Adversarial Schrödinger Bridge Matching

Authors: nikita  gushchin
  skoltech
daniil  selikhanovych
sergei  kholkin
evgeny burnaev skoltech
alexander  korotin skoltech
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-14 15:36:58

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.

Main paper: 
 introduction: 
 Recent generative models based on the Flow Matching [27] and Rectified Flows [30] show great potential as a successor of classical denoising diffusion models such as DDPM [15] . Both these approaches consider the same problem of learning an Ordinary Differential Equation (ODE) that interpolates one given distribution to the other one, e.g., noise to data. Thanks to 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 Diffusion 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 to use many steps of numerical solvers.Contributions. This paper addresses the above-mentioned limitation of the existing Iterative Markovian 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 to solve the Schrödinger Bridge problem without relying on Stochastic Differential Equations. This approach significantly simplifies the inference process, enabling it 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, 4.1). 3. Practice. For general data distributions available by samples, we propose an 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. In the paper, we simultaneously work with the continuous stochastic processes and discrete stochastic processes in the D-dimensional Euclidean space R D . We denote by P(C([0, 1]), R D ) the set of continuous stochastic processes with time t ∈ [0, 1], i.e., the set of distributions on continuous trajectories f : [0, 1] → R D . We use dW t to denote the differential of the standard Wiener process.To establish a link between continuous and discrete stochastic processes, we fix N ≥ 1 intermediate time moments 0 = t 0 < t 1 < • • • < t N < t N +1 = 1 together with t 0 = 0 and t N +1 = 1. We consider discrete stochastic processes with those time-moments as the elements of the set P(R D×(N +2) ) of probability distributions on R D×(N +2) . Among such discrete processes, we are specifically interested in subset P 2,ac (R D×(N +2) ) ⊂ P(R D×(N +2) ) of absolutely continuous distributions on R D×(N +2) which have a finite second moment and entropy. For any such q ∈ P 2,ac (R D×(N +2) ), we write q(x 0 , x t1 , . . . , x t N +1 ) to denote its density at a point (x 0 , x t1 , . . . , x t N , x 1 ) ∈ R D×(N +2) . For continuous process T , we denote by p T ∈ P(R D×(N +2) ) the discrete process which is the finite-dimensional projection of T to time moments 0 = t 0 < t 1 < • • • < t N < t N +1 = 1. For convenience we also use the notation x in = (x t1 , . . . , x t N ) to denote the vector of all intermediate-time variables. In what follows, KL is a short notation for the Kullback-Leibler divergence. 
background: 
 We start with recalling the Bridge Matching and Iterative Propotional Fitting procedures developed for continuous-time stochastic processes ( 2.1) . Next, we discuss the Schrödinger Bridge problem, the solution to which is the unique fixed point of Iterative Markovian Fitting procedure ( 2.2).Modern diffusion and flow generative modeling are mainly about the construction of a model that interpolates one probability distribution p 0 ∈ P 2,ac (R D ) to some another probability distribution p 1 ∈ P 2,ac (R D ). One of the general approaches for this task is the Bridge Matching [29, 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 ϵ with the volatility ϵ which start at p 0 , i.e., the process given by the SDE: dx t = √ ϵdW t , x 0 ∼ p 0 . Let W ϵ |x0,x1 denote the stochastic process W ϵ conditioned on values x 0 , x 1 at times t = 0, 1, respectively. This process W ϵ |x0,x is called the Brownian Bridge [17, Chapter 9] . For some q(x 0 , x 1 ) ∈ P 2,ac (R D×2 ) with q(x 0 ) = p 0 (x 0 ) and q(x 1 ) = p 1 (x 1 ) the process T q def = W ϵ |x0,x1 dq(x 0 , x 1 ) is called the mixture of Brownian Bridges. Following [47] , we say that mixtures of Brownian Bridges form a reciprocal class of processes (for the Brownian Bridge). For brevity, we call these processes just reciprocal processes.Bridge matching [29, 31] . The goal of Bridge Matching (with the Brownian Bridge) is to construct continuous-time Markovian process M from p 0 to p 1 in the form of SDE: dx t = v(x t , t)dt + √ ϵdW t . This is achieved by using the Markovian projection of a reciprocal process T q = W ϵ |x0,x1 dq(x 0 , x 1 ), which aims to find the Markovian process M which is the most similar to T q in the sense of KL:proj M (T q ) def = arg min M ∈M KL (T q ∥M ) ,where M ⊂ P(C([0, 1]), R D ) is the set of all Markovian processes. For the Brownian Bridge W ϵ |x0,x1 it is known [47, 11] that the SDE and the drift v(x t , t) of proj M (T q ) is given by:dx t = v(x t , t)dt + √ ϵdW t , v(x t , t) = x 1 -x t 1 -t p Tq (x 1 |x t )dx 1 ,where p Tq (x 1 |x t ) the conditional distribution of the stochastic process T q at time moments t and 1.The process proj M (T q ) has the same time marginal distributions p Tq (x t ) as the original Brownian bridge mixture T q . However, the joint distribution p Tq (x 0 , x 1 ) of T q and the joint distribution p proj M (Tq) (x 0 , x 1 ) of its projection proj M (T q ) do not coincide in the general case [6] , see Figure 2 . Iterative Markovian Fitting [47, 35, 1] . The Iterative Markovian Fitting procedure introduces a second type of projection of continuous-time stochastic processes called the Reciprocal projection.For a process T , it is is defined by proj R (T ) = W ϵ |x0,x1 dp T (x 0 , x 1 ), see illustrative Figure 3 . T 2l+1 = proj M (T 2l ), T 2l+2 = proj R (T 2l+1 ),It is known that the procedure converges to the unique stochastic process T * , which is known as a solution to the Schrödinger Bridge (SB) problem between p 0 and p 1 . Furthermore, the SB T * is the only process starting at p 0 and ending at p 1 that is both Markovian and reciprocal [25] .Schrödinger Bridge problem. The Schrödinger Bridge problem [44] was proposed in 1931/1932 by Erwin Schrödinger. For the Wiener prior W ϵ Schrödinger Bridge problem between two probability distributions p 0 ∈ P 2,ac (R D ) and p 1 ∈ P 2,ac (R D ) is to minimize the following objective:EQUATIONwhere F(p 0 , p 1 ) ⊂ P(C([0, 1]), R D ) is the subset of stochastic processes which starts at distribution p 0 (at the time t = 0) and end at p 1 (at t = 1). The Scrhödinger Bridge has a unique solution, which is a diffusion process T * described by the SDE: [25] . The process T * is called the Schrödinger Bridge and v * : R D × [0, 1] → R D is called the optimal drift.dX t = v * (X t , t)dt + √ ϵdW tFrom the practical point of view, the solution to the SB problem T * tends to preserve the Euclidean distance between start point x 0 and endpoint x 1 . The equivalent form of SB problem, the static Schrödinger Bridge problem, explains this property more clearly.Static Schrödinger Bridge problem. One may decompose KL(T ||W ϵ ) as [51, Appendix C]:EQUATIONi.e., KL divergence between T and W ϵ is a sum of two terms: the 1st represents the similarity of the processes' joint marginal distributions at start and finish times t = 0, 1, while the 2nd term represents the average similarity of conditional processes T (1) = minEQUATIONwhere Π(p 0 , p 1 ) ⊂ P 2,ac (R D×2 ) is the set of joint probability distributions with marginal distributions p 0 and p 1 . Thus, the initial Schrödinger Bridge problem can be solved by optimizing only over a reciprocal process's joint distribution q(x 0 , x 1 ) at t = 0, 1. This problem is called the Static Schrödinger Bridge problem. In turn, the problem can be rewritten in the following way [12, Eq. 7] :EQUATIONi.e., as finding a joint distribution q(x 0 , x 1 ) which tries to minimize the Euclidian distance ||x-y|| 2 2 between x 0 and x 1 (preserve similarity between x 0 and x 1 ), but with the addition of entropy regularizer ϵ • Entropy(q) with the coefficient ϵ. Thus, the coefficient ϵ > 0, which is the same for all problems considered above, regulates the stochastic or diversity of samples from q(x 0 , x 1 ). The last problem (4) is also known as the entropic optimal transport (EOT) problem [4, 38, 25] .Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes]Justification: Limitations are discussed in Appendix A Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
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 T * [25] . We derive an analogous theoretical result but for processes in discrete time. We provide proofs for all the theorems and propositions in Appendix B.In 3.1, we give preliminaries on discrete processes with Markovian and reciprocal properties. In 3.2, we present the main theorem of our paper, which is the foundation of our Discrete-time Iteratime Markovian Fitting (D-IMF) framework. In 3.3, we describe D-IMF procedure itself and prove that it 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.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 ϵ |x0,x1 , which is given by:EQUATIONEQUATIONThis joint distribution p W ϵ (x t1 , . . . , x t N |x 0 , x 1 ) defines a discrete stochastic process, which we call a discrete Brownian bridge. In turn, we say that a distribution q ∈ P 2,ac (R D×(N ×2) ) is a mixture of discrete Brownian bridges if it satisfies q(x 0 , x t1 , . . . , x t N , x 1 ) = p W ϵ (x t1 , . . . , x t N |x 0 , x 1 )q(x 0 , x 1 ),where q(x 0 , x 1 ) denotes its joint marginal distribution of q at times 0, 1. That is, its "inner" part at times t 1 , . . . , t N is the discrete Brownian Bridge. We denote the set of all such mixtures as R(N ) ⊂ P 2,ac (R D×(N +2) ) and call them discrete reciprocal processes.Discrete Markovian processes. We say that a discrete process q ∈ P 2,ac (R D×(N +2) ) is Markovian if its density can be represented in the following form (recall that t 0 = 0, t N +1 = 1):EQUATIONWe denote the set of all such discrete Markovian processes as M(N ) ⊂ P 2,ac (R D×(N +2) ).Theorem 3.1 (Discrete Markovian and reciprocal process is the solution of static SB). Consider any discrete process q ∈ P 2,ac (R D×(N +2) ), which is simultaneously reciprocal and markovian, i.e. q ∈ R(N ) and q ∈ M(N ) and has marginals q(x 0 ) = p 0 (x 0 ) and q(x 1 ) = p 1 (x 1 ):q(x 0 , x t1 , . . . , x t N , x 1 ) = p W ϵ (x t1 , . . . , x t N |x 0 , x 1 )q(x 0 , x 1 ) = q(x 0 ) N +1 n=1 q(x tn |x tn-1 ),Then q(x 0 , x t1 , . . . , x t N , x 1 ) = p T * (x 0 , x t1 , . . . , x t N , x 1 ), i.e., it is the finite-dimensional projection of the Schrödinger Bridge T * to the considered times. Moreover, its joint marginal q(x 0 , x 1 ) at times t = 0, 1 is the solution to the static SB problem (4) between p 0 and p 1 , i.e., q(x 0 , x 1 ) = p T * (x 0 , x 1 ).Thus, to solve the static SB problem, it is enough to find a Markovian mixture of discrete Brownian bridges. To do so, we propose the Discrete-time Iterative Markovian Fitting (D-IMF) procedure.Similar to the IMF procedure, our proposed Discrete-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 q ∈ P 2,ac (R D×(N +2) ) is a discrete stochastic process. Then the reciprocal projection proj R (q) is a discrete stochastic process with the joint distribution given by: proj R (q) (x 0 , x t1 , . . . , x t N , x 1 ) = p W ϵ (x t1 , . . . , x t N |x 0 , x 1 )q(x 0 , x 1 ).This projection takes the joint distribution of start and end points q(x 0 , x 1 ) and inserts the Brownian Bridge for intermediate time moments, see Figure 4 . The prop. below justifies the projection's name.Figure 4 : Reciprocal projection of a discrete stochastic process q, i.e., r(x 0 , x t1 , ..., x t N , x 1 ) = p W ϵ (x t1 , ..., x t N |x 0 , x 1 )q(x 0 , x 1 ).Under mild assumptions, the reciprocal projection proj R (q) of a stochastic discrete process q ∈ P 2,ac (R D×(N +2) ) is the unique solution for the following optimization problem: proj R (q) = arg min r∈R(N ) KL (q∥r) .Similarly to the discrete reciprocal projection, we introduce discrete Markovian projection. Definition 3.4 (Discrete Markovian Projection). Assume that q ∈ P 2,ac (R D×(N +2) ) is a discrete stochastic process. The Markovian projection of q is a discrete stochastic process proj M (q) ∈ P 2,ac (R D×(N +2) ) whose joint distribution given by: proj M (q) (x 0 , x t1 , ..., x t N , x 1 ) = q(x 0 )EQUATION)Despite it is possible to use any discrete stochastic process q 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 a visualization of the markovian projection in Figure 5 . 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 M (q) of a stochastic discrete process q ∈ P 2,ac (R D×(N +2) ) is a unique solution to the following optimization problem: proj M (q) = arg minEQUATION)Now we are ready to define our D-IMF procedure. For two given distributions p 0 ∈ P 2,ac (R D ) and p 1 ∈ P 2,ac (R D ) at times t = 0 and t = 1, respectively, it starts with any discrete Brownian mixture p W ϵ (x t1 , . . . , x t N |x 0 , x 1 )q(x 0 , x 1 ), where q(x 0 , x 1 ) ∈ Π(p 0 , p 1 ) ∩ P 2,ac (R D×2 ). Then, it constructs the following sequence of discrete stochastic processes: q 2l+1 = proj M (q 2l ), q 2l+2 = proj R (q 2l+1 ). (12) Theorem 3.6 (D-IMF procedure converges to the the Schrödinger Bridge). Under mild assumptions, the sequence q l constructed by our D-IMF procedure converges in KL to p T * . In particular, q l (x 0 , x 1 ) convergence to the solution p T * (x 0 , x 1 ) of the static SB. Namely, we have lim l→∞ KL q l ∥p T * = 0, and lim l→∞ KL q l (x 0 , x 1 )∥p T * (x 0 , x 1 ) = 0.In this section, we show that our D-IMF updates (12) can be derived in the closed form for the Gaussian case. Let p 0 = N (x 0 |µ 0 , Σ 0 ) and p 1 = N (x 1 |µ 1 , Σ 1 ) be Gaussians. Consider any initial discrete Gaussian process q ∈ P 2,ac (R D×(N +2) ) that has joint distribution q(x 0 , x 1 ) ∈ Π(p 0 , p 1 ):EQUATIONwhere Σ ∈ R 2D×2D is positive definite and symmetric and Σ cov is the covariance of x 0 and x 1 . 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 ∈ R N D×2D and K ∈ R N D×N D :U def =     (1 -t1)ID t1ID (1 -t2)ID t2ID . . . . . . (1 -tN )ID tN ID     , K def =     t1(1 -t1)ID t1(1 -t2)ID . . . t1(1 -tN )ID t1(1 -t2)ID t2(1 -t2)ID . . . t2(1 -tN )ID . . . . . . . . . . . . t1(1 -tN )ID t2(1 -tN )ID . . . tN (1 -tN )ID    Here I D is an identity matrix with the shape D × D. Below we present updates for both projections.Theorem 3.7 (Reciprocal projection of a process whose joint marginal distribution is Gaussian).Assume that q ∈ P 2,ac (R D×(N +2) ) has Gaussian joint distribution q(x 0 , x 1 ) given by (13) . ThenEQUATIONTheorem 3.8 (Markovian projection of a discrete Gaussian process). Assume that q ∈ P 2,ac (R D×(N +2) ) is a discrete Gaussian process with q(x 0 , x 1 ) given by (13) and the densityq(x in , x 0 , x 1 ) = N ( x in x 01 | µ in µ 01 , Σ R ), µ in = (µ t1 , . . . , µ t N ),where µ in and Σ R are some parameters of q. Then its Markovian projection is given by:[proj M q](x in , x 0 , x 1 ) = q(x 0 ) N +1 n=1 q(x tn |x tn-1 ), q(x tn |x tn-1 ) = N (x tn | µ tn (x tn-1 ), Σ tn ), µ tn (x tn-1 ) = µ tn + ( Σ R ) tn,tn-1 (( Σ R ) tn-1,tn-1 ) -1 (x tn-1 -µ tn-1 ), Σ tn = ( Σ R ) tn,tn -( Σ R ) tn,tn-1 (( Σ R ) tn-1,tn-1 ) -1 (( Σ R ) tn,tn-1 ) T .In turn, the joint distribution [proj M q](x 0 , x 1 ) is given by[proj M q](x 0 , x 1 ) = N ( x 0 x 1 | µ 0 µ 1 , Σ 0 Σ 01 (Σ 01 ) T Σ 1 ), Σ T 01 = N +1 n=1 ( Σ R ) tn+1,tn (( Σ R ) tn,tn ) -1 Σ 0 .Here ( Σ R ) ti,tj is the submatrix of Σ R denoting the covariance of x ti and x tj , while Σ 0 and Σ 1 are covariance matrices of x 0 and x 1 , respectively.Thus, if we start D-IMF from some discrete process q 0 with marginals q 0 (x 0 ) = p 0 (x 0 ), q 0 (x 1 ) = p 1 (x 1 ) and Gaussian q(x 0 , x 1 ), then at each iteration of our D-IMF procedure q l will be discrete Gaussian process with the same marginals and eventually will converge to q * . In 4.1, we use our derived closed-form to perform an experimental analysis of D-IMF's convergence depending on the number of intermediate time moments N and the value of coefficient ϵ.To implement our D-IMF procedure in practice, one should choose the process q 0 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 x 1 (or trajectories x 0 , x t1 , . . . , x t N , x 1 ) given a starting point x 0 (= x t0 ). Thus, to solve SB between p 0 (x 0 ) and p 1 (x 1 ) one should choose q 0 to have start and end marginals q 0 (x 0 ) = p 0 (x 0 ) and q 0 (x 1 ) = p(x 1 ) accessible by samples.Implementation of the discrete reciprocal projection. The reciprocal projection (8) of a given discrete process q(x 0 , x in , x 1 ) is easy if one can sample from q(x 0 , x 1 ). To sample from proj R (q) it is enough to sample first a pair (x 0 , x 1 ) ∼ q(x 0 , x 1 ) and then sample intermediate points x t1 , . . . , x t N from the Brownian bridge p W ϵ (x t1 , . . . , x t N |x 0 , x 1 ). 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 (10) of a reciprocal process q ∈ R(N ), one just needs to estimate the transition probabilities between sequential time moments, i.e., the set {q(x tn |x tn-1 )} N +1 n=1 and use the starting marginal [proj M q](x 0 ) = q(x 0 ) = p 0 (x 0 ). The natural way to find transition probabilities is to set to parametrize all these distributions as {q θ (x tn |x tn-1 )} N +1 n=1 and solveEQUATIONwhere D adv is some distance or divergence between probability distributions. In this case, a minimum of such loss is achieved when q θ (x tn |x tn-1 ) = q(x tn |x tn-1 ) for each n ∈ {1, 2, . . . , N + 1}.We note that a 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 a given reciprocal process. Overall, the authors show that problems like (15) can be efficiently approached via time-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 q θ (x tn |x tn-1 ) via a time-conditioned generator network G θ (x tn-1 , z, t n-1 ). As in DD-GAN, we use the non-saturating GAN loss [10] as D adv , which optimizes softened reverse KL-divergence [46] . To optimize this loss, an additional conditional discriminator network D(x tn-1 , x tn , t n-1 ) is 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 D.1.Note that after learning {q θ (x tn |x tn-1 )} N +1 n=1 the sampling assumes to take sample from q 0 (x 0 ) = p(x 0 ) and then sample from {q θ (x tn |x tn-1 )} N +1 n=1 . Hence it is guaranteed that q 0 (x 0 ) = p(x 0 ), but there may be an approximation error in estimating q 1 (x 1 ) ≈ p(x 1 ). This is due to the asymmetry of the definition of Markovian projection, i.e., it can be written in two equivalent ways:proj M (q) (x 0 , x t1 , ..., x t N , x 1 ) = q(x 0 ) N +1 n=1 q(x tn |x tn-1 ) = q(x 1 ) N +1 n=1 q(x tn-1 |x tn ).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[proj M q](x 1 ) = p 1 (x 1 ), parametrize {q η (x tn-1 |x tn )} N +1n=1 and solve:EQUATIONIn this case q(x 1 ) = p 1 (x 1 ) is guaranteed, while q(x 0 ) ≈ p 0 (x 0 ).Implementation of the D-IMF procedure (ASBM algorithm). We start with initialization of q 0 by the reciprocal process. Depending on the setup we use initialization with the independent coupling, i.e. q 0 (x 0 , x 1 ) = p 0 (x 0 )p 1 (x 1 ) or a minibatch OT coupling [9, 39] , 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 p 0 → p 1 and in the reverse direction p 1 → p 0 by using functionals ( 15) and ( 16) respectively to avoid the accumulation of errors due to the asymmetry in the definition of the Markovian projection. For details, see Appendix D.2. At the reciprocal projection steps, we use the model q θ (x 0 , x in , x 1 ) or q η (x 0 , x in , x 1 ) learned to approximate q 2l+1 to sample pair (x 0 , x 1 ) and then sample intermediate points from Brownian bridge. We use the term outer iteration (K) for a sequence of two reciprocal projections and two Markovian projections in different directions.There exists a variety of algorithms for learning SB based on different underlying principles: dual form entropic optimal transport algorithms [5, 34, 24, 11, 12, 45] , iterative proportional fitting (IPF) algorithms [51, 7, 2] , bridge matching [49, 29] and iterative Markovian fitting (IMF) algorithms [47, 35, 28, 20] , adversarial algorithms [21] , etc. We refer to [13] for a benchmark and to [24, 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 are not only theoretically well-grounded but also closely connected to the rectified flow approach [30] which works well in large-scale generative modeling [32, 54] . Below we discuss the relation of our contributions ( 1) to the prior works in IMF [47, 35] .Theory I. 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 a misleading viewpoint. Indeed, the Markovian projection in the discrete time, in general, does not match with the continuous time Markovian projection. 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 D] . 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. In our case, at each D-IMF step we learn several transitional 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] .Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Answer: [Yes]• The assumptions made should be given (e.g., Normally distributed errors).• It should be clear whether the error bar is the standard deviation or the standard error of the mean. • It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified. • For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates). • If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 10. Broader Impacts Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed? Answer: [Yes] Justification: Broader impact was discussed in A Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
experiments: 
 We evaluate our adversarial SB matching (ASBM) algorithm, which implements our D-IMF procedure on setups with Gaussian distributions ( 4.1) for which we 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.1, results for the Colored MNIST dataset in Appendix C.3, 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.We analyze the convergence of our D-IMF procedure depending on the number of intermediate time steps N ≥ 1 (we use t n = n/N + 1) and the value ϵ > 0 in the Gaussian case. In this case, the static SB solution p T * (x 0 , x 1 ) is analytically known, see, e.g., [18] . This provides us an opportunity to analyse how fast KL q l (x 0 , x 1 )∥p T * (x 0 , x 1 ) decreases when l → ∞.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 D = 16 and ϵ ∈ {1, 3, 10} for centered Gaussians p 0 = N (0, Σ 0 ) and p 1 = N (0, Σ 1 ). To construct Σ 0 and Σ 1 , 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 p 0 and p 1 for all experiments.We start our D-IMF procedure from the reciprocal process with q 0 (x 0 , x 1 ) = p 0 (x 0 )p 1 (x 1 ), i.e. from the independent joint distribution at times t = 0, 1. 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-IMF procedure empirically shows the exponential rate of convergence in all the cases. As we can see from Figure 6a , the convergence speed dependence on N quickly saturates. Thus, even several time moments, e.g., N = 5, provide quick convergence speed. From Figure 6b , we clearly see that the convergence speed is highly influenced by the chosen value of the parameter ϵ. For instance, the transition from ϵ = 1 to ϵ = 10 requires ten times more D-IMF iterations. Thus, this hyperparameter may be important in practice.To test our approach on real data, we consider the unpaired image-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 ϵ = 1 and ϵ = 10. Following the best practices of DD-GAN [53] , we use N = 3, intermediate timest 1 = 1 4 , t 2 = 2 4, t 3 = 3 4 and K = 5 outer iterations of D-IMF. We provide qualitative results and the FID metric [14] on the test set in Figures 7b and 7e We focus our comparison on the DSBM algorithm based on the IMF-procedure [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 7f . 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 E.Thus, our D-IMF procedure allows us to solve the Schrödinger Bridge efficiently without learning the time-continuous stochastic process, which in turn speeds up inference by an order of magnitude. This aligns with the results obtained in the Gaussian-to-Gaussian 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 of 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, one needs to learn several transition probabilities. We do this via adversarial learning [10] , but actually this can be 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 to 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 are discussed in Appendix A.Adversarial training. It is a 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 [53] 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 we 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 Intelligence, Machine Learning and Generative Modeling. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.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 3.2In our view, the proof of our main Theorem here is the most interesting and insightful (among all the proofs in the paper) as it uses 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 [24, 11] .Proof of Theorem 3.1. We split the proof in 2 stages. The 1st is auxiliary for the 2nd.Here we show that if some q(x 0 , x 1 ) ∈ P 2,ac (R D×2 ) with marginals p 0 (x 0 ) = q(x 0 ) and p 1 (x 1 ) = q(x 1 ) has the density in the formq(x 0 , x 1 ) = q(x 0 ) C(x 0 ) exp - ||x 1 -x 0 || 2 2ϵ ϕ(x 1 ),then it solves the Static SB between distributions p 0 (x 0 ) and p 1 (x 1 ). It is known [25] , that the solution q * (x 0 , x 1 ) def = p T * (x 0 , x 1 ) of Static SB between p 0 and p 1 has the density:q * (x 0 , x 1 ) = ψ * (x 0 ) exp - ||x 1 -x 0 || 2 2ϵ ϕ * (x 1 ).Hence, the conditional density q * (x 1 |x 0 ) is expressed as:q * (x 1 |x 0 ) = ψ * (x 0 ) p 0 (x 0 ) def =C * (x0) exp - ||x 1 -x 0 || 2 2ϵ ϕ * (x 1 ) = C * (x 0 ) exp - ||x 1 -x 0 || 2 2ϵ ϕ * (x 1 ).Thus, both q(x 0 , x 1 ) and q * (x 0 , x 1 ) have their densities in the same functional form and the same marginals q(x 0 ) = q * (x 0 ) = p 0 (x 0 ) and q(x 1 ) = q * (x 1 ) = p 1 (x 1 ). However, we want to prove that in this case q(x 0 , x 1 ) and q * (x 0 , x 1 ) are equal, i.e., KL (q * ∥q) = 0.KL (q * (x 0 , x 1 )∥q(x 0 , x 1 )) = log q * (x 0 , x 1 ) q(x 0 , x 1 ) q * (x 0 , x 1 )dx 0 dx 1 = log ¨p 0 (x 0 )q * (x1 |x 0 ) $ $ $ p 0 (x 0 )q(x 1 |x 0 ) q * (x 0 , x 1 )dx 0 dx 1 = log C * (x 0 ) $ $ $ $ $ $ $ $ exp -||x1-x0|| 2 2ϵ ϕ * (x 1 ) C(x 0 ) $ $ $ $ $ $ $ $ exp -||x1-x0|| 2 2ϵ ϕ(x 1 )q * (x 0 , x 1 )dx 0 dx 1 = (logC * (x 0 ) C(x 0 ) + log ϕ * (x 1 ) ϕ(x 1 ) )q * (x 0 , x 1 )dx 0 dx 1 = log C * (x 0 ) C(x 0 ) q * (x 0 , x 1 )dx 0 dx 1 + log ϕ * (x 1 ) ϕ(x 1 ) q * (x 0 , x 1 )dx 0 dx 1 = log C * (x 0 ) C(x 0 ) p 0 (x 0 )dx 0 + log ϕ * (x 1 ) ϕ(x 1 ) p 1 (x 1 )dx 1 = log C * (x 0 ) C(x 0 ) q(x 0 , x 1 )dx 0 dx 1 + log ϕ * (x 1 ) ϕ(x 1 ) q(x 0 , x 1 )dx 0 dx 1 = (log C * (x 0 ) C(x 0 ) + log ϕ * (x 1 ) ϕ(x 1 ) )q(x 0 , x 1 )dx 0 dx 1 = log C * (x 0 )ϕ * (x 1 ) C(x 0 ) ϕ(x 1 ) q(x 0 , x 1 )dx 0 dx 1 = log C * (x 0 ) exp -||x1-x0|| 2 2ϵ ϕ * (x 1 ) C(x 0 ) exp -||x1-x0|| 2 2ϵ ϕ(x 1 ) q(x 0 , x 1 )dx 0 dx 1 = log q * (x 1 |x 0 ) q(x 1 |x 0 ) q(x 0 , x 1 )dx 0 dx 1 = log p 0 (x 0 )q * (x 1 |x 0 ) p 0 (x 0 )q(x 1 |x 0 ) q(x 0 , x 1 )dx 0 dx 1 = log q * (x 0 , x 1 ) q(x 0 , x 1 ) q(x 0 , x 1 )dx 0 dx 1 =-log q(x 0 , x 1 ) q * (x 0 , x 1 ) q(x 0 , x 1 )dx 0 dx 1 = -KL (q(x 0 , x 1 )∥q * (x 0 , x 1 )) .Thus, KL (q * ∥q) = -KL (q∥q * ). Since the KL divergence is non-negative, we derive that q = q * . Stage 2. In this stage, we prove the theorem itself. First, if N > 1, i.e., there is more than one intermediate time moment, we integrate q(x 0 , x t1 , . . . , x t N , x 1 ) over all intermediate time moments except t 1 . On the one hand, we get q(x 0 , x t1 , x 1 ) = q(x 0 , x t1 , . . . , x t N , x 1 )dx t2 . . . dx t N = p W ϵ (x t1 , . . . , x t N |x 0 , x 1 )q(x 0 , x 1 )dx t2 . . . dx t N = p W ϵ (x t1 |x 0 , x 1 )q(x 1 , x 0 ).On the other hand, we derive q(x 0 , x t1 , x 1 ) = q(x 0 )q(x t1 |x 0 )EQUATIONCombining ( 17) and (18) yields q(x 0 )q(x t1 |x 0 )q(x 1 |x t1 ) = q(x 0 , x t1 , x 1 ) = p W ϵ (x t1 |x 0 , x 1 )q(x 1 , x 0 ).Note that if N = 1, then we already have (19) from the conditions of the theorem. Therefore, p W ϵ (x t1 |x 0 , x 1 )q(x 1 , x 0 ) = q(x 0 )q(x t1 |x 0 )q(x 1 |x t1 ) p W ϵ (x t1 |x 0 , x 1 )q(x 1 |x 0 ) ¨q (x 0 ) = ¨q (x 0 )q(x t1 |x 0 )q(x 1 |x t1 ). From now on, we are interested only in 3 time moments: t 0 = 0, t 1 and t N +1 = 1. To simplify the notation, we will write t instead of t 2 in the following proof. We take the logarithm and getlog q(x 1 |x 0 ) = log q(x t |x 0 ) + log q(x 1 |x t ) -log p W ϵ (x t |x 0 , x 1 )Then, we use the formula for the Brownian Bridge density:log q(x 1 |x 0 ) = log q(x t |x 0 ) + log q(x 1 |x t ) -C + 1 2ϵt(1 -t) ||x t -(tx 1 + (1 -t)x 0 )|| 2 = log q(x t |x 0 ) + log q(x 1 |x t ) -C + 1 2ϵt(1 -t) (||x t || 2 + ||tx 1 || 2 + ||(1 -t)x 0 || 2 -2tx T t x 1 -2(1 -t)x T t x 0 + 2t(1 -t)x T 0 x 1 ) = log q(x t |x 0 ) + log q(x 1 |x t ) -C + ||x t || 2 2ϵt(1 -t) + ||(1 -t)x 0 || 2 2ϵt(1 -t) + ||tx 1 || 2 2ϵt(1 -t) - x T t x 1 ϵ(1 -t) - x T t x 0 ϵt + x T 0 x 1 ϵ = log q(x 1 |x t ) + ||x t || 2 2ϵt(1 -t) + ||tx 1 || 2 2ϵt(1 -t) - x T t x 1 ϵ(1 -t) -C def =f1(xt,x1) + ||(1 -t)x 0 || 2 2ϵt(1 -t) + log q(x t |x 0 ) - x T t x 0 ϵt def =f2(xt,x0) + x T 0 x 1 ϵ .Thus,EQUATIONBelow, we prove that f 3 (x 0 , x 1 ) = g 1 (x 0 ) + g 2 (x 1 ) for some functions g 1 and g 2 . We note thatEQUATIONWe substitute (21) to (20) :EQUATION)Since there is no dependence on x 0 in the right part of ( 22), we conclude that f 3 (x 0 , x 1 ) -f 3 (x 0 , 0) is a function of only x 1 . We define g 1 (x 1 ) def = f 3 (x 0 , x 1 ) -f 3 (x 0 , 0) and g 2 (x 0 ) def = f 3 (x 0 , 0). Now we have the desired result:EQUATIONThus,f 3 (x 0 , x 1 ) = log q(x 1 |x 0 ) - x T 0 x 1 ϵ = g 1 (x 1 ) + g 2 (x 0 ).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.log q(x 1 |x 0 ) = g 1 (x 1 ) + g 2 (x 0 ) + x T 0 x 1 ϵ = g 1 (x 1 ) + ||x 1 || 2 2ϵ + g 2 (x 0 ) + ||x 0 || 2 2ϵ - ||x 0 -x 1 || 2 2ϵ , q(x 1 |x 0 ) = exp g 2 (x 0 ) + ||x 0 || 2 2ϵ def =C(x0) exp - ||x 0 -x 1 || 2 2ϵ exp g 1 (x 1 ) + ||x 1 || 2 2ϵ def =ϕ(x1) = q(x 1 |x 0 ) = C(x 0 ) exp - ||x 0 -x 1 || 2 2ϵ ϕ(x 1 ), q(x 0 , x 1 ) = q(x 0 )C(x 0 ) exp - ||x 0 -x 1 || 2 2ϵ ϕ(x 1 ).Hence, from the first stage of this proof it follows that q(x 0 , x 1 ) is the solution to the Static SB between q(x 0 ) = p 0 (x 0 ) and q(x 1 ) = p 1 (x 1 ) with the coefficient ϵ. That is, p T * (x 0 , x 1 ) = q(x 0 , x 1 ). Since q(x t1 , . . . , x t N |x 0 , x 1 ) = p W ϵ (x t1 , . . . , x t N |x 0 , x 1 ) by the assumptions of the current theorem, we also conclude that q(x 0 , x t1 , . . . , x t N , x 1 ) = p T * (x 0 , x t1 , . . . , x t N , x 1 ), i.e., the discrete processes coincide.The logic of our justification of D-IMF for discrete processes generally follows the respective logic of the justification of IMF for continuous stochastic processes [47] .Proof of Proposition 3.3. The mild assumption here consists in the existence of at least one process r ∈ R(N ) for which KL (q∥r) < ∞. The reciprocal process r ∈ R(N ) has its density in the form r(x 0 , x t1 , . . . , x t N , x 1 ) = p W ϵ (x t1 , . . . , x t N |x 0 , x 1 )r(x 0 , x 1 ) (see (7) ). Thus, we need to optimize only the part r(x 0 , x 1 ). Below we show, that r(x 0 , x 1 ) should be equal q(x 0 , x 1 ) to minimize the functional.KL (q∥r) = log q(x 0 , x in , x 1 ) r(x 0 , x in , x 1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = log q(x in |x 0 , x 1 )q(x 0 , x 1 ) r(x in |x 0 , x 1 )p W ϵ (xin|x0,x1) r(x 0 , x 1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = log q(x in |x 0 , x 1 ) p W ϵ (x in |x 0 , x 1 )q(x 0 , x in , x 1 )dx 0 dx in dx 1 =Const + log q(x 0 , x 1 ) r(x 0 , x 1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = Const + log q(x 0 , x 1 ) r(x 0 , x 1 ) q(x 0 , x 1 )dx 0 dx 1 =KL(q(x0,x1)∥r(x0,x1))= Const + KL (q(x 0 , x 1 )∥r(x 0 , x 1 ))Hence, proj R (q) = arg min r∈R(N ) KL (q∥r) = p W ϵ (x in |x 0 , x 1 )q(x 0 , x 1 ).Proof of Proposition 3.5. Similar to the previous proposition, the mild assumption here consists in the existence of at least one process m ∈ M(N ) for which KL (q∥m) < ∞. This proof is a bit more technical than for the reciprocal projection. We need to define new notation x not tn,tn-1 = (x t0 , . . . , x tn-2 , x tn+1 , . . . , x t N +1 ) for a vector of variables for all time moment except two time moments t n and t n-1 . KL (q∥m) = log q(x 0 , x in , x 1 ) m(x 0 , x in , x 1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = log q(x 0 , x in , x 1 )m(x 0 ) N +1 n=1 m(x tn |x tn-1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = log q(x 0 ) m(x 0 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 + log q(x in , x 1 |x 0 ) N +1 n=1 m(x tn |x tn-1 )q(x 0 , x in , x 1 )dx 0 dx in dx 1 = log q(x 0 ) m(x 0 ) q(x 0 )dx 0 KL(q(x0)∥m(x0))+ log q(x in , x 1 |x 0 ) N +1 n=1 m(x tn |x tn-1 ) q(x 0 , x in , x 1 )dx 0 = KL (q(x 0 )∥m(x 0 )) + log q(x in , x 1 |x 0 ) N +1 n=1 m(x tn |x tn-1 )q(x 0 , x in , x 1 )dx 0 dx in dx 1 + (24)N log q(x 0 , x in , x 1 ) q(x 0 , x in , x 1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1+ log q(x 0 ) q(x 0 ) q(x 0 , x in , x 1 )dx 0 dx in x 1 =0 = (25) KL (q(x 0 )∥m(x 0 )) + logN +1 n=1 q(x 0 , x in , x 1 ) N +1 n=1 m(x tn |x tn-1 )q(x 0 , x in , x 1 )dx 0 dx in dx 1 -N log q(x 0 , x in , x 1 )q(x 0 , x in , x 1 )dx 0 dx in dx 1 + log q(x 0 )q(x 0 )dx 0 dx in dx 1 = KL (q(x 0 )∥m(x 0 )) +N +1 n=1 log q(x 0 , x in , x 1 ) m(x tn |x tn-1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 - N log q(x 0 , x in , x 1 )q(x 0 , x in , x 1 )dx 0 dx in dx 1 + log q(x 0 )q(x 0 )dx 0 dx in dx 1 def =C1 = KL (q(x 0 )∥m(x 0 )) -C 1 + N +1 n=1 log q(x tn |x tn-1 )q(x tn-1 )q(x not tn,tn-1 |x tn , x tn-1 ) m(x tn |x tn-1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = KL (q(x 0 )∥m(x 0 )) -C 1 + N +1 n=1 log q(x tn-1 )q(x not tn,tn-1 |x tn , x tn-1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 def =C2 + N +1 n=1 log q(x tn |x tn-1 ) m(x tn |x tn-1 ) q(x 0 , x in , x 1 )dx 0 dx in dx 1 = KL (q(x 0 )∥m(x 0 )) + C 2 + N +1n=1 log q(x tn |x tn-1 ) m(x tn |x tn-1 ) q(x tn |x tn-1 )dx tn KL(q(xt n |xt n-1 )∥m(xt n |xt n-1 ))q(x tn-1 )dx tn-1 = KL (q(x 0 )∥m(x 0 )) + N +1 n=1 KL q(x tn |x tn-1 )∥m(x tn |x tn-1 ) q(x tn-1 )dx tn-1 + C 2 .In the line (25) , we add terms equal to zero, to match each m(x tn |x tn-1 ) by the separate term q(x 0 , x in , x 1 ) in the line (25) . We need it to as we want to place each term m(x tn |x tn-1 ) in the separate KL-divergence in the final expression. Hence, the minimizer of the objective m * ∈ M(N ) has m * (x 0 ) = q(x 0 ) and all transitional distributions m * (x tn |x tn-1 ) = q(x tn |x tn-1 ), i.e. is given bym * (x 0 , x in , x 1 ) = [proj M (q)](x 0 , x in , x 1 ) = q(x 0 ) N +1 n=1 q(x tn |x tn-1 ).Proposition B.1 (Pythagorean theorems for projections). Assume that r ∈ R(N ) and m ∈ M(N ).If KL (r∥m) < ∞ and KL (r∥projM (r)) < ∞, then KL (r∥m) = KL (r∥proj M (r)) + KL (proj M (r)∥m) (26) and if KL (m∥r) < ∞, KL (m∥proj R (m)) < ∞ then KL (m∥r) = KL (m∥proj R (m)) + KL (proj R (m)∥r)Proof of Proposition B.1. Before proving the first equation ( 26) we prove the additional property of r ∈ R(N ) for any n ∈ [1, 2, . . . , N + 1]: [proj M r](x tn , x tn-1 ) = r(x tn , x tn-1 ).subsequence (r l k ) ∞ k=1 containes a convergent subsequence (r l k j ) ∞ j=1 → r * . Since sets of Markovian M(N ) and reciprocal R(N ) processes are closed under weak convergence, we have m * ∈ M(N ) and r * ∈ R(N ). From the lower semicontinuity of KL divergence in the weak topology [50, Theorem 19] and lim l→∞ KL q l ∥q l+1 = 0 (see Proposition B.2):EQUATION)Thus, m * = r * def = q lim . We know that q lim has the same marginals p 0 (x 0 ) = q(x 0 ) and p 1 (x 1 ) = q(x 1 ) since both Markovian and reciprocal projections preserve marginals. By our Theorem 3.1 since q lim ∈ M(N ) ∩ R(N ), then q lim (x 0 , x in , x 1 ) = p T * (x 0 , x in , x 1 ). Finally, lim l→∞ KL q l (x 0 , x in , x 1 )∥p T * (x 0 , x in , x 1 ) = 0 follows usinglim j→∞ KL r l k j (x 0 , x in , x 1 )∥p T * (x 0 , x in , x 1 ) = 0and the mononotonicity of KL q l ∥q * , see Proposition B.2.The proofs in this subsection are the most technical as there are a lot of manipulations with matrices.Proof of Theorem 3.7. From ( 6) and ( 5) follows that the discrete Brownian Bridge p W ϵ (x in |x 0 , x 1 ) has also a Gaussian distribution. The covariance of the Brownian Bridge with coefficient ϵ at times s < t [17, Eq. 9.14] is ϵs(1 -t). Thus, the matrix ϵK is a covariance matrix for all pairs of time moments t, t ′ ∈ [t 1 , . . . , t N ] of the considered discrete Brownian Bridge p W ϵ (x in |x 0 , x 1 ). The mean value E[x tn |x 0 , x 1 ] of Brownian Bridge at time t n is equal to t n x 1 + (1 -t n )x 0 . Thus, the discrete Brownian Bridge has the following distribution:p W ϵ (x in |x 0 , x 1 ) = N (x in |U x 01 , ϵK).Recall that the reciprocal projection is given by:EQUATION)Since it is a product of two Gaussian distributions, which itself is also a Gaussian distribution, our goal is to find the mean vector and covariance matrix of [proj R q](x in , x 0 , x 1 ). Further we denote [proj R q](x in , x 0 , x 1 ) as r(x 0 , x in , x 1 ) for convenience.The mean vector of [proj R q](x in , x 0 , x 1 ) for each t n is given byE r(xt n ) x tn = E r(xt n |x0,x1) [x tn |x 0 , x 1 ]q(x 0 , x 1 )dx 0 dx 1 = E p W ϵ (xt n |x0,x1) [x tn |x 0 , x 1 ]q(x 0 , x 1 )dx 0 dx 1 = x 0 + t n (x 1 -x 0 ) q(x 0 , x 1 )dx 0 dx 1 = (1 -t n ) x 0 q(x 0 , x 1 )dx 0 dx 1 + t n x 1 q(x 0 , x 1 )dx 0 dx 1 = t n µ 1 + (1 -t n )µ 0 .where µ 0 and µ 1 are the means of q(x 0 ) and q(x 1 ), respectively. Thus, the mean vector of [proj R q](x in , x 0 , x 1 ) is given by (U µ 01 , µ 0 , µ 1 ). Now, we are going to find the covariance matrix Σ R . We will first find the inverse covarianceΣ -1 R = A B B T C of [proj R q](x in , x 0 , x 1 ). Here A has shape N D × N D as the matrix K, while the matrix C has the shape 2D × 2D as the matrix Σ. Matrices A and C are symmetric since they are a part of the inversed symmetric matrix Σ R . 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 x in or x 01 ):log [proj R q](x in , x 0 , x 1 ) = Const - 1 2 ((x in , x 01 ) -(U µ 01 , µ 01 )) T Σ -1 R ((x in , x 01 ) -(U µ 01 , µ 01 )) = Const - 1 2 ((x in , x 01 ) -(U µ 01 , µ 01 )) T A B B T C ((x in , x 01 ) -(U µ 01 , µ 01 )) = Const - 1 2 (x in -U µ 01 ) T A(x in -U µ 01 ) - 1 2 (x 01 -µ 01 ) T C(x 01 -µ 01 ) - (x in -U µ 01 ) T B(x 01 -µ 01 ) = Const - 1 2 x T in Ax in + (U µ 01 ) T Ax in - 1 2 x T 01 Cx 01 + µ T 01 Cx 01 - x T in Bx 01 -x T in Bµ 01 -(U µ 01 ) T Bx 01 -(U µ 01 ) T Bµ 01 = Const - 1 2 x T in Ax in + (U µ 01 ) T Ax in - 1 2 x T 01 Cx 01 + µ T 01 Cx 01 - x T in Bx 01 -x T in Bµ 01 -(U µ 01 ) T Bx 01 .In turn, from (29) we have:log [proj R q](x in , x 0 , x 1 ) = log p W ϵ (x in |x 0 , x 1 ) + log q(x 0 , x 1 ) = Const - 1 2 (x in -U x 01 ) T (ϵK) -1 (x in -U x 01 ) - 1 2 (x 01 -µ 01 ) T Σ -1 (x 01 -µ 01 ) = Const - 1 2 x T in (ϵK) -1 x in + x T in (ϵK) -1 U x 01 - 1 2 (U x 01 ) T (ϵK) -1 U x 01 - 1 2 x T 01 Σ -1 x 01 + x T 01 Σ -1 µ 01 - 1 2 µ 01 Σ -1 µ 01 = Const - 1 2 x T in (ϵK) =A x in + x T in (ϵK) -1 U =B x 01 - 1 2 x T 01 (U T (ϵK) -1 U + Σ -1 ) =C x 01 + x T 01 Σ -1 µ 01 .By matching the formulas above, it follows:EQUATIONThus, we have:Σ -1 R = A B B T C = (ϵK) -1 -(ϵK) -1 U -((ϵK) -1 U ) T U T (ϵK) -1 U + Σ -1By using the formula of block-wise matrix inversion [37, Section 9.1.3] :EQUATIONApplying this formula, we have:(C -B T A -1 B) -1 = (U T (ϵK) -1 U + Σ -1 -U T (ϵK) -1 (ϵK)(ϵK) -1 U ) -1 = (Σ -1 ) -1 = Σ. A -1 + A -1 B(C -B T A -1 B) -1 B T A -1 = ϵK + ϵK(ϵK) -1 U ΣΣ -1 ΣU T ϵK(ϵK) -1 = ϵK + U ΣU T . -A -1 B(C -B T A -1 B) -1 = ϵK(ϵK) -1 U Σ = U Σ.Thus, we obtain the desired result:Σ R = ϵK + U ΣU T U Σ (U Σ) T Σ .Proof of Theorem 3.8. Part 1. Since from the assumptions of the theorem q(x in , x 0 , x 1 ) has Gaussian distribution, it follows that joint distribution of two time moments q(x tn , x tn-1 ) is also Gaussian and is given by:EQUATIONRecall that here ( Σ R ) ti,tj represents submatrix of Σ R with covariance of x ti and x tj . Hence, the conditional distributions are given by [37, Sec 8.1.3]:q(x tn |x tn-1 ) = N (x tn | µ tn (x tn-1 ), Σ tn ),EQUATIONΣ tn def = ( Σ R ) tn,tn -( Σ R ) tn,tn-1 (( Σ R ) tn-1,tn-1 ) -1 (( Σ R ) tn,tn-1 ) T .That concludes the first part of our proof about the whole distribution [proj M q](x 0 , x in , x 1 ) of Markovian projection.Part 2. Next, we find the distribution of [proj M q](x 0 , x 1 ), but before we proceed, we introduce new notation to improve readability:EQUATIONSince the process q M (x 0 , x in , x 1 ) 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. q M (x tn ) = q(x tn ), hence we can already write that q M (x 0 , x 1 ) is given by:EQUATIONwhere µ 0 and µ 1 are the means of q(x 0 ) and q(x 1 ), while Σ 0 and Σ 1 are the covariance matricies of q(x 0 ) and q(x 1 ). Thus, only Σ 01 is unknown. Again, by using the formula for the conditional distributions [37, Sec 8.1.3] we have that:q M (x 1 |x 0 ) = N (x 1 | µ 1 (x 0 ), Σ 1 (x 0 )), µ 1 (x 0 ) def = µ 1 + Σ T 01 Σ -1 0 def =G (x 0 -µ 0 ), Σ 1 def = Σ 1 -Σ T 01 Σ -1 0 Σ 01 .Since the mean µ 1 (x 0 ) of the conditional distribution is a affine map of x 0 with the matrix G we can derive:Σ T 01 = GΣ 0 . Thus, we need to find the expression for G, by considering the expression for µ 1 (x 0 ). To derive the expression of the mean µ 1 (x 0 ) of q M (x 1 |x 0 ) we consider the sequence q M (x tn |x 0 ) for n ∈ [1, . . . , N + 1]. We already know the expression for n = 1 which is given by [proj M q](x t1 |x 0 ) = q(x t1 |x 0 ) in the first part of the proof. For other n, we use the following expression:EQUATIONSince q M (x tn |x 0 ) is Gaussian we denote q M (x tn |x 0 ) = N (x tn | µ tn (x 0 ), Σ tn ). We derive the mean µ tn (x 0 ) by using the properties of conditional expectations as follows: 2 . In all the cases, we run our ASBM algorithm starting from the independent coupling between p 0 and p 1 .EQUATIONAs the baselines, we consider other neural bridge matching methods [49, 47] . The first one (SF 2 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 1 , our algorithm demonstrates superior performance on ϵ = 10, superior performance or comparable performance on ϵ = 1, slightly worse performance w.r.t. SF 2 M-Sink [49] and superior performance w.r.t. DSBM [47] on ϵ = 0.1. Also, from Table 2 one may note that ASBM fits target distribution better then other Bridge Matching SB algorithms. -UVP ↓ (%) between the static SB solution p T (x0, x1) and the learned q θ (x0, x1) on the SB benchmark. The best metric over bridge Matching algorithms is bolded. Results marked with † are taken from [11] . Table 2 : Comparisons of BW 2 2 -UVP ↓ (%) between the ground truth target distribution p1(x1) and learned target distribution q θ (x1). The best metric over bridge matching algorithms is bolded. Results marked with † are taken from [11] .Remark. There exist recent light SB algorithms [24, 11] 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 it is also constructed using Gaussian mixtures. Therefore, we exclude them from comparison, see the comments of the authors in [24, 5.2 ] and [11, 5.2] Here we 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 ϵ ∈ {1, 10}. We learn ASBM and DSBM on train set of digits and show the translated test images in Figures 10 and 11 along with calcualted test FID in Table 3 .For ϵ = 1 the color stays almost exactly the same through translation and there are minor shape diversity for both ASBM and DSBM, see Figures (10b, 10c, 11b, 11c) . In turn, ϵ = 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 (10d, 10e, 11d, 11e ). As one can see from Table 3 , ASBM has better FID on both ϵ ∈ {1, 10}. However DSBM experiences a notable increase in FID with ϵ = 10. We conjecture that this is due to the FID unstability w.r.t. slightly noisy images which may appear in DSBM due to the neccesity to integrate noisy trajectories (for large ϵ).Below, we discuss the parametrization of the discriminator and generator in detail. In general we 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] we use a time-conditional discriminator D ξ (x tn , x tn-1 , t n-1 ):R D × R D × [0, 1] → [0, 1].For each time moment t and object x tn-1 , the role of this discriminator is to check whether the sample x tn is from the distribution q(x tn |x tn-1 ). As well as in the DD-GAN paper [53] , we train this discriminator by optimizing the following objective:EQUATION+E q θ (xt n |xt n-1 ) [-log 1 -D ξ (x t , x tn-1 , t n-1 ) ]]Here, the samples from q(x tn |x tn-1 ) play the role of true samples, while the samples obtained from the parametrized distribution q θ (x tn |x tn-1 ) play role of fake samples in terms of original GANs.To estimate the first expectation E q(xt n-1 ) E q(xt n |xt n-1 ) = E q(xt n ,xt n-1 ) one should sample from q(x tn , x tn-1 ). To sample a pair (x tn , x tn-1 ) ∼ q(x tn , x tn-1 ), we use the properties ( 5) and (6) ofFor all the experiments, Discrete Markovian Projection is conducted using the DD-GAN code [53] :https://github.com/NVlabs/denoising-diffusion-ganThe only thing that we modify is the replacement of the DDPM [15] [42] and discriminator by a ResNet-like architectures with addition of positional time encoding as in [53] . Neural networks are optimized with the Adam optimizer [22] and apply the Exponential Moving Averaging (EMA) on generator'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 q 0 (x 0 , x 1 ) may be either Ind, i.e. q 0 (x 0 , x 1 ) = p 0 (x 0 )p 1 (x 1 ), or Mini Batch Optimal Transport coupling (MB), i.e. discrete Optimal Transport solved on mini-batch samples [49] .The hyperparameters which we use in the experiments are summarized in Table 4 Other details & pre-processing. Test FID is calculated using pytorch-fid package. Working with CelabA dataset [33] , we use all 84434 male and 118165 female samples (90% train, 10% test of each class). Each sample is 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 [53] (42M Generator parameters and 27M Discriminator parameters). Working with Colorized MNIST [12] , we pick digits of classes "2" and "3" (we use the default MNIST train/test split), resize them to 32 × 32 and normalize by 0.5 mean and 0.5 std. We use the same generator and discriminator as DDGAN uses in CIFAR10 [53] .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 2 days of training on GPU A100. Toy2D and Schrödinger Bridge benchmark experiments take several hours on GPU A100.DSBM [47] implementation is taken from the official code:https://github.com/yuyang-shi/dsbm-pytorchFor CelebA experiment all the hyperparameters, except for 200k training iterations for the first IMF iteration (Bridge Matching pretrain, Appx I.3 [47] ) and number of overall IMF iterations (that is taken the same as for corresponding ASBM experiment, see Table 4 ), were taken from [47] . As a 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, 6] . Starting coupling is exactly the same as for ASBM in corresponding experiments (Table 4 ).Model ϵ = 1 ϵ = 10 DSBM 0.246 0.386 ASBM 0.242 0.294 Table 9 : Perceptual similarity for male→female translation for DSBM and ASBM models with ϵ = 1 and ϵ = 10 measured using LPIPS↓ [55] between inputs from CelebA test and generated outputs. We include additional analysis on dynamics of model samples with D-IMF iterations for ASBM (ours) and IMF iterations for DSBM with ϵ = 1. As one can see from Figure 15a ASBM visually almost converges after 5 iterations in terms of similarity of generated sample w.r.t. to input data, i.e., the transport cost. From plot in Figure 14 we see that ASBM's FID does not change through subsequent 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 it can be seen at plot in Figure 14 , FID stops to improve after outer iteration 9 and does not improve drastically from outer iteration 5. Hence, we take ASBM and DSBM with 5 outer D-IMF/IMF iterations as a balance point for our comparison. Justification: For every contribution in introduction there are links to sections about them.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA]Justification: Presented research doesn't need safeguards Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: All the used assets are cited.Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. 
: 
 Figure 1 : Our D-IMF approach performs unpaired image-to-image translation in just a few steps, achieving results comparable to the hundred-step IMF [47] . Celeba [33] , male→female (128 × 128). 
 
 Appendix:  


Decision: 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.

Reviews: review: paper_summary: paper_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. 

[1] Shi et al. (2023) -- Diffusion Schrodinger Bridge Matching

[2] Peluchetti (2023) -- Diffusion Bridge Mixture Transports, Schrödinger Bridge Problems and Generative Modeling

[2] Xiao et al. (2021) -- Tackling the Generative Learning Trilemma with Denoising Diffusion GANs
strength_weakness: strength_weakness: * 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. 

* 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. 

* 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. 

* 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.

[1] Shi et al. (2023) -- Diffusion Schrodinger Bridge Matching

[2] Jolicoeur et al. (2020) -- Adversarial score matching and improved sampling for image generation
 
[3] Dieleman (2024) -- The paradox of diffusion distillation
questions: 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]. 

* Some typos:

* l.280 "Schrodigner" --> "Schrodinger"

[1] Xu et al. (2023) --UFOGen: You Forward Once Large Scale Text-to-Image Generation via Diffusion GANs

[2] Sauer et al. (2023) -- Adversarial Diffusion Distillation

[3] Song et al. (2023) -- Consistency Models

[4] Song et al. (2023) -- Improved Techniques for Training Consistency Models
limitations: 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. 

[1] Dieleman (2024) -- The paradox of diffusion distillation
paper_summary: paper_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.
strength_weakness: strength_weakness: - 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.

- In L220, the non-saturating GAN loss should be provided in the main paper rather than being postponed to the Appendix.
questions: 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: Limitations were addressed in Appendix A.
paper_summary: paper_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).
strength_weakness: strength_weakness: - 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.

- 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.

- 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.

Minor comments:
There is a typo in Line 224: the term $q(x_{t_n}|x_{t_{n-1}})$.
questions: questions: - Can the authors provide a comparison of the computational burden between your method and DSBM, particularly in terms of time and resource usage?
- Is the time discretization handled as like DDPM style or Uniformly?
- Did you use any pretrained model (e.g. pretrained DSBM) on I2I task?
limitations: limitations: The limitation is adequately discussed in the paper.
paper_summary: paper_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.
strength_weakness: strength_weakness: - 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. 
- This model does not always yield state-of-the-art scores for syntactic datasets (see Table 1&2).
- 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

Typo
- L213: Denoising Denoising GANs

[1]
@inproceedings{
korotin2024light,
title={Light Schr\"odinger Bridge},
author={Alexander Korotin and Nikita Gushchin and Evgeny Burnaev},
booktitle={The Twelfth International Conference on Learning Representations},
year={2024},
url={https://openreview.net/forum?id=WhZoCLRWYJ}
}
questions: 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?   

[1] https://arxiv.org/pdf/2401.09603
limitations: limitations: The authors adequately addressed the limitations

Comments: title: Thank you for the response
comment: I would like to thank the authors for their thorough rebuttal. While the authors have answered to my main questions I would like to keep my score to 6. In case the paper is accepted I think the discussion on diversity of the samples should be included. This is actually a really good starting point for a wider discussion on the use of GANs methods within diffusion models.
I would like to thank the authors for their clarifications and answers. After reading through other reviews and considering the pointed-out weaknesses, I decided to keep my original score.
I thank the authors for the reply. I've decided to keep my score.
Dear Reviewer LCNa,

We thank you for your review and appreciate your time reviewing our paper.

The end of the discussion period is close. We would be grateful if we could hear your feedback regarding our answers to the reviews. We are happy to address any remaining points during the remaining discussion period.

Thanks in advance,

Paper authors
Dear Reviewer xGf5,

We thank you for your review and appreciate your time reviewing our paper.

The end of the discussion period is close. We would be grateful if we could hear your feedback regarding our answers to the reviews. We are happy to address any remaining points during the remaining discussion period.

Thanks in advance,

Paper authors
Dear Reviewer xtGA,

We thank you for your review and appreciate your time reviewing our paper.

The end of the discussion period is close. We would be grateful if we could hear your feedback regarding our answers to the reviews. We are happy to address any remaining points during the remaining discussion period.

Thanks in advance,

Paper authors
I appreciate authors for clarification and for the additional ablation studies. I would like to raise my score to 6.