Adversarial Schrödinger Bridge Matching
Authors: Nikita Gushchin
Daniil Selikhanovych
Sergei Kholkin
Evgeny Burnaev
Alexander Korotin
Venue: NeurIPS 2024 poster (2024)
Keywords: Schrödinger Bridge
Bridge Matching
GANs
Optimal Transport
Entropy regularized OT
Unpaired Learning
Primary area: generative_models

Abstract: The Schrödinger Bridge (SB) problem offers a powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem is the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-time stochastic processes. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations. To address this limitation, we propose a novel Discrete-time IMF (D-IMF) procedure in which learning of stochastic processes is replaced by learning just a few transition probabilities in discrete time. Its great advantage is that in practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), an already well-established adversarial generative modeling technique. We show that our D-IMF procedure can provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds.

Adversarial Schrödinger Bridge Matching
Nikita Gushchin"
Daniil Selikhanovych Skoltech Moscow, Russia sel seiihaaorryyndannigmmaa.... com
Skoltech Moscow, Russia
n. www.ghhhinnskkltehhhh....
Sergei Kholkin" Skoltech
Evgeny Burnaev Skoltech', AIRI Moscow, Russia
Alexander Korotin
Skoltech", AIRI Moscow, Russia a. korotiinskolteeh.. ru
Moscow, Russia
s. kkookkin@akklteehh.mu
e. wewnnaeev@slootec..r...
Abstract
The Schrödinger Bridge (SB) problem offers powerful framework for combining optimal transport and diffusion models. A promising recent approach to solve the SB problem the Iterative Markovian Fitting (IMF) procedure, which alternates between Markovian and reciprocal projections of continuous-ting stochastic pro- cesses. However, the model built by the IMF procedure has a long inference time due to using many steps of numerical solvers for stochastic differential equations. address this limitation, we propose novel Discrete-time IMF (D-IMF) proce- dure in which learning of stochastic processes is replaced by learning just few transition probabilities in discrete time. Its great advantage that practice it can be naturally implemented using the Denoising Diffusion GAN (DD-GAN), already well-established adversarial generative modeling technique. We show that our D-IMF procedure provide the same quality of unpaired domain translation as the IMF, using only several generation steps instead of hundreds. We provide chanovychhASS!!
the code at //ttts:://giiiiiii wwt.bi...ommaaannieaeaaaa
Input
Result
D-IMF (ours)
NFE=4
NFE=100
IMF
-80 more steps
Figure 1: Our D-IMF approach performs unpaired image-to-iman translation just few steps,
achieving results comparable to the hundred-step IMF [7]. Celeba 33], male-xfemale (128 x 128).
"Equal contribution
1 Skolkovo Institute of Science Technology Artificial Intelligence Research Institute
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Introduction
Recent generative models based on the Flow Matching [2 and Rectified Flows 30 show great potential as successor of classical denoising diffusion models such as DDPM [5].
Both these approaches consider the same problem of learning an Ordinary Differential Equation (ODE) that interpolates one given distribution the other one, e.g., noise data. Thanks the close connection to the theory of Optimal Transport (OT) problem [52), Flow Matching and Rectified Flows approaches typically have faster inference compared to classical diffusion models [32 39]. Also, it was shown that they can outperform diffusion models on the high-resolution text-to-image synthesis: they even
lie in the foundation of the recent Stable Diffusson 3 model (8).
The extension of Flow Matching and Rectified Flow approaches to the SDE are Bridge Matching
(Markovian projection) and Iterative Markovian fitting (IMF) procedures 36 47
35 respectively.
They also have a close connection with the OT theory. Specifically, it is known 47
35 that IMF
converges to the solution of the dynamic formulation of entropic optimal transport (EOT), also known
as the Schrödinger Bridge (SB). However, learning continuous-time SDEs in IMF is non-trivial and, unfortunately, leads to long inference due to the necessity use many steps of numerical solvers.
Contributions. This paper addresses the above-mentioned limitation of the existing Iterative Marko- vian Fitting (IMF) framework by introducing a novel approach to learn the Schrödinger Bridge.
1. Theory I. We introduce a Discrete Iterative Markovian Fitting (D-IMF) procedure 3.2 3.3).
which innovatively applies discrete Markovian projection solve the Schrödinger Bridge problem without relying on Stochastic Differential Equations. This approach significantly simplifies the inference process, enabling to be accomplished (theoretically) in just a few evaluation steps.
2. Theory II. We derive closed-form update formulas for the D-IMF procedure when dealing with high-dimensional Gaussian distributions. This advancement permits a detailed empirical analysis of our method's convergence rate and enhances its theoretical foundation ($3.4)
3. Practice. For general data distributions available by samples, propose algorithm (ASBM) to implement the discrete Markovian projection and our D-IMF procedure in practice ($4.2) Our
algorithm is based on adversarial learning and Denoising Diffusion GAN [53]. Our learned SB
model uses just 4 evaluation steps for inference ($3.5) instead of hundreds of the basic IMF 47
Notations.
the paper, we simultaneously work with the continuous stochastic processes and discrete stochastic processes in the D-dimensional Euclidean space RD. We denote by P(C((O, 1)), R P)) the set of continuous stochastic processes with time t E [o, 1). i.e., the set of distributions on continuous trajectories ʃ [0, 1] + R.P. We use dW, to denote the differential of the standard Wiener process.
To establish link between continuous and discrete stochastic processes, we fix N > 1 intermediate time moments 0 = to < ( < ... < Lx < (N+11 = 1 together with 4g = 0 and NN+1 = 1. We con- sider discrete stochastic processes with those time-moments as the elements of the set (TR (N++)) of probability distributions on (N++)) Among such discrete processes, we are specifically C P(N of absolutely continuous distributions on
interested in subset (I (N+2) which have a finite second moment and entropy. For any such E (2 (RDX(N+22) :tt Ttt: E (D++)
we write q( "t, to denote its density at point (ro,
For continuous process T. we denote by p" E PPRRDx finite-dimensional projection of T to time moments venience we also use the notation Fin (Ii)..... &iN to denote the vector of all intermediate-time variables. In what follows, KL is a short notation for the Kullback-Leibbe divergence.
the discrete process which is the to < 4 <... < IN < (N411 = 1. For con-
2 Background
We start with recalling the Bridge Matching and Iterative Propotional Fitting procedures developed for continuous-time stochastic processes ($2.I). Next, we discuss the Schrödinger Bridge problem, the solution which is the unique fixed point of Iterative Markovian Fitting procedure ($2.)
2.1 Bridge Matching and Iterative Markovian Fitting Procedures
Modern diffusion and flow generative modeling are mainly about the construction of a model that interpolates one probability distribution Po E P2... to some another probability distribution
Pi E Pa (R. One of the general approaches for this task is the Bridge Matching 22 31 3.
Reciprocal processes. The Bridge Matching procedure is applied to the processes, which are represented as a mixture of Brownian Bridges. Consider the Wiener process W with the volatility e which start at Po. i.c., the process given by the SDE: dz, = VedWe, 20 Po- Let We denote
(mp.0)
2

---

the stochastic process W conditioned on values Fo, 21 at times t = 0, 1, respectively. This process W called the Brownian Bridge [17 Chapter 9]. For some g 21) E P2 (R Dx2) with
g(ro)) (20) and q(æ1) Pi (21) the process T, def J W (no... dq( x1) is called the mixture of Brownian Bridges. Following [47], say that mixtures of Brownian Bridges form reciprocal class of processes (for the Brownian Bridge). For brevity, we call these processes just reciprocal processes.
Bridge matching 229 31 The goal of Bridge Matching (with the Brownian Bridge) is construct continuous-ting Markovian process M from po to p1 in the form of SDE: dx, = v( + V VedW,- This is achieved by using the Markovian projection of reciprocal process T f W mo.ar (q( T1), which aims to find the Markovian process M which is the most similar to T in the sense of KL:
Mr
def arg min KL MM), MEM
where M C P(C 1), RD) is the set of all Markovian processes. For the Brownian Bridge We [20,3, is known 47 that the SDE and the drift være, t) of proj (Tg) is given by:
/ 21 2t TTo (xx (x112))d23
dr = vlz,, t)dt + VedW,, v( 1)
where (2 (xt) the conditional distribution of the stochastic process T, at time moments and 1. The process pr has the same time marginal distributions yd (arz) as the original Brownian bridge mixture TG. However, the joint distribution pra (xo, 21) of T, and the joint distribution prmis((T)) (x) z1) of its projection p (Tg) do not coincide in the general case [6, see Figure
Input: reciprocal process T
Output: markovian process Ehd (r))
Ta
Winn P p'((x, xo)c
TM dxp (xx,, t)dt + VedW,
Xo
p' (x))
X
p (x,
P
P
p (x,
Xo
Xo
Figure 2: Markovian projection of reciprocal stochastic process Tq.
The Iterative Markovian Fitting procedure introduces a second type of projection of continuous- time stochastic processes called the Reciprocal projection. For process T, it is is defined by proj (T) = = W 1zo. dp" (ro, #1), see illustrative Figure
Iterative Markovian Fitting 47 35
Input:
distribution pi(,,, x,) of process T at times t=0,1
reciprocal process with distribution p(Xar x,) at times t=
Output:
The Brownian Bridge W is
pT(x: (xa) P
(x,(xx))
process conditioned Xo;
P
P
(x,
X1
(x.18))
Xo
N [xx[(1 t)x bq 1(1 t)clo)
Figure 3: Reciprocal projection of stochastic process T, i.e., p (T) f We 2, dp (æo, #1).
The process p (T) is called a projection, since: proj (T) arg min KL (T] REE
where C P(C((0, R))
))P)) the set of all reciprocal processes. The Iterative Markovian Fitting procedure is alternation between Markovian and Reciprocal projections:
2
(T
2
proj 2T
It is known that the procedure converges the unique stochastic process 7 which is known solution to the Schrödinger Bridge (SB) problem between Po and Pt. Furthermore, the SB 7 the only process starting at po and ending at p1 that is both Markovian and reciprocal [25).
3

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2.2 Schrödinger Bridge (SB) Problem
Schrödinger Bridge problem. The Schrödinger Bridge problem 44 was proposed in 1931/1932 by Erwin Schrödinger. For the Wiener prior W Schrödinger Bridge problem between two probability distributions po ac (IR D) and p1 (2, is to minimize the following objective:
min TEF(PO-P1)
KL (T])W>),
(I)
where F(po. P1) C ((C 1)), IR 2) ) is the subset of stochastic processes which starts at distribution Po (at the time = 0) and end at p1 (at t = 1). The Scrhödinger Bridge has a unique solution, which is diffusion process 7 described by the SDE: dX, v" (Xt, t)dt + vedW, 155. The process T
is called the Schrödinger Bridge and V² R x [o, 1] + RD is called the optimal drift.
From the practical point of view, the solution to the SB problem T tends to preserve the Euclidean distance between start point 2n and endpoint 21, The equivalent form of SB problem, the static Schrödinger Bridge problem, explains this property more clearly.
Static Schrödinger Bridge problem. One may decompose KIL(TIIW>>) as [1 Appendix C}:
KL(TIJW>>) (I. (zo, 31 x1)llp" (Io, =1))
[I [ ddp 21),
i... KL divergence between 7 and W is sum of two terms: the 1st represents the similarity of the processes' joint marginal distributions at start and finish times f = 0, 1, while the 2nd term represents the average similarity of conditional processes T I20..: and W =. In [25] Proposition 2.3], the authors show that if T solves (T) then 12g.21 WT
1 Hence, one may optimize (T) over for for every 2o, 21, L.e., over reciprocal processes T:
which T Mm = W [20,21
min
(L(p"(20,22)) (10, (1))
min KL(g(zo, =1)((=" (0o, (x))))")))))""")x) (3)
gEll(po.P1)
where [[pa P1) C P2 (P Dx2) is the set of joint probability distributions with marginal distribu- tions po and P1. Thus, the initial Schrödinger Bridge problem can be solved by optimizing only over reciprocal process's ' joint distribution g z1) at t = 0, 1. This problem is called the Static
Schrödinger Bridge problem. In turn, the problem can rewritten in the following way [2 Eq. 7]:
| - x1|1² dq(z0. z1) e Entropy (4) + C, (4)
min <LL(ellp" (zo, = min (ETI(p..P1)
qEII(po.PI)
2
ie., as finding a joint distribution q( Ø1) which tries to minimize the Euclidian distance ||>
between zo and Ij (preserve similarity between zo and I1), but with the addition of entropy regularizer e Entropy(g) with the coefficient e. Thus, the coefficient >> 0, which is the same for all problems considered above, regulates the stochastic diversity of samples from (rg.. The last problem 44 is also known as the entropic optimal transport (EOT) problem
38 25
3 Adversarial Schrödinger Bridge Matching (ASBM)
The IMF framework [35] 47 works with continuous time stochastic processes: it is built on the
well-celebrated result that the only process which is both Markovian and reciprocal is the Schrödinger bridge 7 25]. We derive an analogous thcoretical result but for processes discrete time. We provide proofs for all the theorems and propositions in Appendix B
In 33.1 we give preliminaries on discrete processes with Markovian and reciprocal properties. In H3.2] we present the main theorem of our paper. which the foundation of our Discrete-time lteratime Markovian Fitting (D-IMF) framework. In $3.3 we describe D-IMF procedure itself and prove that allows us to solve the Schrödinger Bridge problem. In $3.4 we provide an analysis of applying our D-IMF for solving the Schrödinger Bridge between Gaussian distributions. In $3.5
we present the practical implementation of our D-IMF procedure using adversarial learning.
3.1 Discrete Markovian and reciprocal stochastic processes
Discrete reciprocal processes. We define the discrete reciprocal processes similarly to the continuous case by considering the finite-time projection of the Brownian bridge W Wirp.ar which is given by:
N
p (ot Tay (oo, 2I
II n=1
p W (xe, pet, 211; (5)

---

w (st, (stt, Fr, 11) = (Irr, Fr in tn-1 (x) Ita 1),
(tn bn- bn-1)(1 En)
(6)
hn-1
1--1
This joint distribution y : (ro, 21) defines discrete stochastic process, which we call discrete Brownian bridge. In turn, we say that distribution aa DX(XX)))) is mixture of discrete Brownian bridges if satisfies
g Rt Fex x1)= =p" (It.....,; .ty [o )g(z0))(000) 21),
where g3 mi) denotes its joint marginal distribution of q times 0, 1. That is, its "inner" part at times f1,...,,y by is the discrete Brownian Bridge. We denote the set of all such mixtures as R(N) C P2 (BDx and call them discrete reciprocal processes.
Discrete Markovian processes. We say that a discrete process q E P2. ( (IR2((N+)) Markovian its density can be represented in the following form (recall that ln = 0, 1N+1 = 1):
N+1 II n=1
x1) g 2t, Pt........ Ttyy = ((xo)
( (=..)))
(7)
We denote the of all such discrete Markovian processes as M(N) C a2 (DDx()+))).
3.2 Main Theorem
Theorem 3.1 (Discrete Markovian and reciprocal process is the solution of static SB). Consider any discrete process E (RR which is simultaneously reciprocal and markovian, i.e. q E RRN) and q E M(N) and has marginals g(ao)) po(ro) and q(x1) pi (x1):
N+1
g(Fo, Tets ttys 01) p= W' (Et Ten To. x1 (1)4(F0, g(
g (tn-.)> n=1
II
Then q( Sty..... Tex , 21) = DT" (Fo, Tt....... TtN. 11), i.e., it the finite coriteddimesiinaa projection of the Schrödinger Bridge T to the considered times. Moreover, joint marginal q( x1) times
1 0, the solution to the static SB problem between po and pt, i.e. g(200 x1) p" (x0,
Thus, to solve the static SB problem, it is enough to find Markovian mixture of discrete Brownian bridges. To so, we propose the Discrete-time ticrete-time Iterative Markovian Fitting (D-IMF) procedure.
3.3 Discrete-tima Iterative Markovian Fitting (D-IMF)) procedure
Similar to the IMF procedure, our proposed Discrete-ting time IMF is based on two alternating projections of discrete stochastic processes: reciprocal and Markovian. We start with the reciprocal projection.
Definition 3.2 (Discrete Reciprocal Projection). Assume that E P2 Dx is discrete stochastic process. Then the reciprocal projection proj (q) is discrete stochastic process with the joint distribution given by:
[proj (4) (ro, :ty.... Tty: : v1) W ( Ioo 2n (1
This projection takes the joint distribution of start and end points g( x. and inserts the Brownian Bridge for intermediate time moments, see Figure 4 The prop. below justifies the projection's name.
Input: distribution q(x,, x,) of discrete process at times t=0,1
Output: discrete reciprocal process with distribution q(x,r x,) at times t=0,1
Discrete Brownian Bridge W'
(Xtr Xty Xty 1x0, X1).
X
1xx P ((x,1xa)
X
q(x, P q(x,(18))
X
X1
P
X 4
X
X
Xo
X
M(xa,11 - t.)xo (ta)s,, to(1- t.)elb)
Figure 4: Reciprocal projection of discrete stochastic process g, i.e.,
r r(zo, Lt, Tex, = W (Iz, .... DeN Jo. L1 )q(2), z1).
5

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Proposition 3.3 (Discrete Reciprocal projection minimizes KL divergence with reciprocal processes). Under mild assumptions, the reciprocal projection projæ of stochastic discrete process q P2. (RDXNN22))) is the unique solution for the following optimization problem:
projæ (q) = arg min KL (gllr).
(9)
(ER(N)
Similarly the discrete reciprocal projection, we introduce discrete Markovian projection.
Definition 3.4 (Discrete Markovian Projection). Assume that E ( DR+ is discrete
stochastic process. The Markovian projection of q is a discrete stochastic process projm( (R (RDX(N+2)) whose joint distribution given by:
N+1
[proj_[(4] (Fo, St, .t.,..., TAN 21) ((r)) II g 12)
(10)
R=
Despite is possible to use any discrete stochastic process g as an input to a discrete markovian projection, in the rest of the paper only discrete reciprocal processes are considered as an input. For such cases, we provide visualization of the markovian projection in Figure
Input: reciprocal process
Output: markovian process m
9(x; (xo)
X
m(x, (xo)
P
P
P
P mxx
a(x,1x))
Xo
Xo
Figure 5: Markovian projection of a reciprocal discrete stochastic process q.
As with the reciprocal projection, our following proposition justifies the name of the projection. Proposition 3.5 (Discrete Markovian projection minimizes KL divergence with Markovian processes). Under mild assumptions, the Markovian projection proj proj (q) of a stochastic discrete process q P2.c DX(N+2)) is a unique solution to the following optimization problem:
proj M()) arg min KL (q|(m)
(11)
mEM(N)
Now we are ready to define our D-IMF procedure. For two given distributions po P2 (R
and pi (RR at times / = 0 and 1, respectively, it starts with any discrete Brownian mixture p w (Itz....,, Bty (r (1lr/1ggg (x(2)))))))))))))))))) =1). where g( 21) E II(po. P1) P2,00 Dx) Then,
constructs the following sequence of discrete stochastic processes:
21+1 = (q")
q2 ( (12
(12)
Theorem 3.6 (D-IMF procedure converges to the the Schrödinger Bridge). Under mild assumptions,
the sequence o' constructed by our D-IMF procedure converges in KL to p" In particular, o' (xo x1)
convergence the solution p" (Io, of the static SB. Namely, we have
lim KL (g'11²"") [-yo
=0, 0,
and
lim KL 21)=l"= 21). = 0.
[-x
3.4 Closed form Updates of D-IMF for Gaussian Distributions
In this section, we show that our D-IMF updates (22 can be derived in the closed form for the
Gaussian case. Let Po (Nooloo Eo) and p1 ~N Σn) be Gaussians. Consider any initial
discrete Gaussian process g E P2.00 (R (IDxx())) that has joint distribution g(zo, x1) II(po, PI):
def
TD
101
def
HO 11
Σ
DT cov
Exar Σ
(1)
def N p0n, E)
(13)
where Σ 2Dx22 is positive definite and symmetric and Egov is the covariance of zo and 2 In this case, the result of updates (12) is always a discrete Gaussian processes with specific parameters. To show this, we introduce two auxiliary matrices U E Dx2D and K RNDNND:
TD
Tp
(1 tp) 41 t2) ... (1 - (T) 61 Tp)
11
Tn
ta
t2(1 In
tz(1 - IN
der
Kdder
(1
ID Lx
41(1 - IND tz(1 - ID) In - ID)
Here Ip is identity matrix with the shape D D. Below present updates for both projections.

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Theorem 3.7 (Reciprocal projection of a process whose joint marginal distribution is Gaussian).
Assume that E P2. { (2)) has Gaussian joint distribution q(Io, x1) given by 13 Then
[preja q. (Ime 20, 21) ) = NV(
Tin F01
(
(UHo1) Poi
En), ,En),
def (cK + UEUT ( +
En
(UE)T
UE
(14)
Σ
Theorem 3.8 (Markovian projection of discrete Gaussian process). Assume that q Pa Prac ( (R)) is discrete Gaussian process with g(3o,, v1) given by (13) and the density
(aan (O, xi) -N(
Fin zo1
(Mm (oo
ER), Pin ( pty):
where Um and En are some parameters of Then its Markovian projection is given by:
N+1
To. 21) = q(z0)
II
g 1x ((tt,.eea-.) = -N(xx, Pe. (2tt- 1), Eee.:
==1
Pha (#ta-1) )= Me. + (ER)emitn (ER)) - -1 Men-1)
Et, = (ER)tnito - (En)endn- (aaeaa(((R)))))) (DR) ((aR)) )²
In turn, the joint distribution (proj. x1) is given by
N+1 (E) ((ER)ende) (
[proj Mal 22) = -N( =N((20)
20 201
1m)
En
Ex
= II 1=
(1)
(Eo1)
)
Σ
Here (ER),, is the submatrix of ΣR denoting the covariance of Ft, and Bi while Eo and En are covariance matrices of To and Ii, respectively.
Thus, if we start D-IMF from some discrete process qd with marginals (00) po()) q (xr)) P1 (1x) and Gaussian ( 21). then each iteration of our D-IMF procedure o' will be discrete Gaussian process with the same marginals and eventually will converge to g". In 1 we use our derived closed-form perform an experimental analysis of D-IMF's convergence depending on the number of intermediate time moments N and the value of coefficient
3.5
Practical Implemetation of D-IMF: ASBM Algorithm
To implement our D-IMF procedure in practice, one should choose the process gd and implement both discrete Markovian and reciprocal projections. Note that one is usually not interested in the processes' density but only needs the ability to sample endpoints 21 (or trajectories To. Tt...... Fiy x1) given starting point 2y (= ). Thus, to solve SB between (o(2)) and p1 P1(21) one should choose q" to have start and end marginals qº (xo) = (o())) and (x11 = p(21) accessible by samples.
Implementation of the discrete reciprocal projection. The reciprocal projection ( of given discrete process g(Zig Tin, x1) ) is easy if one can sample from g( 21). To sample from proj (1) it enough to sample first pair (1), ~ q(z0, 21) and then sample intermediate points Jt; ................................ Bay from the Brownian bridge p (.r,,.... Zt [ro, 21). This can be straightforwardly done using the formula 5 where the involved distributions (6) are simple Gaussians which are easy to sample from.
Implementation of the discrete Markovian projection via DD-GAN. To find the Markovian projection (TOp of reciprocal process g E (N), one just needs estimate the transition prob- abilities between sequential time moments, i.e., the set {g(#t, [xt,. N+ and use the starting marginal [proj (2o)) ( = (o)) The natural way to find transition probabilities is to set to parametrize all these distributions as {go (It,,, Ite, N and solve
N+1
min
E Dudv Duav (g(2,,, ( )) 1ggo (F,,, (2).-)));
(15)
where Duv is some distance divergence between probability distributions. In this case, minimum of such loss achieved when go (I [ ) = g( (2....) for each n E 21, N + 1}.
We note that related setting is considered in the Denoising Diffusion GANs (DD-GAN), see 53 Eq. 4]. The difference is in the nature of q: there q comes from the standard noising diffusion process, while in our case it is given reciprocal process. Overall, the authors show that problems like [15)
7

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can be efficiently approached via line-conditioned GANs. Therefore, we naturally pick DD-GAN approach as the backbone to learn our discrete Markovian projection and use their best practices.
In short, following DD-GAN, we parameterize qo (Tt,, [1, via a time-conditioned generator network (1 2, 1n- As in DD-GAN, we the non-saturating GAN loss [10] Datv, which optimizes softened reverse KL-divergence [46.. To optimize this loss, an additional conditional discriminator network ( IIt -n-1) ) needed. We do not recall technical details here as they are the same as in DD-GAN. For further details on DD-GAN learning, we refer to Appendix [.I
Note that after learning (10 Ist,, n) the sampling assumes to take sample from go = p(zo) and then sample from (1,, [ 1 Hence it is guaranteed that go = p(zo), ) but there may be an approximation error in estimating qı (21) = p(21). This is due to the asymmetry of the definition of Markovian projection, i.e., it can be written in two equivalent ways:
N+1
N+1
[ (zo, Iit ... TaN x1) g(20)
II n=1
g(t,,, (xtn,)) = q(x1) =
( Itt,) n=1
II
Analogously to the implementation of IMF 47 Algorithm 1], we address this asymmetry in our
D-IMF by alternatively learning Markovian projection in forward and reverse directions. To learn Markovian projection in the reverse direction, we just need to use starting marginal (r)) (1(1), ) parametrize (I I2t,) 1 and solve:
N+1
min
Σ Byteer Dudr (g( (s )l My (Ft, (k2,)))
(16)
10
In this case ((11) = Pi (21) is guaranteed, while ((x)) Rs (2))..
Implementation of the D-IMF procedure (ASBM algorithm). We start with initialization of & by the reciprocal process. Depending on the setup we use initialization with the independent coupling,
i.. q (2(2,, po(zo)P1 (I1) or a minibatch OT coupling O [3], see Appendix D.3 for details.
We follow the best practices of IMF 47 and in the Markovian projection steps, we alternately learn models in the direction Po- P1 and in the reverse direction p1 Po by using functionals (15) and (66) respectively to avoid the accumulation of errors due to the asymmetry
the definition of the Markovian projection. For details, see Appendix D.2 At the reciprocal projection steps, we use the model (0(20, Zin, x1) or gn (zg, Tin, x1 learned to approximate q² 2 to sample pair (ro, X1) and then sample intermediate points from Brownian bridge. We use the term outer iteration (K) for sequence of two reciprocal projections and two Markovian projections in different directions.
3.6 Relation to Prior Works
There exists a variety of algorithms for learning SB based on different underlying principles: dual form entropic optimal transport algorithms [5] 34 (24
12 45). iterative proportional fitting (IPF) algorithms [51 2). bridge matching [49 29 and iterative Markovian fitting (IMF) algorithms
47 35 28 20 adversarial algorithms 211. etc. We refer to [ for a benchmark and to 124 Table
1] for a quick survey of many of them. In turn, in our paper, we specifically focus on the advancement of IMF-type algorithms (47 35 as it they not only theoretically well-grounded but also closely connected to the rectified flow approach (30) which works well in large-scale generative modeling Below discuss the relation of our contributions (1 the prior works in IMF (47 B5].
32 54.
Theory As we detailed in 2 basic IMF operates with stochastic processes in continuous time and iteratively performs Markovian and reciprocal projections. Our D-IMF procedure ($3 does the same but in the discrete time, so it might deceptively seem like our D-IMF is just an approximation of IMF. However, this is misleading viewpoint. Indeed, the Markovian projection in the discrete time, in general, does not match with the continuous time Markovian projection. Still our D-IMF procedure provably converges to SB. Furthermore, D-IMF procedure can theoretically work with just one intermediate time step (when N = 1). Overall, its convergence rate varies depending on the number of intermediate points, see $4.1 Naturally, we conjecture that in the limit N + oo (when the time steps f,....,ty ty densely fill [o, 1l) our D-IMF behaves the same as IMF since the discrete and continuous Markovian projections start be close, see discussion in 47 Appendix E].
Theory II. In 3.4 we derive the closed-form expression for our D-IMF updates in the Gaussian case. For the continuous IMF, there exists an analogous result [35 16.1 However, unlike our result,
8

---

D-IMF iteration (D 16, E=1.0)
KL D-IMF iteration (D=16, N 10)
KLiq/Do,x x.lxxlllo" (Xo. x)))
10
KLLq KLtq'xx..X k:lllo "Illo"tw Ke)xx))
10
N=1
c-1
10-2
N
10-2
< 10
N 10
threshold
10-1
10-
N 100
threshold
10-3
10-9
0
60
80
100
20
40
a0
100
D-IMF iterations
D-IMF iterations (1)
(a) Dependence on the number of time steps N. (b) Dependence on the variance e of the prior process.
Figure 6: Dependence of convergence of our D-IMF procedure on N and c.
that one is not explicit in the sence that it requires solving the matrix-valued ODE [35 Eq. 39] to get the actual projection. The analytical solution is known only when D 1, i.e., 1-dimensional case, see also 47 Appendix DJ. In contrast, our Gaussian D-IMF updates work in any dimension D.
Practice. Default continuous-time IMF 47 35 in practice is naturally implemented via the Bridge Matching approach which learns an SDE. our case, at each D-IMF step we learn several transi- tional probabilities and do this via also well-established adversarial techniques. In this sense, our practical implementation differs each approach is based on its own backbone bridge matching vs. adversarial learning- and naturally inherits the benefits/drawbacks of the respective backbone. They are fairly well stated in the discussion of the generative learning trilemma in [53].
4 Experiments
We evaluate our adversarial SB matching (ASBM) algorithm, which implements our D-IMF procedure on setups with Gaussian distributions ($4.1) for which have closed form update formulas (:3.4) and real image data distributions ($4.2). We additionally provide results for an illustrative 2D example in Appendix C.I] results for the Colored MNIST dataset in Appendix C33 and results on the standard SB benchmark in Appendix C.2 The code for our algorithm and all experiments with it is written in Pytorch, is available in the supplementary materials, and will be made public. We provide all the technical details in Appendix(D
4.1 Caussian-to-Gausse
Schrödinger Bridge
We analyze the convergence of our D-IMF procedure depending on the number of intermediate time steps N 2 (we use in = n/N + 1) and the value e > in the Gaussian case. In this case, the static SB solution p" (zo, 21) is analytically known, see, c.g., [8]. This provides us an opportunity to analyse how fast KL (g' (z z1 1pp" (1)) decreases when - oc.
We conduct experiments by using our analytical formulas for D-IMF from $3.4 We follow setup from 12 and consider Schrödinger Bridge problem with the dimensionality for centered Gaussians pa (O, En) and p1 (0, E.). To construct En and Σ,, sample their eigenvectors from the uniform distribution on the unit sphere and sample their eigenvalues from the log uniform distribution on [- log 2, log 2). We use the same po and P1 for all experiments.
16 and e E (1,3, 10)
We start our D-IMF procedure from the reciprocal process with qd (x)0
po (=())) (I1), i.c. from the independent joint distribution at times t = 0, We present the convergence plots in Figures (6a and 6b In both plots, we use 10-10 as a threshold corresponding to the exact matching of distributions to prevent numerical instabilities. We see that our D- procedure empirically shows the exponential rate of convergence in all the cases. As we can see from Figure baa the convergence speed dependence on N quickly saturates. Thus, even several time moments, e.g., N = 5, provide quick convergence speed. From Figure 6bb we clearly see that the convergence speed is highly influenced by the chosen value of the parameter e. For instance, the transition from e = 1 to = 10 requires ten times more D-IMF iterations. Thus, this hyperparameter may be important practice.
4.2 Unpaired Image-to-image Translation
To test our approach on real data, we consider the unpaired image-to toage-to-image translation setup of learning male -> female faces of Celeba dataset [33]. We use 10% of male and female images as the test set for evaluation. We train our ASBM algorithm based on the D-IMF procedure with c = 1 and 6 = 10. Following the best practices of DD-GAN (53), we use N = 3, intermediate times and K 5 outer iterations of D-IME. We provide qualitative results and intermediate time moments, our algorithm requires only 4 number of function evaluations (NFE) at the inference stage.
4 = 7, the FID metric [4] on the test set in Figures 7h and Te Since we use
2.

---

(a) x (b) ASBM (ours), 1 (lower diversity)
(c) DSBM 47 1 (lower diversity)
FID 16.08, NEE
4.
FID 37.8. NFE
100.
(d) x ~ Po (c) ASBM (ours), 6 = 10 (higher diversity) (f) DSBM 47 E = 10 (higher diversity) FID = 17.44, NFE = 4.
FID 89.19, NEE = 100.
Figure 7: Results Celeba, male -feemae translation learned with ASBM (ours), and DSBM learned on Celeba dataset with 128 resolution size for E {1, 103.
We focus comparison the DSBM algorithm based on the IMF- MMF-prcceurre 47 since it is closely related to our method. We train DSBM following the authors 47 and use NFE 100. As well as for ASBM, we use 5 outer iterations of IMF, corresponding to the same number of reciprocal and Markovian projections, but for continuous processes. We use approximately the same number of parameters of neural networks used for models in Markovian projections for ASBM and DSBM (see Appendix D.3) For other details, see Appendix D.4] We present results for DSBM in Figure 7c and Figure 7 Our algorithm provides better results while using only 4 evaluation steps. Further additional results and measurements for ASBM and DSBM algorithms on the Celeba dataset are presented in Appendix
Thus, our D-IMF procedure allows us to solve the Schrödinger Bridge efficiently without learning the continuuss stochastic process, which in turn speeds up inference by order of magnitude. This aligns with the results obtained the Gaussian-to to--tussiann setups about exponentially fast convergence of D-IMF even with several intermediate time moments.
Discussion
Potential impact. Beside the pure speed up of the inference IMF, we want to point to another great advantage of our developed D-IMF framework. In the continuous IMF, one is forced to do Markovian projection via time-consuming learning of continuous-time SDEs (using procedures like bridge matching). In our D-IMF framework,
needs to learn several transition probabilities. We do this via adversarial learning [00, but actually this can done using almost any other generative modeling technique (moment matching (26), normalizing flows 23 41), energy- based models (56), score-based models [48], etc.). We believe that this observation opens great possibilities for ML community further explore and improve generative modeling algorithms based on Schrödinger Bridges, Markovian projections (bridge matching) and related techniques, e.g., flow matching 27]
Limitations and broader impact discussed in Appendix
A
10

---

Acknowledgements
The work was supported by the Analytical center under the RF Government (subsidy agreement 000000D73003215000002 Grant No. 70-2021-00145 02. 11.221....
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A Limitations and Future Work
Adversarial training. It is generic knowledge that the adversarial training may be non trivial to conduct due to instabilities, mode collapse and related issues. Fortunately, our ASBM algorithm relies on the already well-established and carefully tuned DD-GAN [33 technique as a backbone. The latter is specifically designed to address many such limitations and is known to score good metrics in generative modeling.
Theoretical convergence rate. We derive the generic convergence result for our D-IMF procedure (Theorem 3.6 but without the particular convergence rate. Empirically observe the exponentially fast convergence ($4.1) but theoretically proving this rate is an important task for the future work.
Broader Impact. This paper presents work whose goal is to advance the field of Artificial Intelli- gence, Machine Learning and Generative Modeling. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
B
Proofs
Here we provide the proof of our theoretical results one-by-one. Additionally, we introduce and prove several auxiliary results to simplify the derivation of the main results.
B.1 Proofs for Statements in Section 3.2
In our view, the proof of our main Theorem here is the most interesting and insightful (among all the proofs in the paper) as it some tricky mathematics, especially in its stage 2. In turn, stage 1 of the proof is inspired by the recent insights of [13, Theorem 3.2] about the characterization of static Schrödinger Bridge solutions
25 and manipulations with KL for SB in 224
Proof of Theorem 3.7
We split the proof in 2 stages. The 1st is auxiliary for the 2nd.
Stage 1. Here show that if some ( .11 (I Dx22) with marginals po(zo) q(zu) and pi (111) ((3))) has the density in the form
||||| 1 xol1 ((x1))
a( mi) al [ exp
2r
then it solves the Static SB between distributions (o() and p1 (x1). It is known 25 that the solution q² (xo, def pTd (za, 21) of Static SB between po and pı has the density:
||=1 xull3
q" (Jnn m1) vp* (mo) exp
o5 (31).
2c
Hence, the conditional density q" (x1 (zo) is expressed as:
4/* (2ro)
|| - zo 12
(11101 soll?) (x)))
(0(1100)) =
exp
8 (x1) CCo
Pu(zo)
2c
2e
Thus, both q( 21) and qº (zo, x1) have their densities in the same functional form and the same marginals ((z0) = 8" (xo) (xoo and g(z1) = q" (xx) (x)). However, we want to prove that in this case q(r. =1) and q² (zo, are equal, i.e., KL (q (lq) = =0
[los qº (2o, v1)
KL (q" (No, (1))
q- (No, x1 )drddr
q( 21)
f log BaKTo )9" (2 (zu)
o- (xo, vn )daqdær
pafon) of (2 on (oo)
('(2o) exp
||=1=00111
Φ (xx)
f log ((xx) exp
2x
(vo, xi )dxode
||
((x1)
2c
14

---

(log (a(2o)
log (-(21) )a" (ro, 21 )dxodxi
((1o)
Φ(x1)
J log (=(z0) ((1))
+ log (x1x1) a' (JO, x1 )doodx
o' (To, xi )dxodo
2(x1)
f log C(xo)
(a( Po (1r) + / log 8" (x1) p1 (ar )dæı
o(x1)
/ log ('(xo)) (( x1 )dæoder + / log (x1) (000, ti ddooox
((xo)
((11)
/ (log (200
log &- (x11) )q( 11 d
C(xo)
Φ(I1)
(zozo (x1)
log
q 11 )doodz =
((zo)o(11)
(*(rro) exp
||0 zaal"
Φ (1(11)
f log
((F0, x2 )doddy =
((xo) exp
20 - 20ll
((11)
/ log q- (2))
q(20, T1 )dxodx
g (2))
I log po(oo)d" (Jr (Fo)
( xn )drodz
( (zo)
I log qº (20,
q(00. x1) q ((To, r1ddddd
-floss 1lg g(20, x1) (" x1 (w00 x1 )doadr = KL (g((0. (co.
Thus, KL (g"1lg) (g Since the KL divergence is non-negative, we derive that o². Stage 2. In this stage, prove the theorem itself. First, if N > 1. i.c., there is more than one intermediate time moment, integrate g(z0, Ity ".,,", 21) over all intermediate time moments except f1. On the one hand, we get
( It, x1, = / q( F( Biy 2 dddd ... daan
f 1 On the other hand, we derive
Tty to, In ))(())((o zn ei)der drax = =p" W (ot, loo, an 20)
(17)
N+1
q( Ie, x1, / ((z)))((1,, (o))
II q( (x)).)) drez... dxex = ((20)@(21,100g(00)))) (18)
n=2
Combining and 18 yields
(()))((F, (oo) ( x1) = W (otz (oo xn (1)g(111
(19)
Note that if = 1, then we already have (99 from conditions of the theorem. Therefore,
W (xt, (1) mn) = |eoll
W (x,, Io, v (g( (m)) g(25) = g(~2) g( (20)q(21(=)))
From now on, interested only in 3 time moments: 0. t and f 1. To simplify the notation, we will write & instead of t2 in the following proof. We take the logarithm and get
log q(<1)1(00)) (zo) = log q(x)/20) + log (x2)
log W 21)
15

---

Then, we use the formula for the Brownian Bridge density:
1
log (<01 log (x) + logg(x11<z)
C+
2et(1 12 (tx1 + (1 t)xo)112=
log (2' (2)) + logg(<11<;) C+
2et(1 t) (1|2211 + | 11 + [I(1 t)zoll 2222t 2(1 -)a 2t t) 21)
log (2 (T)) + log ( - C+
|2,11
[111 t)
| 1|
x Φ1
2 20
2 11
+
2et [1 - t)
2et(1 (1 t)
2et(1 -t) 1)
E(1 - t)
et
|
||t11] 1|2
It
log ( (2xz) +
+
0+
2cl(1
2el(1 -
e(1 (1-
5 (are
| 1 2ct - t)
3y xo J
T
log ( (20)
et
#r ff
Thus,
log [2)) = fi 01) + f2(2 + xf a
e (1( + f2(21, 20 T1
log (<)1(2)))
fa(2n:01)
fs( 01) f1(22,01) + fe(x
(20)
Below, we prove that fa x1) = 91 (2o) + 92(21) 1 for some functions Φ1 and g2. We note that
fa( 0) = (zz, f2 (z), 20)
f2(11, zo) = fa(vo. 0) fi 0).
(21)
We substitute 21 to (0)
fs(20, = fi(21,01) + fa(xo, 0) fi
=f2(#(00)
fa(2o,2)) - fa(xo, 0) = (x1,
[i((z,, 0).
(22)
Since there is no dependence zo in the right part of (22) we conclude that fz(10, - fa 0)
is a function of only æ1- We define 91 (x1) fa(2o. #1 - fs(wo, 0) and (2( def
def
fa(2o, 0). Now
we have the desired result:
fa(zo, x1) = (x1) + fz(20, 0) = 91 (x1) + 92(2))-
(23)
Thus,
x1 x1 = (x11 + 92(x))
fa x1) log ((11)
Now, we can use this result about the separation of variables together with the result from the first stage to conclude the proof of the theorem.
2 I1
log g( (20) = 91 (x1) ) + +2(2(()))
+
|
|22l|
|1:00
112
(xx) +
+ g2(x1)
+
2c
2e
26
||
|
x 1//12
||
((11/10) exp (2(20)
exp
exp
91 (1()))
2e
2c
2e
#el =0(wo)
del
6(x1)
16

---

200 x 1
((01 - ((ro) exp
((x)),
2e
220 2:
g æ1) = ((zo) exp
((x1).
2e
Hence, from the first stage of this proof it follows that q x1) ) is the solution the Static SB between g(Tg) = (o( and ((x1) P1 (x1) with the coefficient That is, p² (Zo, 21) ((Fo, #1). Since g( ................................ Fix [To = p"" (Ieys-.., (oo x1) by the assumptions of the current theorem, we also conclude that g( Ity. ..s stys 21) = p² (Fo, discrete processes coincide.
F1), i.e. the
.......
B.2 Proofs for Statements in Section
The logic of our justification D-IMF for discrete processes generally follows the respective logic of the justification of IMF for continuous stochastic processes 47
The mild assumption here consists in the existence of at least one process (g][r) oo. The reciprocal process r R(N) has its density in the form W (Zt / 21 11) (1 (see (7). Thus, we need to optimize only the part r(zo, T1). Below we show, that r(Zo, x1) should be equal I() to minimize the functional.
Proof of Proposition 3.3
r E R[N) for which KL
r(Ig, T(Ig. Fty...., Tiy: 21)
= = - g( Fin, T1)
(Il())))
((D,, Tin; 21 )dzgdridd
r(20, Vins F1)
f log (fin .oo 21 (1( 21)
q (To, (Tp, 21 drgdrmdry
r( Lo, x1) r( pWr (mal=o:=1)
/ log pw (00 21)
g0om x1)
(2 Fins Tı dn + /
log g( 21)
q( Tin drr
r( x1)
log 21)) 211
Const +
((00, x1) g(00, 21 21) ddxddr
Const + KL (g(#0. 21 21)l:))))))(0)...)) 21))
Hence, pr (g) arg min KL (gllr) = ( To 21 (4(2,)
Similar to the previous proposition, the mild assumption here consists in the existence of at least one process E .M(N) for which KL (gllm) < co. This proof is bit more technical than for the reciprocal projection. We need to define new notation Ea. (Itoz.... -2 2 Itwer) for a vector of variables for all time moment except two time moments t, and 1-
Proof of Proposition 3.5
not
f flog log Fi
Fin, .1)
KL (gllm)
(000 Fin 11 ddzodeidzzz
m( Tin, x1) q( #11
f108 m(xo)
(T Tin )dzodzmda
II m(Fe, (
/ [log ((20) gl00, Fin xn dad + log m(xo)
g ( I1 x1 (2o) II m(rre, ( (000, Dins In dd
(( )dro f log m(xx)
((50)
qfFin 21 (20) N m m(Fe, fre, ,)
log
((50, x1 )
[lg(zo))lm(zo)

---

KL (1(zo))lm(20)) + / log q(xin;2110 ( Vin xt Diddrgdrdde, (24)
II m(3e, (een-))
N / log g( Tin F1) (10, Fins ddæodwiidzr + log ((FO) g Tins (ddddddg (25)
g1200 win: x1
((20)
KL (9(zo)]lm((z) f log N+1 m( (zen-1)
I g( Tiny x1) o (z Lin 21 ddgdddddr
(N / log q( Fm, 21 21)0((0 Vin 21 )drgderindr + / log
N+1 Σ
Fin ((x0 win, a di)dggdrdd m( lure..
KL (L((o)))lm(())
log
log q Iin x1 (2(0 Tin, Ii ddrgdrndn (N f
/ log flog
c
KL (1(zo)]l((ZO) - C1 +
N+11
g( (en- (g(cen (g(2 tnitais
De Ila-1) (00, Tin ti )drodein
Σ
log
m(I (een-1)
n=1
N+1 Σ
KL (c(o))l(((())
O1 +
log (g( )g tnite Fe Fia (F0, Tim, x1 ddrddddd +
Sc
log g ( of (o, Finy xn )doddedd2
m(Ire, (
N+1
log q (Dem-1) ql Den- ddre )a(aen )dxte-
KL (g(20)]lm(zo) C2 +
Σ
m(xe, (laa-1)
KI. (1( losn Mim(aren (1e
N+1 Σ R=1
KL (g((z))]l((z))
+
KL (g(0e, k 1 [m( (zn- )) g( dd + Cz.
In the line 255 we add terms equal to zero, to match each m(Fe, (xt,-1) ) by the separate term g(ro, Tin 11) in the line (25). We need to as we want to place each term m (zt,-1) in the separate KL-divergence in the final expression. Hence, the minimizer of the objective m² M(N)
has m² (ro) = ((x)) and all transitional distributions m (Ite ((,_)) = q(x,, (t,_ i.c. is given by
N+1
m" (xo, Tin, x1) (q) Tin, x1))
q(xo)
II n=1
a(x,, (xt
Proposition B.1 (Pythagorean theorems for projections). Assume that RRNN) and m E M(N). If KL (rl[m) < oo and KL (rilproj (r)) < 0o, then
KL (rl KL (r((projxx(r)) + KL (proj (m)
(26)
and if KL (m/]r) < co, KL (mllprooj (m)) < 0o then
KL (mllr) KL (mlpproj KL (projæ (m)]lr)
Proof of Proposition B.1 Before proving the first equation 26 we prove the additional property of r E R(N) for any n [1, ..,,, N 1]:
pl It,..)) r rIze, It.-)).
18

---

[ -(a_,)) = (
= r(ze, (
(27)
Since [proj و (2,,, (z,h_)) = r(z,, ( by the definition and since Markovian projection pre-
serve all intermediate time marginals. Now we prove the first equation (26).
/ log m( Tin, 21
r(wo, Tin, 31)
KL (r]lm))
r (00, Tim 1n ddggdridd
/ log m( Tin+ T1
r(Ja, Fin: 2'1
r (To Fin 21 ddddddinddy +
prooj Mr (T)] Fin 21)
log
(20, Tins Ii diddgddddd
[proj (T) Fins 21)
r( Fin F1)
log
T (2o, Tin In dnd +
(proj. (r) Tin, 21)
(L(r|lproj,x
pproj M (Z) Tins T1)
log
rwvo Tin, In ) ddrgdsiddr
mi Kin, 21)
KL (rlpproo + f1o0, log (r))
N+1
( (r) (oien-)) r(00, Tins I ddrgddddd
m(xa)
N+1
m(x (ztn-1)
1==1
KL (r (rllproi (r)) + KL M (r)( +
[r ( |itn 1), (zzg, I'm In d1)dddddd
log
m(Ft, (tm--1)
[proj (r)i(())](T,, (=m--)) KL (r(pprojjm()) + KL (+)
+
N+1
log
r r(Ita Fta-i
dr dr
m( (ta-s)
n=l
=/proj (rr) ite
KL (rilprojjum)) (r)) + KL ([proj (r) + (r) (Dem-1))
N+1
Σ
log
(proi Tte ) dxt.
m m( (
KL (r(lproj (r)) KL. ([ (+
N+1
(r) (Tm-
log
oy di
m( (
log [ (4) + m(xo)
KL (r))
(m)
J log
N+1 [proj (n)o (2te--.) I N+1 m(Iee, (t
(proj (r)) Win, dddradddde
KL (r(lprooj + / log m( Vin, 201)
(r) Fin x1)
Zin, x ddrodrindd
KL (r(( + KL (pro),u(7)llm)
That concludes the proof the first equation (26). The proof for the second equation (7 similar.
KL (mllr) = f
log m( Finy L.1) r(#g, Fin x1)
m m((o Tin x1 ) d +
19

---

log [projR (m)] (20: Zin x1)
m m( Win, 21 )droddindr
(proj (m) (Tro Vin 261)
(( Tin- 21)
log
m m( Jim T1 d1)dd +
[proj (m) (To, Fin, wi)
(L(m (m:)
(m) 01)
log
m( Fins I1 ddrodoan
r( Pin: x1)
KI. (mojprooj (m)) +
J log
(Fintto 21 [proj (m) (20. 21)
m (2o, Tins Tı ddxddadd
p (2in 21 (r(()((())))) (2. x1)
/ log
prooj (m) (zo. 21)
m(
dwode
r r(vn, 11)
=lpmoj (m)))(
KL (mill fmllppoojz (mi) + flog log [proj (m) x11
[proj (m)) 2n )drodr
r r(00, F1)
KL (ml projæ (m)) + / log ( (1mn .11)
prroj (m)](20, Iim Ty d
r(zo, x1)
KL (m|lproj (m)) + / log p¹ W: (Jin) po. x1 ( F1)
pproj (m)] (x)) win, 1i dd
pW, (Din kro x1 )r (x(z0, 21)
KI. (mllprrr)) + / log [proiq ( Tin
[proja (m)]((oo Vins xn ddrgdamndy
r(zo, Fin, 01)
= [[mooi (rtn
KI. KL. (m/lprojj (m)) + KI. (projx
That concludes the proof of the second equation
Proposition B.2. Assume that we have sequence of processes {q' 1 D-IMF procedure start- ing from d for which KL (q" lq") < co. Assume that for each reciprocal and Markovian projection in sequence KL (1) < co. Then KL (g'+²lla") S KL (g'llg') and lim; KL (g'llg/+)) 0.
Proof of Proposition B.2] We use the same technique was used in the proof of IMF procedure [47 Proposition 7], and for forward KL in 43]. We apply Proposition B. and for every we have:
KL. (d'lla") = KI. (1 + KL. (1'*"))"")
Since the KI. divergence non-negative, follows that KI. (1'+"114") S KI. (1'll4") Applying this proposition for each / & L E N, we have
KL (q"lla") KL (9"lla)) KL (q'lla") KL (a'lla/ + KL (1a""lla").
Since KL is non-negative and KL (1º114*) < co, we get lim KL (g'l = 0.
Proof of Theorem 3.6 The mild assumptions here are the assumptions of the Propositon B.2 i.c.
KL (q'1lg'* +11 < 30. To prove the current theorem, we follow the proof [47] Theorem 8] but do the
derivations for discrete stochastic processes instead of continuous. By our previous Proposition B. B.2
holds that KL (q'll4") < KL (q"lla") < oo for every l. Hence the sequence (q (q)) and its
subsequences of markovian (m' )ee (q ) and reciprocal processes (r') = (q )
are subsets of set {q E P2 Px(N+2)) KL (gllg") < KL (q"ll9")) which is compact 50
Theorem 20). Hence, (m))e) contains convergent subsequence (m/* )g0 - m². In turn, the
20

---

subsequence (r")) containes a convergent subsequence (r )ーا +r". Since sets of Markovian M(N) and reciprocal (NN) processes are closed under weak convergence, we have m (M(N) and r" E RNN. From the lower semicontinuity of KL divergence the weak topology [50 Theorem
19] and lim KL (q 11 (see Proposition B.2
S KI. (m (m"llr") & lim inf KI. (m's: = 0. 80000
(28)
Thus, m = r" der glt.. We know that d has the same marginals (= = g( and (x1) g(x) since both Markovian and reciprocal projections preserve marginals. By our Theorem 3.1 since qll .M(N) n R(N), then ( Ving x1) pd (20, Vin; v1). Finally,
lim KL (zo, Win x1)llp" Tio x1) 0 follows using
lim KL (1 (Zo: Pin 21) 21pppp" (zo, Vin 211) = 0
5-400
and the mononotonicity of KL (g"ll")), see Proposition B.2
B.3 Proofs of the Statements in 3.4
The proofs in this subsection are the most technical as there lot of manipulations with matrices.
Proof of Theorem 3.7 From 66 and ( follows that the discrete Brownian Bridge n (zin ( x has also Gaussian distribution. The covariance of the Brownian Bridge with coefficient e times < t [17 Eq. 9.14] is cs(1 t). Thus, the matrix eK is covariance matrix for all pairs of time moments t, t' [f1...... [N] of the considered discrete Brownian Bridge p [Zin| x1 ). The mean value Exx,,, (ro xi] Brownian Bridge at time i, is equal to tn + (1 (n) Thus, the discrete Brownian Bridge has the following distribution: p (zin (n, x1 = N (Jin eK).
Recall that the reciprocal projection is given by:
[proj (2im, To. vo. .1) p" (Im/ 21)g(20; 21).
(29)
Since it is a product of two Gaussian distributions, which itself is also a Gaussian distribution, our goal is find the mean vector and covariance matrix of [proj (Tim Ta #1). Further we denote [ (2m, Fo, x1) as r(zo, Pin 21) for convenience.
The mean vector of proje q (Iin: 2o, 21) for each t, is given by
Br(x,.) Ft.
By(ee, (1o [Ten Io, x 21]g(00. dd
J Epw (ze,, (ao (ate, lwa xn (a( In ddrdde / [Tro + tn mo)] ( 21 dd
(1 (1- tn)) / Rog(Eo, dd + tn xig(20, xn )droda = tn/l1 + (1 f, )/
where μi and μ1 are the means of q(xo) and q(x1), respectively. Thus, the mean vector of g v1) is given by (Upm), Ho, p1).
Now, we are going find the covariance matrix Σn. We will first find the inverse covariance
A B
MR
C)
of 9 (i Zn. 21). Here has shape ND ND the matrix K, while the matrix C/ has the shape 27 27 the matrix Σ. Matrices A and C are symmetric since they are part of the inversed symmetric matrix Σn. We exploit the fact that the logarithm of a Gaussian distribution has the form (by Const we denote all terms that does not depend on Zin Zo1):
log ([projv (Tin, No, 2o, 211)
Const 1 ((in Fo1) (U Mmi 2 Σx ((mm: Fo1) (U (01o1Hoo))
21

---

1
A
B
Const 2
((Imm Fuo) pU1 Mox))
you.
B
C
((Fms Fu1) (U poz: Mui))
Const
1
(sim UNOI)* A(tin Upo))
1
(201 moi) C( Coro) M111
2
(Tin U U0i) BB(oo m01)
Const
2 in
Arin + (0)))) ( Ar Poi
T Cr + Mol Cro1
Fin Br00
T
2 Bpo1 (UNo1) (Ho1)"BBIII" BB00 po1) Bpo1
1
Const
- A (0)1)) 3 Amin 2o Croi + H Croi
I Buo - I Buor - (U Buoo-
In turn, from 29 we have:
log
vo, 21)) = log
(Imn Tp x1) + log ( σ1) =
1
Const
2 (ii
(ek)) (zin U2001
(Jm
m01 m01)² ) -- (101 (101))
1 T 2d (cK)) :in + x (cK) -UZOI
1 (U<01)) (cK)) -UUx0
Const
2
2 01
T Σ-
r00
+
Σー 441 - 201 3.01 00111011
1 T' (eK) Vin + Jin (eK)- 201 2"tn
1 T
Const
2
v01
(c")
Σー r01 + x --ン P01-
By matching the formulas above, it follows:
A (EK)-t B = (eK) C -
(30)
Thus, have:
A
(-
B
(EK)-1
-EKK)-
MR =
BT
C)
=
(eKKK)- -1(
//++ +-1
By using the formula of block-wise matrix inversion
37 Section 9.1.3)
A
-CC Bd (
B 1
(C- BT BC- (31)
DT
C
Applying this formula, we have:
CC-B"e (C-
(cK) 1
A- 4"""""""C--33"=000"01"""""
eK +
- - US.
Thus, we obtain the desired result:
Σn (K + + UΣ (UE)"
Σ
Proof of Theorem 3.8 Part 1, Since from the assumptions of the theorem g(Tin, To, 21) has Gaussian distribution, follows that joint distribution of two time moments q It,, also Gaussian and is given by:
(tt,,, -V(
Ite x1n-1
)
( (
Pin ) ( ERdtan
) (ER)
ER)tnsnn (ER) 1it--
(32)
22

---

Recall that here (ER),.,,, represents submatrix of Σn with covariance of Iı, and Tt,, Hence, the conditional distributions are given by 337 Sec 8.1.3]:
(T Itt 1) = (a Men, (2te 1), Ecr))
The (#tn-1) def Men + (ER)snnn ((2R)) 1
(en-1))
(33)
E def (ER)eneen - (ER)cne ((ER), in .)=² ((ER)enn .)²
of That concludes the first part of our proof about the whole distribution [projuu Fin x1) Markovian projection.
Part Next, find the distribution of [proj x1), but before we proceed, introduce new notation to improve readability:
9M (Jo. Tin: 21) def
[rojjm9l(2pp) Tin, 21).
(34)
Since the process qu (Zo, Zin: 21) is Gaussian, all its joint and conditional distributions are also Gaussian. Moreover, we know that from the definition of the Markovian projection (10) follows that it preserve all marginal distributions, i.e. (x,,).) q( hence we can already write that (2 21) is given by:
21 (200
20
( Eo En1 ΣΙ
QM (M. x1) = -N(
(35)
(Eo1))
where μif and are the means of (00)) and ((x1). while Eo and E1 are the covariance matricies of g( and q(x1). Thus. only Yoi unknown, Again, by using the formula for the conditional distributions [37 See 8.1.3] we have that:
am (vo) = N(nn [ (zo), Ei (zo)); (x0)
def
#1 + 2011 2n 'x
(xo H0),
"G
Σ Σ 2/1 - E01-
Since the mean 7i (Ir)) of the conditional distribution is affine map of zo with the matrix G we can derive:
01 GEn-
Thus, we need to find the expression for G, by considering the expression for T (zo). ). To derive consider the sequence qu (Ire,, (m)) for n [l..... N + 1]. We already know the expression for n = which is given by [proj (T,, ( = q(r,, (20) in the first part of the proof. For other n, we use the following expression:
the expression of the mean Ai (zo)) of q. (2 (2o)
aM (Zt,, (mo) g( (z (Z (o)dd
(36)
Since (x, (mr) is Gaussian we denote (1 (mo) (x li (zo), Eem.). ). We derive the mean Men, (2o) by using the properties of conditional expectations as follows:
/
iic (zo) = Egga [Er
(gg(eee (a (Trca) (on(I)._1z0)daa
Den (m
/ (pe. (ER)en (aat:,,ttn(EERrnam... (ER) (2ta-1 Men-1) (20
He, + (Endendory (EE))tn ((/
Fla- and (Ft (o)dd(t.-)) - Men-1)
pt, + (En)eentn ((
(tn-1 M
= (ao)
-.b-))) Pt. (ER)). An- (1t ( (20) M(m-1) pee, (Mim- (oon--1(y (2O)).
(37)
23

---

Note that in the line (37.. we use equation (33) for Pen, (Itm- with sta-1 pi (ro) to simplify the expression. Since Men (20) Pt.. - (In)) we can derive Ti (2n) recursively follows:
11 (20) Mew+a (2o) PeNt1 (Pex (ro) Pexes (Mew (... Po(z))...
where each Pt., (X ,) is affine map given by 33) with the matrix given by
(ER) (En)
Hence, 71 (2o) is composition of affine maps, and its matrix is given by the product of matrices Pt, (Itn-1) as follows:
N+1 II
G
En ER)nnn 1 (to
in turn 201 is given by:
N+1
G
πI
(n) ((ER) itt 1 En.
01
This concludes the proof.
C Additional Experiments
C.1
Illustrative 2D Example
Here we consider the SB problem with Po as a 2D Gaussian distribution and P1 as the Swiss-roll distribution. We use independent (mro, Po(2)))1 (I1), . = (t,, N+1) and K 20 ASBM algorithm with different values of parameter and present
outer iterations. We run results in Figure In all the cases, we observe the convergence to the target distribution. Overall, the trajectories similar to the Brownian bridge and the closeness of start and endpoints is preserved. In Figure we show the evolution of trajectories for different D-IMF iterations, which become more straight when number of iterations increase.
C.2 Benchmark
We use the SB mixtures benchmark proposed by 13 34] to experimentally verify that our ASBM algorithm indeed able to solve the Schrödinger Bridge between Po and P1. The benchmark provides continuous probability distribution pairs Po. Pi for dimensions D E {2, 16, 64, 128] with {0.1, 1.10). To evaluate the quality
the known static SB solution pT (Io, for parameters of our recovered SB solution, we use cBW2--VV metric as suggested by the authors [3] $5] and provide results in Table [
Additionally, we study how our approach learns the target distribution
topetery stnntr bancier ewitiiiee)
Tuestery bueccer mas ritted
Topeta's SARE--- Sapetary 1S7A8)
(a) To Po. I1 P1.
(b) e = 0.03.
(c) c 0.1.
(d) 0.3.
Figure 8: The final process qu learned with ASBM (ours) in Gaussian + Swiss roll example.
24

---

Rdee Ban Trantory ietrmeeeo aredittor
Frint du twaes
Pitne distett Iremee are prodecions
Tiie coutaaain Piermaaion prodectors
Pretwcian predetion
(a) Outer iteration 0.
(b) Outer iteration 1.
(c) Outer iteration 10.
(d) Outer iteration 19.
Figure 9: Evolution of our learned discrete process qo depending on D-IMF iteration in Gaussian Swiss roll example with 0.03.
P1 Table 2 In all the cases, we run our ASBM algorithm starting from the independent coupling between Pu and P1-
As the baselines, we consider other neural bridge matching methods 40 47
The first one (SF M-
Sink) is based on minibatch OT approximations, while the latter implements continuous IMF (DSBM). Additionally, we include the results of the best algorithm (for each setup) from the benchmark (13]
As shown in the Table [ our algorithm demonstrates superior performance on E = 10, superior performance or comparable performance on = 1, slightly worse performance w.r.t. SF²M-Sinh [4 and superior performance w.r.t. DSBM 47 on 0.1. Also, from Table oone may note that ASBM fits target distribution better then other Bridge Matching SB algorithms.
10
Algorithm Type
64
D
D
D
Rest algorithi on benchmark
Varies
1.94
13.67
11.74
11.4
1.04
9.08
18.05
15.23
1.40
1.27
2.36
1.81
DSBM
1.21
4.61
9.81
10.8
0.68
0.63
5.8
29.5
0.23
5.45
68.0
362
SF M Sink
Bridge motching
0.54
3.7
9.5
10.9
0.2
1.1
9
0.31
4.9
319
819
ASBM tours)
0.89
8.2
13.5
63.7
0.19
1.6
6.8
10.5
0.18
0.4
1.9
4.7
Table Comparisons of cBW? UVP J. between the static SB solution p (ito at and the learned (u(xo at
the SB benchmark.
The best metric over bridge Matching algorithons is holded. Results marked with i are taken from [I]
0.1
Algorithm Type
12=16
D=
64
D=
128
D=2
D=
D=
D=2
D= 64
D= 128
Best algorilhm on benchmurk
Varies
0.016
0.05
0.25
0.22
0.005
0.09
0.56
0.12
0.01
0.02
0.15
0.23
DSBM
0.1
0.14
0.44
3.2
0.13
0.1
0.91
6.67
0.1
5.17
66.7
366
MFSM-
Bridge matching
0.04
0.18
0.39
0.07
0.3
4.5
17.7
0.17
4.7
316
812
ASRM (ours)
0.016
0.1
0.85
11.05
0.02
0.34
1.57
3.8
0.018
0.25
1.7
4.7
Table 2: Comparisons of -WV- J (%) between the ground truth target distribution p1(#) (#11) and leamnd target distribution qa (21).
The best mettic over bridge matching algorithms is bolded. Results marked with f are taken from [[]]
Remark. There exist recent light SB algorithms [24 [T] which do not use neural parameterization and rely on the Gaussian mixtures instead. However, these methods have very strong inductive bias towards the benchmark as also constructed using Gaussian mixtures. Therefore, we exclude them
from comparison, see the comments of the authors in 24 $5.2] and 11 35.2]
C.3 Colored MNIST
Here test ASBM (ours, NFE=4) and DSBM (NFE=100) algorithms starting from mini-batch OT
coupling (49 on transfer between colorized MNIST digits of classes "2" and "3" with e E {1, 10).
We learn ASBM and DSBM train set of digits and show the translated test images in Figures T0
and along with calcualted test FID in Table 3
For = 1 the color stays almost exactly the same through translation and there are minor shape diversity for both ASBM and DSBM, see Figures [1bb] 10c 11b TTc.. In turn, e 10 introduces more stochastisity to the solutions, and expectedly the color and shape vary a bit but overall stays similar to input data for both ASBM and DSBM, see Figures (Tod) T0e TTd [Te). As one can see from Table 3 ASBM has better FID on both {1) However DSBM experiences notable
25

---

2
3
3
3 3
3 3
3 3
3 3
3 3
3
3
3
3
2
3
3
3
3
3
3
3
3
3
3 3
3
3
3
3
3
3 3 3
2
3
3 3
8 3
3
3
3
3
3
3
3
3
2
3 3 3
3
3 3
3
3
3 3
3 3
3
3
3
(a)
(b) ASBM (ours),
(c) DSBM 47
(d) ASBM (ours), 10
(e) DSBM (47)
Po
1
(=1
10
Figure 10: Samples from ASBM (ours) and DSBM learned on Colored MNIST 2- (32 x 32)
translation for {1, 10).
8
2
2
2
2
2 2
2
2
2
2
2
2
2
2 2 2 2
2 2 2 2
3
2
2
2
2
2 2 2 2
2
2
2
2
2
2 2 2
3
2
2
2
2
2 1 2 2
1
1
2
2
2
2
2
3
2
2
2
2
2
2
2
2
2
2
2
(a)
(b) ASBM (ours),
(c) DSBM 47
(d) ASBM (ours).
(c) DSBM (47)
~ Po
10
< 10
Figure 11: Samples from ASBM (ours) and DSBM learned on Colored MNIST 3-+2 (32 X 32)
translation for c E {1, 10).
Model
FID 3)
FID
+
ASBM (ours)
2.7
2.8
DSBM
6.2
5.3
ASBM (ours)
4.3
4.53
DSBM
10
58.7
59.9
Table 3: C-MNIST FID values for ASBM and DSBB with E {1,
increase in FID with c 10. We conjecture that this is due the FID unstability w.r.t slightly noisy images which may appear in DSBM due to the neccesity to integrate noisy trajectories (for large e).
D Experimental Details
D.1 Details of DDGAN Implementation for Learning Markovian Projection
Below, we discuss the parametrization of the discriminator and generator in detail. In general follow [53], but we change their DDPM diffusion inner process on the Brownian bridge process.
Parametrization and objective for the discriminator. As in the DD-GAN paper 53
time-conditional discriminator De (Iz (tt.-1,1n-11) R x IR x [0, 1] - [0, 1]. For each time moment and object -t,-,, the role of this diseriminator is to check whether the sample 2r,, is from the distribution ( [ )). As well as in the DD-GAN paper [53 optimizing the following objective:
we train this discriminator by
N+1
min
E Eg(ss ( (= )l log De Ft x(.-11
(38)
+ Eygo (ne, Ie (1 De ( St, (n)1)
Here, the samples from g(z,, (=t,_.) play the role of true samples, while the samples obtained from the parametrized distribution go (Zt, (zt,-1) play role of fake samples in terms of original GANs.
To estimate the first expectation Ey(ss, ,)BBylr let, 1) = Eg(z,,, "t, 1) one should sample from
q(It,,, It.-.). ). To sample a pair (z,,, xen-1) q(Ien F we use the properties and ( of
26

---

the reciprocal process
/ p (It, (Pta-1::1)) (2t, 21 20)
q _f Fto-1)
Sampling from ( 10 (To (2...) for estimation of second expectation is given in detail below.
Parametrization and objective for the generator. We follow the same setup the authors of
DD-GAN [53 and parametrize go (,, [t, .) implicitly through the generator Go (o 12 ()
x
R
10. 1]
IR as follows:
qo (x (tan-)) def { 2o
(at, 1, 11
[ R2
W (2t, 1: Go Go( t))p
where ( | [t) should match q(x) (xt,-1) and pz(z) is the auxiliary probability distribution for the generator Gø model samples from (0 1 Thus, for given _t,_ sample Tt,, qo (2,,,, ) is obtained by first sampling 21 from the generator Gø and then using sampling from the Brownian bridge p" (zr, [Trn-- 111 While in the DD-GAN, the authors use the intermediate time distribution g( [2o x1) from DDPM [5] and it is the main difference between our Markovian projection and which the authors of DD-GAN used. As in the non-saturation GANs 10 train the generator by optimizing the following objective:
N+1
E Egise ,) Buo loen Doog (h
D.2 Details of D-IMF Implementation
General description of the ASBM algorithm. D-IMF algorithm is parametrized by the number K of outer D-IMF iterations, number of inner D-IMF iterations (number of generator gradient optimization steps inside one IMF iteration), ASBM number of inner steps N and starting coupling (2o, 21) used in the initial reciprocal process qº (x 21)
p (xin (2 xi )q ( ( 21). Our
ASBM Algorithm for D-IMF procedure is analog of DSBM 47
Algorithm 1] for IMF procedure.
Algorithm 1: Adversarial SB matching (ASBM).
Input :number of intermediate steps N;
initial process gd (mo, Ft,,.... Ttx, 21) accessible by samples; number of outer iteration K N;
forward transitional density network (go (xtm-,)) ) N-N+1
n=1
backward transitional density network { (2) (1 (e,, ) N)
Output:po(2z II
n=1 (2,,,, 18t., .) Rs (1(1)) I on (Tt, (1tt.)) ss p²" (io, Din x1).
for k = 0 to K 1 do
Learn (gr( (x. W+ using with o
Let tt= be given by (o(o))
go (Iz, xtn-
Let th- be given by W" (oin To. Tz gg (2o. x1):
Learn (gg(#th-,12re 1 using 16 with o 4k+2
Let g 1-2-3 be given by pi (x1)
IT On (3 |xte,):
Let q g be given by W (#m [ro, 21 (10, 21);
We do not reinitialize neural networks during the ASBM algorithm.
Special pretraining on the O-th outer iteration. While, in general, Algorithm [ implements our scheme, in our experiments, we slightly modify the initial outer iteration based on purely empirical reasons. We train both forward and backward models
and {g,, (3, W ). We use more gradient setups on this iteration than on the further outer iterations. We do that to "pretrain" both processes qo and (n to model pı and Po respectively. Then we proceed other iterations as described in Algorithm
with gd and the let q² be (o(2)) II + go (x+,, |itt,
27

---

D.3 Hyperparameters of ASBM
For all the experiments, Discrete Markovian Projection is conducted using the DD-GAN code
53
https://github... http:/uwww..omm//iaaaa//dnaaaaaa diffusion gan
The only thing that we modify is the replacement of the DDPM 13
posterior sampling for generator
with our Brownian Bridge posterior sampling, see Appendix D.I In all the experiments we use uniform time discretization, i.e., for the number of inner times points N, Ln = N+1
n for n E [0, N +1].
In Toy 2D (Appendix C.1 and SB Benchmark (Appendix C.2 experiments, both generator and dis- crimintor are parametrized by MLPs with inner layer widths [256, 256, 256]. LeakyReLU activations and 2-dimensional time embeddings using torch, nn. Embeddings. In CelebA ($4.2) and Colored MNIST (Appendix C.3) experiments, generator is parametrized by U-Net 42 and discriminator by
a ResNet-like architectures with addition of positional time encoding as in are optimized with the Adam optimizer 22 and apply the Exponential Moving Averaging (EMA) on generator's 's weights. At the start of a new D-IMF iteration, both the generator, generator (EMA), discriminator and optimizers are initialized using checkpoints from the end of the previous D-IMF iteration. Inside each D-IMF iteration (except the initial one), EMA generator weights are used for sampling from previous Discrete Markovian Projections. Starting coupling 4 ( (z0,x1) may be either Ind, i.e. (20, 21) = Po(zo Pp (21), or Mini Batch Optimal Transport coupling (MB), i.c. discrete Optimal Transport solved on mini-batch samples [49].
5 Neural networks
The hyperparameters which we use in the experiments are summarized in Table
Start couping q² (xo, 21) Ind
D-IMF
D-IMF=0
D-IMF
DIG opt ratio
EMA
Experiment
N
Batch Size
Lr G
Lr D
outer iters
grad updates
grad updates
decay
2D Toy
20
400000
40000
3
512
1:1
0.999
le44
le-4
SB Bench
Ind
133000
67000
31
128
3:1
0.999
lc-4
lc-4
C-MNIST
MB
100000
50000
3
64
1:1
0,999
1.25e-4
1.6e-4
CelebA
MB
5
1000000
40000
3
32
1:1
0.9999
1.25e-4
1.6e-4
Table 4: Hyperparameters for experiments. stands for Discriminator and stands for Generator.
Ratio of Discriminator optimization steps w.r.t. Generator optimization steps is denoted by DIG opt ratio. Lr stands for learning rate.
Other details & pre-processing. Test FID calculated using pytooch-fid fid package) Working with CelabA dataset K3 we use all 84434 male and 118165 female samples (90% train, 10% test of each class). Each sample resized to 128 128 and normalized by 0.5 mean and 0.5 std. Generator and discriminator are the same as for CelebA-HQ in DDGAN [33 (42M Generator parameters and 27M Discriminator parameters). Working with Colorized MNIST [2), we pick digits of classes "2" and "3" (we use the default MNIST train/test split), resize them to 32 x 32 and normalize by 0.5 mean and 0.5 std. We use the same generator and discriminator as DDGAN uses CIFARIO (3)
Computational time. The most time challenging experiment on CelebA runs for approximately 7 days on 1 GPUs A100. Experiment with Colored MNIST takes less then days of training on GPU A100. Toy2D and Schrödinger Bridge benchmark experiments take several hours on GPU A 100.
D.4 Details of DSBM Baseline
DSBM 47
implementation taken from the official code:
https://gittuu..comm www./guyynnggaaee hhtp:/annnss//dd-m-ppppa
For CelebA experiment all the hyperparameters, except for 200k training iterations for the first IMF iteration (Bridge Matching pretrain, Appx I.3 [7]) and number of overall IMF iterations (that is taken the same as for corresponding ASBM experiment, see Table 4 were taken from 447. As neural network time conditional U-Net model (38M parameters) was used. Hyperparameters and neural network for Colored MNIST experiment were taken from MNIST <> E-MNIST experiment 47 S6]. Starting coupling is exactly the same for ASBM in corresponding experiments (Table
28

---

E Additional results on CelebA
E.1 Extended Evaluation using Other Metrics
FID for female->male We evaluate the backward model (female-tmale) trained for unpaired CelebA (128x 128) image-to-image translation and present the test FID in Table 5
Model
10
DSBM
24.06
92.15
ASBM (ours)
16.86
17.44
Table 5: Test FID,, values for CelebA female female-Smale Smale image-to-image translation.
CMMD. To strengthen the unpaired CelebA (128x128) male-sfemale image-to-imane -maee translation (4.2) experimental results, we add CMMD 19 metric. CMMD is recent analogue of the FID that enjoys unbiased estimation and rich CLIP 40
embeddings. We estimate CMMD on CelebA for the same DSBM and ASBM models as for the FIDD calculation ($7) using all available female test samples and present results Table It can be seen that the CMMD values correlate with the FID values.
Model
=
6 10
DSBM
0.365
1.140
ASBM (ours)
0.216
0.231
Table 6: CMMD] 19 metric for unpaired CelebA (128 x 128) male male->female image-to-ime to-image
translation estimated on female test
Training with different NFE. In the unpaired CelebA (128x 128) male-tfemale Amage-to-imagee translation (8 4.2), number of inner steps = 3 is considered. However, it is possible to train the model with different values of N, which correspond to the model NFE minus one. For completeness, we provide experimental results with training and evaluation at N = and N 7 (NFE= 2 and
NFE= 8). Here all training hyperparameters are the same for N 3, see Appendix D Samples and test FII) are shown in Figure 12
NFR=2, FID=11.829
(b) NFF=4, FID=16.08
(c) NFE=8, FID=29.58
Figure 12: Unpaired CelebA (128x) x 128) male- male-xfemale FID., values for ASBM trained with different NFE E {2, 42,488) with 1.
translation samples and
Inference with different NFE. Although in practice models are trained with fixed NFE (see Appendix [D. it is possible to use different NFE at the inference stage by exploiting the continuity of the time-conditional module, see Algorithm 2 We take the model for male->female trained on NFE=3 with
and evaluate it with different NFE E {1, 2, 3, 4, 8, 16, 32]. see the results in Figure 13 and do quantitative evaluation using FID and MSE cost (MSE between inputs and outputs)
in Table 7 As can be seen, the MSE cost increases with NFE and the FID is optimal at NFE=4.
29

---

Algorithm 2:
Inference of forward ASBM model.
Input :number of intermediate steps N; sample 20
forward EN generator network Go:
Output sample from (a I (2t, (2tn-1).
for n O to N do Ea+1 ~ p" (Itn41 Ft, 21 = Go 2,t))
Input
NFE
NFE
MMM
01
Figure 13: CelebA male- -female translation samples of ASBM trained with NFE= and evaluated with NFE E 21 3, 4, 8, 16, 32).
NFE
2
3
4
16
32
FID
58.71
32.27
17.67
16.62
55.72
86.97
MSB cost
0.009
0.023
0.047
0.113
0.288
0.354
0.50
Table 7: Quantitative evaluation of ASBM model trained with NFE= 4 and evaluated with
NFEE (1,2, 4, 16, 32). FID, and MSE cost calculated on the test
LPIPS diversity. To measure the generation diversity of our model on CelebA male male-xfemale translation, compute the LPIPS variance [16]. Specifically, we take subset of 500 images from the test part of the Celeba dataset and sample a batch
generated images for each input image. We then compute the average LPIPS (55) distance between all possible pairs of these images and average these values. We present the results in the Table for DSBM and ASBM with different values of the coefficient 1 and 10.
Model
1
10
DSBM
0.1047
0.1909
ASBM (ours)
0.0933
0.1878
Table 8: Average diversity of DSBM and ASBM generative models for male xfemalee translation measured by using I.PIS variance [16]
LPIPS perceptual similarity. To evaluate the content preservation during the unpaired image-to- image male tile-e translation on CelebA, we calculate the perceptual similarity. Namely, we take the test samples from CelebA dataset, translate them using learned DSBM and ASBM models with parameters and e = and then calculate I.IIS (55) between inputs and generated outputs and average results. One can results in the Table 9
30

---

Model
10
DSBM
0.246
0.386
ASBM
0.242
0.294
Table 9: Perceptual similarity for male ffeale translation for DSBM and ASBM models with 1
and c = 10 measured using LPIPS! [53]
between inputs from CelebA test and generated outputs.
E.2 Analysis on D-IME/IMF iterations dynamics
CCBN 24PD-100
Joom (nP2-1) 41
We include additional analysis on dynamics of model sam- ples with D-IMF iterations for ASBM (ours) and IMF iterations for DSBM with = 1. As one can see from Figure 15a ASBM visually almost converges after iter- ations in terms of similarity generated sample w.r.t. to input data, i.e. the transport cost. From plot in Figure [4 we see that ASBM's FID does not change through subse- quent D-IMF iterations; ASBM fits target on the iteration 5 rather well. Looking at Figure 15b] one can conclude that for DSBM visual similarity along side with transport cost starts to diverge after 5th outer IMF iteration. Also, as can be seen at plot Figure 14 FID stops improve after outer iteration 9 and does not improve drastically from outer iteration 5. Hence, we take ASBM and DSBM with outer D-IMF/IMF iterations as balance point for our comparison.
10
15
Figure 14: ASBM and DSBM FID w.r.t.
IMF iterations.
meraton
meration
Eeration
Earazion
cteration
iteration
Iteration
Itertooo
merabon
Iteraiio
Input Data
X=B
(a) ASBM (ours)
Input Data
[-1
1-2
3
Jmd
S
/=6
(-8
A
(b) DSBM
Figure 15: Samples dependence on D-IMF/IMF outer iterations number k e 1.
31

---

(a) I Po
(b) ASBM (ours), 1
(c) DSBM 47 <
FID: 16.87
FID: 24.06
(d) x Po
(c) ASBM (ours), é 10
(f) DSBM 47 < 10
FID: 14.73
92.16
Figure 16: Samples from ASBM (ours) and DSBM learned on Celeba femaale-male (128 X 128) for E {1, 10}
E.3 ASBM (ours) and DSBM samples for female mmale (128 128)
we provide additional examples for female- (128 x 128) setting with E (1, 10) for ASBM (Figures 16b 16eb and DSBM (Figures [6e 16f along with quantitative evaluation of FID values. Both ASBM and DSBM models were evaluated at D-IMF/IMF iteration number 4. As one can see ASBM (NFE=4) outperforms DSBM (NFE=100) in FID using only 4 evaluation steps.
In Figure 16
E,4 Extra (uncurated) samples for ASBM (ours) CelebA male male+>female (128 x 128)
In Figures 17 and g we provide additional samples for ASBM CclebA malet-female (128 x 128) experiment with (1, 10).
32

---

C
(a) Input
(b) Output for
(c) Output for 10
Figure 17: ASBM (ours) Celeba male -female (128 128) samples for e E {1,
33

---

(a) Input
cb) 10
6c) 10
Figure 18: ASBM female-male (128 128) samples for E 10)
34

---

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Reviews:
[{"review_id": "U4daWSy7FE", "reviewer": "Reviewer_xtGA", "summary": "In this paper, the authors study the Schrodinger Bridge problem and propose an improvement over the Diffusion Schrodinger Bridge Matching (DSBM) methodology [1, 2]. In particular, they propose a discrete counterpart formulation of DSBM. Instead of relying on stochastic processes, they rely on a discrete-time version of the Markovian and reciprocal projections which are the key components of DSBM. By doing so, they have a better control on the number of steps needed in DSBM. They next leverage advances in the field of diffusion models to reduce the number of stepsizes needed in order to solve the Schrodinger Bridge. In particular, they focus on larger jumps using the approach of Denoising Diffusion GAN (DD-GAN), introduced in [3]. The method is illustrated on translation tasks on the Celeba dataset. \n\n[1] Shi et al. (2023) -- Diffusion Schrodinger Bridge Matching\n\n[2] Peluchetti (2023) -- Diffusion Bridge Mixture Transports, Schrödinger Bridge Problems and Generative Modeling\n\n[2] Xiao et al. (2021) -- Tackling the Generative Learning Trilemma with Denoising Diffusion GANs", "questions": "* I understand that the main goal of the author is the speed-up of DSBM-like algorithms. However, why choose DD-GAN? This doesn't seem to me like the obvious candidate for speeding up DSBM. Other techniques like distillation (adversarial distillation if one really wants to use GANs [1, 2]) or consistency models seem like potential better candidate. If the finetuning strategy is what the authors want to avoid, consistency models can be trained from scratch [3, 4]. \n\n* Some typos:\n\n* l.280 \"Schrodigner\" --> \"Schrodinger\"\n\n[1] Xu et al. (2023) --UFOGen: You Forward Once Large Scale Text-to-Image Generation via Diffusion GANs\n\n[2] Sauer et al. (2023) -- Adversarial Diffusion Distillation\n\n[3] Song et al. (2023) -- Consistency Models\n\n[4] Song et al. (2023) -- Improved Techniques for Training Consistency Models", "limitations": "The limitations are discussed in the Appendix A. However, the authors are quick to dismiss these limitations. For example, regarding the adversarial training \"While we mention this aspect, we do not treat it to a be a serious limitation.\" I think there are serious problems regarding adversarial training regarding the training (suffer from training instabilities) and the results themselves (less diversity). These issues are usually reported in the field of adversarial distillation of diffusion models, see [1] for instance. I think this is doing a disservice to the paper and the community by trying to minimise them instead of exposing them for what they are. \n\n[1] Dieleman (2024) -- The paradox of diffusion distillation", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is overall clearly written and easy to follow. It relies on the notions of Markovian and reciprocal projections that were already introduced in [1]. All the required notions are properly introduced. \n\n* I enjoy reading the full description of the DSBM algorithm in discrete time. I agree that this should allow for 1) easier spreading of DSBM-related methodology 2) more flexible algorithm (like the introduction of GAN jumps as proposed).\n\n* The experimental results seem to compare favorably compared to DSBM (more on that later). \n\n[1] Shi et al. (2023) -- Diffusion Schrodinger Bridge Matching", "weaknesses": "* Part of the theory was already done in [1]. Especially the discrete Markovian projection was already identified as the minimizer of the Kullback-Leibler divergence in [1, Appendix E]. I think the claims of novelty of the paper regarding the introduction of the discrete scheme should be tamed down. \n\n* I am not clear regarding what is the purpose of the Gaussian study. I understand that the authors use it in their experimental section (Section 4.1) to study the convergence of their algorithm but I think Section 3.4 is distracting the reader from what I consider to be the main points of the paper which is 1) full discrete study of DSBM 2) GAN jumps. Section 3.4 could easily be considered in the Appendix which would leave more room for additional experiments. \n\n* The claim of exponential convergence should be tamed down. Reading the paper it seems like this is proven in the current paper (I am talking especially about the bold \"exponential rate of convergence\" in l.294 which can be misleading). To my understanding this is only experimentally shown. \n\n* There is something quite striking in Figure 3. (for example see the last row of (d) and (f)). While the quality of ASBM is better (by the way, one of the reason the FID score is so high for DSBM must be because of the remaining noise in the output samples, an easy way to remove this noise would be to consider a deterministic last step as a postprocessing as is routinely done in diffusion models, see [2] for instance), it seems that the diversity is much lower than DSBM. I would like to see a proper investigation of this phenomenon which is also reported in the adversarial distillation of diffusion models. According to [3], there is an incompressible trade-off between high-NFE/good diversity and low-NFE/low-diversity.\n\n[1] Shi et al. (2023) -- Diffusion Schrodinger Bridge Matching\n\n[2] Jolicoeur et al. (2020) -- Adversarial score matching and improved sampling for image generation\n \n[3] Dieleman (2024) -- The paradox of diffusion distillation", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer xtGA, thank you for your comments. Here are the answers to your questions and comments.\n\n**(1) Part of the theory was already done in [1]. ...**\n\nIndeed, the authors of [1] analyze the discrete-time Markovian projection and show that it converges to continuous-time projection in certain limiting cases. We cite this result in line 260. We will add additional citations and discussion in Section 3.3 after defining Markovian projection to highlight some aspects that have already been introduced in [1].\n\nPlease note that our Proposition 3.5 on KL-minimization extends the results from [1, Appendix E]. The authors of [1] considered the Markovian projection for a mixture of **Markovian** bridges. In turn, in our Proposition 3.5 we consider arbitrary distribution $\\pi(x_0, x_{t_{1}}, \\dots, x_{t_{N}}, x_1)$ without any assumptions on its conditional distribution $\\pi(x_{t_{1}}, \\dots, x_{t_{N}}|x_0, x_1)$. It makes the proof technically more complex but provides a more general result. \n\n**(2) ... the purpose of the Gaussian study. ...**\n\nThere are two purposes: to experimentally study the rate of convergence and to demonstrate that a closed-form solution for the multivariate Gaussian case can be derived within the discrete-IMF framework. In the final version, we will shorten Section 3.4 by moving detailed technical definitions (such as matrices $U$ and $K$) to the Appendix but retain the key points in the main text.\n \n**(3) The claim of exponential convergence should be tamed down. ...**\n\nWe agree that the phrasing in line 294 could be misleading. This observation is based solely on our experimental results, and the whole discussion on convergence rate is placed inside the experimental section (Section 4), while all of our theory is placed in Section 3. We also directly mentioned that there is no theoretical proof in the limitation section (lines 465-467). To avoid possible misunderstanding, we will revise the bolded text.\n \n**(4) ... remove this noise ... consider a deterministic last step  as a postprocessing ...**\n\nWe use the original published code from the DSBM authors [1] to ensure a fair comparison. In this code, noise is not added at the last sampling step. As you requested, we additionally used the drift function for the last time-moment $t=0.99$ to add an extra denoising step at the final step of the sampling process. It indeed removes the remaining noise, (see the samples in Figure 4 of the **attached pdf**). For the Celeba setup with $\\epsilon=1.0$, we observe the improvement of the FID metric from $37.8$ to $29.7$.   We will add this additional result in the experimental section.\n\n**(5) ... it seems that the diversity is much lower than DSBM. I would like to see a proper investigation of this phenomenon ...**\n\nAs requested, we provide a study on the diversity of images produced. We measure the diversity of images produced by DSBM and ASBM on the Celeba setup by utilizing the LPIPS diversity as was done in [3, Table 1]. We take a subset of $500$ images from the test part of the Celeba dataset. We sample a batch of generated images of size $16$ for each input image, compute the average distance between all possible pairs of these images, and average these values over all inputs. We present the results below for DSBM and ASBM and different values of the coefficient $\\epsilon=1$ and $\\epsilon=10$.\n\n|| $\\epsilon=1$ | $\\epsilon=10$ |\n|-------------|--------------|---------------|\n|DSBM [1]| 0.1047| 0.1909|\n|ASBM (ours) | 0.0933| 0.1878|\n\nIndeed, the LPIPS diversity is lower for ASBM. However, without a known ground truth for diversity, it is difficult to determine definitively whether ASBM provides insufficient diversity or if DSBM produces too much. We will include these findings in the final version of the paper and discuss this trade-off.\n\n**(6) .... However, why choose DD-GAN? ...**\n\nThe mentioned adversarial distillation techniques [1,2] are based on tractable forward process in diffusion models for simulation-free sampling, which is not feasible for the unpaired Schrödinger Bridge (SB). Additionally, these methods require an already trained model for distillation, whereas our approach does not need a pretrained diffusion model.\n\nRegarding consistency models, it is unclear how to use them in an unpaired image-to-image setup. Works [3, 4] do not address this. The work [3] discusses generative models and zero-shot translation with known degradations, applicable only to specific tasks like colorization. Similarly, [4] focuses on generative modeling. While consistency models can potentially accelerate arbitrary ODEs, the Schrödinger Bridge (SB) problem's solution is an SDE, which makes it challenging to use consistency models.\n \n**(7) ...the authors are quick to dismiss these limitations. ... (Appendix A)**\n\nWe understand that adversarial training is a limitation. We intended to say that we use the DD-GAN approach, designed to address this limitation to some extent (mode coverage [2, Section 5.3], training stability [2, Appendix D]). \n\nFurthermore, to show that our DDGAN-based ASBM procedure is stable, we additionally provide the plot of FID during the training of all D-IMF iterations in Figure 3 of the **attached pdf**. We provide FID vs the number of generator gradients updated during the training. We highlight the starts of each D-IMF iteration by the vertical lines. We observe the stable decrease of FID over the generator gradient update steps. \n\nWe will remove the phrase \"we do not treat it to a be a serious limitation\" in the final version.\n\n**Concluding remarks**. \nWe would be grateful if you could let us know if the explanations we gave have been satisfactory in addressing your concerns about our work. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.\n\n**References** are the same as you used in the corresponding weaknesses and questions.", "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": "m9UkaznOoO", "reviewer": "Reviewer_xGf5", "summary": "The paper extends the recently proposed Schrödinger Bridge method, DSBM or IMF, to a discrete-time setup. This extension is non-trivial, requiring appropriate notation for discrete reciprocal and Markovian projections. By leveraging the structure of Brownian bridges used in the original IMF, the paper achieves efficient sampling and inference processes. The discrete-time formulation also establishes a natural connection to other generative model classes, such as GANs. Experiments were conducted on low-dimensional Gaussians and unsupervised image-to-image translation.", "questions": "- The notions of $T$ and $M$ are slightly confusing. For example, In L73 and L76, both represent Markovian processes. Is there a typo?", "limitations": "Limitations were addressed in Appendix A.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The discrete-time construction of IMF is an elegant piece derived from the DSBM, which heavily relies on Brownian bridges. The analytic solutions for Gaussians may also be of interest to readers from other domains.\n\n- The connection to DD-GANs is both straightforward and clever. It's exciting to see how different classes of generative models are starting to overlap and combine, leveraging the strengths of each approach.", "weaknesses": "- I enjoyed reading the majority of the paper—except, perhaps oddly, the paper title. Over 85% of the main technical contributions (Sec 3) stem from the construction of the discrete IMF formulation. The concept of \"adversarial\" appears only as one (of many plausible) implementation of Eq (15). I feel like the \"discrete\" aspect isn't highlighted enough, or at all, in the title. This is, of course, just a personal comment rather than a criticism.\n\n- In L220, the non-saturating GAN loss should be provided in the main paper rather than being postponed to the Appendix.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer xGf5, thank you for your comments. Here are the answers to your questions and comments.\n\n**(1) I enjoyed reading the majority of the paper—except, perhaps oddly, the paper title. Over 85\\% of the main technical contributions (Sec 3) stem from the construction of the discrete IMF formulation. The concept of \"adversarial\" appears only as one (of many plausible) implementation of Eq (15). I feel like the \"discrete\" aspect isn't highlighted enough, or at all, in the title. This is, of course, just a personal comment rather than a criticism.**\n\nThe title of the work which introduces DSBM algorithm is \"Diffusion Schrödinger Bridge Matching\".  Our work is based on it but, at the same time, proposes to use adversarial training as a current main competitor of diffusion models in the field of generative models. To highlight both the relation with the prior work and the usage of adversarial training in our approach, we decide to use the title \"Adversarial Schrödinger Bridge Matching\".\n\n**(2) In L220, the non-saturating GAN loss should be provided in the main paper rather than being postponed to the Appendix.**\n\nThank you for this suggestion. We initially placed the non-saturating GAN loss in the Appendix to keep the main text focused and avoid introducing technical details too early. Furthermore, the non-saturating GAN loss is rather standard approach in GANs, so we did not pay much attention to it. However, we agree that it would be beneficial to mention the non-saturating GAN loss in the main paper for clarity. We will move it to the main text in the final version and refer readers to the Appendix for a more detailed description of our adversarial procedure. \n\n**(3) The notions of T and M are slightly confusing. For example, In L73 and L76, both represent Markovian processes. Is there a typo?**\n\nIn this context, the process $T$ and $M$ are the same Markovian process. We will replace $T$ with $M$ in the final version. Thank you for mentioning it. \n\n**Concluding remarks**. \nWe would be grateful if you could let us know if the explanations we gave have been satisfactory in addressing your concerns and questions about our work.  We are also open to discussing any other questions you may have.", "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": "pERAJgtKmg", "reviewer": "Reviewer_RgPk", "summary": "This paper proposes the Discrete Iterative Markovian Fitting (D-IMF) method. Specifically, this work introduces discretized reciprocal properties and Markovian processes, showing that the optimal plan matches the solution of the static Schrödinger Bridge (SB) problem. Additionally, this work demonstrate that the D-IMF procedure theoretically converges to the (static) SB solution. For the implementation of the Markovian projection, the authors employ a GAN structure (DD-GAN).", "questions": "- Can the authors provide a comparison of the computational burden between your method and DSBM, particularly in terms of time and resource usage?\n- Is the time discretization handled as like DDPM style or Uniformly?\n- Did you use any pretrained model (e.g. pretrained DSBM) on I2I task?", "limitations": "The limitation is adequately discussed in the paper.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is theoretically fruitful. This work extends the reciprocal and Markovian properties into discrete sense, and also derive that the D-IMF method converges to static SB solution. This theoretical result is non-trivial and theoretically significant.\n- Integrating these concepts into an adversarial framework is an innovative idea, adding value to the research.", "weaknesses": "- The algorithm seems to be ineffective. It requires two discriminator and two generator networks, which seems to require significant computational resource as well as significant computational time. It would be beneficial to compare the computational burden (both training time and gpu occupancy) with that of DSBM. Moreover, since it employs two adversarial networks, it would be hard to stabilize. It would be valuable to show its (empirical sense) stability through various ablation studies.\n\n- The paper lacks of empirical studies. The practical experiment is only conducted on Male-Female data. Moreover, the paper lacks ablation studies. For example, the important hyperparameter such as the initial distribution setup (simple distribution produce vs minibatch matching), the number of function evaluations (NFE, DDGAN steps), and the number of phase (K) are not explored.\n\n- The FID for Male-to-Female translation is reported, but not for the reverse direction. Since bidirectional learning is being conducted, it would be valuable to report results for both directions. Moreover, there is no metric provided to measure if the images are transformed into ones that are perceptually close. It would be useful to compare c-FID or Wasserstein distance with DSBM.\n\nMinor comments:\nThere is a typo in Line 224: the term $q(x_{t_n}|x_{t_{n-1}})$.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer RgPk, thank you for your comments. Here are the answers to your questions and comments.\n\n**(1) Computational burden and stability study.**\n\nAs requested, we provide a study on the effectiveness of ASBM and compare it with the DSBM. \n\n**Number of parameters.** Our ASBM generator has about 42 million parameters, while each discriminator has 27 million. The generator of DSBM has 38 million parameters. Thus, generators have similar capacities.\n\n**Training time.** We mentioned (lines 714-716) that for Celeba setups, our ASBM uses 7 days of training on a A100 GPU, while the DSBM uses 5 days of training. The time is comparable since both methods use the same number of IMF and D-IMF iterations.\n\nPlease note that we provide an analysis of ASBM (ours) and DSBM algorithms in Appendix E, where we show that additional iterations only slightly change the results.\n\n**Inference time.** We measure the inference time of both ASBM (ours) and DSBM algorithms on the same hardware with one A100 GPU. We measure the mean and standard deviation over 16 runs. For a batch of 100 images, ASBM requires about 842 ms (std 97 ms), while DSBM requires 33870 ms (std 47 ms). **Thus, our algorithm provides up to 40x speedup compared to DSBM.**\n\n**GPU memory consumption.** On the Celeba setup, our algorithm requires about 25 GB of GPU for training with a batch size of 32 and about 12 GB on the inference stage to translate a batch of 100 images. Meanwhile, DSBM requires 34 GB of GPU for training with batch size 32 and about 26.7 GB on the inference stage for a batch of 100. Thus, our algorithm utilizes significantly less GPU memory.\n\n**Stability of training.**\nTo show that our procedure is stable, we provide the plot of FID during the training of all D-IMF iterations in Figure 3 of the **attached pdf**. On the plot, we provide FID vs the number of generator gradients updated during the training. We highlight the starts of each D-IMF iteration by the vertical lines. We observe stable decreases of FID.\n\n**(2) Lack of empirical and ablation studies.**\n\n**The number of phases K.** We provide an analysis of D-IMF/IMF iterations (K) in Appendix E.1. of our work. \n\n**Independent vs minibatch coupling.**\nAs per request, we provide a study on running ASBM with different coupling. We provide the additional results for the Colored MNIST setup with $\\epsilon=10$ (which we discuss in Appendix C.3 of our paper). In Figure 1 of **attached pdf**, we provide qualitative generation results for our ASBM model with independent and minibatch couplings after 3 D-IMF iterations, which we used in our work for this setup. We also provide FID values in Table 1 of **attached pdf**. We see comparable visual quality as well as similarity for both results; the FID metric is slightly better for minibatch case.\n\n**Study on different NFE.**\nAs requested, we present the study's results on different NFE of our ASBM model. In our study, we use our ASBM model trained with $4$ steps for Celeba male to female with $\\epsilon=1.0$. We test this model with different numbers of NFE $\\in [1, 2, 3, 4, 8, 16, 32]$. In all the cases, we use a uniform schedule. (See answer 4 for details). We present qualitative results of generation for several inputs in Figure 2 of the **attached pdf**. We also present the FID values computed on the test set and average $L_2$ cost $c(x_0,x_1) = \\frac{||x_0-x_1||^2}{2}$ between the input image $x_0$ and generated image $x_1$ to assess similarity in Table 2 of the **attached pdf**. Interestingly, our model with $\\text{NFE}=3$ provides almost the same FID value as for $\\text{NFE}=4$, which we use at the training, but significantly lower $L_2$ cost.\n\n**Training with different NFE.**\nSince the time of the rebuttle and our resources are limited we train ASBM with $NFE=2$ and $NFE=8$ with $K=2$ D-IMF iterations and $\\epsilon=1.0$ using the same procedure as before. We present samples in the Figure 5 together with ASBM model with $NFE=4$ after $K=2$ D-IMF iteration trained before. Male-to-female metrics:\n\n||NFE=2|NFE=4|NFE=8|\n|-|-|-|-|\n|FID|11.829|16.08|29.58|\n|LPIPS|0.214|0.253|0.241|\n\nWe observe better results for lower NFE, possibly because the model needs to learn fewer transitional distributions. We will add all the results to the final version.\n\n**(3) The FID of the reverse direction.**\n\nThank you for this suggestion. The values are given below.\n\n||$\\epsilon=1$|$\\epsilon=10$|\n|-|-|-|\n|DSBM [2]|24.06|92.15|\n|ASBM (ours)|16.86 |17.44|\n\nThus, the FID values for the reverse direction show the same behavior as for the forward direction.\n\n**(4) Measurement of perceptual quality**\n\nAs requested, we measure perceptual similarity between input and translated images using standard LPIPS metric [1] (as used in DSBM). For our Celeba male-to-female setup, we measure LPIPS between input and translated images from the test. We present the results below (the lower the better):\n\n|| $\\epsilon=1$ | $\\epsilon=10$ |\n|-|-|-|\n|DSBM [2]|0.246| 0.386|\n|ASBM (ours)|0.242|0.294|\n\nWe observe almost the same similarity in the case of $\\epsilon=1$ and better similarity for ASBM (ours) in the case of $\\epsilon=10$. We will add these results in the final version.\n\n**(5) Time discretization.**\n\nWe use uniform time discretization in all our experiments, i.e., for the number of inner times points $N$, we use time discretization $t_{n} = \\frac{n}{N+1}$ for $n \\in [0, N+1]$. \n\n**(6) Did you use any pretrained model?**\n\nNo, we do not use any pretrained models in our experiments.\n\n**Concluding remarks**. \nWe would be grateful if you could let us know if the explanations we gave have been satisfactory in addressing your concerns and questions about our work. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.\n\nReferences:\n\n[1] Zhang R. The unreasonable effectiveness of deep features as a perceptual metric. CVPR, 2018.\n\n[2] Shi Y. Diffusion Schrödinger bridge matching. NeurIPS, 2024.", "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": "sSmyfXM6V6", "reviewer": "Reviewer_LCNa", "summary": "This paper proposes Discrete-time iterative Markovian Fitting, which is an efficient alternative that greatly reduces NFE from IMF (Shi et al. 2023). This was possible due to the tractability of Brownian bridges for any subintervals, so discretization of IMF does not hurt the performance of if the model can be trained on it. For training a generative model, the authors adopted a time-dependent GAN called Denoising Diffusion Generative Adversarial Models (DD-GAN). Compared to DSBM, the proposed ASBM shows significantly low computational costs with good FID scores.", "questions": "How do the authors suspect that the FID score is very low for ASBM? Following the arguments of [1], could the authors report CMMD scores or other metrics for results in Fig. 3?   \n\n[1] https://arxiv.org/pdf/2401.09603", "limitations": "The authors adequately addressed the limitations", "rating": 6, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, I think this paper is well written and quality of presenting idea and theoretical statements are very good. This work also contains certain contributions for probabilistic generative models.\n- This work broadens our understanding of IMF, reference measures, and the application of Brownian bridges. \n- The results verified that GANs hold as a good nonlinear probabilistic generative model training method. \n- Theoretical statements were clear and completely proved.", "weaknesses": "- I strongly believe the performance gains are mostly from applying GAN architecture, since D-IMF and IMF indicates essentially the same learning scheme.  Therefore, it is indeed an improvement from DSBM, but novelty of this work as a distinct SB study might not be significant from the perspective of diffusion models. \n- This model does not always yield state-of-the-art scores for syntactic datasets (see Table 1&2).\n- While I understand this work for some extent, I do not clearly understand the motivation why do we need a sampling-based approach when we have sampling-free method such as SF^2M and LightSB [1] models\n\nTypo\n- L213: Denoising Denoising GANs\n\n[1]\n@inproceedings{\nkorotin2024light,\ntitle={Light Schr\\\"odinger Bridge},\nauthor={Alexander Korotin and Nikita Gushchin and Evgeny Burnaev},\nbooktitle={The Twelfth International Conference on Learning Representations},\nyear={2024},\nurl={https://openreview.net/forum?id=WhZoCLRWYJ}\n}", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer LCNa, thank you for your comments. Here are the answers to your questions and comments.\n\n**(1) I strongly believe the performance gains are mostly from applying GAN architecture, since D-IMF and IMF indicates essentially the same learning scheme. Therefore, it is indeed an improvement from DSBM, but the novelty of this work as a distinct SB study might not be significant from the perspective of diffusion models.**\n\nIt is not possible to just apply GAN architecture in the original continuous IMF framework [2] since their Markovization objective is based on bridge-matching and implies learning a drift function of SDE and not transitional densities of a discrete process. The main cause of it is that the continuous IMF framework is based on the known property that SB is a unique **continuous** Markovian and reciprocal process. \n\nThe novelty of our work as a distinct SB study comes from our main theorem (Theorem 3.1). We show that to solve the SB problem, it is enough to find a probability distribution $\\pi(x_0, x_{t_{1}}, \\dots, x_{t_{N}}, x_1)$ with **any** number $N\\geq 1$ of intermediate points, which is Markovian, discrete reciprocal and shares the marginals $p(x_0)$ and $p(x_1)$. Even $N=1$ is enough in our framework, i.e., it is enough to find a joint distribution with three variables $\\pi(x_0, x_t, x_1)$ and $t \\in (0, 1)$, which is Markovian and discrete reciprocal to solve SB. \n\nWe believe that our main theorem opens the door to a novel family of algorithms (including ASBM) for solving SB, based on finding a discrete probability distribution $\\pi(x_0, x_{t_{1}}, \\dots, x_{t_{N}}, x_1)$ that is Markovian and discrete reciprocal to solve the SB problem. \n\nWe also believe that this theorem may facilitate the theoretical study of this problem, including the study on the convergence speed of D-IMF and IMF algorithms since it allows us to work with a more simple object — joint probability distribution instead of continuous stochastic processes. For instance, we show that it is possible to derive closed-form Markovian and reciprocal projections for a multivariate Gaussian case in our framework. In contrast, the continuous counterpart, form solution, is derived only for a 1-dimensional case [2].\n\n**(2) This model does not always yield state-of-the-art scores for syntactic datasets (see Table 1\\&2).**\n\nOur algorithm does not beat all other competitors on all the setups. However, it provides the best results on most considered setups by both plan and target metrics (Tables 1 and 2).\n\n**(3) While I understand this work for some extent, I do not clearly understand the motivation why do we need a sampling-based approach when we have sampling-free method such as SF$^2$M [3] and LightSB [1] models.**\n\nIn the limitation section, the LightSB [1] authors state that their algorithm is not designed for large-scale generative problems. Furthermore, they do not provide experiments in the image space, and they only consider image-to-image translation in the latent space of the autoencoder. Other experiments with real data include only single-cell data, which has lower dimensionality than images and does not lie on the low-dimensional manifold-like image distribution. \n\nThe authors of SF2M [3] also consider only single-cell problems. Furthermore, while SF$^2$M [3] is indeed simulation-free, it theoretically guarantees convergence to SB only if samples from the ground-truth static SB solution $\\pi^*(x_0, x_1)$ are used for training. However, if one has access to the $\\pi^*(x_0, x_1)$, the problem itself is no longer unpaired domain translation since there is access to the ground-truth paired data. The authors of [3] address this problem by approximation of $\\pi^*(x_0, x_1)$ by minibatch Optimal Transport techniques, which seems to provide a good approximation in the considered single-cell setups but is known to have a bias growing with the dimensionality of the considered space. For instance, no experiments show that this approximation works well in the case of high-dimensional unpaired image-to-image setups. \n\nThus, while each of these simulation-free algorithms has its advantages and works well in the setups for which they were designed, there is still no simulation-free algorithm that can demonstrate good quality on the unpaired image-to-image translation without using any autoencoders. \n\n**(4) How do the authors suspect that the FID score is very low for ASBM? Following the arguments of [1], could the authors report CMMD scores or other metrics for results in Fig. 3?**\n\nSince it is hard to judge quality given only by the absolute values of FID, we compare our method with the DSBM and see that it provides better FID. As per your request, we measure the CMMD score and present the results below:\n\n|             | $\\epsilon=1$ | $\\epsilon=10$ |\n|-------------|--------------|---------------|\n| DSBM [2]    | 0.365        | 1.140         |\n| ASBM (ours) | 0.216        | 0.231         |\n\nWe observe that CMMD scores align with FID scores presented in our work.\n\n**(5) Typo L213: Denoising Denoising GANs**\n\nThank you for pointing out this. We will fix it in the final version.\n\n**Concluding remarks**. \nWe would be grateful if you could let us know if the explanations we gave have been satisfactory in addressing your concerns about our work. If so, we kindly ask that you consider increasing your rating. We are also open to discussing any other questions you may have.\n\nReferences:\n\n[1] Alexander Korotin. Light schrödinger bridge. ICLR, 2024.\n\n[2] Shi et al. (2023) -- Diffusion Schrodinger Bridge Matching\n\n[3] Tong A. Y. et al. Simulation-Free Schrödinger Bridges via Score and Flow Matching AISTATS 2024.", "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:
Dear reviewers, thank you all for taking the time to review our paper and for your thoughtful feedback. We are delighted that you found our work to be well-written and that it effectively introduces key concepts (xtga, LCNa). It is encouraging to hear that you see it as theoretically fruitful (RgPk), that it broadens our understanding of IMF (LCNa), and that it introduces a straightforward and clever connection between IMF and DD-GANs (xGf5). We are also pleased that you believe our work allows for easier spreading of DSBM-related methodology and developing more flexible algorithms (xtga).

Please find the answers to your questions below. **Please note that we have added tables and figures in the attached pdf to support our responses to the reviewers xtGA and RgPk.**

Meta-review: This paper proposes a novel Discrete-time Iterative Markovian Fitting (D-IMF) procedure for solving the Schrödinger Bridge (SB) problem, aiming to improve the efficiency of the existing Iterative Markovian Fitting (IMF) method. By leveraging the Denoising Diffusion GAN (DD-GAN) framework, the authors achieve comparable results to IMF with fewer steps.

Reviewers generally agree that the paper presents an interesting and principled approach, and it is well-written. However, to maximize the paper's impact, the authors should carefully address the remaining concerns raised by reviewers in the final version. The experimental section should be strengthened to include detailed comparisons with DSBM, ablation studies, and experiments beyond the Celeba dataset. Additionally, the computational complexity of the method, particularly the use of two discriminator and two generator networks, should be discussed more thoroughly, as this might be practically costly and potentially lead to instability during adversarial training.
Decision: Accept (poster)