proof. During IPF steps, keep the conditional distribution and change the marginal. For the first IPF, we keep the inner part 2y 1i for all 21 E R
N TolPo + Ps- (z1 (z1-1)) v), Eo - PS-1 PT) p = N + (x1 100 PPT
This is equivalent to the system of equations:
Σn P5- PT
Q- 1-P'D-1(P"))
(29)
P'E-1 -
(30)
14g PS-1y
11 P' pl.
(31)
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Similarly, after the second IPF step, we have equations:
E - -T S' (P")
T Ig p" Ij'P"
(32)
(P") (0)
(33)
11 (Q) J' (P") Exp 14-
(34)
Covariance matrices. Combining equations
30
and 33 32
together, we obtain:
Eg Q
//029).
30
(35)
ID Eo(Q) Ps- (E1-S)5''PPCC
//
1/35)
(Q)
(36)
Σ1 5
(I
- E)(Q)))),,
// 32).
333
(37)
Σ 5
PT(Q)-1P---(E, - (Q)
36
insert to
37
Σ - 5
/change using
33
"S ID = ( $ Ps-
(S t5 Ip) + Σの 20 p" (S')
The matrices 29 and 32
must be SPD to be covariance matrices:
No P-PS--PT 2
= Ip -
20 1/1
S' S'-(P")
Ia - (1)))
1/2
In other words, denoting matrices Pn. == Σg 1/1/25 1/2 and P == Σg 1"' (S') we can bound their spectral norms as ||Pr112 S. and covariance matrices:
| < 1. We write down the final transaction for
-n (P:)) P. (S ビ ID) P.T P"
(38)
Hence, the spectral norm of the difference between ground truth Σ₁ and current S' drops exponen- tially as:
- Tpll S P.
P 2
i - 1p12
Means. Combining equations
31
30
and
34
33 together, we obtain:
Po " = ---tv- 41 = P' (v (1-11)
//31
30
(39)
V - f1 (P")TIZ114 11 (Q) PT (T())= -(00 n),
34
33
(40)
d --M P1
n1).
/nnsert
339
to
(4
XT mi) =.. (Q) P'
-3.
( -z))) m1)-
The matrices 29 and 322
must be SPD be covariance matrices:
Q P'E
Ip ž (PP))
=
Σ PT (T)(TQQ) (Q)
In b (0)-T((Q)-10 1/1
Denoting matrices P = Q (Q) P 1 and P. 11) we can bound their spectral
norms ||P/ < 1 and Palla < 1. We use this to estimate the lz- -orm of the difference between the ground truth and the current mean:
3 (v' - - PA M 2 (v u1),
(41)
EΣ } (v/ m1)l2 - P1
|P2 | 3 (v - -1ill2
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Lemma D.2 (Marginals norm bound during IPMF procedure). Consider initial 2D -Ddimnnnso
Gaussian joint distribution N
Ho Vo
Eo Po
56))
Po
E (T. x PR.). We run
So
IPMF
step between distributions
NHo, Eo) and (N,,, E1) and obtain new joint distribution
Eg
Ps
N
Then the norm
|Sxll2 can be bounded independently of k by:
Pk
S
|| 1|2
|| 112 S max (amas 15051,0))))1..11 1311 (42)
|| ll
S
min (amin (S) S ")
This statement also implies the invertibility of all matrices Sk.
For matrices Qk, the results are analogous.
Proof. Consider the last IPMF step. We denote symmetric matrices As == S +2
S 5 E max (0, (A)). Next, we estimate spectral norm of St-1 as follows:
TD, A4-1
Io and Amin(A) (A) == min (0, Amin (A)] Amar
A = 5 E1S ID ł Amin ID) = Amin (D:)
Sp E = (Amin (At) + 1)ID,
Σ1
(mmi (As) + 1)Sx
Note, that by design we have A 2 - In = -1 S Ami)(Ae) $0= 0 (Amin (Ax) + 1) S and can obtain
E1 2 (Amin (A)) + 1)5k.
Sx
En,
Amin (Ax) +1
||> l|2
||1l|2
S
(43)
Aman (Ak)
Similarly, we prove that
1
Sa Amaz (At) +1
E, ||S 112 (Amaz(Au) 1) 112-
Now, we prove that Amin (Ax) 2 Smin (Ab-)). We denote by P" and P., normalized matrices after the second IPF step and the first IMF step on the last iteration, respectively (see Lemma D.I. For any x R ||0||| S calculate the bilinear form:
A
x TPPN P. (S (S ID) P.T P"
(P,T
xTx) A P=xx),
Omin(Di)
min
0, min
x' A
2
I
min
0. min (PT x) T D 1t-1(P)) pk*))
||alaa=1
M M
PP
P" min {0, Amin (At-1)}
P. 2 PPPP Amin (Ak-1) 2 7min (Ax-1)
Hence, after each IPMF step Åmii (Ax) increases and can be lower bounded by the initial value Amin ( (x)) > Amen(Au using math induction. It implies the invertibility of all matrices S₃ and boundness of norms
|E ll2
|E
| 1|
||S* 2
S
Amin (x) +1
Xmii (A)) +1 min Minn (8 1 Σ )
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Similarly, prove that
Amar (Ax) I Amar(Ax--) ( 7 ||s v (Amar(Ao) + +
2 (5) EiS ",,)),11171 1H|1111
Lemma D.3 (IPF step does not change optimality matrix A). Consider an initial 2D- dimensional Gaussian joint distribution N (ww
Pg
Eo
P 5)
P2 RP). We
P
S
run IPF step between distributions N Eo) and ((1,, 21) and obtain new joint distribution
Q P (P')
- Then, IPF step does not chunge optimality matrix A, i.e.
N
A A= 5))
For the second IPF step, the results are analogous.
Proof. The explicit formulas for E(P, and E('' are
E(P, E, = S-1PT (Eo - PP--1T) E(P',Q.E, En) = (P) (Q - -'',
The first terms are equal due to equation (30). and the second terms are equal due to (29
We prove this lemma in general way. We derive the formula 25 for A only from the shape of the conditional distribution (00 .1) (24). During IPF step, this distribution remains the same by design, while parameters S, P change. Hence, IPF step has no effect on the optimality matrix.
For the second IPF step, the proof is similar.
D.3 DISCRETE IMF STEP ANALYSIS: MULTIDIMENSIONAL CASE FOR LARGE
(
(?) PT ()))
We run discrete IMF 5))
Consider 2D-ddimeniionll Gaussian distribution A step consisting of reciprocal and Markovian projections with N intermediate timesteps 0 to < ..... IN < 1N+1 and volatility parameter e.
Following (Gushchin et al. 2024b), we have an explicit formula for the reciprocal step. For any S i,j S N ++11 we have marginal covariance Eq,t, time moment f, and joint covariance Ex,t, between time moments f; and f;:
En,,, = - (1-t))
(1 (1-
+ + t,)e -
( Q2
+ tt,5 +t(1-,t,,,,
- pt) t
(1 -
Q22) + +
Eo.t - 1-4) -1-41-1--4100 -1=011100+(:"))7.1111
ENN.1 INS+(1 + P = 1 (1 (0
where P. = Q PPPP-- 1, Marginals Σo,0 Q and E S at time moments 0 and 1 and covariance Eto.AN41 P do not change.
For the Markovian step, we write down an analytical formula for the new correlation P in the resulting process /V (τ
PT 8)) 5)
Q
p)
namely:
N
p = En,0
II
[m Etadeer = Eo,t, Enj,, Etyds... Eny Ely.1-
0=
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For the normalized correlation Pa = PP = Σ0,0 p. we can simplify the formula:
Pa (b=I(PP))
--
20.0 Do,t
21,41 211 2 Din.e
1/2
21 ... E-122 --1/2
1/2
LNN LiNdN Lty,
11,
N
=
II
21, bi
1/2
1-1/2
- II
(44)
1=0
==0
2012 de+1 denotes normalized correlation between marginals at where Endtiii = 20,, 3:
1/2
time moments 1 and 4i+1 and satisfies |EEnstate .1l2 1.
Lemma D.4 (IMF step correlation transition properties). Let matrices Q,S > 0 be the marginals of 5)). The function f from (44) defined
2D-dimensional Gaussian distribution N (2)
on the ball {Pn || 22 S 13 transforms the normalized correlation Pn == Q-- PS 1 new one after discrete IMF step. Then f(Pr) is Lipschitz on the unit ball with constant
((),S,e) = 10 |S (V
tN
tN tN)
N-1 4,) + 1
VE
(1-t))
Velli+1 (1 - 1:)(1 4+11
(45)
Moreover,
1 t1) tN)e
1S(()))2 31
(46)
(IQ + | + v0)²
Proof. We differentiate f(P,) w.r.t. Pn and obtain
N
df = d
0000
1/2
II 1=0
E E Sandt))) 2T,11 1/2
N
11/2
II K
Digh, d'tent Pii Enneen
II
ET,1/2
-0.0
1=0
f21
N
(mstdda (Eijle di 1441 be
I-1/2 1/2
1=0
K
N
Σ
(Emt ler
(Ergle dE
II [[Ensty(gHn) 1/
0=
321
Since all normalized correlations are bounded by ||Enstr ll2 S 1, we can also bound df by
N
|/
<
(1 di 2441 llz + |[ dis... S7.122 1=122112)
1/2
X-1/2
N
<
1/2 (11 E 1/ 122ldd dan ||2|| |2
1/2
|/2 + 11211 ( leaeell2212212 12a122,,,,23111
1=0
Since En,t, > 441144)), Vi E N,N. we get estimate dE we have explicit formula and bound
|E 1/2
5 1/ 1/1144411
(:(1 - (,)e. For differential
d = (1 - (2 0222 (1- ) Q1/2
t < ((1 t1) - t 11 |2||
In total, we can bound ||df']2 & 7(Q, S, || dd where -Q, =
|[Q 1/2 //2///||||//// ( (a-)t)) (1)
t 11Q
1/2 |1Q-1/21 + V (1-tx)
||s 1/2 ||S-1/21|2
N
(1 - t,)tt41 + (1- - t441)ts
Ve
V Vettt41 t:)(1 -441)
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Now prove 46).
We bound the norm of f using formula (44)
N
N
llf(P))l2 -
II
(mite. ) 112 - II (1
1 112) S Dmmton ll2 |End
2=0
S ||Q- 10-1 S0, 54 11212 | 1/2 Ixx, 1-111/212
[1((1 11)Q1/2 + P S/22//1111 //) 2 2 (1-
We note that - Pr P. > - > and
(Q- (Q-
+
(1 1= P.
t1
En - (1 p,) (1 - 116el
En - t
Similarly, we have
MiydN - -(1-t)) ( tx(1 tN)el
ΣiNdAN tty11 tyl
Next, we consider
[I((1 (1 ttt P, ) 1/2
1/2112
| 1/2 (Etg,t 1//51/211
1/2
S [ 1/2 (2er,ts - -11 -116))27,,1111
|| - t1 11) 1 112 S f1 t ( ( (Frg)h))
I 1- 12) / (E,,,t)) t1 [
(47)
We also can see that
< (1 -
11 +4411111 - 4.)e < ((1 ((1-4)) +|] 4, 1,)e.
Thus, conclude that
41 (1 Li)e
[I((1 1/) + +1 1/2
1 1 1
((1 t 1nl+ttlll(11111 11 11 t
(1 - 11)c
I +//////|++1$+1231 V2)2
Similarly, we have
IN I )e
E (tNS + +1 -1 (1-
(1QQ//112 + S + v2)2
The final result follows:
(P(()))
II((1 (((1-4)0 (((14)Q1/2 P,,, 2 1/22 1 (12 + (x2
1 116x(1 - (1)( IN)E
(IQQ1/212 + //112112 V
Next, we switch from tracking the changes of normalized correlation matrices to tracking the changes of the optimality matrices. Recall that, for a 2D-dimensional Gaussian process
Q p)) PT ))) the optimality matrix A from the definition I0 calculated
w
A(P) = - 1pS)
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Functions E(P,, Q.S) and A(Pr) can take the normalized correlation P. Q PS-SS- first argument. this case, the formulas and notations are
as the
(P... = (T PPPP Q-1/3 A(P,) (P. Q.S).
(48)
Lemma D.5 (Optimality matrix map properties), Ler matrices Q.S x be the marginals of (0) P ()) with the normalized correlation
5))
2D 2D-iieensional Gaussian distribution N
Pn :- Q- P Then the map from normalized correlations to optimality matrices A(Pr) = is bi-Lipschitz on the set {Po E RDDD ||Palll vI-wW for any < w < Specifically, for any Pn and Pu, from this set, the following inequalities hold
S-1/2p,T (PP.P.)) - Q-
P -Pll2 - [|((p)) - A(Px)ll S Ma Pr
Pall2
where
M, = 1--1111 1/2 110-111
1/2
( + w
L
22 2
1/22
1Q1
1/2
Before proving the lemma, we introduce some notations. Let h be a scalar function. For any diagonal matrix A = ....).)) we define
h(A) diag (h(A))...... h(AD))
Next, given a symmetric matrix B E DxD with spectral decomposition B = ZAZT, we set
h(B) Zh(A)ZT.
Proof. To estimate M.. we differentiate A(Pa) w.r.t. P. that
dA s-1/2p, -I- (PPP. (dPrP.T + (I- Q-1/2
SP (1 P..P.P Q-1/2,
(49)
By the conditions of the lemma, x P.P.. * (1 w)1, hence I1 -I PTPP 1 S and
[|Px < Thus,
- 2 w2 w
|dAl|, S 1|1111121 || 1/211
|dPall2
Since the ball (Pn |Pull22 S v1-w) is convex, this yields the bound Ms on the Lipschitz constant.
To estimate L, we define B = S1/2 1Q = PTT (I (I- P and note that
B = -1-P.P P.P.T (I-P
(I P P.P... -(I-PaPT)"
Next, we define h(x)) = 1+V1+4z" I 2 0, so that h - = - - b 1 < Therefore, we have
I - P.P. = h B). Bh(BBB) B).
(50)
P B[=B(I-P.P..)
For now, consider B such that its singular values are positive and distinct (note that the set of such matrices is dense R DxD). Then the SVD map B+ (U, A, V) such that B UAV* is differen- tiable at B (see Magnus & Neudecker 2019 Section 3.8.8). thus so is the polar decomposition map B + (Q, S) such that B = K KC, where K is orthogonal and C' is PSD matrices. As
p
BL T B) KCh(C²) U
and wh(2²) (2)² is diffcrentiable, we obtain that
dP dKCN(S + Kd(Ch(s²)).
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Furthermore, 0 h(œ) S and (xh(2²)) DivI+rr"y/LE (0. 11. hence 0 h(52)
and Ch((2) is 1-Lipschitz w.r.t the Frobenius norm (Wihler 2009 Thm. 1.1). Note that
| = Tr[(KdC) dkC] = Tr((CdC) dK)] 0 since dK is skew-symmetric.
It can be shown from the orthogonality of K:
1
0= 0= dk K dK + dK K - dk.
Thus, we have
|dB] || KdclE |aka |KdaC| = |dkC| + ||ac||
Therefore,
P [[ + Kd(Cc((C)))l S ( Id(CWC))) S ||dKO| |IdOO S v2]
(51)
In particular,
|ddP - ||dPall S d2] F S 22DD S 122
By continuity of the SVD and thus of the map Bh 3 B), this yields that
L = 2 / 1Q//
Now we can show that the function, changing the optimality matrix during IMF step, is Lipschitz. This function is constructed as follows: first, it transforms optimality matrix into the normalized correlation via =,1 then it makes IMF step obtain new normalized correlation via f from (4). finally transforms new correlation back to new optimality matrix via E
Corollary D.6 (IMF step optimality matrix transition properties). Let matrices Q,S be the marginals of 2D-dimensional Gaussian distribution N
(()
PT 5)) with the optimality
matrix A defined in (48). Set the function g(A) == Ex ( (A, Q) where En and f
defined in (48) and (44). and (1; Q, denotes the inverse map En w.r.. the first argument.
Then g is Lipschitz continuous with constant Mpyy on the set (Al 0 < w < 1.
| S vT ww- w for any
Proof. Lipschitz constant of the functions composition is the product of Lipschitz constants of the combined functions. From Lemma D.5 we know that the constant for E,, is M.,, for E is 1/L. as inverse of E,, For transition function J, the constant y comes from Lemma D.4
We also prove the upper bound for the normalized correlation for further proofs. Let matrices Q, 0 be the marginals of 2D-dimensional
Corollary D.7 (Bound for [.P..
Gaussian distribution N
?
Q
2))
with optimality matrix A = E(P, Q) and normal-
v
PT
ized correlation P., = Q-
à PS- 4. Then the following bound holds true:
2
| 11, 2
(52)
1 1+
4 |
Proof. We recall the explicit formula 50 connecting Ps. and A:
I T-P.P. h(B B).
(53)
where matrix B = 51/2AQ /2 and scalar function h(z) == I+vI+1 2 0. Given a D x D
symmetric positive definite matrix C with spectral decomposition C. = UAU* we set ((C))
Uh(AJU* We start with estimate
Amin Amin(h(B (N(BT B)) >miin PTP = Xmaz (P.P..) 1 P
(54)
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Since function h is monotonously decreasing on [0, +oo) and matrix B B has non-negative eigen- values, we have Amin ( B)) = h(Amaz [ B)) and continue with:
Amin (h(B = h(Amaz (B B)) = h(1 B Bl
2 h(lIA 12 E2 (E
2
1 + 4 Q 2 11
Combining bounds together, we conclude:
2
Pr 2
+ + 4 Q 2 S11 |
Finally, we are ready to demonstrate convergence of the optimality matrix to the desired solution A" c- Ia after IMF step.
Lemma D.8. IMF step convergence Let matrices Q, > 0 be the marginals of a 2D-dimensional
Gaussian distribution N (!).
Q PT
5)) with the optimality matrix Then after IMF step,
we obtain new optimality matrix A = g(A) (see Corollary D.6. satisfying the inequality
M
1| - -Tll2 S (Q, c "Tall2:
where
7 110 |2
1111Q 1
S 11
(1 4,)t411 + (1 ti+1)ts
1 (1-t1) +
Ve
tN)
/ettt+1 - t,) - tst1)
41x(1 - 4 (1 IN)E
min
1 || 12
1/1 2 + //// + v2
L-
/22/11//1/211
M ||-"11??
11 11-111122
w2
Proof. The IMF method with volatility parameter can be viewed an iterative application of the transition function Since IMF converges to A c --Ia. it follows that A is stationary point of g, i.e. ((A*) A". Hence, we apply Corollary D.6 to get
M L
A A (0((4) 9(1')12
>11 A" 12,
where explicit values for y and w, Ma, L are taken from Lemmas D.4 and D.5 respectively. We only need to satisfy condition on w from Corollary fo6 matrices A, A, A":
||)] S vI VI-w.
In terms of normalized correlations Pn. Pn f(P)) from and P = E,"((")) from 48 the conditions are
Pa
1
P w.
(55)
For the second inequality f:
55 we use bound (46) from Lemma D.4] for the result of applying
h1 41(1 tx)e
-
= -
IP(())) N
(1Q |S ve)2
Finally, combine all the bounds under the single minimum.
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D.4 PROOF OF D-IPMF CONVERGENCE THEOREM 3.2 D>1
Proof. We denote by Qo marginal matrix at / 0 after the first IPF step. First, we note that all marginal matrices Q at and S at / emerging during IPMF procedure bounded by the initial ones (Lemma D.2
E l|
=: ug, ||5- S mmax(Amol (Sp $ S1 b), 13|2 '12 rf56)
||
5
min Amin (S)
1),
||0o 2
||Q]]2
ue ue 11Q L112 S max max (Q -00 1312 =:
min { Amin @Q Q0 )).1
Optimality convergence and condition on e, Consider any IMF step during IPMF procedure which we denote by
P (
IMF
Q P
P S
Pa
Q
We want to find such e that new optimality matrix Ā E(P, Q, becomes close solution A = =n than starting A = Q,, This transition from á satisfies (Lemma D.8
| ll s
2D
1
2
>Q, x(Q))n(5)))
A 12:
(58)
w
w2
is defined in 45
tyt 11 tN
w = min
1.
(59)
(1|Qu/2 + || + vE)²
where x(-) is condition number of a matrix.
Estimate w. The second term of w in (59) can be lower bounded by
t1t (1 11) tN)e (1/2//2112 + || ve)²
- 41-44/111110 In).
(60)
To estimate 1 - || in the second term, we use lower bound (Corollary D.7
2
1 P
1I+ 12 A
Hence, we have lower bound for
(1 t1- t1) - tx))
2 min VI+ 4| 1- t1tn(1 (1))(11 -t)(L
VI+ Q 212551111
The change of difference norm after one IMF step is
(1 1|2
|| A,ll2 GVD- (({))())((($tt
-1- (1 (N))² ((),),e) 1|A 11A-AA11 A. (61)
:=(QQ.S||A(]2.))
Now need make this map contractive, i.e., bound the coefficient I(Q, (l],,,)) for all matrices Q, S, A appearing during IPMF procedure.
Universal bounds (57) and (56) state that matrices Q and S lie on matrix compacts Bę {Q 0|||[||2 < uq.,
ro} and Bs {s O|||S||2 < us, || rs), respectively. Moreover, the function 1(Q, S. ||A]|,, e) is continuous w.r.t. all its parameters on these compacts. Hence, we can get rid of Q, dependency, since the following maximum is attained
max [(I[A]),,)) QEBq.SEB, Bg
S, el)
||Al||2 dependency. IPF steps do not change optimality matrices (Lemma D.3 hence, we consider only IMF steps here. We prove by induction that if at the first IMF step with initial optimality matrix
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Au coefficient [[IlAb] |22 2c < 1 is less than then all optimality matrices {A,} during IPMF procedure will be bounded by
||A:l12 _ ||-ooll + 2,-,, ||A, - A- I Al --112-
First, we note that the coefficient [[]]_A]2, e) is increasing w.r.t. ||A||2 As the base, we show that after the first IMF step matrix A is bounded:
|| 2|
s A A" ||2 + || A.112 1(11-Aoll,, e) A-1 + ||A-||2
< [[]l_o|l2 + 2 , Ab A + ||A S ||Ao - + |
< Aolla + || Anll2 + 26
Moreover, we have
||A1 - A 12 - ||A - A" 112.
Assume that the bounds ||4:122 I || + 2c- and |A; - A - |40 - 2 hold for the i-th
matrix, then, for the next matrix A;+1, we prove:
A 1
S
[(11 A, 2, <) |) - A S Aoll 2e ,,)]A, - A
||A; A < ||Ao- - 12
|4411l2 ||:411 A + ||44|12 S |l - A 12 + || 11 - || + 2c-
Thus, we take maximal possible norm among all matrices
||Ail12 S || + 2c 1 to upper bound
the coefficient ([|A: e) S Aool + ,e) <1
The final condition 62 on is
((D)x()=)n(S$)) (1+ (1+1)11011)))) + 26
-1Q.5500) <1.
B(Qo, So, co,
QeBq.SEBg
(1- (1-tit(( - +1((1 tN))2²
We see from the definition
|| 11
4 4111-= V
I SN 211
(1 (1-4 1-44)441 (1 14+144
((, (,c)
+
Ve
(1-tz)
(1 (l-tx)
/ettt+1(1 t.) t(+1)
that the largest value of y is achieved when Q/22122,, /|/1//11122/ 11Q 1/2 22, |||||| 1/222 are the largest. Since these values are bounded by ||Q I vug. /|5 I /us and 1|Q-1/2 1/2 s
/r..11s-1/ S Tr. we can estimate the maximum and get lower bound for e:
(1 + suqus (1l Aollz + 1))
3(Qo. So, Po. s /DD. uurousts.
(1 titn(1 +1)(1 tx))²
Vuqus Ve
tuQ
tNts + Σ
N-1
(1 - fi)tt+1 + (1 - t
+ (I-
S
(1-t))
Vettit t,)(1 t,+1)
= ( rQ's
4 "QuS ||
(62)
If the above e-condition 62) holds true, then A₁ exponentially converges to A (square appears since IPMF step includes two IMF steps):
||4 2 - B(Qo. So. Po. (2k -1
Marginals convergence. Furthermore, we prove that marginals converge to ground truth Σ, as well.
Q
P
We note that, during any condition
of IPMF procedure, the norm of the normalized
PT
S
matrix P. = PSP- 1 is bounded:
2
|| <
1+ / + 4 Q12
Since ||Q]]2 S un holds from 57p. ||S|| < us holds from (56) and || I | + 2c (due to contractivity of A), we can upper bound the normalized correlation
2
P, 1 < 1
a..
So <1. 1.
1+ V 4u Aolle 2 + 22-11y
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Finally, apply bounds from IPF steps Lemma D.1 at k-thh step and put maximal norm value a aQu,, So Po. e
| k : S IDl2 I
a(Qu, So, Po. c)² 1| + IDll2.
EE (uk= m1)l2 s a(Qu, So P), ||= (vo -m1ll2
D.5 IMF STEP ANALYSIS IN LD
Preliminaries. In case D 1, we change notation from matrices to scalars:
0 P IMF F
P
Pn := p\ p 33, Pn. == p/ /sq.
8)
Using these notations, formula (48) for optimality coefficient E R (instead of matrix A) can be expressed as
Pn
En 1.8)
= -XE((-os +00).
(63)
ssq(1 - Ph)
The function En monotonously increasing w.r.t. On E (-1,1) and, thus, invertible, i.e., there exists function E,1 (-00, +00) x R x R - (-1,1) such that
xx² + 1/4 - 1/2
E (x.8,) 8,
(64)
X/sq
The inverse function is calculated via solving quadratic equation w.r.t. Pn-
In our paper, we consider both discrete and continuous IMF. By construction, IMF step does change
marginals of the process it works with. Moreover, for both continuous and discrete IMF, the new correlation converges to the correlation of the e-EOT between marginals.
Lemma D.9 (Correlation improvement after (D)MМF step). Consider 2-dimensional Gaussian distribation with marginals N q) and N (w, 8) and normalized correlation Pn. E (-1,1) between its components. After continuous IMF DIMF with single time point correlation Pn. The distance between Pn. and EOT correlation p" = =-1
obtain normalized = (3/e, q, decreases as:
P0 - P'l S 7. -Prlb
where factor 7 for continuous and discrete IMF (with = is, respectively,
2c²qs f(0)
2-22
2s2
V2
492
s)
tanh-1
+ tanh-1
(65)
(62 -2)
Ve V=2202
V 192
yalg,8,)) s,t)
(66)
+ P1-00g---11 (1-00t (1-
Proof. Continuous case. Following (Peluchetti
2023a Eq. 42), we have the formula for Pn:
tanh
tanh -1
Pn. f(px)
=exp
exp
cg 0,
(67)
C3
C = c 2s(png - 8), 1
V (c+ 2(pr. + 1)qs)(E + 2(pm. - 1)gs),
22 + 2q(pns q).
Note that the function f(x) is positive and concave on (-1, 1), i.c., its derivative is decreasing
(-1, 1). Hence, in negative segment (-1,0), 0]. the distance until the fixed point P > 0 is decreasing faster, than in positive segment, and need deal only with the positive segment [0. 1). We will show that the function f has a derivative norm bounded by on [o, 1). and, hence, is contrac- tive. Due to concavity, its derivative is decreasing [0, 1), and we can check the bound only for derivative at the point Pn. = O. Direct calculation gives us:
f()) 2egs
E - 29
Ve- 4 ]
2g2 2-2s2
1-2
(0) = (2 - 02-4492270 2223
EE 29
tanh -1
+ ttnh -1
-
492
/ 49
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(c(9,8) [f(()) < 1.
Thus, we can bound 1f (pa)) 1 Yol 8), Vpn E [0, 1) and get on the whole interval (-1,1)
Iin - -PPl = (f(p)) - f(p%) S Volg, s)pp Pzll
Discrete case (N = 1). We use explicit formula (44) for a new correlation Pr. = (pr) after D-IMF step from (Gushchin et al. 2024b) provided in the beginning of Section D.3
In the case of single point = & (N 1), we prove that the function f(,)) is a contraction map. The sufficient condition for the map to contraction is have derivative's norm bounded by "a < 1. First, we can write down the simplified formula (p.):
(11 (1-441 + 10 (LV) + (1 1)
f(pm) (1 (1- + 2 + ((1
(68)
12
l)e
Next, we simplify derivative (p()::
00.t (1 (1- -
0t1 8 (1-t).p.
det (1 11=)..... 2(1 t)t p 8......77 8 t)c () 0o,t + 0,,1 +1(11 t)c,
(1 - t)oo.
to 2. to (1 (1- -t)o0.
f(())
+
Tt,t
Tt,t Oz,t
We define new variables o., del (1 1)00, Et,1 def lo 1(1 l)e and restate f' as:
δ1,
200.001.1
5 =
ōu, σ1 +e + ōu, δ1 tē (ōo, + δ1,t 2)2
(69)
(ōo,: + ( + + +) - 200,
(ōo,t 01,t 8)2
ab. + of. + (0o.t + δ1.t)ē
(70)
(ōo. + 1 + 6)2
0 + o.. + (ōo.t + δ1,1)
(71)
σh + 200 0 1,t o.. 2(00.t + 01,1)++
(72)
1 25 (i (70 t J8+4
06 tof, (00.
We note that all terms in 70p are greater than 0 and, thus, f is monotone:
<<> (pa), Pn (-1,1).
(73)
In the negative segment Pn. (-1, 0j, the derivative norm If' is greater than in the positive segment [0, 1). and value of the function always larger than its argument. Thus, in negative segment, the distance until the fixed point P% is decreasing faster, than in positive segment.
For P.. E [0, 1), we can bound the fraction in denominator of (22) by taking its numerator's minimum at P.. = 0 and its denominator's maximum at p.. i,i.e
0 < f < 7a(g. s, t) < 1,
va(g,811)
21 1 1 (1=
We note that 7a(g. 8, 1) is increasing function w.r.t. g,
we put into the function argument p", corresponding to the <FOOT correlation, DIMF does not change it Hence, Ph is the fixed point of (p.), and we have
Pn - p..l = (f - f(p.)) < 7a(g, s, t))p P.l-
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Lemma D.10 (X improvement after (D)MMF step). Consider a 2- 2-dimensional Gaussian distribution
with marginals N [1, q) and N 8) and normalized correlation Pn E (-1,1) 1) between its compo- nents. After continuous IMF or DIMF with a single time point we obtain new correlation Pa. such that Pm Pi.. S spppn - Pal where Pi = -/- q, 8) and Ty < is from (65) for IMF and from
166) for DIMF We have bound in terms of X En(pp: 9, and X (pn: 9, 8):
1x 3/cl S 1(pms 7) :)x1)) -1x 3/el,
(74)
[ (1 -7) (1 max [Pn: |P 13)2
UPn, Pa:
< 1.
1+ max |
Proof. Monotone. The function f(P)) from (67) for continuous IMF and from (68) for DIMF is monotonously increasing on (-1,1). The monotone means that the value Or. always remains from the same side from ph:
( Pn Pm
((a) > Pro
(75)
Pn S Ph = f(pa) S P.,
The same inequalities hold true for X = (p.(pr q. 8), X E(pn. 9, 8) and X+ 1/c as well: < Xx, then X < Xx and vice versa, since (m 9, 8) is monotonously increasing w.r.t. Pr..
q. s) and En (p 9, because they E, Properties. In this proof, we omit arguments q, of =, " do not change during IMF step. The second derivative of the function E. (Pn) is
d²=, (pn) dph
2pm Ph))
/sq(1 (1 - )3
Hence, we have d (pn) S 0 for P.. E (-1,0) and d= (pn) 2 0 for Pr. E [0, 1). It means that the dp function En (Pr) concave on (-1,0) and convex on [0, 1).
The function (px) is monotonously increasing w.r.t. Pe. thus, decreasing of the radius
h def
Ipn - -P%, around of causes the decreasing of Ix - X+1 around Xx. We consider two cases: X > X+ and x X..
Case X > Xx- We have Pa. = P% + h, = (ph + h) = (pn) and En(ph + Th) 2 X. We compare the difference using convexity on [0, 1):
dEn dpn
x- x 2 5n + h)) - En(p% + +h))= 2 (pr h (ph + h7))
(p 7h)
= (1 - - d=n
dpn (p% + 7h)-
Since the derivative of E,, is always positive, we continue the bound:
den (p')) (1 - (1->))pm -Pal
(., + h) - En (p% + 7/h)= 2
min Pa M.EEL
dpn
Next, we use Lipschitz property of En, i.e.,
d=n (pn) Ipn - PPll dpn
Ix - x-| = (p.) En S
max 1% o +h]
and combine it with the previous bound
min n/ e..
dpn. (ph)
d=n
X x2 En ht) -n + +7h)
(1 - (1--)]x- xol-
max
d=n dpm.
(ph))
,4 E(P
Case X < X.. We have Pn = P/1 h x = En - h) En (pn) and En(ri, 7/) S X. There
three subcases for X, X positions around 0:
1. For positions obtain
X. > X En - - x 2 0, we use convexity of E,, on (0, and
dEn
- 2 5, -h) -E, - h) 2 (1 - 7)h
(Pa h) dpn
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d=n dpm
min 6 Ele - m.
(p/) (1 - 7) - -Pil-
2. For positions X+ > 2 X 2 En - >h) > x and X+ 2 1202E.xp2 - -2h) > X. we
use concavity of En on (-1,0) and obtain
b dpn
X- N
En 7/) - -En En - -h) 2 (1 - 7)h-
(ph 711)
d=n
min
- 7Mmn - -Pal. (pha)
Rg Elp N.PPPl
dpn
3. For positions X+ > X 2 En - -hh) > 0 X. we use concavity of En on [-1,0] and
convexity of E. on [0, 1) and obtain
--x 2 En (pn - --h) En - h) = 1 - -h) -x + - (, - h)]
dEn (0) (h- -%-p) d=n (0) = (1 3)
dEn (0)
(0h -hh)
dpn
dpn
dpm
d=n (ph) (1 pn -Pal.
min P. Eln h.nt]]
dpn
Overall, we make the bound
d=n (p/)
--x X >
min p4. Elph - h.pil
(1 71)p - P%
dp
min 0% (0m
J=n
(p'x)
don
(1 (1-1) -))x- -x-l-
d=n (p'))
max 0% [()
For the function En (pn) =
Pn
the centrally symmetrical derivative is
v=g(1
p.j.
d (pm) dpn
1 1+ P V 22)
The derivative dp.
& has its global minimum at Pr. = It grows as Pn. -> +1, hence, the maximum value is achieved at points which are farthest from 0:
d=n
dEn
(P:))
(pn).
max P .l
dpn
dpn
d=n
S max { den, d=n((tt) (Ph): dEn (1pxl) }
max
(Ph))
P% Vl
dpn
dpn
dpn
dEn (P'n)
p min
11+
dpm
/sq
Thus, we prove the bound
xx x.1 1 18 - x=l= x/2 (1 max Pr: [pa 22
(1 (1-7)lx- -))x- - -xll.
+ max (p:, 1pm
1 (1 7) (1 - max max(pin |pa 1p 1292
1 -
Ix
+ max |pm |2
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D.6
PROOF OF IPMF CONVERGENCE THEOREM
3.2
D
Proof. Notations. We introduce the notations for a k-th IPMF step in terms of scalars
Jo
Pk
IMF
Pk
IPF
Pk
Pk
Skc
Pk:
8k
Pk
01
IME
IPP
Oo
Pk+1
Pk:
σ
pk+1
8k+11
and means
IMF
Ho
IPF
TJk [1
IMF
IPF
Po
Vk
P1
Vk++1
We denote the variance of the 0-th marginal after the k-th IPMF step as Qk- For the first one, we have formula 33 Qo
where Pn.o = po/ Vanso is the normalized correlation after the first IMF step. More explicitly, Pn,o
op
ooP 23
def f(P.,)). where Pn.o is taken from (67 for contin-
uous IMF and from (68) for DIMF. We denote optimality coefficients X En
def
Pk Voosz"
Co. 8k) and
X. 1/c.
Ranges. We note that IMF step keeps qk Sk, Ukk Vk, while IPF keeps XA- Due con- tractive update equations for Xx (75) and for 5h (38). the parameters 8k: Xk remain on the same side from 01, E. respectively. Namely, we have ranges for the variances Sg E
man [min su), maxfo1, su]l def [ominn ,o7"= 9k E [minfous fou, qu), max Holl def min 0 and parameters Xx [minn(x; (xol], (oxxxxx,Xxo)) def min :x" mmaa..
Update bounds. We use update bounds for need limit above the coefficients |x, q. 8) considered ranges of the parameters
74 twice, for
(38 and for 41 however, we
and and ((= (x. 9: 8), =-1 (x+: 9. 8), 7. 8)) over the
(a main
mee 8.88 E (arnin o maaa and E mem ;X" maz].
The functions E,, .4, are defined (64).
74).
655 (or (66) with fixed t), respectively.
Since the function [x, 9,5) s)1 is increasing w.r.t q, and X (growing symmetrically around 0 for X). we take maximal values og of and xmaa Similarly, the function ((,,' (X,, s), s)) is increasing w.r.t. all arguments symmetrically around O.
Hence, we maximize the function E and the function y, which is also increasing w.r.t. q and s.
Final bounds. The final bound after l step of IPMF are:
1 oi
a S 0i
26 2
1/
11
a p/ 11l,
Ix 1/4
k2h Ixo 1/cl
where def 1(= maz 0b man o En (Xa, o maz , _amaz)) ), (amaaz ofmo)) and a der
E (x aama _ maaa)) fixed for discrete IMF.
taking from
774 from 655) for continuous IMF and from (66) with
D.7 PROOF OF IPMF GENERAL CONVERGENCE THEOREM 3.3
Proof. We split the proof into two parts. First, consider the discrete case.
Discrete case. Let l 1. Note that the transition probabilities gA+++ (z1, (rr)) can be bounded from below with af((z,), where a E (0,1) and μ depend only on &1. and supports of pa and P1- Thus, we can bound o AR+1 (oo) 2 app' (x1), with μ (x11 depending on 4k+1
( = / proimla 4++ (x (r,, )g(x,, (mo)dRr,
2 a
(r )dp (p))
(76)
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Similar statement holds for 4k+3 40
Thus, the IPMF step is contracting. Specifically.
11 pollev (1 a) pılliv,
|1
1k+1_
pillrv S (1 a) 110 2kt2 po ITV.
where TV denotes Total Variation distance. Thus,
LV po. q1 4k+2 1 pt.
40
(77)
Since Po and Pi have compact supports, Prokhorov's 's theorem ensures the existence of weakly sk; " 40.1 Moreover, (77 ensures that T0, E Ll(po.P1).
converging subsequence J
Let IMF(g] be the result of the IMF-step applied to g, and let IPMF[g] be the result of IPMF-step applied to g- Note that the IMF step is continuous w.r.t. weak convergence, since all intermediate
steps have smooth transition (i.e., conditional) densities. Combining the above results, we get that
IPMF 46g 3 1 = IMF(IMF[00.]]
(78)
The equality holds due to do. [T p1). Note that also use the fact that convergence in TV stronger than weak convergence.
is
Recall that 4k IPMF IPMF(10.1) [. fo.1
78) ensures that for any fixed > 0 it holds do.1
48,+4n +4n
[. Moreover, by Theorem 3.6 in [ASBM), it holds that 7M F2 [ūo,1] -t,, 40.1
Thus, there exists weakly converging subsequence
41 "i 9 90,1
(79)
Finally, we argue by contradiction: if 90.1
4 we can select weakly converging subsequence
40,1
"
d6.1 * g- But by 799 %6 = 46 This finishes the proof.
Continuous case. We now explain how to extend the above argument to the continuous- set- ting. The key point is verify Doeblin minorization condition for the Markovian process obtained after the projection step (see, e.g., Section 2 in Stroock (2005)).
Fix some E (0, 1/2). For each l E N. let (x, (x,4+2) )re(o.1] denote the Markov diffusion correspond- ing to the law g g and let
=x), (a A) = PP 2x E A X4 x).
x E TRR, A E B(RP),
be its transition kernel from time 0 to time 1. This admits a decomposition of the evolution on [0, 1]
into three subintervals [0,5], [5, 1 8) and [1 6, 1].
Disspaptivity. Next, we show that, by construction of the Markovian projection, the drift on [ő, 1-5] is Lipschitz and dissipative. Set
X1 t
mx(x)
(t,
x)
X, x
m+(x) == E[X, X, xx,
1-t
and recall that since the supports of po(x) and pr (I) are bounded, ||x||| < M1 and (x:)] s Mi for any Then
(b+ (1, (m)()),
|
M, |
||
||-|| + M2
<
2(1 - 1)
2(1 - -t)
This yields uniform coercivity (Lyapunov-type dissipativity) for any t E 16, 8). The drift is Lipshitz. Consider
(.)(()) -
V Vab+
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and recall that the transition density is
(-
|| (-1|=r =12
Tz (x1)) 4r, (x0 1x) x g(x1) exp
2: t)
Consequently,
/ f
= V (x))(x)= = =
V dri
= =((nn (on Vx (
Cov=, [X1]-
26 1)
Thus,
diam 2 (supp
( S C
c²(1 -t)
This ensures
1 diam(supppi)
IV (t, 1 1 E²(1 t)²
A similar result holds for the backward drift
Xo
m1(x)
b (t, x))
E
X1
m4(x) == E[Xo Xx x])
Small-set condition on [6, 1 16,1-6,, 5]. Fix and consider the transition from time 1 8. Denote by P& (a: y) the transition density of X1=8 given X-A4+2 x. Since the diffusion coefficient is constant non degeneegeer (E)Ip) and the drift is locally Lipschitz (with a uniform bound on || (-)ll on [8, 1 - o)), heat- lower bounds for uniformly elliptic diffusions yield that yı: is continuous and strictly positive, and admits a lower bound on compact sets with constants independent of k; see, e.g., Theorem in Aronson 1968).
In particular, for any fixed r > there exists YAR such that
inf inf (1 y) 2 mR, > KEN #EBa, B,
Let Uy be the uniform probability measure on B, and set 3n = mR, Leb(B,). Then for all e N, all E Br, and all measurable A,
P(x 4h+2 E A x x) 2 (R(x(A)
(80)
By the Markov property, for any measurable A,
P(x E A || +k2 = LT) IP 4k+2 E A 2) P E dz x = x).
x4
Combining this with 88). we obtain
P E A x = x) 2 (n)r,,(A),
x E Bn.
where pr, is the probability measure defined by
(A) = |(xx EA |
X-1+2 x/xx===+=- =) vr (dz).
(81)
Next, we control the tails of distribution of x-I++2 uniformly in k. By the definition of the reciprocal
projection, the segment [0, 1] between Xo and X, (conditionally on (Xo.X,)) X2)) distributed
Brownian bridge with variance parameter o² = =22 - 8). Hence, the marginal at time 8 is mixture of Gaussian laws with covariance matrix a I and mean
ms (No, (1) (1 5)zm 8x1,
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where (zu, .1) ranges over the support of the endpoint coupling. Since the supports of Po and p1 bounded, there exists Ry > 0 such that || I1) I Ry for all (Zo: 21 in this support. Standard Gaussian tail bounds then imply that, for any 7 E (0, 1),
choose R >> 0 large enough that
P(X & Bn X x) s v.
sup sup KEN zEsupp(po)
Equivalently,
P F(X BR X =x) 2 1 7
e supp(pp)) E N.
(82)
Combining 81
and 82
we obtain, for x E supp(po) and any measurable CC RD,
Pa A)
E IP X ak+2 E A x
X
xdb+2
E ( (x 2k+2 A X
(xfett2
X
BRVR (
P( x E BR X XAh+2 =
BR(1 11) (A).
Thus, for all x E supp(po) and all &s E N
(x 2 a (r,((() with a: - E (0,1), :== un-
That is, the family of kernels (Pa)a satisfies uniform Doeblin minorization on supp(po).
It is well known that such minorization implies total-variation contraction: for any probability measures X, X' on supp(po),
|AP& XPPlIIV & 1A- XIIVv, E
Applying this with 40 1k+1 and X' = po yields
l 4++1 pillIV
1 P. PoPAIITV s (1 ax) |1
Polliv-
The convergence 9o changing the roles of Pa and P1), and we conclude that
4k+11 TVV Po is shown by the same argument applied backward in time (inter-
TV
q1
P1.
k-000
In particular, the continuous-time analogue of (77 holds.
Having established the convergence of the marginals ensure the well-posedness and stability of the iterative procedure. Next, we recall that the drift b+(()) locally Lipschitz and dissipative.
Doeblin minorization, remains to
Next, by the symmetry of the IMF procedure, it suffices to consider the transitions to the mid- point, (0, 1/2) and 1/2). By standard diffusion theory, the regularity of the drift ensures that the corresponding Markovian transition kernels smooth. Consequently, because the updated joint distribution o (mo, 21 1/2 x1) is constructed by integrating these transition kernels against the pre- vious distribution q', the mapping of + qi+1 preserves weak convergence. Therefore, the IMF iteration is continuous with respect to the weak topology. The rest of the proof is similar to the discrete case.
E EXPERIMENTAL SUPPLEMENTARY
E.1 ILLUSTRATIVE 2D EXAMPLE VISUALIZATION.
We provide the visualization of the starting processes and corresponding learned processes for Gaus- sian + Swiss roll translation in Fig. / One can visually observe that all the particle trajectories or relatively straight and therefore close the Schrödinger Bridge problem solution.
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Table Datasets and code used in our work along with their licenses.
Name
URL
Citation
License
Colored MNIST
GitHub Link
Gushchin et al.
2023b
MIT
CelebA
Dataset Link
Liu al. 2015a
Non-commercial research only
SB Benchmark
GitHub Link
Gushchin al.
2023b
MIT
ASBM Code
GitHub Link
Gushchin et al.
2024b
MIT
DSBM Code
GitHub Link
Shi et al. 2002
MIT
E.2 SB BENCHMARK BW2-UV -VV
We additionally study how well implementations of IPMF procedure starting from different starting processes map initial distribution Po into by measuring the metric BW also proposed by the authors of the benchmark (Gushchin et al. 2023b). We present the results in Table One can observe that DSBM initialized from different starting processes has quite close results and so is the case for ASBM experiments with E {1, 10). but with 0.1 one can notice that ASBM starting from IPF and Identity experience decline in BW3 metric.
Algonihm Tyre
D=2
7= 16
P==61
70 128
7
f= 10
D= 64
D= 126
D=2
D= 16
D=04
D= 126
Best ulposithm benchmaskk
Varies
0,106
0.15
11:25
0.22
0.005
01.00
01.01
0,01
DSBM-IME
0.1
0.14
0.44
8.2
0.15
0.1
0.91
6.67
0.1
5.17
66.7
836
DSBMM-IF
0.33
0.6
0.6
1,62
0.01
0.18
0.2
3.75
204
DSAM-6lemiyy
ASBM IME
IPME
0.016
0.13
0.01
0.35
2.67
11.06
7.12
0.12
0.34
6.67
0.013
0.00
0.25
3.8
58.4
913
ASBM
0.75
32.05
10.67
0.33
4.19
0.002
0.18
5:08
ASDM 6onnity
0.32
2.65
00.3
2.02
0.00
0.2
2.71
M-Sikk Bridge
Table 6: Comparisens of BWR- (%) between the ground truth static SB solution >" (ito, and the learned solution on the SB
benchmark. The best metric over bolded. Results marked with are taken from (uushchin et al. 2024b) or (Gushchin et al. (2023b).
E.3 CELEBA SDEDIT STARTING PROCESSES DESCRIPTION
The IPMF framework does not require the starting process to have po. P1 marginals or to be a Schrödinger bridge. One can then try other starting processes that would improve the practical performance of the IPMF algorithm. Properties of the starting process that would be desirable are (1) q(x)) = (a() and marginal q(x,) to be close to Pi (xx) and (2) g( 21) be close SB. In the IMF or IPF, we had to choose one of these properties because we can not easily satisfy them both.
We propose to take basic image-to-image translation method and use it as coupling to induce a starting process for the IPMF procedure. Such a coupling could provide the two properties men- tioned above. We use SDEdit (Meng al. (022) which requires already trained diffusion model (SDE prior). Given an input image x, SDEdit first adds noise to the input and then denoises the resulting image by the SDE prior make it closer to the target distribution of the SDE prior. Vari- ous models can be used as an SDE prior. We explore two options: trainable and train-free. As the first option, we train the DDPM (Ho al. (020)) model on the CelebA 64x64 size female only part. As the second option we take an already trained Stable Diffusion (SD) V1.5 model (Rombach al. 2022) with text prompts conditioned on which model generates 512x 512 images similar to the CelebA female part. We then apply SDEdit with the CelebA male images as input to produce similar female images using trainable DDPM and train-free SDv1.5 approaches, we call the starting processes generated by these SDEdit induced couplings DDPM-SDEdit and SD-SDEdit. Hyperpa- rameters of SDEdit, DDPM and SDv1.5 are provided in Appendix E.9
The visualization of the DSBM and ASBM implementations of the IPMF procedure starting from DDPM-SDEdi and SD-SDEdit processes is in Figure
E.4 CELEBA EXPERIMENT ADDITIONAL QUANTITATIVE STUDY
In Table 8, we report the final CMMD (Jayasumana et al. 2024) values for IPMF, while Figure 9 illustrates how this metric evolves over TPMF iterations.
Both evaluations performed the
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Tepettry -Phonaal poin
Trandhay (94162) Warreaatege
Treedie and -ohtrna warwainspein
(a) Pl.
(b) IMF starting process
(c) IPF starting process
(d) Identity
Hive sxjector sant Insernaiins poins
moandryy incrediate poir
Thpatty ef
(0) Po-
(f) DSBM-IMF
(g) DSBM-IPF
(h) DSBM-Identiry
Telestrr poec
Tepoery senki
Tagaden soPtte stererdee ande
Topst COEOKONT
boren late panki bians
(i) 21 P1.
ii) ASBM-IMF
ASBM-IPF
ASBM-Ldentity
Figure Visualization learned processes with DSBM and ASBM solvers Gaussian Swiss translation using IMF, IPF, Identity starting processes for = 0.1.
(a) 22 Poo
(b) DSBM-Identity
(c) ASBM-Identily
Figure 8: Results on CelebA dataset for the male - female translation task, where zo ~ Po represents samples from the souree distribution. DSBM-Identity and ASBM-Identiry refers to the outputs generated using trained DSBM/ASBM with the Identity initialization. The model was trained with 10.
same test set as in 44.4 Notably, the resulting CMMD curve closely mirrors the behavior observed for FID Figure 5 Additionally, Figure & and Table 7 present results obtained using DSBM and ASBM with the Identity initialization process the CelebA dataset, with 10.
E.5 GENERAL EXPERIMENTAL DETAILS
Authors of ASBM Gushchin al. 2024b) kindly provided the code for all the experiments. All the hyperparameters including neural networks architectures were chosen as close as possible to the used by the authors of ASBM in their experimental section. Particularly, it is described in
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Initialisation (coupling)
DSBM
ASBM
IMF SDEdit
DDPM
SD SDEdit
Identity
IMF
SDEdit
DDPM
SD SDEdit
Identity
Identity
c=10
IMF
DDPM
SDEdit
SD SDEdi
Identity
Identity
10
FIDU
35.23
28.77
61.56
13.65
33.11
65.50
19.32
21.84
20.64
27.47
MSB( 2,)4
0.16
0.02
0.02
0.0
0.16
0.04
0.03
0.16
0.17
0.07
0.08
0.11
Table Extended e = 10 qualitative results on CelebA (64 61) for male- translation with ASBM and DSBM across different starting processes. Generative quality (FID.) and similarity (MSE( 31 (4) are reported on the test set. Best and second-hest values for solvers are marked in bold and underline, respectively.
Initialisation (coupling)
DSBM
ASBM
DDPM
SD
DDPM
SD SDEdit
DDPM SDEdit
SD SDEdit
IMF
SDEdit
SDEdit
Identity
IMF
SDEdit
Identity
IMF
Identity
CMMD
0.0
0.31
0.69
0.84
0.32
0.46
0.34
0.33
0.28
0.42
0.32
0.51
Table 8: Qualitative results CelebA (61 x 64) for male ofee-ae translation with ASBM and DSBM across different starting processes. Generative quality (CMMD.) is reported the test set. Best and becond-best values for solvers are marked in bold and underline, respectively.
IMF DGPM
1.2
SDE:S /GDM-identty
DSBM identity
0.7
0.4
75
1.
(a) CMMD for DSBM with various couplings.
(b) CMMD for ASBM with various couplings.
Figure 9: CMMD metric in CelebA male male-female (64 x 64) as a function of IPMF iteration for various starting couplings.
Gushchin et al. 2024b Appendix D), authors used DD-GAN (Xiao et al.) with Brownian Bridge posterior sampling instead of DDPM's one and implementation from:
https: http http::////ftthhoooc.m//////a//sr
DSBM (Shi et al.
2023 implementation is taken from the official code repository:
https://gitie com www.//oo//y/../eeeee shi/ dsbm.-pyorrch
Sampling on the inference stage is done by Euler Maryama SDE numerical solver Kloeden
1992)
with indicated in Table
The Exponential Moving Average (EMA) has been used to enhance generator's training stability of both ASBM and DSBM. The parameters the EMA are provided in Table 9 in case the EMA decay is set to "N/A" no averaging has been applied.
E.6 ILLUSTRATIVE 2D EXAMPLES DETAILS
ASBM For toy experiments the MLP with hidden layers [256, 256, 256) has been chosen for both discriminator and generator. The generator takes vector of (dim+11 +1+) length with data, latent vari- able and embedding (a simple lookup table torch. .n. Embedding) dimensions, respectively. The networks have torch. nn. LeakyReLU activation layer with 0.2 angle of negative slope. The optimization has been conducted using torch. opt im. Adam with running averages coef- ficients 0.5 and 0.9. Additionally, the Cosi neAnnea IgL scheduler has been used only
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Model
Datasct
Start process
IPMF iters
IPMF-0 Grad Updates
IPMF-k Grad Updates
ASBM
CelebA
All
20
200,000
20,000
DSBM
CelebA
All
20
100,000
20,000
ASBM
Swiss Roll
All
20
400,000
40,000
DSBM
Swiss Roll
All
20
20,000
20,000
ASBM
EMNIST
All
20
75,000
38,000
DSBM
CMNIST
All
20
100,000
20,000
ASBM
Bench
All
20
133,000
67,000
DSBM
SB Bench
All
20
20,000
20,000
Model
Dataset
Start process
NFE
EMA decay
Batch size
DIG opt ratio
LrG
Ir
ASBM
CelebA
All
4
0.999
32
1:1
1.6e-4
1.25e-4
DSBM
CelebA
All
100
0.999
64
N/A
1c-4
N/A
ASBM
Swiss Roll
All
4
0,999
512
1:1
le-4
le-4
DSBM
Swiss
N/A
128
N/A
le-4
N/A
ASBM
cMNIST
All
0,999
64
2:1
1.6e44
1.25e-4
DSBM
cMNIST
All
30
0,999
128
N/A
lc-4
N/A
ASBM
SB Bench
All
32
0.999
128
3:1
lc-4
lc4
DSBM
SB Bench
All
N/A
128
N/A
N/A
Table 9: Hyperparameters of models from CelebA ($4.4), SwissRoll (:4.2). cMNIST ($4.4) and Benchmark
experiments. In "Start process", the column "All" states for all the used options. "N/A" corresponds
either not used or not applicable, corresponding option.
pretraining iteration with minimal learning rate set to le-5 and no restarting. To stabilize GAN training RI regularizer with coefficient 0.01 (Mescheder et al., 2018) has been used.
DSBM. MLP with [dim + 12, 128, 128, 128, 128, 128, dim] number of hidden neurons, torch. mn. S.S.ID actívation functions, residual connections between 2nd/4th and 4th/6th layers and Sinusoidal Positional Embedding has been used.
E.7 SB BENCIMARK DETAILS
Scrödinger Bridges/Entropic Optimal Transport Benchmark (Gushchin al. 2023b) and
cBW3-UVP, -VP-UVP metric implementation was taken from the official code repository:
https:
Conditional plan metric cBW3-UVP. see Table I was calculated over predefined test set and condi-
tional expectation per each set sample estimated via Monte Carlo integration with 1000 samples. Target distribution fitting metric, BVP,UVP, see Table 6 was estimated using Monte Carlo method and 10000 samples.
ASBM. The same architecture and optimizer have been used as in toy experiments E.6 but without the scheduler.
DSBM. MLP with [dim + 12, 128, 128, 128, 128, 128, dim] number of hidden neurons, torch. nn SiLU activation functions, residual connections between 2nd/4th and 4th/6th layers and Sinusoidal Positional Embedding has been used.
E.8 CMNIST DETAILS
Working with the MNIST dataset, we use regular train/test split with 60000 images and 10000 images respectively. We RGB color train and test digits of classes "2" and "3" Each sample is resized to 32 32 and normalized by 0.5 mean and 0.5 std. ASBM. The eMNIST setup mainly differs by the architecture used. The generator model is built upon the NCSN+ architecture (Song et al.), following the approach in (Xiao et al.) and (Gushchin et al. 2024b). We use 2 residual and attention blocks, 128 base channels, and (1, 2, 2, 2, feature multiplications per corresponding resolution level. The dimension of the latent vector has been set to 100. Following the best practices of time-dependent neural networks sinusoidal embeddings employed to condition the integer
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time steps, with dimensionality equal to 2x the number of initial channel, resulting in 256- dimensional embedding. The discriminator adopts ResNet-like architecture with 4 resolution levels. The same optimizer with the same parameters as in and SB benchmark|E.. |. have been used except ones that are presented in Table 9 No scheduler has been applied. Additionally, R1 regularization is applied the discriminator with coefficient of 0.02, in line with (Xiao et al.) and (Gushchin et al. 2024b).
X Po t= 00011 t=0.5 0.99 t= 1.0
DSBM The model based on the U-Net architecture [Ronneberger et al., 2015) with attention blocks, 2 resid- blocks per level. attention heads, 128 base channels, 2, 2) feature multiplications per resolution level. Training was held by Adam (Kingma & Ba 2014 op- timizer.
7
7
3
3
E.9 CELEBA DETAILS
3
7
7
Test FID, see Figure is calculated using pytorch-fid package, test CMMD is calculated using unofficial im- plementation in PyTorch. Working with CelabA dataset (Liu et al. (2015b), we use all 84434 male and 118165 fe- male samples (90% train, 10% test of each class). Each sample is resized to 64 x 64 and normalized by 0.5 mean and 0.5 std.
Figure 10: Inverted 7 starting process, i.e.
process in the reciprocal class with marginals pe and p dv7 visualization.
ASBM. As in cMNIST experiments E.8 the generator model is built upon the NCSN++ architecture (Song et al.) but with small parameter changes. The number of initial channels has been lowered to 64, but the number of resolution levels has been increased with the following changes in fea- ture multiplication, which were set to (1, 1, 2, 2, 4). The discriminator also has been upgraded by growing the number of resolution levels up to 6. No other changes were proposed.
DSBM. Following Colored MNIST translation experiment exactly the same neural network and optimizer was used.
SDEdit coupling. DDPM (Ho et al. 2020) was trained on CclebA female train part processed in the same way as for other CelebA experiments. Number of diffusion steps is equal to 1000 with linear Se noise schedule, number of training steps is equal to 1M, UNet (Ronneberger et al. 2015) was used neural network with 78M parameters, EMA used during training with rate 0.9999. The DDPM code was taken from the official DDIM (Song et al. 2021) github repository:
com www.m//rmongrouu/oooo ddimm
The SDEdit method (Meng et al. 2022) for DDPM model was used with 400 steps of noising and 400 steps of denoising. The code for SDEdit method was taken from the official github repository:
http:ihhub.comm///moonnoooooo SDEdit
The Stable Diffusson V1.5 (Rombach et al. 2022 model was taken from the Huggingface (Wolf
et al., 2020) model hub with the tag htwpaaayym/staabeldifffiioooooo
The text prompt used is "A female celebrity from CelebA". The SDEdit method implementation for the SDv1.5 model was taken from the Huggingface library (Wolf et al. 2020), i.e. SiatleDDiiiiiiommmmmmmpiieie
with hyperparameters: strength 0.75, guidance scale 7.5, number of inference steps 50. The output of SDEdit pipeline has been downscaled from 512x 512 size to 64 64x64 size using bicubic interpolation.
E. AFHQ DETAILS
We first pretrain the networks using Bridge Matching for 100000 steps, then run DSBM for 20 iterations with 25000 steps per outer iteration. We follow (Shi al. 2023 and use the same U-Net architecture. The batch size is 4, and the EMA rate is 0.999. We choose 2 = 5, and again we use 100 sampling steps with constant stepsizes.
43

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Published as a conference paper at ICLR 2026
E.11 COMPUTATIONAL RESOURCES
The experiment on CelebA for each of the starting processes takes approximately 5 days and days on Nvidia A00 for DSBM and ASBM, respectively. Experiments with Colored MNIST take less than days of training on an A 100 GPU for ASBM or DSBM, and for each starting process. Illustrative 2D examples and Schrödinger Bridge benchmark experiments take several hours on GPU A100 each for ASBM or DSBM and for each starting process.
44


Reviews:
[{"review_id": "EVOeObxhgv", "reviewer": "Reviewer_hGUD", "summary": "The paper introduces **Iterative Proportional Markovian Fitting (IPMF)**, a unified framework connecting **Iterative Proportional Fitting (IPF)** and **Iterative Moment Fitting (IMF)** in stochastic bridge learning.  \nThe authors show that practical bidirectional IMF procedures implicitly perform IPF-like proportional updates, and formalize this equivalence into IPMF with convergence guarantees in Gaussian settings.  \nExperiments on synthetic and low-resolution image domains demonstrate stable coupling and controllable trade-offs, situating existing DSBM and ASBM methods as special cases.", "questions": "1. Beyond theoretical unification, does IPMF yield measurable training or sampling improvements over IMF or DSBM?  \n2. Is the convergence guarantee valid for non-Gaussian or continuous-time bridges?  \n3. How sensitive is IPMF to the choice of starting coupling (e.g., SDEdit vs. identity)?  \n4. Can IPMF incorporate stochastic regularization (e.g., adversarial or entropy terms) without breaking convergence?  \n5. Are there cases where separate IPF or IMF updates remain preferable in practice?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- Provides a **theoretical unification** between IPF and IMF with clear mathematical derivation.  \n- Clarifies the conceptual foundation of recent bridge-based diffusion methods.  \n- Demonstrates improved stability under diverse initial couplings.", "weaknesses": "*Limited novelty:** IPMF mainly reinterprets existing IMF practices under an IPF perspective; lacks a genuinely new algorithm.  \n- **Narrow empirical scope:** Evaluations are confined to Gaussian and 64×64 image settings without large-scale or continuous-time experiments.  \n- **Unclear practical benefit:** No evidence of faster convergence or lower compute cost compared to IMF or DSBM.\n\n## Minor\n- Heavy notation reduces accessibility.  \n- Missing comparison with recent OT/consistency bridge baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "VGzFKWIHYC", "reviewer": "Reviewer_6f73", "summary": "This paper proposes a new Schrodinger Bridge training method for combining IMF and IPF.", "questions": "See weakness", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "It combines the two processes of IMF and IPF, with alternating method.\nIt theoretically combined the IPF and IMF with thorough analysis and also proposed to apply proposed method to both of DSBM and ASBM.", "weaknesses": "1. Although it proved convergence in gaussian and bounded support, it does not contain any generalization or guarantee on other general distrubutions. \n\n2. To apply this proposed method, it requires alternating process for IMF and IPF. It takes heavier computation burden compared to previous processes. Although the paper proposes faster convergence, it does not contain any comparison between previous methods in terms of computation efficiency and time complexity.\n\n3. The performance heavily rely on initial hyperparameter setting such as initial coupling choice.\n\n4. The most important and critiral issues is the practical usage of proposed method. Although the theoretical analysis and proposed methods is quite new, but the shown experiments are only limited on simple dataset such as toy data , colored MNIST and CelebA. Also the used dataset has relatively low resolution. To prove the practical generalizability of proposed method, the paper must include experiments on more dataset with higher resolution.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "yF3Ej7QxzY", "reviewer": "Reviewer_aDs4", "summary": "Schr\\\"odinger Bridge techniques have been widely used to address unpaired data-to-data translation tasks. This paper focuses on a practical heuristic implementation of the Iterative Markovian Fitting (IMF) procedure that has been proposed in the literature because it improves empirical performance. The central observation in this paper is that the ``bidirectional'' implementations of IMF used in practice (namely Diffusion Schr\\\"odinger Bridge Matching (DSBM) and Adversarial Schr\\\"odinger Bridge Matching (ASBM) implicitly interleave both IMF and Iterative Proportional Fitting (IPF) updates. The authors formalize this approach as IPMF and analyze its convergence properties. Conceptually, IPMF alternates (1) reciprocal/Markovian projections (IMF), with (2) projections that enforce boundary distributions (IPF).\n\nIn the Gaussian case, the paper introduces an optimality matrix $A(q)$ for couplings and shows that any two-variable Gaussian coupling with D-dimensional marginals plan is the entropic Optimal Transport (OT) solution for a suitable bilinear cost $c_A(x_0,x_1)=-x_1^\\top A x_0$ (Theorem 3.1). It then proves exponential convergence (Theorem 3.2;) for various cases : $D=1$ (discrete/continuous, any $\\varepsilon>0$) and $D>1$ (discrete time, sufficiently large $\\varepsilon$)} of IPMF to the SB (1D in continuous/discrete time; $D>1$ in discrete time under a large-$\\varepsilon$ regime). Beyond the Gaussian settings, it establishes weak convergence under boundness assumptions (Theorem 3.3). \n\nFinally, the paper provides numerical experiments: convergence on high-dimensional Gaussians, a 2D toy problem, quantitative results on the SB benchmark (Table~1) , and qualitative unpaired image-to-image translation on Colored MNIST and CelebA. Beyond unification, the paper shows that selecting different initial couplings yields a controllable trade-off between input-output similarity and generative fidelity.", "questions": "Tightness and Practicality of the Large-$\\varepsilon$, Can you quantify how large $\\varepsilon$ needs to be in the discrete $D>1$ proof for typical image resolutions (i.e., dimension $D$), and how this compares to $\\varepsilon$ used in DSBM practice? \n\nContinuous-Time Guarantees. Is there any way you could extendthe Gaussian contraction argument to continuous time beyond $D=1$ (e.g., by controlling the Markovian projection via stability of SDE discretizations)?\n\nNumber of IPMF Rounds and Stopping Criteria. How sensitive are results to the number of  IPMF iterations? Can you think of a reliable practical stopping criterion?  A small experiment (quality/similarity vs.\\ rounds) on the CelebA dataset would be good.\n\nTrade-off via Starting Couplings. The Identity and SDEdit couplings improve similarity, sometimes at the expense of FID. Could you relate the similarity/quality outcome to properties of the initial coupling (e.g. entropy or cost)?\n\nRelation to Rectified Flows. It is correctly suggested that bidirectional IPMF could mitigate error accumulation in rectified flow. Could you provide some results on a small dataset (e.g. CIFAR10) showing that IPMF (with \\epsilon=0) stabilizes Rectified Flow training? I conjecture using \\epsilon decreasing to zero across iterations might even work better.\n\nPositioning. Given the novelty relative to prior bidirectional heuristics (DSBM/ASBM), I think the paper would benefit from a short table in the appendix clarifying the  precise technical advances of IPMF (e.g., convergence rates, new unifying formulation, starting coupling analysis) over these related works.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "Clarity: The paper is well-written and the arguments are presented with excellent clarity. Theoretical concepts are well-motivated and well illustrate; e.g. Figure1 illustrates neatly how IPF and IMF projections combine within the IPMF framework.\n\nOriginality: As clearly acknowledged by the authors, the proposed method is not new but was presented as some heuristic implementation of IMF in the literature. The originality of the paper is to formalize it as an explicit technique alternating IMF and IPF.  This is significant as it connects two separate approaches in the literature, i.e. Sinkhorn/IPF and IMF/flow matching-type ideas, into a unified procedure. In particular, it helps understanding why bidirectionality helps mitigate the \"prior forgetting\" behaviour of the diffusion implementation of  IPF alone) and error accumulation of the naive implementation of IMF alone (which motivated the introduction of the bidirectional scheme). This will be of interest to people working on unpaired data-to-data translation.\n\nQuality: Beyond formalizing the bidirectional method as a rigorous combination of IPF/IMF, the paper presents some interesting theoretical results. In particular, it provides (to the best of my knowledge) the first theoretical analysis of the bidirectional variant used in practice. While limited in scope, the analysis of the Gaussian case is neat and rigorous. The empirical section is fairly complete, including high-dimensional Gaussians, a 2D toy example, the SB benchmark, and real images (Colored MNIST, CelebA) . It also includes convergence diagnostics (KL forward/reverse etc.). \nThe literature in this domain is plagued with over-the-top claims, this paper does not make any such claim and remains very factual which I really appreciate.\n\nSignificance and Practical Impact: The experiments convincingly illustrate that  different couplings induce a principled, tunable trade-off between input-output similarity and generative fidelity. This is useful for practitioners working on unpaired translation.", "weaknesses": "Theory results: They remain limited. Exponential convergence is only proved in the Gaussian case for (a) $D=1$ continuous/discrete settings and (b) $D>1$ in discrete time under a fairly restrictive large-$\\varepsilon$ condition. For applications, we care about $D$ large and non-Gaussian distributions. In this case, exponential convergence is only conjectured.  The authors should try to clarify the practical implications of the large-$\\varepsilon$ assumption. It  would be helpful to quantify how large $\\varepsilon$ must be for the proof to hold in typical image dimensions, and whether this aligns with the values used in practical DSBM implementations.\n\nContinuous-Time IPMF: The continuous-time version of IPMF is given in (22)-(23) but theoretical guarantees for that case are not provided beyond $D=1$ for the Gaussian case. The authors should discuss the specific challenges in extending the contraction arguments to the continuous-time and high-dimensional setting.\n\nBenchmarking & Ablations: On the SB benchmark, performance is sometimes on par with prior matching solvers and sometimes worse (e.g., DSBM variants at higher $D$/$\\varepsilon$ show large $\\mathrm{cBW}^2_2$ errors). The paper argues that different initializations converge to similar outcomes within each solver, but stronger head-to-head comparisons would make the empirical case more compelling.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "MXoO280vNg", "reviewer": "Reviewer_N3B2", "summary": "This paper shows that a heuristic bidirectional modification of the Iterative Markovian Fitting (IMF) procedure such as Diffusion Schrodinger Bridge Matching or Adversarial Schrodinger Bridge Matching is also a valid algorithm for solving the Schrodinger Bridge Problem (SBP). The authors prove that this heuristic modification of IMF, dubbed Iterative Proportional Markovian Fitting (IPMF), converges exponentially to the solution when the marginals are Gaussian. IPMF is also shown to converge to the solution under a weaker condition, when the marginals have bounded supports. In contrast to IPF or IMF, IPMF can be initialized with any starting process. The authors show the benefits of flexible initialization with experiments on Gaussians, SB benchmarks, unpaired image-to-image translation, etc.", "questions": "- **[Q1] In Table 2, can the authors provide FIDs for initial processes as well?** I believe this FID is necessary in order to judge whether there are performance gains after running IPMF on the given initial process such as SDEdit or Identity.\n\n- **[Q2] In Table 2, can the authors provide LPIPS between $x_0$ and $\\widehat{x}_1$?**\n\n- **[Q3] Can the authors provide results with larger $\\epsilon$ when starting with the identity process?** I am curious whether this improves FID by increasing diversity in the outputs.\n\n- **[Q4] Do observations made on CelebA 64x64 scale to higher resolution images?**", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **[S1] This paper is significant in the aspect that it bridges a gap between theory and practice.** Specifically, the paper provides several proofs regarding the convergence of a heuristic modification of the IMF procedure, while proofs in previous works [1,2] show the convergence of the original IMF procedure.\n\n- **[S2] This paper is original in the aspect that it provides a novel insight into better training of SBs.** The authors also show that unlike IMF or IPF, IPMF admits any initial starting process. Hence, one may potentially achieve faster training of SBs via IPMF by using a well-designed starting process.\n\n[1] Diffusion Schrödinger Bridge Matching\n\n[2] Adversarial Schrödinger Bridge Matching", "weaknesses": "- **[W1] Experimental results only weakly support the benefits of using arbitrary starting processes.** Experiments in Section 4 do not really show the strength of using arbitrary starting processes in terms of both scale and performance. In terms of scale, the only non-trivial task is male to female on CelebA-64. In terms of performance, non-IMF initializations such as SD SDEdit, DDPM SDEdit, or Identity all suffer from FID degradation at the cost of smaller MSE between $x_0$ and $\\widehat{x}_1$. Previous works on SB such as DSBM or ASBM demonstrate consistent performance improvements on more difficult data with resolution $\\geq 128$, so it would be nice to see results on similar data in this paper as well.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)