Fractional Diffusion Bridge Models
Authors: Gabriel Nobis
Maximilian Springenberg
Arina Belova
Rembert Daems
Christoph Knochenhauer
Manfred Opper
Tolga Birdal
Wojciech Samek
Venue: NeurIPS 2025 poster ()
Keywords: Schroedinger bridge
stochastic interpolant
data translation
fractional diffusion process
conformational changes in proteins
bridge matching
diffusion models
Primary area: deep_learning
TL;DR: We present Fractional Diffusion Bridge Models (FDBM), a novel generative diffusion bridge framework that enables generative diffusion bridge modeling with fractional noise for both paired and unpaired training data.

Abstract: We present *Fractional Diffusion Bridge Models* (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). 
As a remedy, leveraging a recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non-Markovian nature of fBM. We prove the existence of a coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared to the Brownian baselines, yielding lower root mean squared deviation (RMSD) of C$_\alpha$ atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.

Fractional Diffusion Bridge Models
Gabriel Nobis + Fraunhofer HHI
Maximilian Springenberg Fraunhofer HHI
Arina Belova Fraunhofer HHI
Rembert Daems
Christoph Knochenhauer
Manfred Opper
Ghent University-imee
Technical University of Munich
Technical University of Berlin University Potsdam University of Birmingham
FlandersMake-MMRRO
Tolga Birdal Imperial College London
Wojciech Samek
Fraunhofer HHI Technical University of Berlin
Abstract
We present Fractional Diffusion Bridge Models (FDBM), a novel generative diffu- sion bridge framework driven by an approximation of the rich and Markovann fractional Brownian motion (fBM). Real stochastic processes exhibit a degree memory effects (correlations in time), long-range dependencies, roughness and anomalous diffusion phenomena that are not captured in standard diffusion or bridge modeling due to the use of Brownian motion (BM). As remedy, leverag- ing recent Markovian approximation of fBM (MA-fBM), we construct FDBM that enable tractable inference while preserving the non Markkoaan nature fBM. We prove the existence of coupling coupling-preserving generative diffusion bridge and leverage it for future state prediction from paired training data. We then extend our formulation to the Schrödinger bridge problem and derive a principled loss function to learn the unpaired data translation. We evaluate FDBM on both tasks: predicting future protein conformations from aligned data, and unpaired image translation. In both settings, FDBM achieves superior performance compared the Brownian baselines. yielding lower root mean squared deviation (RMSD) of C. atomic positions in protein structure prediction and lower Fréchet Inception Distance (FID) in unpaired image translation.
Introduction
Stochastic differential equations (SDEs) offer a natural framework for modeling the inherent ran- domness and continuous-time dynamics real-world systems [1, 2]. This precisely why they serve as the backbone of state-of-the-art generative diffusion models [3-5]. Traditionally, these models assume noise driven by standard Brownian motion (BM) [6-8], which Markovian with independent increments [9]. However, this choice is motivated by mathematical tractability and simplicity rather than faithfulness and fidelity real-world data. Empirical data, particularly in complex systems such as proteins, often exhibit long-range temporal dependencies, heavy-tailed be- haviors, and intricate dynamics that are poorly captured by memoryless processes [10]. A generative process, lacking temporal dependencies, may lead to insufficient approximations of such intricate data, due the absence of modeled memory effects. These limitations have motivated recent ef- forts to explore generative models with non-standard noise sources [11-18]. Our work extends
Equual contribution; Shared senior authorship
corresponding authors: (gabriel.nobir nobis, maximilian apringenberg)@thh. htw.aaeerrrreiit..ffammm.ree..dd
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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this line of research to generative diffusion bridge models [19-21], where the goal is to transform a structured, non- Con-Gausiian source distribution into a complex target distribution. We specifically investigate stochastic bridges driven by fractional Brownian motion (fBM) [22, 23], a generaliza- tion of BM with dependent increments, characterized by the Hurst index H, which governs both roughness (i.e., pathwise regularity) and long-range dependence. However, directly using fBM as the driving noise in a stochastic bridge introduces an intractable drift [24]. To address this, we adopt Markov approximation of IBM (MA-IBM) [25, 26] that enables efficient simulation. By using MA-IBM as the driving process, we introduce a more expressive and flexible framework for building bridges: when H = 0.5, fBM recovers classical BM, whereas other values of H flexibly allow us to model a broader range of temporal behaviors, as demonstrated in our experiments. Our framework, Fractional Diffusion Bridge Models (FDBM), enables generative bridge modeling with fractional noise for both paired and unpaired training data, applicable across a broad range of ma- chine learning tasks. In this work, we focus on predicting conformational changes in proteins to explore effects in paired-data problems, as well as unpaired image translation. In the context of pro- tein generation, diffusion processes driven by MA-fBM have proven effective in their superdiffusive regime, showing improvements in both sample fidelity and diversity [17], potentially due to a better capture of long-range correlations in protein structures. Building on this observation, we propose MA-IBM-driven diffusion bridges as a principled extension for modeling conformational changes in proteins. To the best of our knowledge, our framework is the first to incopporate fractional noise into generative bridge modeling within machine learning. Our contributions are:
We propose method for learning generative diffusion bridges that interpolate between two un- known distributions via a non-Markovian trajectory with controllable correlation of increments and long-range dependencies, enabling more flexible modeling of real-world variability and bio- logical dynamics.
We prove that, for these generalized stochastic dynamics, there exists a process solving stochastic differential equation that preserves the coupling given in the training data.
We formulate the Schrödinger bridge problem with a reference process approximating fractional Brownian motion and propose method to learn stochastic transport trajectories, whose roughness and long-range dependencies are controlled by the Hurst index.
We apply our framework broadly to common use cases of stochastic bridges in machine learning, including inferring conformational changes in proteins and performing unpaired image transla- tion, achieving lower root mean squared deviation (RMSD) of C., atomic positions in protein prediction, and improved Fréchet Inception Distance (FID) scores for image translation.
We accompany our work with several publicly available implementations to facilitate the adoption of our framework in both paired and unpaired settings, as well as a stand-alone reimplementation of the method proposed by Bortoli et al. [27]." 2.3
Background
Stochastic bridges interpolate between two given data points by conditioning a prior reference pro- cess to start and end at prescribed values. A common choice for this reference process in machine learning is scaled BM X = VEB with E > O. Conditioning on the endpoints (zo, x1) E Rd x Rd yields the scaled Brownian bridge (BB) 1 which starts at zo and ends at æ1, while evolving for t E 10, according to the stochastic dynamics [28]
(1) dt + VedBi, X0) = zg.
p. (t) = 1
(1)
This scaled BB, or a generalization thereof, serves as the starting point for many machine learning applications [20, 21, 27, 29-34], where the goal is to learn a stochastic process X* that interpolates not only between the fixed endpoints (Io, 21), but in law between two unknown distributions Ho and Tī on Rdd, Since the drift of such a stochastic process is generally intractable, the drift term in eq. (1) serves as a target for neural network, which is optimized by minimizing conditional expectation.
Coupling porserig- data translation. Data translation aims map between two unknown dis- tributions. In the setting where training data is provided in puihs-such as the unbound and bound
http://hhhhcccctaaaaaaaaaa//a/// paireed
aatppe /tt////ttttbbbb www.//spriigg//ccrreeeegee Bridde-

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11
022 I1 0.3 0.3
H
0.6
0B8
Figure 1: Trajectories from the approximate 2d-fractional Brownian bridge for different Hurst indices H.
states of a protein [31, 35], a distorted and a clean image [36, 37], or two snapshots of cell dif- ferentiation recorded different days [31]-the additional objective is preserve the coupling given in the training data. We build our framework for paired data translation on Augmented Bridge Matching (ABM) [32], where stochastic process X is learned that transports an unknown distribution NT on R.d to another unknown distribution II on RRd while preserving the coupling ( oo,1 IRd x Rd. Additionally, X* solves SDE such that can sample from the coupling (1) ~ 1o..1 by first sampling X = To ~ Ho according to the first marginal of ll.. and then simulating the SDE forward in time arrive at sample Xi I II, Bortoli et al. [32, Proposition 3] show that for the scaled Brownian reference process X = vEB there exists such a coupling preserving process X with associated path measure IP* that solves
Xi
Xi x; Xi dt + VEdB,, X6 Ilg
dx;; = EEp, ijo.r
(2)
1-t
The drift of X* is intractable and approximated by a time-dependent neural network of resulting in process X with associated path measure P Minimizing now the KL divergence DKL (IT (Po) with respect to the weight vector 0 yields the loss function
[(0 Ep- || [e((xx
|
Xi X
LABM(() =
df.
(3)
Given paired training data sampled from the unknown coupling 1o, we can approximate the above loss function since, by construction, P = Io.1 where Q denotes the path measure of the scaled BB X 0,1 solving eq. (1). Consequently, to compute the loss during training, first sample (Ro, 21) ~ 1o and then sample
(- Zo, 20,21). 21).
the Schrödinger bridge. On the other hand, in unpaired data translation via the Schrödinger bridge, the objective is to find the coupling that corresponds to the optimal transport [38] between two unknown distributions. Here, we aim to learn the stochastic process X corresponding to the solution of the dynamic Schrödinger bridge problem [39-42]
Unpaired data translation
JBBB arg min (DLL (T)Q) ; To = llo Ti = 1111, TEP(C")
where the minimization is taken over path measures T. defined on the set of continuous func- tions Cd. from the unit interval [0. 1] to IRd. We build our framework for unpaired data transla- tion on Schrödinger Bridge Flow (SBFlow) [27], whose unique stationary point corresponds to the Schrödinger bridge. See Section E for a detailed summary.
In the following, we incorporate fractional noise into generative diffusion bridge models in order to control the roughness and long-range dependencies of the interpolating stochastic trajectories, replacing the BM used as the driving noise in traditional diffusion bridge models. Our work builds directly on Daems et al. [26] for the approximation of fBM, on Somnath et al. [31] and Bortoli et al. [32] for the paired-data setting, and on Peluchetti [33], Shi et [34], and Bortoli et al. [27] for the unpaired-data setting. See Section D) for detailed discussion of related work.
3 A stochastic bridge driven by fractional noise
We first define and characterize the fractional noise that serves as the driving process replacing BM. For mathematical details, we refer the reader to Section B, along with the notational conventions in Section
3.1 Fractional noise
We begin with the definition of Riemann-LLouuee (Type II) fBM, non-Markovian, centered Gaus- sian process with non-stationary and correlated increments.
3

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Definition 1 (Type II Fractional Brownian motion [22]). Let B = (Bi)120 be (multiddmenniooaa) standard Brownian motion (BM) and I the Gamma function. The centered Gaussian process
1
J (L-s)"=AdB. 12 0,
B T(
(5)
is called Type II fractional Brownian motion (fBM) with Hurst index H E (0,
Compared to BM with independent increments (diffusion), the paths of (BM become smoother for H > 0.5 due to positively correlated increments (super-diftusion) and rougher for < 0.5 due to negatively correlated increments (sub-diffusion), while H 0.5 recovers BM. A stochastic bridge can be derived for Gaussian processes, including fBM; however, the drift of the fractional Brown- ian bridge (IBB) is intractable [24] and therefore unsuitable both for sampling from its marginals and as a loss-function target analogous to eq. (3). Rather than introducing an additional approxi- mation error by attempting to approximate the drift of the fBB, follow Harms and Stefanovits [25), Daems et al. [26] and first approximate fBM by linear superposition of (OU) processes. These augmenting OU processes are all driven by the same standard BM, thereby approximating the time-correlated behavior of fBM.
Definition 2 (Markov approximation of IBM [25, 26]). Choose E N Orastein-UUllnberr (OU) processes
Y f
k l. K, K KEN, 220. 0,
(6)
with speeds of mean reversion ?1,-.. "K and dynamics d = 7 & + +dB. Given a Hurst index H E (0, 1) and geometrically spaced grid 7ik = , with r > and n = K+1 we call the process
: ===>> W.Y. H (0,1), 2
(7)
(multidimensional) Markov-aprroximatee fractional Brownian motion (MA-BBM) with approximation coefficients WKK E TR.
While the choice of approximation coefficients in Harms [43] enables strong convergence to fBM with high polynomial order in K for H 0.5, we opt for the computationally more efficient method proposed by Daems et al. [26]. This method selects the (P) optimal approximation coefficients for given K, achieving empirically good results in approximating fBM, even with small number of OU processes, See Daems, Rembert [44, Figures 3.13-3.15] for the approximation error of Type II fBM. We fix K throughout all experiments presented in the main text.
Proposition 3 (Optimal Approximation Coefficients [26)). The optimal approximation coefficients (w.,.... .... wx) E R. for given Hurst index H E (0, 1), terminal time T O and fixed geometrically spaced grid to minimize the (e))rrrrr
E(w) = ["u
(T
B - }
2
dt
given in closed form by the linear system Aw b, where R.K. and E RK are known.
We now use MA-fBM, equipped with the optimal approximation coefficients, as reference process to approximate iBB, thereby enabling efficient simulation and closed-form drift computation in the stochastic bridge derived in the next section.
3.2 A Markov approximate fractional Brownian bridge
Towards the goal of defining a stochastic bridge driven by fractional noise we fix the reference for the vector of the OU processes and Z (X, Y) for the augmented reference process. The reference process X non-Markovian (see Theorem 8) and becomes Markovian only after augmenting it with the OU processes, resulting the Markovian process Z. To define a stochastic bridge connecting two given data points Zu Ho and ~ II, via X, we only need to steer the first dimension of Z towards 21, while the terminal values of Y are not required to attain a specific value. The dynamics of the resulting stochastic bridge Z, can be derived directly from Daems, Rembert [44, Chapter 4], where posterior SDE steered towards 21 is constructed. In Section B, we present an alternative derivation using Doob's h-transform [2]. Both approaches yield the dynamics stated in the following proposition.
process to X = " with e > O, and write Y = (Y² "y.",..,"""" Y

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Proposition 4 (Markov approximation of fractional Brownian bridge [44, 45]). The partially pinned process (so.m] == (X, = In. X1 = 21) solves for the SDE
dz (so.an (t) F' aa (t)t + GG uut, Z mn (t)) + GdB(t), 2, (0) -Io (x), (9)
u(t, 2) = (1, (,, 1), WK 1)]² 21 M1)e(2)
(10)
ilp
where F E RK+IK++ and G E gK+ are known, Ca (t, + 8) = VE(e 1) and (1j(2z) and Lie denote the mean and the variance of the conditional terminal XıI(Zi = 2), respectively. We call the process Aza.i scaled Markov-approximate fractional Brownian bridge (MA-fBB).
See Figure I for visualization of two-dimensional MA-fBB trajectories for different Hurst indices. We now incorporate fractional noise into generative diffusion bridge models by using the defindd MA-IBB for both paired and unpaired data translation.
Fractional diffusion bridge models
Paired data translation. Paired training data arises in tasks such as predicting conformational
changes in proteins, where the unbound and bound states of the same protein form a p [31, 35, 46];
SBALIGN
ABM
forecasting the future state of a cell, where two snapshots of cell differentiation are recorded different days [31]; or reconstructing a clean image from its distorted counter- part [36, 37J. We assume access paired training data (xi, x isign independently sampled from the unknown coupling (#), +1) ~ Io, on Rd x Rd, with unknown marginals ΠTg and IT, on Rdd The goal is to transport Th to II via stochastic trajectories driven by MA-fBM, while preserving the coupling 1l..1 To this end, we construct in the following proposition stochastic process X that preserves the coupling in the sense that (Xo: Xi) ) ~ ll.
PORM (H
FNRM (H
and that solves an SDE, generalizing the result of Bortoli et al. [32] to driving MA-IBM. See Section B.2 for the proof.
Proposition' 5. Fix the -ar-Markovian reference process X V H with associated path measure Q and denote by Z (X,Y) the augmented reference process with associated path measure S. We write 8 for the condi- tional distribution of 11| Recall that Qjo.1 denotes the path measure of the references process X conditioned on (2o, x1) E Rd x Rd, and define P = Io.11 Q by integrating (zo. with respect to llo.1. Assum- ing that absolutely continuous with respect Q we can lift the path measure P coupling preserving path measure Tp* on the augmented space. Under the additional Assumption 2, the SDE
Illustration of FDBM coupling- preserving property shown in Theorem 5: ABM and FDBM preserve the intended pling, unlike SBALIGN, while FDBM of- fers a broader range of trajectories.
Figure 2:
azi FZide GGT E>; [V, (X111
2j GdBr,
(11)
with initial vector Zú OXo, ....) admits a pathwise unique strong solution Z" (X*, Y") with distribution P In particular, X* preserves the coupling Ho, that is, (Xz, X() ~ Ho,
Given a data point Xū = Zn ~ Io, and assuming we could simulate the coupling preserving process Z", we could sample from the coupling Ia,1 by simulating the SDE in eq. (11) forward in time on [0, 1] to arrive at a sample X = Ø1. As X" preserves the coupling, it follows that (Iy. x1) ) is drawn from 1o However, the expectation in the drift of Z* is intractable and hence we approximate this expectation by time-dependent neural network u.. We now define Fractional Diffusion Bridge Models (FDBM) for paired data translation as the stochastic process z associated with the path measure Td solving
dz = FZ at + GGT ve (t, No, Z))dt+ GdB, 2 = (Xo. O.....)).
(12)
u (t, Fo. 2) = [L,w7Gi (t. .......Wkkkk (t. 1))²T (t, (m)())) n = (u.... ..... 4
(13)
5

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where u² = (ūf...... (.) is a time-dependent neural network that takes the starting value xo and the mean (1)((2) of the conditional terminal X (2, = 2) as an input. Note that the output dimensional- ity of the neural network it, trained in the following, correspond to the data dimension d. It is only scaled via eq. (94) to obtain u., which has the output dimensionality of the augmented space. Hence, for FDBM, we can employ exactly the same model architectures as in ABM and simply transform the network input and output according to eq. (94). As a result, replacing BM with MA-fBM incurs minimal additional computational cost compared to ABM, as shown in Section H. To train FDBM for paired data translation we derive in Section B.2 the KI.-divergence DRI. [P'), which yields the loss function
Caaieed (0) = L. Ex.
Xi (in((Z;) ii (1, (o.MM)e(22)) Ile
(TDBM(
dl.
(14)
To compute the above loss during training, we first sample (xo, x1) ~ 1o. and ~ 410 1], and then sample 2, Stixo, (- (1). This is justified since IP* = -lo, Sixo, by Corollary 12.
To provide a first proof of concept of FDBM in the paired data setting, and in particular to illustrate the practical implications of Theorem 5, we replicate the toy experiment from Bortoli et al. [32, Figure 1]. Initial samples from a Gaussian centered at (-2, -2) are paired with Gaussian centered at (2, 2). and samples from Gaussian centered at (-2, 2) are paired with one centered at (2, -2). In Figure 2, we observe that this coupling is not preserved by SBALIGN, in contrast to ABM. Consistent with Theorem 5, FDBM preserves the intended coupling while offering a broader range of trajectories. In the rough regime (H = 0.2), trajectories explore a larger portion of the space, whereas in the smooth regime (H 0.9), nearly straight-line paths emerge.
Unpaired data translation via optimal transport. For unpaired data translation, the goal is again to transport Hy to II, but the training data consist of unpaired samples from II, and II; without given coupling. The dynamic formulation of Entropic Optimal Transport (EOT) seeks the transport plan between Hg and II, as the solution to the Schrödinger Bridge (SB) problem [42], which induces the corresponding optimal coupling. In the SB problem, the reference process defines the underlying stochastic dynamics that regularize the transport, determining how probability mass evolves between Th and IT, We replace in the following the BM commonly used as a reference process in the formulation of SB problems in machine learning [19-21, 27, 29, 30, 33, 34j Let X = v be our scaled MA-IBM reference process associated with the non-Markovian path measure Q. We seek solution to the dynamic Schrödinger Bridge problem
TSB =
arg min (Tk((T)Q) ; To = No, 11) TEP(C*)
(15)
We assume that TSB denotes a solution to eq. (15), inducing the coupling 1 := T. Assuming that P == 16 _ is absolutely continuous with respect to Q and under Assumption 2, we can, via Proposition 5, construct the -couplin preserving path measure P associated to the process Z' (X*, Y*) following the dynamies eq. (11). On the other hand, letting S be the path measure associated with the augmented reference process Z, we define using the marginals of IP* the SB problem on the augmented space via
vSB arg VEP(CE((*)))
(VIS) Vo P.. PP))
(16)
Since Z is a Markov process, the path measure solving the lifted SB problem in eq. (16) is associated with Markovian process [42], whereas 2 in eq. (11) is non-Markovian due to its dependency on X, in the drift function. Motivated by this observation, we generalize in the following the definition of reciprocal class [34, 47] and the notation of Markovian projection [21, 34, 48] our setting of a scaled MA-fBM reference process. We define the augmented reciprocal class RS) of S as the set of path measures V on the augmented space whose marginals can be sampled by first drawing (vo, .1) ~ V Xx,,x, and then sampling žt ~ Sexxo.x,( (- zo, 21).
Definition 6. We say that Ve E (((C4((++1) is in the augmented reciprocal class R. of if
S|xo.X (. (Zo, xz )dW Xo.x,
X (Io, =: X,
(17)
3dxB"
For any V E (K(d)))))) we define the augmented reciprocal projection by
(roj := V Vx..x, 3 X1x
(18)

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(a) ABM(WWD. 0.015 + 0.019)
(b) FDBM 0.7; WSD: 0.012 0.002)
(WSD: 0.082 + 0.028)
(d) 0.2; WSD: 0.048 0.039)
Figure 3: Qualitative comparison on Moons and T-Shape. Plots and datasets design follow Somnath et al. [31].
Since we know that the solution to the lifted SB problem in eq. (16) a Markovian measure, we project any element of the augmented reciprocal class to Markovian path measure by the following definition.
Definition 7. For E P(c)((K+)) with R. (S) we define the augmented Markovian projection
projm, (V) by the path measure associated to M (M²,
22 ..... ......++... solving for Mb V Mg
dM,- PM + GG Ev,
Vm, log l (M M.IIM.] dt GdB.. Mo (M. 0k) (19)
Finally, define FDBM for unpaired data translation stochastic process Z associated with the path measure pp" solving
az F d2 GGT o ) GdB,, z (t, [1, (... (t, (t.
(20)
v (uf (f, .
(21)
where, in contrast to the paired setting in eq. (12), we do not provide the starting value Xo as an
input to the neural network of. We conjecture that the results of Peluchetti [33] and Shi et [34]
generalize to our setting, such that the path measure solving the lifted SB problem in eq. (16) is
the only Markovian path measure in the augmented reciprocal class (S) and that a solution to the lifted SB problem give in its first marginal a solution to the SB problem in eq. (15). Following
Bortoli et al. [27] we define for our scaled MA-IBM reference process a flow of path measures (P= P )=20 recursively by 1 =
Xo. 0.P² = projR, (8) (P))) i= = (s; (22)
and propose the generalized loss function
Appoaired (o.P) Je [.a Janya 8 (e, ((e(l))))))))) M)je(() of
dl xo (Fo. 21)8, X (sel jde. (23)
We define ateteratvve Markovian Fitting (a-IMF) with respect to scaled MA-fBM reference pro- cess using the loss function in eq. (23), following Bortoli et al. [27, Algorithm 1] with a two-stage training procedure consisting of pretraining and finetuning. As discussed in Section B.5, simulating the time reversal of eq. (20) is generally intractable, since the terminal value of the noise process de- pends information from the initial distribution llg. We therefore adopt the forward wwward-forward train- ing strategy described Bortoli et al. [27, Appendix I], and mitigate error accumulation through the loss scaling proposed in Section B.5.
We emphasize that we do not claim convergence of the resulting algorithm to the solution of the Schrödinger bridge problem in eq. (15). Empirically, we observe that the finetuning stage with an MA-fBM reference process performs reliably only in regimes close to H 0.5. We hypothesize that this limitation arises from discrepancies between the Schrödinger bridge transforming II, - II, and the Schrödinger bridge transformation IT, > Πo. See Section B.3 for more details on challenges and limitations of FDBM in the unpaired data setting.

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RMSD(A) J
% RMSD(A) <
RMSD(A)
Table 1: D3PM Con- formational changes, re- sults marked with as-
DAPM Test Set [3]]
Median
Mean
Stdd
7-
7
10
Median
Menn
Sid
EGNN [I]]
19:50
21.37
8.21
156
11
3%
SBALIGNis.a (osaoo [31]
3.80
4.98
3.95
6976
90%%
terisk (*) are obtained
SBALIGN [1]]
3.81
5.02
3.96
0%%
70%
93%
SBALIGN [35]
3.67
4.82
3.93
05%
71%
90%%
1,30
1.92
2.59
from the specified refer-
ABM T-] (eiranne
Sesame" [S5]
233
2.87
3540
3.65
2.95
38%
8276
9696
96%
2.15
---T""""
3.41
4.26
ence. Metrics for FDBM
FDBM 0.3: (ours)
2.35
3332
3.12"
1950
8550
9796
2.52
3.49
3.39
and ABM are averaged
FDRMCH M 0.2) (ours)
2.12
3.34
3.50
48%
64%
960
2.44
3.39
4.28
FDBM 11
2,20
3.44
3.57
40%
85%
97%
2:47
3:45
1.29
over 5 training trials.
Experiments
We evaluate the performance of FDBM both paired and unpaired data translation tasks; see Section for a detailed description of the evaluation metrics. In the paired setting, we first show in a proof-of-concept synthetic data that the alignment of training data is preserved and then predict conformational changes in proteins. In the unpaired setting, we consider image-to-image translation across visually distinct domains. Detailed architectural specifications, compute resources, training protocols, and dataset descriptions are provided in Sections and G, additional experiments are reported in Section K, and additional use case on cell differentiation is presented in Section J.
5.1 Experiments on paired data translation
Synthetic data. We evaluate FDBM on the Moons and T-Shape datasets introduced by Somnath et al. [31], and depicted in Figure 7, where the goal is to transport the initial distribution (blue) to the target distribution (red) while preserving training data alignment. Quantitative performance is assessed using the Wasserstein-1 distance (WSD) between the generated and true target distribu- tions, averaged across the two data dimensions and over ten training trials, where for each training trial 10,000 trajectories are sampled for evaluation. We first do in Table 8 an ablation on the best performing diffusion coefficient V for our baseline ABM, where we find the best performance for ve = 0.8 on Moons and VE = 0.2 on T-shape. In Table 9 we observe that FDBM improves the quantiative performance on both datasets. For the Moons dataset, optimal performance is achieved for V = 0.8 trajectories. Conversely, for the T-shape dataset, rougher dynamics with H = 0.2 and vE 0.1 yield the lowest WSD. In Figure 3, we observe that both ABM and FDBM preserve the training data alignment, with FDBM showing qualitatively better performance on T-Shape.
0.8 in the smoother regime with H E {0.6, 0.7), suggesting benefits from more regular
Conformational changes in proteins. Following the training and evaluation setup of Somnath et al. [31], we use their curated subset of the D3PM dataset [49] evaluate the ability of FDBM predict 3D ligand-bound (holo) structures from given 3D ligand-free (apo) unbound protein conformations. Performance is quantified using the not-mean-square deviation (RMSD) over carbon atom coordi- nates. To assess whether predicted structure is closer to the target holo conformation than to the initial apo conformation, we further compute the ARMSD, where positive values indicate better per- formance [46]. We first optimize our baseline ABM with respect to the diffusion coefficient /E and find that low value of ve = 0.2 yields the best performance for ABM (see Table 10), We then use the same training configuration and a diffusion coefficient of V 0.2, to train ABM and FDBM five times and report the averaged scores over sampled trajectories from these trials. We first observe in Table that ABM outperforms SBALIGN [31] across all variants and metrics, and Sesame [35] in all but one metric, highlighting the strength of our bascline. For our FDBM, we find in Table that all configurations in the rough regime (H 0.3, 0.2, 0.1) of MA-IBM achieve equal or better performance across all but one metric compared to the best-performing baseline, ABM. The best overall performance for the ARMSD metric is achieved for H 0.3, indicating that FDBM gener- ated structures are closer to the target holo conformations--rlatin those produced by ABM or Sesame. For H = 0.2 and H = 0.3. FDBM matches or exceeds ABM and Sesame across all evaluated metrics. In particular, an RMSD below 2Å is commonly used as threshold for correct bound structure prediction [50] and structural discernibility [31, 35J. Accord- ingly, the proportion of predictions falling below this threshold is direct indicator of the model's ability to generate physically realistic conformations. FDBM increases the proportion of correct and discernible predictions (RMSD < 2Å ) from 43% with ABM to 48% while also improving the median RMSD from 2.40A to 2. 2.1 This indicates that, in the rough regime of MA-fBM, FDBM produces average slightly higher fraction of naar-native structures compared to ABM.
to their apo starting points-than
8

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Input
Outputs Maaler-aaaaammaaa
Input
Outputs
(a) FDBM (I=0.44, K=5; FID: 30.11) for AFHQ-512
(b) FDBM (H=0.6, K=5; FID: 19.42) for AFIQQ-256
Imput
Outputs (Euler-Mae Murysmaa
Input
Outputs (EalerMaal Meryamaa
(c) FDBM (H-0.4, K=5; FID: 14.27) AFHQ-512 (d) FDBM (H=0.6, K-5: FID: 11.62) for AFHQ-256
Figure 4: Exemplary FDBM samples (ours) for wild cat (a. b) and cat + wild (c. d) using DiT- L/2 on AFHQ-512 and AFHQ-256. Left: inputs; right: Euler-Maruyama samples (distinct seeds).
5.2 Unpaired data translations
Unpaired data translation evaluated for the and wild subsets of the AFHQ dataset [51]. Ex- periments range from low-resolution pixel space (32 x 32) to high-resolution latent space settings (256 x 256, 512 512) [52]. Following the regime Bortoli al. [27], we report Fréchet Inception Distance (FID) [53] and Learned Perceptual Image Patch Similarity (LPIPS) [54] scores. Given the sensitivity of metrics essecially at low resolutions where pixel-level perturbations dominate each configuration is evaluated at ten distinct seeds, with mean and standard deviation (or error bands) reported. To ensure comparability, pixel data is normalized by the standard deviation of AFHQ-32, while latent representations are scaled using the standard deviation of the latent space. This harmo- nization enables consistent settings for ε in both domains, leading to consistent performance trends (Figures 5a and 5d). We use Diffusion Transformer (DiT) [55] backbone, where DiT-B/2 is used for ablations and DiT-L/2 for final evaluations. Pretraining is conducted for 100K steps, followed by 4K finetuning steps, samplings follow the Euler-Maruye Maruyama method [1]. We compare to SBFlow and adopt an SBFlow- -Boooioopti entropic regularization parameter for FDBM experiments. Further, evaluate Hurst indices H (0.1, ... 0.9) and the number of OU processes K E {1,....6) 6) to analyze sensitivity in sparse (AFHQ-32) and dense (AFHQ-256 and AFHQ-512) features.
Results for unpaired data translation. The ablation study reveals stable generation performance for H > 0.4 and K < 5, with instabilities and accuracy degradation observed for K > 5 and H < 0.3, Figures 5b, 5c, 5e and 5f. Our method remains stable for high dimensional data, such as AFHQ-512 even for 0.4 < H < 0.5 (see Figure 4). Across various configurations, our method consistently outperforms the SBFlow pretraining- and online finetuning baseline (see Table as well as Figure 5). Notably, with K we do not recover BM, as we fix 71..... 7KK. even when / = 0.5. MA-IBM with H 0.5 and K 5 is non-Markovian, though its distribution is empirically close to BM. This subtle differences may be the reason why FDBM performs better than SBFLow on 0.5 and
AFHQ when
5. Bortoli et al. [27] propose a finetuning method for processes driven by BM, which can yield significant improvements over their proposed pretraining for natural images. The online finetuning assumes the bidirectional processes transition on the same bridge with matching pairings and respective terminal distributions. In our framework, we can not assume a shared Schrödinger bridge for the transformation I. - IT and IT, - Th. general, two distinct bridges are learned. Improvements during fine-tuning were observed only for MA-fBM

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Table Results for AFHQ-32 and AFHQ-256 (10 runs average). Standard deviations are reported beside each score. Bold indicates the best result and those within one standard deviation.
(a) AFHQ-32 results with hyperparameters 5 and H 0.5, K = 5.
Pretraining
Online Finctuning
Method
Architectuee
cats wild
cons
cats
wild
wild
FTD
LPIPS
FID
LPIPS
LPIPS
LPIPS
SRFFlo
DIT-R/2
50.04
0,104
74.36
0.151
43.85
0.083
64.77
0.107 10:000
DIEL22
50.68
10.7a
0.10G
10.00
71.77
10.17
0.162 1300
38.92 10.50 0.001
10.000
64.10
2..72
0.099 1.0000
HD (ours)
0.007
45. Fovdo
25.06 ta.a
0.075
28.33 -0.35
0.078 50.501
FDBM (ours)
D/F-22
35.99
0,101
48.84 0.165
20.26
0,079
26.79
0.065
#2.001
(b) AFHQ-256 results with hyperparamcters 1 and H 0.6, K 5.
Pretraining
Online Finetuning
Method
Archisecture
cats
cats
cats
wild
wild
FID
LPPSS
RID
IPPIS
4
LPIPS
IPIIS
SHFiow
DIT-B/2
15.67
0.578
30.75
0.594
17.60
0.628
10.001
25.86
0.687
DIFL2
16.02 62 to.as
0.804
20.001
33.96
20.87
0.600 +3.002
16.98 +0.39
0.500 =0.001
-0.43
0.547 +0.003
TDBM (ours)
DiF
0.330
18. I4 Fo.5s
0. 585 =0.001
FDBM (ours)
DIT-L/2
11.62 20.7
0.548
19.42
#0.41
0.561
140
501
501F
01
15
(a) AFHQ-32- - V SBFlow
(b) AFFQ-33 H in FDBM
(c) AFHQ-32 32 K FDBM
6.50
1:00
175
1150
32
(d) AFHQ-226- V in SBFlow
(c) AFHQ-HQ220 256 H in FDBM
(f) AFHQ-256 K in FDBM
Figure 5: DiT-B/2 ablation for AFHQ-3 32 (a-c) and AFHQ-256 (d-f); (c): H = 0.5, and (f): H = 0.6. We show error bands with averages over 10 runs. SBFlow baselines marked in (b. c, c), and (f).
with H = 0.5, likely because the forward (Tl- - IT and backward bridge (TT, + II)) are close- though not identical-to the Brownian case with bidirectional bridge. However, this effect does not generalize to other H, and developing a principled finetuning strategy for FDBMs distinct bridges remains important direction for future work. Table 2 shows that our method can significantly improve the fidelity of generated samples, while maintaining data alignment. Figure 4 highlights that we can obtain cohesive data alignment without online finetuning for H 0.4 and H 0.6 at scale. See Section K and in particular Figures 5 and 6 for samplings at scale.
6 Conclusion
We introduced Fractional Diffusion Bridge Models (FDBM), a new generative framework that ex- tends diffusion bridges beyond the Markovian assumptions by incorporating a Markovian approx- imate fractional Brownian motion to retain computational tractability while preserving long-range dependencies or roughness that are absent in Brownian generative models. Our fractional generative diffusion bridge is coupling-presse preserving in the paired case and generalizes the Schrödinger bridge formulation for unpaired settings. In the paired regime, FDBM improved the near-native structures of predicted protein conformations potentially by capturing non-local dependencies; in the unpaired regime, it achieved superior quality in image translation scaling robustly across high- nigh-dimensional domains and image resolutions.
FDBM opens a broader avenue for generative modeling, bridging fractional stochastic dynamics and machine learning, and poses a foundation for learning from the correlated, memory-rich phenomena in real-world. Future work includes theoretical guarantees for fractional Schrödinger bridges, fine- tuning of asymmetric bridges, and extensions to manifold-valued fractional processes.
10

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Acknowledgments and Disclosure of Funding
This research received funding from the Flemish Government under the 'Onderzocksprogramma Ar- tificiële Intelligentie (AI) Vlaanderen' programme. Furthermore under the SBO project CADAIVISION. This work also received funding from imee-PROSPECT project ADAPT ('Affinity and Developability through AI for Protein Therapeutics', This work was also supported by the German Research Foundation (DFG) through research unit DeSBi [KI- FOR 5363] (project ID: 459422098). 'T. Birdal acknowledges support from the Engineering and Physical Sciences Research Council [grant EP/X01 364/1]. Birdal was supported by a UKRI Fu- ture Leaders Fellowship [grant number MR/Y01 8818/1] as well as Royal Society Research Grant RG/R1/241402.
supported by Flanders Make
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---

Appendix of Fractional Diffusion Bridge Models
Notational conventions
20
B Mathematical framework of fractional diffusion bridge models
21
B.1 A Markov approximate fractional Brownian bridge.
21
B.2
Theoretical framework for paired training data
24
B.3
Theoretical framework for unpaired data
28
B.4 Sampling from partially pinned process
29
B.5 Loss regularization via the reverse pinned process
C Broader impact
34
Related work
34
E The Schrödinger bridge problem for unpaired data translation
35
Implementation details for paired data translation
36
G Implementation details for unpaired data translation
38
H Computational efficiency
38
I Evaluation metrics
39
J Cell Differentiation
40
K Extended Experiments
41
19

---

A
Notational conventions
Rm
m dimensional Euclidean space
B (R)
Borel-o-algebra on IR
(S.)
Stochastic process taking values in IR"
Set of continuous functions (paths) C.D = C( R") from the unit time interval [0, 1] to R. Borel-o-algebra on C
B (CCM)
P((
Set of probability measures (C (C")))
B
P(C
Path measure
Distribution (S,,.... S...) under the path measure
P Prinan
Path measure IP is associated with process S and Pl.... denotes the distribution of (St, St..)
Conditional distribution of (St..-.. St..) given (Sm.....Sp, Sy.)
Pir
Conditional distribution of S given (.. S.) Data dimension
d L
Il II
Source and target distribution Rd Joint (coupling) distribution on Rd Rd
Io,11
B
(Multidimensional) standard Brownian motion
BH
(Multidimensional) Riemann-Liouville (Type II) fractional Brownian motion (fBM)
yr
(Multidimensional) OrnsteinUhlenbeck (OU) process with speed mean reversion ? E IR
K, N
Number augmenting processes K and S & KSK
7,.. 7K
Geometrically spaced grid
Wi...., WK
Approximation coefficients
3"
(Multidimensional) Markov-ap fractional Brownian motion (MA-IBM)
x
Scaled MA-fBM reference process X = VeAz with & > 0
Q
Path measure of the reference process
1o. Q..QQn 0,1
Mixture of bridge measures JR Qu, ( x )dlll, (20,
y
Augmenting process y Y"
y
Stacked augmenting OU processes Y = (Y' .y.,,.., y taking values in BaK
Z
Augmented process Z (X, (K++(+1))
Path measure of the augmented process
F
Drift matrix (((())))))((() of the augmented forward process
G
Diffusion vector G E dR((++)) of the augmented forward process Partially pinned process Z (i),
Zixo.1
Spxo,X
Path measure associated with the partially pinned process Zix
TSB
Solution to the dynamic Schrödinger bridge problem
Path measure the augmented path space P(Cd(K+1)
20

---

B Mathematical framework of fractional diffusion bridge models
In this section, we present the mathematical details of Fractional Diffusion Bridge Models (FDBM). The main contribution of this section is the proof of Theorem 5 in Section B.2, which generalizes the construction coupling- presservngg stochastic process by Bortoli et al. [32] to our fractional noise setting.
Notation. For any m E N. we equip the Euclidean space R with its Borel-o-algebra (R. Below, will typically be either equal to 1, d, dK, 1).+ Next, write C = C( 1], ) for the set of continuous functions (or continuous "paths") from the unit time interval [0, 1] to Th and equip this set with its Borel-o-algebra B(( where open sets understood with respect to the topology of uniform convergence. The set of probability measures on (C (B(²)) is denoted by P(C"),, ), and refer to the elements of this as path measures. If X is stochastic process and F E PP(CT) denotes the distribution of X, we subsequently say that the path measure F is associated with the process X. Observe that any P P(C")) is associated with some stochastic process X, as ), IP) as our probability space and let X be the canonical process
we may take the space (C"h, ((TT)
given by
[0, 1], E E (x(w) == Wt. E
(24)
Given a path measure E P(CT)) associated with a process and time points , [0, 1] for some E N, we write Ft, Fi for the joint distribution of (Xe....., Xe,,) that is
Pr P == P(fw CD
+)).
(25)
In particular, Pr denotes the marginal distribution of (()) for any 1 E [0, More- over, given $1....,5E
[0, and :.,...., J E RT. we write Pr....... 181....8 and for the (regular) conditional distribution of (X xx,.) given
P fale.ne (115s1 [is.....,,,
(K......,. . Xx.) and {X,, Tsi:..., Xs, J. respectively. In the same spirit, write al and P. PIst..,se ( -[ ) for the (regular) conditional distribution of the process X given (X.,....,, ) and (X., Es ,...,..,,,,
X = J.c respectively.
B.1 A Markov approximate fractional Brownian bridge.
We fix d-dimensional Brownian motion and define the Beemann-LLouville (Type II) fractional Brownian motion (fBM) [22] with Hurst index H E 10, via
B T(H + 2) [ (( "c-ss)-bdBB, 3dB,, 12
(26)
For given Hurst index H E (0. 1), we consider Markovian approximation of fBM 265, For K E and geometrically-saaeed speed of mean reversion parameters ?,,-..,?K >> 0, we consider Mustein-UUlenbee (OÚ) processes of the form
Y = 1 - e
t E [0, 1), k:
K.
(27)
With this, for given scaling parameter E >
0 and suitably chosen approximation weights wi....,wK wK E R, the process X == VeB" defined in terms of the weighted superposition
DH
Y
(28)
of the OU processes scaled Markovian approximation of fBM (MA-fBM), While the choice of approximation coefficients in Harms [43] enables strong convergence to fBM with high polynomial order in K for И < 0.5, we opt for the computationally more efficient method proposed by Daems et al. [26]. This method selects the (P( optimal approximation coefficients for given Hurst index H (0, 1) and given K IN by minimizing
27 dt B, ),²
(Wi,..., wK) = arg min
B (
(29)
21

---

Following [26, Proposition 5], the so defined optimal approximation coefficients w = (wi...... ...,,K) solve the system Aw b, where A and are given closed form [26, eq. (19), eq. (21) and hence we choose w == b- Note that these optimal approximation coefficients depend on the Hurst index H E 10,1) and the number of OU processes K E N, since the matrix A and the vector b are functions of these parameters. We subsequently refer to X = VEB" as the reference process, Y == ....., y*k as the vector of OU processes, and Z == (X,Y) as the augmented reference process, respectively. Note that the dynamics of the augmented reference process are given by [16, 26]
dz = FZ,dt + GdB,
(30)
for a matrix F E ()(KK)) and avector G E R((K+1) [16, 26]. The path measure associ- ated with the reference process X is denoted by Q E P(CC), whereas the path measure associated with the augmented reference process is denoted by S E P((C((+1)) and we write 8/ for the conditional distribution of XilZ.. Note that the reference process X, well as its corresponding path measure Q is non-Markovian and becomes Markovian only after augmenting it with the OU processes, resulting in the Markovian augmented reference process Z.
Proposition 8. The reference process X for K > 1 and all H E (0, 1) the non man-Maakovian process
K
(t,t s)Y + VE VE>ur wk [
frve
pts
Xe+s X, +
Σ
e-
(31)
where for each hs == 1,...., K, and 1,8 E [0, 1] with 1 + 8 S
(t :>==))>=))) f. e- Vee(e -1).
Velc"ue
(32)
In particular, we see that S stolee ( (2) is Gaussian and hence for d = 1, with s = 1 - t.
vx log 8 (=112) [1, 1), wwL.(tt 1)1 -1 (2j{(z)
(33)
of +
where
(1p(2) := x +
Σ
wt, 2 (2, V1. yk)
(34)
k
denotes the conditional mean and
WaWe (1 e -10000mtt))))
(35)
7k + 7/6
k
the conditional variance of the reference process X21(2, = =))
Proof. For the scaled MA-IBM we have for s,t E [0.1] with t + I 1 by the Stochastic Fubini Theorem [25]
K Σ
Wkitk ('' dr + Σ B. Σ
K
Wkikk L. Y + Σ wa (B44. -B))
X4+.
(36)
Wkyk phte le le- S-" "T dB. /
Σ
wa Wu(Bi++ B.)
(37)
1=1
pto {- c WD
= BM
We Yet (6- 705 - 1) + (Bets - Bi) - 7x Se
(38)
pto geta e u)
= B
Σ We y,* 77 -1) + (Bets - Bi) 7h
(39)
K
76 S.t pts
D
v. (e - +
Σ
Wk
e c
(40)
h-1
K Σ
K
pho
X.
wxCult,t + +s)Y,² + Ve
Wk
e w
(41)
k=
22

---

Additionally, we calculate via eq. (41) with 1 - t and = (I, Vi.....Kk the conditional mean
Pi = B = 2 = x + Whyn Ca (4,1), = - (2r, ,1 5K)
(42)
and the conditional variance
while (1 e1
i = Cov (X,, 2, = =
(43)
A,8=1 7ke + ne
where we use Ito's isometry. To see that X is non-Markovian, we note that the future Xt+s depends not only on X, but also on Y ,,..,, Y which depend on the path B up to time t. For a more precise argument, we have by definition
B
whe Te(t-u) dB
(44)
and note that process X = (Xe)ee(0.1) with
x, S nol,, u)dB..
(45)
is a Markov process, if and only if we can find functions f and g such that [56, Theorem II.1]
x(t, u) = f(t)g(u).
(46)
Since have n # 7/ # 7k for the defined MA-fBM, functions f and g satisfying
wye Tul-Ta((-u) = f(t)
(47)
exist for K >>> 1 if and only if w/ o 0 for at most one 1 < j S K. Hence, MA-BMM and therefore our reference process X-is not Markov-process for K > and any choice H (0, 1).
To define a stochastic bridge with respect to connecting two given points 2g Rd and aı E Rd observe that we only have to steer the first dimension of the augmented reference process Z XX towards 21, while the terminal values Y are not required attain a specific value.
Proposition 9 (Markov approximation of a fractional Brownian bridge [45]). Let X = VEB" be scaled MA-fBM, >> O and Z (X, Y) the augmented reference process. The partially pinned process Z ; (1 = Io. X1 = œ1) associated to the path measure S|s0.3, follows the dynamics
d7 12 (1) Pe (1)dd + GGT ul1, Mpo.os (1))dl + GdB,,
(48)
M1j((2)
2t,, (1, WK. 1))"
u (ui,.... ua)
(49)
Proof. Daems et al. [45] use a Gaussian expression for the reference process to construct the pos- terior SDE that is steered towards I1. We derive for a fixed data pair (zo, the dynamies of the partially pinned process Z (X,, Fo. X1 = 21) using Doob's s h-transform [2]. resulting in the same dynamics as in Daems et al. [45]. Towards that goal, we define the transform
h: [0, 1) x ((+1) + [0, (t,z) + (2112)
(50)
where s satisfies
P(XI E AlZ. =)== 8 (1r(d)) A c cd.
(51)
Denote by (2) = (z, y) the density of Z, such that
(x) = | S1 yddy
(52)
23

---

(a) Trajectories from the ap- proximate 1d- id-fractional Brown-
(b) Variance of MA-fBM with normalized terminal variance.
(c) Variance of the augmenting OU processes Y1 Yi, Ys approxi- mating fBM as a weighted sum.
lan bridge for different Hurst in-
dices.
Figure 6: Evolution of variance in MA-IBM.
and write (=)2)o the transition density of Z from time / to + s. To show that h defined in eq. (50) satisfies the space-time regularity property we mimic the proof of [2, Theorem 7.11]. We
write with Bayes rule
I Slicto, (x) l ())())((((1a)))
I Siito (1)
Setale (1)
(53)
1 (xi|z)
S (Z]
where we use for the second equation that a Markov process. Hence, equivalently
S (=)=)( (x1 (3)2,
(54)
such that
S S(+s|o(E|2)1(() =,= S ( (ar (11|=)d
(55)
J Sepoleers (3)2, x1 21) 22)d
(56)
/
Setale. (i)
(57)
-t(t,z)
(58)
Hence, by Särkkä and Solin eq. (7.73)-eq. eq. (7.78), we conclude that the partially pinned process 2i satisfies
dz 2: (t) = FZ 20 (t)dt + GGT Vg log 8 (Ir 1
// 20 (t))dt + GdB,
(59)
Moreover, from eq. (33), obtain V, S V log S
V log 1ogg
with
V. log 111, i10,
(z11z)
aattt 2).
(60)
oi
See Figure 6 for visualization of id-trajectories and Figure for 2d-trajecotires of the above defined Markov approximate fractional Brownian bridge (MA-IBB).
B.2 Theoretical framework for paired training data
Fix probability measure Io Rd x Rd, which refer to as the coupling measure. The marginals of this measure are denoted by Ho and II, respectively, which means that
(A) = { fAves (2o, and [₂(A) = / dI (1o.
A B(R)).
JR
Our goal is to construct a stochastic process X* that preserves the coupling in the sense that (X6,Xi) lo., and that X solves a stochastic differential equation (SDE). If that is achieved,

---

we can sample from the coupling Io by first sampling Xi = 2o ~ Iy according to the first marginal of 1. and then simulating the SDE forward in time on [0.1] to arrive at a sample Xi = I1. As X preserves the coupling, it follows that (æo, x1) is drawn from No,1- Recall
that Q ( To, v1) E PP(()) denotes the path measure of the reference process X conditioned on (Xo, (zg. E R x RR. We define new path measure P P(C")) by integrating (Io, z1)
with respect to 1I that is
P
Q [oo xn )d (Zo, 21). B4XB"
(61)
To wit, the process X associated with IP is the reference process conditioned on xi) ~ oo.1 Indeed, this is seen immediately as for any Borel sets Au, A1 C R we have
P([w E Cd w() E Ao. (1) E A1))
Jee xRd
Qoo.(ffw Ca w(0) e (o. E 2 1
(62)
/ A (z) (d)ddll. (1))
(63)
JR
1 A1)
(64)
key assumption for establishing the existence of an SDE whose solution X has distribution is that IP is absolutely continuous with respect Q
Assumption 1. The path measure P & P(C4) is absolutely continuous with respect to the path measure Q E P(C")) of the reference process X. In particular, there exists a density
dP dQ
c (0)
(65)
The density dPP /dQ allows to lift the measure P path measure Fpt on the augmented path space (1(K+1) via the Mndon-Nikodým density
dP
dP
(w)
(wx,wy) cd(K+1).
(66)
ds
dQ
As first step, we show that IP* still preserves the coupling. Lemma 10. For any Borel sets Au, A1 Rd, it holds that
Ipt (p E cd(K+)), (() Ao BRKK ww(1) Ai x R) IAo A1).
(67)
In other words, T preserves the coupling No,1.
Proof. Any E (d(K+1) decomposes uniquely into pair i = wwx,wy) with wx E Cd and wy E cak Next, we subsequently write ( (wx) for the (regular) conditional distribution of the OU process Y conditional the path of the reference process X being wx Cd. Using the disintegration theorem, it therefore follows that
P'((w E cd(K+)) w()) E A X R.cK .w(1) E A1 x (dd1))
(68)
(HW E cd(K+t) (0) E Ax, E AI))
(69)
dP J dQ
(4x))(())))
(70)
|(a dQ
dlP
(wx [wx) (wx (4x) (wx(1)) /
dQ (dd()))))
(71)
CaN
I.s dIP
(wx) ( (0)) A (wx (1))dQ(wx)
(72)
TP( E Cd (()) E Ao, (1) A1)
(73)
= (Au x A1),
(74)
showing that P* preserves the coupling No,1-
25

---

For any E Rd(++1) we subsequently denote by
1i 10 (- |=0) [d((-1)
+ (0, (0,
(75)
1$1
the density of Fijo -12) with respect to S $0p0
(0). In the same spirit, given zo R we write dlP dlp
(- (mo) IRd + [0,
(76)
dQuo
for the density of Piju( (20) with respect to Qıpl( (z)). By eq. (66), it follows that
dlP 110
dP 110 (#1 (m)
(2) 20) = 1 (0) (3b)
(77)
d3 10
d0
for all (Io. yo), 21 (01, v1) Rd(K+)), Now introduce two functions
(z0. + (20. x1) == [o) (vv) dPij0 (21
h1 : Id((+1) x Rdd + [0, oc),
(78)
dQu
and, with this, Id((+)) [0. Rd(K-1) [0. 00) given by
1 (,= == J In 01 (1) (z0, E Rd(++)) x (0, 1] Bd((+))
(79)
Observe that
h(z0,4, (, Es [1 Xi) =0, Zi = = [11 (=o, Xi) 20. Ze z]:
(80)
where S/ denotes the conditional distribution of X1 given Z. In particular, 1(20, 1, = h1 (2)
whenever (x,y). v) In what follows, enforee the following assumptions on h.
Assumption 2. The function h defined in eq. (79) is jointly measurable. Moreover, for all fixed 2o (K++)) the mapping (t, 2) + h(20, 2, satisfies
inf (h 2) (t,z) E [0, x R(KK+1) >0
(81)
and member of C[0, 1| x Rd((+1), o0)), the space of bounded and twice continuously differentiable functions with bounded first- and second-order derivatives.
Under these assumptions, it is possible to show that the coupling preserving augmented measure is the distribution of a solution of stochastic differential equation.
Proposition 11. The SDE
dz; FZ, GG Ez; GGT V log (x1127))))) dt GdBG, = OX..))...0), (82)
admits a pathwise unique strong solution Z* = "X,Y") with distribution F*. In particular, X* preserves the coupling 1o,,1, that is. XX) I1.11.
Proof. For 2n E +1) and E 1], consider the linear differential operator Lt mapping functions tp E ((( to
Le0 (F2+ (GG* V h(z. 2), Vp(2)) to
(+p(2)) VAp(2)) 2 Rd(K+1) (83)
Due to the assumptions imposed on h, it follows from Lemma 3.1 in Palmowski and Rolski [57] that the local martingale problem associated with the operator Le30 and initial distribution 8z, is solved by (- Thus, by Theorem 18.7 in Kallenberg [58], follows that the stochastic differential equation
az* = FZ* (1)dt + GG V2 log 1, )t + GdB, Z =
(84)
admits weak solution in Z 2g with associated path measure P.( (z0). Next, since h(zo: :) C[( 1] x pd(K+)) [0, oo)) implies that 2t + V, log h(20,,,, 2) is Lipschitz continuous and
26

---

therefore the solution of the SDE is even strong and pathwise unique. Finally, it follows that the pathwise unique strong solution Z* of
dz = FZ* + GG V log h Zg. t, dt) + GdB,,
Z = (Xu, (X..0......)
(85)
has distribution Px as P Ll.. We conclude since
lo (Z t, 2 Ep ijo. V log
2,1
(86)
using eq. (80) and following the arguments in Bortoli et al. [32, Proof of Proposition 3].
In Theorem 11 we constructed the coupling-preservee path measure F* associated with the stochas- tic process wish to learn. The following corollary establishes that can obtaín samples zt P by first sampling (fo, v1) ~ 1 and subsequently sampling žt ~ S (: Io, 21).
Corollary 12. For the coupling-preserving preserving process Z* constructed Theorem 11, the associated path measure satisfies IP* x|x
Proof. Since Fx* preserves the coupling Io.1 we have
[
I
Pix..x, (-120, x1) )dP Xo. x, (ro, x1)
(87)
Rd RdxRd
Pix..x (-k0,, x siddlll, (x0,
(88)
J Ra
1o, Pc.PP'oo XX.x..
(89)
For Xi we find
V. log Ep i
(1 41x;=== -Ei V log S (o) 2;)]
(90)
V. log S (z))
(91)
since V. log S (x1 (2') is measurable with respect to oZii Z2). Therefore 2xz,x; solves the SDE eq. (48) of the partially pinned process and we conclude Z Zxo.x, such that
Sx
(92)
Given data point X3 = To Ho. and assuming we could simulate the coupling preserving process Z", we could sample from the coupling Io, by simulating the SDE in eq. (11) forward in time on [0, to arrive a sample Xi = z1. As X* preserves the coupling. follows that (ro, drawn from Il, I. However, the expectation in the drift of Z* is intractable and hence we approximate this expectation by time-dependent neural network uf We now define Fractional Diffussin Bridge Models (FDBM) for paired data translation as the stochastic process 2 associated with the path measure 1e solving
d FZZ dt + GGT ve (t, (t, No, 2 GdB., 28 (Xo, O...,,,.. 0),
(93)
u (t, To, 2) [1, Wi (t.1))...,,K WKCK(t (t, 1)1² ut (t, ro, (z)())), u = (u)
(94)
where v (ūf ....... i time-dependent neural network that takes the starting value xo and the mean M1pp 22) of the conditional terminal X2 X11(Z, 2) as an input. Denote
ū(t) = [1, wi (.wiGG( 1), SK (t, 1))" E RKK+
(95)
and define
o(t)
0
(t)
0
(t) =
E Rd(K+1)
(96)
((t)
27

---

Since t + ||G b(t)l is continuous, it attains its maximum on the compact interval [0, 1]. Hence, we find ||G 11)) S c for some constant e > 0. Parameterizing the learnable process 2 associated with the path measure Te according to eq. (93) we aim to minimize the KL-divergence (KL(P" (Px IP*).
We calculate using Girsanovs theorem (See Blessing et al. [59, eq. (30)) for our setting), together with the stochastic Fubini theorem and Jensens inequality
DKL (P" (F"IF") = Boo,
| (Erijo. [v. log S| (Xi (i zi] u² Xo, zi)
2 dt
(97)
2 / Erg.
1
[II"" { (Eei o.t
Iv log S
u (1, Xu, Zi) Zū. -i1} 11 di (98)
1
L.
| (CPoco)12
Xi 11-1i1i17 (Zi) ū (t, 2i11 Zö. Zi
$
2
0
Erg.
Bey Bei
dt
(99)
Xi (ii(
E2g.
Esinao
u (t, Xo, Zi))
Zo, Zi
dt
(100)
2 0
Hie
Xi Mije(Zi)
$
Erg.
Ex
ui Xo, Zz
1/0 4;
dt
(101)
2
iie,0
oic
c
Ep.
X1 Mije (ir(((t)
u (t, Xo. z))
dt.
(102)
J
Hence, we aim to minimize Equation (102) in order to learn the stochastic process Z- During training, the loss is computed by first sampling (zo, .11 ~ Пo. and subsequently sampling 2t
Sejxoox, (. No, 21). This procedure is justified since TP' = 1 by Corollary 12.
B.3 Theoretical framework for unpaired data
Given two unknown distributions IT, and IT and the reference process X VeBB we seek to find a solution to the dynamic Schrödinger Bridge problem [39, 40, 42]
TSB arg min (DkL(TQQ) : To = Il, To 111]
(103)
TEP(Ca)
By Föllmer [41]. Léonard [42, Proposition 2.3] there is at most one solution TSB to the dynamic Schrödinger bridge problem in eq. (103) and if the solution TSB exists, then Th is the solution to the static Schrödinger bridge problem. Assume there exists solution TSB for the above dynamik Schrödinger bridge w.r.t. Q such that n == To. is the solution to the corresponding static Schrödinger bridge problem. By the above Theorem we can construct a process Z' (X*, Y*) with path measure I and dynamics
dz; = FZ, dt + GG
"G"EEjj.. Erjo. Vv So (xi17))) Z7] + GdB(t)
(104)
that preserves the coupling 1g In contrast to the setting of paired training data, we have access to samples of 11 On the other hand, letting & be the path measure associated with the augmented reference process Z, we define using the marginals of 1* the SB problem on the augmented space via
SB arg min
(VIS) Vo P7 V1 = PT].
(105)
(EP(Ca
Since 7 is Markov process, the path measure solving the lifted SB problem in eq. (105) is associ- ated with a Markovian process [42], whereas Z* in eq. (11) is non-Markovian due to its dependency on Xo in the drift function. Motivated by this observation, we generalize in the following the defi- nition of reciprocal class [34, 47J and the notation of Markovian projection [21, 34, 48] to setting of a scaled MA-IBM reference process. We define the augmented reciprocal class R2 (S) below the set of path measures V the augmented space whose marginals can be sampled by first drawing (1) ~ Vx..,, and then sampling 2 ~ S (- 21).
Definition 13. We say that V E ((Cd.(K+1) is in the augmented reciprocal class (S) of
V
Sixo,x ( (ro, 21 )dW xo.x, (20, 211) =: =: Xx,,x,
(106)
Jad
28

---

For any Ve E ((Cd)(K+))) we define the augmented reciprocal projection by
(s) V Xx.X, X1
(107)
Since we know that the solution to the lifted SB problem in eq. (105) is a Markovian measure, we project any element of the augmented reciprocal class to Markovian path measure by the following definition.
Definition 14. For V E ((()((+1)) with V E (S) we define the augmented Markovian pro- jection projm. (V) by the path measure associated to M = (M', ............ MK*1) solving for M3 Mg
dM, FMdd + G. 1x V log
I (M dt + GdB, Mo (Mg. 0k). (108)
Bortoli et al. [27] introduce a flow of path measures (P*))") IP))220 and show that, for reference process driven by BM, time discretization of this flow with step size E (0, 1] yields a family of procedures called a-IMF, all of which converge to the Schrödinger bridge. For a reference process driven by MA-fBm, we propose to define a flow of path measures (IP= IP*) ).20 recursively by
p = (Il & I1) 0,I = (erooi.cepprroimm.co
(IP*) T,
1 proj.. (PP)) (109)
Both procedures a-IMF and IMF are based on the loss function [27, 34]
#1 - Re C(e,P) Li(v, P)dt J. Saur [P(te))
1- (2x
(110)
We propose to replace the above loss function with CYDBM (0, P))
[ Ine Seepp
oe (mna(z))) - (1]]((() (Rdg2
di (xo, 2ı )d3 (o, æn dtt.
die
(111)
define a-IMF with respect scaled MA-IBM reference process.
Challenges & Limitations. The dynamic Schrödinger bridge problem can be formulated with scaled fBM as the reference process, since Léonard [42] includes non-Markovian processes with continuous paths. To sample paths from the resulting solution, one must draw from a fractional Brownian bridge (fBB). Janak [24] constructs such bridge by leveraging the fact that fBM is a Gaussian process and additionally derives an integral equation characterizing the fBB [24, Theo- rem 5]. However, the drift of the derived bridge involves an integral that is not available in closed form [24, eq. (17)), necessitating an approximation of this drift term when sampling from an (ap- proximate) solution to the dynamic Schrödinger bridge problem. Hence, we first approximate fBM using a Markovian approximation [25, 26] to enable simulation-up to discretization error-of the exact bridge, which corresponds to a partially pinned process. We leave the analysis of how well the solution to the thus-defined dynamic Schrödinger bridge problem approximates the solution of the corresponding problem with a scaled iBM as the reference process for future work. We empha- size that in the unpaired training data setting, we only propose method for using FDBM and do not prove convergence of the algorithm to the corresponding solution of the dynamic Schrödinger bridge problem. To the best of our knowledge, the setting of Léonard et al. [47, Theorem 2. 14] is not applicable here, as our pinned path measure refers to a partially pinned process, rather than a fully pinned process. As a result, proving the convergence of our method would require an adapta- tion of Léonard et al. [47, Theorem 14), which is beyond the scope of this work. Additionally we point out that we are only able to simulate the learned bridges forward in time, since the terminal distribution of the augmenting processes of the learned stochastic bridge depends on the initial data distribution, see Section B.5 for details.
B.4 Sampling from partially pinned process
In this section, we derive the marginal distribution of the partially pinned process for any E (0, 1), enabling simulation-free sampling. For t < 1 we know that (Xe, Yr, X1) XZ) 2) is Gaussian [2] with
YY ..... v.K X1 = =)) ~ (mib(z)
((mi(tt)
Ene (11s))
Ene(tls))
,....
of.
(112)
29

---

with
mm]((2) = (z), (2),
Me/ K+ (z))²
(113)
and
E12(t|s) (coo((x; X1), (Y), coo (x)))* =
(114)
Hence, the process partially pinned at (I:; 21) follows the distribution
Zi((X, = Is, X1 = x1) ~ N (ftaa,, E(ls,)):
(115)
with
1
M (2) [1)((2))+ + E1z(t|s)(21 (11(((()))
(116)
o
==0 (zo, 0,... 0)T + (1)((()))) - zo)
(117)
110
and
Enj.11 = Ee/a- [ns(t)))Een(((s) = Edjs (1)()))))7a(()s))
(118)
oi
oip
We further calculate for constant diffusion coefficient g(() = ER
tu(s,t) = { mnoluue va(u->) -aluueee -
- S: e du g 7u(t-o) -
(119)
and
Maj((2) + wnn + Σ (00-11))))) - -1ykk.
(120)
Left to calculate are the entries of Eges and E12(t|s). With <t 1 we calculate
WHWhy I
Cov(X,, = 2) =
Σ
9)) du
(121)
1.j=1
WiWj J' ((e + ((e-)(1-0)- + du
i==1
(122)
Σ WiWl '' (e (e-ilt-w)) (0-101--) du
K
g²
(123)
i,-=
/' 6 - - du - (6 (o7t
Cov(Y?,yy/1z, =) =
(124)
95 + 7s
and for 0
K
(' -n(l-wu (Ca(u, 1) g(u))du
Cov(Y! X1) = Σ Wk
(125)
10
x=1
K
9
W/
c e- - (1)) du
(126)
00
k
(e*(xtTu)) - 1) -t
g
Wi
(127)
71 + 7
k=1
30

---

B.5 Loss regularization via the reverse pinned process
In the derivation of the previous section (Section B.4), we see from eq. (117) that the terminal values of the noise process Y directly depend on za, i.e., on information from the initial distribution Ny. Hence, initializing the time reversal of the partially pinned process only feasible when the desired endpoint is already known, which makes simulating the time reversal of FDBM impractical in general. However, we derive below the time reversal of the partially pinned process, which allows us to use the drift of the reversed process to regularize the loss function during training in the unpaired setting, where we condition on both initial and terminal state. Whenever X
(X (e).)) is a stochastic process and g is function on [0, 1], we write (t) X(1 t) for the reverse-time model and g(t = g(1 -) for the reverse-time function. In Bortoli et al. [27] the reverse pinned process connecting 21 and Ze is again Brownian bridge. For our reference process,
the reverse model of the partially pinned process follows [60]
d2 a1 (1)
[2 =1 (1) + GG
a "(1 2 an (0) GGTV. log prl on (0] m de GdB(t) (128)
{ 2 =0 =1 (1) GG [u- =0 =1 (1)) v. log (2 mm)) dt Gd
(129)
= Fz mp (1) + GG [V. lo (21) 10 (0) V. log Pe (2 (egn (1) ] dt + GdB())
(130)
where (-]= == (11) is the density of (2 = 2), pt := 3 is the marginal density of the
augmented reference process 7, Pt Pe (-1200 21) = Xo (- [ .1) is the marginal density of the
partially pinned process defined in eq. (9) and u == u according to eq. (10). We find with Bayes' theorem
p= No, 211)
pe(2 21)
(131)
Ta, (To, 21)
where To.1 is the joint density associated to 1o,,1 and Pr is the joint density of (Z,, X1). Since Z is Markov with Zo = (Xo, 0ak). we have
q1jt. (x1 |2, zo) = qu (21 20) ) = (1)e(2112))
(132)
and
( (2) (1) (1, 20) T0,
Pe No, T1)
(133)
p(2,, No)
Pe(Ze (20) (x)))))
where To corresponds to Hlg Hence, by the above equations
log (21222)))2) - log Pe (22
(134)
20, x1)
x1
log
(135)
pr (r(2 (o)
Pe (2t, No, 20,
Mzo, x1)
loog
(136)
log A(ro, - log Pr (2) - to
(137)
and we find for the gradient
v. [log (10((1()))=)))) - (0000 - V. (o)
(138)
such that
dž ar (t) [F 1o. (t) GG log (21 dt + GdB(t).
(139)
Hence, the reverse dynamics of the partially pinned process coincide with the reverse dynamics of the reference process conditioned on xo. In addition, we have
log pr (=alar) = Vy(log Fi (calul...., ...,
+ log (vl (vl, [o))
(140)
v, vyllog pf (salut, .... vo) log (u.,... vf))
(141)
where we use the independence of (Y ..... )xK) and Xo. To calculate further, we note that Xr|(Y? = " YKK = yd Xo = IO) ~ NNu æo), o² is normal distributed with
(1) = [x (Y, = y² ..., Y.K = ,xX,0 = =mo)]
(142)
K
(11)
Eu +
Ch (0, (t)
(143)
k=1
=M40(2)
+
(144)
31

---

where 2 = (a, "i,.... yik and
oqr
V [Y,l yy .... ...... .x zu)
(145)
K
Ve
whyj (1 e -((++)))))
(146)
7/ + 75
30
110
(147)
Therefore
2₂ log (Ze(o0))
(148)
dz log Pi (I Vi.....
y/ z0) + +og Wo v1)]
(149)
dr log Pe (Iell (zelv)
, To)
(150)
It 1r0 Σk= S4 (0, ()y²
(151)
To It + Σ
k= SA t)x
(152)
and
V log (oppe(r(2 = V v log pi (xt (oelvt..
bi......... ..... K .o) + log p8 (vl
V6 )l
(153)
Mel(Y,mo)
V yyHeeey + V, log pK (yb..... y6
(154)
%
0
(0,1))) - Mel(Y,
yi
(155)
0010
A₂
ye
(156)
such that, in total
d7 10 (f) = FFF mm (f) (GTT""( b0,0 (8)) dt GaB(t).
(157)
with
0
> (0, (0,
at =t, -.x.
0110
A-1(1)
(158)
We use the above calculations to derive backward loss. Let (1) ~ NN (i)) with
1 (12(1())=11-
ee (Zo, O,...,
0) +
(159)
and
Σ = Enjo
E12(t (1)
(160)
2 T|
according to the derivations in Section B.4. Since by the calculations of this section
V. log (1)((12)))))
V. log p. (2 21 == (t, 2) = 0a(K+1)>
(161)
we aim to enforce
0ạcK+1) (1,, 1). WKC 1))" of (()))(())) V. Jog (1)wa, 21) a- (1,
(162)
32

---

for the neural network oy learned to approximation the forward dynamics transforming to W1- Moreover, since
V V2logmm(2oo1(0000 =
mte):
(163)
we aim for
OK+1 Ee wilt, wxx(,, oi (2m)(((0,2(1))) - (t,,)))000) (uz(1)) - (Z)a, - jie)
(164)
and define
c (0, vo. Ex (l. D. (x.c.tttt (10001((1111100000 =² (a (o.11)))) (0)0.110) meil
with
0
wxck.0000. 2o x Ets
we (0, 3)ye
31
x" (6,2)=
(1)
(166)
MK
minimize for some X [0, 1]
0f,
(167)
SSaKen Jowup 1o (mae((z))
E. WeCa(t 1)v (1) ?
dl xaix (#o, (18 (=
Jeep Co (0, (0:44 )aP x (mo. on )dS.
(#t (ooom)))
(168)
incorporating, for X >> O, the drift of the time-reversal of the partially pinned process.
33

---

C Broader impact
Fractional Diffusion Bridge Models (FDBM) introduce non-Markovian stochastic dynamics into generative modeling, enabling the learning of long-range dependencies and memory effects ob- served in real systems. This is largely a theoretical contribution. This framework can benefit scientific domains where temporal correlations are fundamental, including molecular design, pro- tein dynamies, materials discovery, and biological simulation, by improving the physical fidelity of generative models.
By bridging stochastic physics and machine learning, FDBM contributes to more interpretable and physically grounded generative tools, potentially reducing experimental costs and accelerating dis- covery. Nevertheless, as with any generative model, misuse for fabricating deceptive unsafe data is possible. To mitigate this, our open-source release emphasizes research and educational use with clear documentation.
D Related work
Diffusion based generative modeling. Diffusion models [61, 62] have achieved remarkable suc- cess in generative modeling, setting state-of-the-arr performance across image [52, 63] and molecule generation [64, 65]. They have had a major impact across a broad range of domains, includ- ing materials and drug discovery [66, 67], realistic audio synthesis [68, 691. 3D object and tex- ture generation [4, 5, 70], medical imaging [71, 72], aerospace design [73], and DNA sequence modeling [74, 75]. Building on the seminal contribution of Song et al. [3], who introduced continuous-time framework for score-based diffusion models via stochastic processes with an ex- act reverse-time model, a large body of subsequent work has expanded this perspective by analyzing its properties [76-78] and generalizing it to subspaces [79], Riemannian manifolds [80, 81]. alter- native stochastic dynamics such as non-linear drifts [82], general corruptions [12] and reflecting
processes [83, 84], as well as by learning the drift of the forward process [85]. A unifying per- spective on diffusion and diffusion bridge models has been proposed through mixtures of diffusion bridges [21], optimal control [86], and the generalized Schrödinger bridge problem [59, 87], with applications to sampling from unnormalized densities. Recent methods incorporated non non-Gaussian priors, as well as non-Gaussian conditioning into diffusion modeling and considered the boundary value problem through diffusion bridges [88-90]. In line with our research, non-standard noise sources for continuous-timg diffusion models have been explored, including heavy-tailed Lévy pro- cesses [14, 15], and non- Markovian fractional Brownian motion [1 16, 17].
Fractional Brownian motion in machine learning. Memory-aware fractional Brownian motion has been employed in machine learning for generative modeling [11, 16, 17, 91], variational in- ference [26], and stochastic optimal control [45]. Our work builds directly on the Markovian ap- proximation of fractional Brownian motion (MA-fBM) introduced by Harms and Stefanovits [25] and further refined through the derivation of optimal approximation coefficients by Daems al. [26]. Daems et al. [26] demonstrate how variational inference can be performed for SDEs driven by MA-fBM, framework later enhanced by Daems al. [45] using techniques from stochastic optimal control. Nobis et al. [16] introduced a continuous-time score-based diffusion model driven by MA-fBM, which Liang al. [17] extended to protein generation.
The Schrödinger bridge problem. The Schrödinger bridge problem [39-42] is stochastic opti- mal control formulation that serves an entropy-reeulaaizee generalization of the optimal transport problem on path spaces. It offers a principled alternative to Diffusion Models[3, 61, 62] and Flow Matching approaches [30, 92]. by directly interpolating between marginal distributions via maxi- mum entropy dynamics [19, 20J. While the algorithm proposed by Bortoli et al. [19] was based on Iterative Proportional Fitting (IPF) [93-97], Shi al. [34] and Peluchetti [33] concurrently in- troduced Iterative Markovian Fitting (IMF) for Brownian-ding driven diffusion processes, which directly learns the time- dependent drift of stochastic process solving an SDE. Specifically, Shi et al. [34] considered a scalar, positive diffusion function, whereas Peluchetti [33] formulated their approach for matrix-valuee diffusion functions that may depend on the state of the process. For unpaired data translation, we build upon the framework of Bortoli et al. [27]. where IMF is extended to a-IME, an online variant of IMF, summarized detail in Section E. See also Peyré and Cuturi [38] for a comprchensive overview of optimal transport methods.
34

---

Stochastic bridges for paired data translation. Recent studies have extended stochastic bridges to the paired data settings. Liu al. [30] proposed a structured diffusion framework for constrained domains, alongside a task-specific training loss. Liu et al. [36] propose generative bridge model for image-to-image translation and Somnath et al. [31] introduced aligned diffusion bridges that in- terpolate between matched samples and evaluated the method on toy datasets, cell differentiation, and predicting conformational changes in proteins. Bortoli et al. [32] identified limitations in pre- serving the coupling of the training data in the approach of Somnath et al. [31] and Liu et al. [36], which they resolved by augmenting the drift of the learned process with the starting value. Our framework FDBM in the paired setting is built upon the repository provided by Somnath et al. [31]*, including the training setup, model architectures, data visualization, and all used datasets. Concep- tually, we adopt the viewpoint of Bortoli et al. [32], providing the initial value to the neural network, approximating the drift, at all points in time.
E The Schrödinger bridge problem for unpaired data translation
In this section, we summarize the Schrödinger Bridge Flow (SBFlow) introduced by Bortoli et al. [27], which our FDBM builds upon for unpaired data translation. Adopting the perspective of En- tropic Optimal Transport (EOT) and assuming unpaired data samples from the distributions Ho and 111 on Rdd Bortoli et al. [19] seek to find the coupling distribution
1 I1l=0 Io - x11 (Il/ddl((zo x1) -H)))
II' arg Hl (EP((*
min
(169)
where the differential entropy HIII) can be controlled by regularization parameter E > 0, and P(R" x (Rd) is the set of coupling probability measures on IR x Rd. Adopting EOT rather than optimal transport (OT) restrred when & = alallows degree of regularizing stochasticity when solving for a transport map. The formulation of EOT in eq. (169) can he understood as a static version of the dynamic formulation of the Schrödinger bridge problem described eq. (4). We refer the reader to Léonard [42] for a detailed discussion of the relation between the static and dynamic Schrödinger bridge Problem
T arg min TEP(Cd)
{DkL ; To = Il, Ti = 111)
(170)
where we now seek path measure pSB with marginal distributions II and I1, The reference path measure Q in Bortoli et al. [27] is associated with a scaled Brownian motion vēB, with e > 0.
Remarkably, under some assumptions, eq. (169) and eq. (170) share the same unique solution [42] for the coupling distribution in the sense that TSB = 1h
The difficulty of solving eq. (170) stems from the need to optimize over the infinite-dimensional space of path measures. Traditional approaches like Iterative Proportional Fitting (IPF) [93-97] become computationally costly in high dimensions as they require simulating complex conditioned processes. The Iterative Markov Fitting (IMF), concurrently introduced by Peluchetti [33], Shi et al. [34], bypasses this bottleneck by operating directly on learning the time-evolving drift of a stochastic process solving an SDE. It operates by iteratively alternating between fitting a forward-time process and backward-time process. Bortoli et al. [27] introduced online version of IMF called a-IMF that is described in the following.
a-IMF, much like IMF, builds on reciprocal projections and Markovian projections [33, 34, 47]. These projections accomplish two key objectives. Projections to the reciprocal class ensure match- ing terminal distributions HI. II, while Markovian projections ensure that the drift of the learned process depend only in expectation on X, and that the learned process satisfies an SDE. path measure P is in the reciprocal class of some other path measure Q if
f JRdxRd
P
Q(11z0, x1 ddF (Zo, 21) =: 1/..1
(171)
Now, when we assume that Q is induced by the scaled Brownian Motion (V) then follow- ing Bortoli et al. [27, Definition 2.2] and Shi et al. [34, Definition 1] the Markovian projection of the path measure II is the Markovian path measure MI associated with X' solving
dx; U/ 1 dx + (VedB:), X Xo
(172)
35

---

and the intractable drift function
(x4) Bun [X] X; = 24/] 2t
(173)
being learned by neural network. In the following we will refer to the projection for the recipro- cal class (see eq. (171)) of Q as p-ojç(-) and to the Markovian projection associated to the SDE eq. (172) as projea()). Bortoli et al. [27] consider IMF from the perspective of a flow of path measures (IP,,1Pa) )22)),0. describing Markovian and reciprocal class states respectively
p = (IIo @ 11)
(174)
T projx (P*).
(175)
Oi = (pois()roxx(P"")
P
(176)
where the only fixed point w.r.t. the vector field of the flow of path measures in (176) is the Schrödinger bridge. Finaly Bortoli et al. [27] propose a novel discretization approach
po+t = (1 a)) + a projq((P)))
(177)
which converges to the Schrödinger bridge [27, Theorem 3.1] and recovers IMF for 1.
To counteract error accumulation issues, a bidirectional online procedure can be implemented to achieve a-IMF. This involves concurrently training two models or a single direction-conditionn
model: one approximating the forward drift for the IIg - Il1 process, and another approximating the backward drift for the IT To process. Bortoli al. [27] first pretrain bridge matching model r for both directions following DSBM [34] w.r.t. eq. (174), where samples are drawn from Ho 11, such that of (x) (Big [X x] 21 t). Furthermore, they propose
bidirectional loss formulation of the online procedure of a- n-DSBM, where samples are drawn from the opposing directional processes
,,, 11)) = / C,(7, ,
31 - Iz
1/ (Ix)
dIP (To, xz 1 x1)
(Rajs
1 1-6
31 3t
v (xt)
dP (zo. xn a1ddddd(.o..... 1 (2, .1)
(3dj3
(178)
with associated forward and backward SDEs following the Markovian projection, as described by eq. (172), in respective directions.
F Implementation details for paired data translation
In the followign we will provide implementation details for all experiments with paired data transla- tions. We emphasize here again that the implementation of FDBM in the paired setting built upon the repository provided by Somnath et al. [31]s, including the training setup, model architectures, data visualization, and all used datasets.
E. Network architectures
Toy experiments and cell differentiation. For SBALIGN, we use two multilayer perceptrons (MLPs) to approximate the drift b2 and Doobs h-score m² For ABM and FDBM, we use only the MLP employed in SBALIGN to approximate the drift 10, but the initial state 2o is additionally provided to the network by concatenating it with the input, following Bortoli al. [32]. This setup is used in the experiments shown in Figures 2 and 7, on the Moons and T-shape datasets, as well as in the cell differentiation task, with the respective number of parameters reported Table 4.
Conformational changes in proteins. We use the GNN architecture from Somnath et al. [31]. However, following Bortoli et al. [32], the initial state Io additionally provided to the network by concatenating it with the input. See Table for comparison of the number of parameters.
'nttps://githut htt.aa/......eeoooaaaaaaaggggeee diffusion__biii brrdgggeennbbddeesssee
36

---

F.2 Training & Sampling
Toy experiment. We follow precisely the training of Somnath et al. [31]. For sampling, we use 100 steps of the Buler-Maruyama method and generate single trajectory for each test starting point. This procedure is used both for calculating the WSD and for the visualization in Figure 3, whereas Somnath et al. [31, Figure 2] report trajectories averaged over multiple trials.
Cell differentiation. We follow precisely the training and sampling setup of Somnath et al. [31].
Conformational changes in proteins. The results reported in Table 1 were obtained by averaging over 5 training trials, each run for 300 epochs, and performing one sampling trial per trained model, generating single path over 100 time steps. The remaining training set-up closely follows Somnath et al. [31]. We use the AdamW [98] optimizer with an initial learning rate of 0.001 and training batch size of 2. During validation, inference performed using the exponential moving average of the model parameters, which is updated at every optimization step with decay rate of 0.9. After each epoch, we simulate trajectories on the validation set and compute the mean RMSD. The model achieving the lowest mean RMSD on the validation set is selected for final evaluation on the test set. We observe that the best model was saved for ABM and FDBM towards the end of training, indicating that longer training could further improve the overall results.
F.3
Compute
The toy experiments were locally on CPU and completed within minutes. Each trial of 300 training epochs for the protein conformational change task was completed within 24 hours on a single NVIDIA A100 GPU (40 GB VRAM).
F.4
Datasets
Toy datasets. The Moons dataset is obtained by generating two moons to produce samples from Ho and then rotating them clockwise 90 degrees around the center to produce samples from 11, The T-Shape dataset is produced by a bi-modal distribution, where 1lg is supported on two of the four extremes of imaginary T-shaped area. The target distribution Ig is created by shifting ΠI the opposite side. The rotations and shifts imply paired data, since there a one-to-one correspondence between samples in HI and For detailed description of the datasets, refer the reader to Somnath et al. [31], who designed both datasets. See Figure for a visualization of the dataset marginals.
Figure 7: Marginals of the Moons dataset and the T-shape dataset introduced by Somnath et al. [31].
Cell differentiation. We use a dataset of genetically traced cells during the process of blood forma- tion, created by Weinreb et al. [99] and curated by Somnath et al. [31]. The dataset consists two snapshots: recorded day 2, when most cells remain undifferrntiated, and another on day 4, which includes diverse set of mature cell types. For a detailed descirption of the dataset we refer the reader to Somnath et al. [31].
Conformational changes proteins. We use the curated subset from Somnath et al. [31] of the D3PM dataset [49], which focuses on structure pairs with C. RMSD 3A. This subset initially comprises of 2, 370 ligand-free -fre (apo)- ligand-boume (holo) pairs. To ensure high-quality alignment,
37

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Somnath et al. [31] compute the C. RMSD between pairs of proteins common residues superim- posed using the Kabsch algorithm [100] and retain only those examples where the computed RMSD closely matches the original D3PM value. This results in cleaned dataset of 1, 1,591 pairs, which is split into training, validation, and test sets of 19,21/150 /150/150 examples, respectively. All structures are kabsch-superinppsse to remove global translational and rotational artifacts, ensuring that the model focuses solely on internal conformational changes. For more details see Somnath et al. [31].
G Implementation details for unpaired data translation
In the followign we will provide implementation details for all experiments with upaired data trans- lations AFHQ [51].
G.1 Experiments unpaired data translation
Network architecture. The Diffusion Transformer (DiT) [55] is scalable architecture that adapts the Vision Transformer (ViT) [101] for generative modeling with diffusion processes. Unlike convolution-based U-Nets commonly used in image diffusion models, DiT model treats denois- ing as a sequence modeling task by operating directly on (latent) patches of an image, capturing long-term dependencies via Attention [102]. DiT architectures
grouped into small (DiT-S), base (DiT-B), large (DiT-L), and extra large (DiT-XL) variants, where Peebles and Xic [55] observed di- minishing returns after scaling from DiT-L to DīT-XL. Notably, Peebles and Xie [55] show that the model scales with FLOPs, rather than parameter size. Therefore, a smaller model with more tokens (i.e., smaller patches) can achieve identical performance lowing this finding, we used the models with the most tokens for respective parameter sizes. Hence, we selected the variants DiT-B/2 and DiT-L/2-wwere the "/2" indicates a patch size of 2 x 2 for respective tokens-as suitable backbone architectures for all experiments on imaging data.
larger model with fewer patches. Fol-
Training & sampling parameterization. We used the same training and sampling parameteri- zations for all datasets and experiments. Parameterizations for DiT-B/2 and DīT-L/2 were kept identical. See Table for detailed parameterizations of all experiments.
Model
Optimizer Learning Rate EMA Rate
Linear Warmup
Cosine Decay
Online Finetuning
Fuler Fuler- Manuyama Steps Parameters
DIT-B/2 lion [103]
0.0001
0,999
10K
90K
4K
200
130M
DIT-1/2
[103]
0.0001
4K
458M
Table 3: Hyperparameters for experiments with Diffusion Transformers.
Compute. Experiments were conducted in single- and mutli-GPU settings, using full precision (FP32) for all runs. Computation times are denoted in an equivalent of A100 GPU (40GB VRAM) hours, as a common reference for scientific compute time. All pretrainings of 100K steps for the AFHQ-32 and AFHQ-256 datasets were completed in 16 hours (A100) for the DiT-B/2 variant and 54 hours (A 100) for the DiT-L/2 variant. The online finetunings of 4K steps were completed in 12 hours (A 100) for the DIT-B/2 variant and 43 hours (A100) for the DÍT-L/2 variant. Samplings experiments were completed in 0.5 hours (A100) for the DiT-B/2 variant and 1.5 hours (A100) for the DiT-L/2 variant. All pretrainings of 100K steps for the AFHQ-512 122 datasets were completed in 256 hours (A100) for the DiT-L/2 variant. Samplings experiments were completed in 5 hours (A100) for the DiT-L/2 variant.
H Computational efficiency
Number of learnable parameters. We use the GNN architecture from Somnath al. [31], but fol- lowing Bortoli et al. [32] the initial state Zo is additionally provided to the network by concatenating with the input. Nevertheless, the GNN we use for ABM and FDBM has fewer parameters, since Somnath et al. [31] approximate two functions (b, and V, log h.) with a single GNN resulting in more parameters in the output layer. We emphasize that ABM and FDBM deploy the same model architecture and summarize the number of learnable parameters in Table 4. Throughout all unpaired
38

---

data translation experiments, we use the same model architecture for both SBFlow and FDBM with the same number of learnable parameters.
purameters per lask
SBALIGN [31]
ABM [32]
FDBM
Coupling-preseringg (Figure 2)
58, 692
31, 618
31, 618
Moons
58,
31, 618
31,
T-Shape
19, 204
10, 754
10, 754
Cell Differemtiaionn
310, 372
177,970
177,970
Predicting Conformations
545,220
537,
537,900
Table 4: Number of learnable parameters in SBALIGN, ABM, and FDBM.
Runtime comparison. We provide a runtime comparison FDBM in the paired setting in Ta- ble 5. The runtime per training step is averaged over 1000 training steps, and the runtime to sample one conformation averaged over the 150 test samples of the D3PM test set. Training times for ABM and FDBM are nearly identical and both outperform SBALIGN, which requires approximat- ing two functions and thus involves a larger model. For sampling, ABM and FDBM again show an advantage over SBALIGN. FDBM requires, average, only 0.0422 seconds more than ABM to sample a conformation over 100 Euler-Maruyama steps. This slight increase is due to simulating higher-dimensional stochastic process. However, the effect is minor, as the dominant computational cost during sampling comes from forward passes through the GNN, which are identical for both ABM and FDBM. Throughout all unpaired data translation experiments, we use the same model architecture for both SBFlow and FDBM.
The differing components during training are the sampling from the (partially) pinned process and the loss computation, both showing nearly identical runtime in Table 6. The sampling algorithm of FDBM during inference is identical for the paired and unpaired settings. All computations of this section were performed on an NVIDIA A100 GPU (40 GB VRAM).
Average Runtime [s]
SBALIGN(1]]
ABM[32]
FDBM
Triiing step
0.0159 + 0.0075
0.01 + 0,0065
0,01412 = 0. 0.0063
Sampling one conformation over 100 sampling steps
0.7078 1 0.3409
0.6424 1 0.2992
0.6846 0.3021
Table 5: Runtime comparison of SBALIGN, ABM, and FDBM.
Average Runtime [s]
SBFlow(27]
FDBM
Sampling from (partially) pinned process
0.0010 + 0.0002
0.0011 + 0.0003
Calculation of loss term
0.0132 + 0.0074
0.0132 0.0018
Table Runtime comparison of SBFlow and FDBM. The runtimes are averaged over 1000 compu- tations. All computations were performed NVIDIA A100 GPU (40 GB VRAM).
I
Evaluation metries
Wasserstein distance. To measure the distance from the original data distribution from the pre- dicted data distribution use Wasserstein-1 distance [104]. The Wasserstein-1 distance between ground truth data distribution pt and sampled data distribution ps is defined as
W Wi Ps) = inf
[1ke - =[]]-
(179)
The lower the Wasserstein distance, the better are the distributions pt and ps aligned.
Root Mean Square Deviation. Root mean square deviation of C. atomic positions a distance between two superimposed molecules/proteins, If x is an observed 3D structure/configurrtioo of the
39

---

protein and X is predicted configuration of the protein then
1
RMSD(x, =
x - -
(180)
nt
i=1
The lower the RMSD, the lower their 1.2-distance w.r.f. some unit of measure. In our example, the unit of the measure is Angstrom, Å.
Fréchet Inception Distance (FID). The Fréchet Inception Distance (FID) [53] measures the dis- tance between the feature distributions of real and generated images, typically using embeddings from pretrained Inception network. Given the empirical mean and covariance of real images (pr; 2,) and generated images (Mg, Eg) in this feature space, FID is defined as
FID(X,Y) = ||,, Mall + Tr [,, + -i, -2(
(181)
where X and Y denote the sets of real and generated images, respectively. The first term captures differences in mean features (style/content shifts), while the second term accounts for differences in variability. FID is widely used in image generation and style transfer as it correlates well with human judgment of realism and diversity.
Learned Perceptual Image Patch Similarity (LPIPS). The LPIPS metrie [54] quantifies percep- tual similarity between two images by comparing deep features extracted from a pretrained network (e.g., VGG, AlexNet). Let and y be two images. The LPIPS score is computed by comparing their normalized feature maps f1(x)), fi(y) at multiple layers
H:
Wi
LPIPS( v) Σ H.Wi
ΣE
wi O (f -i
(182)
where W: are learned per-channel weights, and Hi, Wr denote the spatial dimensions of layer I. LPIPS has been shown align well with human perceptual similarity judgments, making it valuable for evaluating and training generative models, especially in style transfer tasks where pixel-wise metrics fall short.
J Cell Differentiation
Methods
MMD 4
W. 4
(a(PS) 4
RMSD
FBSB"
1.55e
12.50
4.08
9.64e-1
FBSB WITH SBALIGN"
5,31e-3
10.54
0.99
9,85e-1
SBALIGN
1.7e-2
11.11
9.21e11
ABM
4100.-
99.0
0.87
8.726-1
FDRM 0.3) (ours)
5334-2
9.32
0.89
8.16-
FDBMCH (ours)
4,520-2
9.35
0.85
8.21e-1
Table 7: Comparison of performance on the cell differentiation task. Results marked with an asterisk (*) are obtained from Somnath et al. [31].
We evaluate FDBM on the cell differentiation task introduced by Somnath et al. [31]. We fix the across all retrained methods SBALIGN, ABM and FDBM. All
diffusion coefficient to v scores are averaged over 10 training trials and 10 sampling trials for each trained model. We follow the approach of Somnath al. [31] and average for each prediction over 20 sampled paths. This task allows us to assess FDBM for cell differentiation prediction on both the distributional quality and perturbation accuracy of the generated data using distributional metrics such as Wasserstein-2 distance (Wz) ) [105] and kernel maximum mean discrepancy (MMD) [106], as well as the Pertur- bation signature (PS) [107] and RMSD. The dataset consists of two snapshots: one recorded on day 2, when most cells remain undifferentiated, and another on day 4, which includes a diverse set of mature cell types. We assess the performance of FDBM against forward-backwart Schrödinger bridge models (FBSB) [108], SBALIGN, and ABM. Consistent with our findings on protein con- formational changes, we observe in Table that ABM shows superior performance compared to all other Brownian baselines in all metrics except MMD. FDBM achieves the best performance in the rough regime (H 0.3 and H 0.4), with slightly better average W. and RMSD scores, while ABM remains superior in terms of MMD.
40

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K Extended Experiments
In the following we provide more details on the results reported in the main paper. Detailed scores of all ablations are listed in Tables 12 to 14. Additional evaluations of AFHQ-256 Dogs + Wild and Dogs ++ Cats are listed in Table 15. Additional visual examples for AFQH-512 samples are displayed in Figures 10 and 11 and for AFHQ-256 in Figures 8 and 9. Additional results for the D3PM dataset are listed Table 10, as well as additional experiments with toy data in Tables and 9.
Table 8: Average Wasserstein distance over 10 runs between samples generated by the Brownian- driven baseline and the target distribution, for varying diffusion coefficient Ve.
driven
Distance
V2
V8 0.8
V8
v2=
V2 0.2
0.1
0.05
V8 0.01
Moons
0. 0.020 008
0.015 005
0. 0.01940.000
O. 0.03 0.0
0.206 008
0.12:=0.016 0. 0.206 019
T-shaped
O. 0..055
0.346:1.0299
0.25110.008
0.154:0.010
0.052:1.002
0.178=0.049
0.029=0.007
0.670=0.099
Table 9: Wasserstein distance (10 runs average) between generated samples and target distribution.
Wasserslein Distunce
H 0.8
H u.7
H G.G
ABM [32]
H U.4
H 0.3
H U.2
Moons
0.0172:0.005 002
0.012$0.000 002
0.012::0.00 003 0.015=0.019
0.029::0 006
0..00
0.048$000016
T-shaped
0.08210.043
1.0011
0.08310.031
0.082 0.002
0.06810.0015
0.002 10.013
0.048:10.039
Table 10: Ablation of the diffusion coefficient VE of our Brownian driven baseline ABM [32]. Additionally compared to the scores reported in Somnath et al. [31].
RMSD(A)
t RMSD(A) <
D3PM Test [31]
Median
Mean
Std
2
5
10
EGNN [31, [09]
19.99
21.37
8.21
1%
1%
3%
(10.1))(10.10)) [31]
3.80
4.98
3.95
0%
69%
93%
SBALIGN, (100, [31]
5.02
3.96
0%
70%
93%
ABM(E = 1.0) [32] (1 trial)
3.14
4.11
3.32
1%
79%
97%
ABM(E 0.8) (321 ttal)
3.93
23%
79%
96%
ABM(E 0.6) [32] (1 trial)
2.47
3.65
3.59
35%
85%
97%
ABM(= 0.4) (32) (1 trial)
2.47
3.60
3.66
43%
86%
93%
ABM(= 0.2) [32] (1 trial)
2.20
3.58
3.45
45%
81%
97%
ABM(: 0.1) (1 trial)
2.70
3.67
3.54
43%
83%
96%
ABM(E 0.05) [32] trial)
2.69
3.59
3.83
35%
82%
95%
ABM(E 0.01) [32] (1 trial)
2.96
3.78
4.08
30%
77%
93%
ABM(E 0.2) (5 trials)
2.40
3.49
3.54
433
84%
96%
FDBM IT 0.4, 0.2) (5 trials)
2.24
3.39
3.57
45%
84%
97%
FDBM H 0.3.
0.2) triak)
2.33
3.42
3.42
43%
85%
97%
FDBM H
0.2. 0.2) (5 trials)
2.12
3.34
3.59
48%
86%
96%
#DBMCH 0.1, 0.2) trials)
2.20
3.44
46%
97%
Table 11: Comparison of FID and LPIPS for AFHQ-512 across Cats - Wild and Wild - Cats translation tasks.
AFHQ-512
FID
SBFlow
17.39 d: 0.66
24.17 0.81
(TI=0.4)
41

---

Table 12: Pretraining ablation for entropic regularization of the SBFlow [27] baseline.
(a) AFHQ-32 32 with DIT-B/2.
(b) AFHQ- 256 with DIT-B/2.
Method
wild
wild
Method
cats
wild
cat wild
FID
LPPPS 4
RDJ
LPIPS 4
FID 4
LPIPS
FDD
LPIPS g
SDFlow 0.75
161.
=2.0
0,159 20.00
138. 20
#2.27
0.135 20.001
SBPlow 0.75
42.67 40.75 0.659
00:01
46. 42 #3.0 0.586 +0.001
SBFw
59.04
21:14
0.104
20.001
74.36
21-02
0.151
20.001
SBPlow
15.67
4:
0.578
10 00
30.
75 10.58 0.594 1:0
S0Flow
77.
=0.00
#0.001
SBFlow
1.125
46.46-1
+1.25
0.592
+0.001
37.
+295 0.609 +0.202
SBFlww
1.25
96.66
$1.37
0.110
20.001
88.77
21.00
0.172
20.001
SBPlow 1.25
54.05
#1.10
0.623
00:001
48. 63 #1.10
0.629 20.201
Table 13: Pretraining ablation for hurst index H related parameterization of our method. K 5 was fixed for all experiments. The best results and results where the mean is within the standard deviation of the best result are highlighted in boldface.
(a) AFHQ-32 with DiT-B/2 and K = 5 for FDBM (ours).
(b) AFHQ-256 with DiT-B/2 and K 5 for FDBM (ours).
Method
wild
wild
Method
wild
wild
FID 4
1PPIPS 1
FTD
E.PIPS
FID J
IPIPS
FID J
I.PIPS J
FDBM
0.9
47.03
11.53
0.099
20.001
02.38
13.06
0.155
20.002
FORM
0..
21.15
13:36
0,622
00 002
19.50
10.36
0.639
20.002
FDBM
0.5
45.18
0.095
50.59
19.65
0.523
19.58
0.542
FDRM
0.7
48.36
10:52
0.085
_0.001
51.65
16.71
0.156
10.002
FDRM
0.7
18:64
0.529
10:002
19.46
10.16
0.547
1.0000
FD8M
0.6
43:45
10:53
0.097 20.001
48.79
16.75
0.155
10.001
FDBM
0.6
16.77
10.71
0.530
10.202
19.14
20.35
0.551
20.00
FDBM
0.5
40
+1.15
0.097
=0.001
45.74
+0.69
+0.49
0,154
+0.002
FDBM
0.5
16.19
#0.83
0.534
+0.002
21.91
+0.35
0.565
=0.002
FDRM
0.4
0.096
47.65
17.02
0.542
24.32
0.677
FDHM
0.3
58.27
10:07
0.090
10.001
54.59
16.78
0.153
20.001
FISM 013
13:08
10.002 0.549
30.63
19.79
0.591
20.003
FDBM
0.2
83.0
62
+1.45
68.05
+1.21
PDBM
0.2
59.83
#2.36
37.17 +0.62
FDRM
0.1
131.04
11.51
123.20 L1.92
FDRM
0.1
81.36
13:38
43.69 13:00
Table 14: Pretraining ablation for hurst index H related parameterization of our method. == was fixed for all experiments. The best results and results where the mean is within the standard deviation of the best result are highlighted in boldface.
(a) AFHQ-32 32 with D/T-B/2, E and H 0.5.
(b) AFAQ-HH--00 256 with DIT-B/2, e 1 and H 0.6.
Method
cats
cats+ wild
Method
wild
wild
HD
LPIPS J
FID
LPIPS
FID 4
LPIPS
FID
LPIPS
FDBM
62.19-2.15
0.081
20.001
53.46
0.150
#0.001
FDBM
6
59.08
+1:00
0.547
#0.001
38.54
+1.11
0.592
=0.002
FDBM 40.21
s0:0:
45.
20.002
FDBM
19.14
20.001
FDDM
41.13
+:.19
0.097
=0.0:
46.92
=3.20
0.155
+0.002
FDBM
4
18.67
+0.75
0.528
+0.002
19.77
+0.02
0.555
=0.002
FDBM
3 41.32
+1as 0.096
#0:0:
47.82
0.154
+0.001
FDBM
3
17.89
+0:58
0.528
+0.002
20. 27
#0.45
0.555
=0.001
FDRM
42.14
0.095
44.61
20.05
0.154
FDBM
2
19.20
21.04
0.527
20.002
74
20
0.555
20.009
FDBM
41.15
0.097
+0.003
48.64
=0.57
0.153
+0.052
FDBM
19.98
+0.50
0.550
+0.052
+1.20
0.594
=0.001
Table 15: Additional evaluations of AFHQ-256 Dogs + Wild and Dogs + Cats. The best results and results where the mean is within the standard deviation of the best result are highlighted in boldface.
(a) AFHQ-256 with DIT-B/2 and K 5 for
(b) AFHQ-256 with DIT-B/2 and K 5 for
FDBM (ours).
FDBM (ours).
dogs wild
dogs + wild
dogs + cats
dogs + cats
Method
H
Method
FTD
LPIPS 1
FID
IPPPS 4
FID
LPIPS 1
FID L
IPPPS
SBFllow
20.74 0:6 0.53 : no0 47.07
0:300 0.56
0002
SBFlow
18.38 n.36 0.56 com 50.08
1.58 0.56 0:00
FDRM
0.9
20.37 1:0 0.52 : JuMz 43.11
1111::000
0.6
0.54
am2
FDRM
0.9 19.86 0.07 0.55 :: 6M0 45.19
074
0.55 0,02
FDBM
0.8
19.22 0.3 0.53 : 008 41.76
0.51 0.55
:: 0002
FDBM
0.8
20.14
1064 0.56 4 000
45.08
uw
0.55
1 0,001
FDBM
0.7 18.11 0:55 0.53 0.02
40.08
0.56
0002
FDBM
U.7 21.12
0.56 0.02
43,44
0.56
0.00
FDBM
0.6 18.43 0.72 0.53
# 0.02
39.84
0.80 0.57 0.02
FDBM
0.6 22.06 0.5.7 0.57 # 0:01
41.35
0.00 0.57
+ 0.002
FDRM
0.5
14.74
0.55
37.68
0.58
FDRM
0.5 25.
0.58 0om 42.36
0.58 0:000
FDBM
0.4 15.78
0:5 0,56
+ 0822 38.51
0.044
0.59 000
FDBM
0.4 24.79
4.6 0.59 a 0,01 41.21
1.12 0.59 à 0,000
42

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Input
Output
Input
Output
(a) AFHQ-256 cats - wild
(b) AF6Q-256 wild + cats
Figure A detailed look at exemplary samplings with our method with H=0.4, K=5 for AFHQ-256.
Input
Output
Input
Output
Input
Output
Input
Output
(a) AFHQ-256666 cats + wild
(b) AFHQ-256 wild + cats
Figure Overview of exemplary samplings with method with H=0.4, K=5 for AFHQ-256.
43

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Input
Output
Input
Output
(a) AFHQ-512 cats + wild
(b) AFHQ-512 wild + cats
Figure 10: detailed look at exemplary samplings with our method with H=0.4, K=5 for AFHQ 512.
Input
Output
Input
Output
Input
Output
Input
Output
(a) AFHQ-512 cats - wild
(b) AFHQ-512 wild + cats
Figure 11: Overview of exemplary samplings with method with H=0.4, K=5 for AFHQ-512.
44

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Reviews:
[{"review_id": "soJCG1e6yy", "reviewer": "Reviewer_fb9f", "summary": "The authors introduce Fractional Diffusion Bridge Models (FBDMs), a novel generative diffusion bridge framework based on non-Markovian fractional Brownian motion (fBM), which captures memory effects and long-range dependencies that standard Brownian motion cannot. The authors use a Markovian approximation for the fBM (MA-fBM) to enable tractable inference of the fBM while preserving its non-Markovian properties. The authors provide a principled theoretical framework for FBDMs based on the MA-fBM and demonstrate its use case through a set of empirical experiments on synthetic and real data. They evaluate FBDMs for the tasks of paired and unpaired data translation. Specifically, the authors demonstrated that their novel framework outperforms Brownian-based methods in both protein conformation prediction and unpaired image translation.", "questions": "What is still missing for me after reading the paper is intuition for why defining the stochastic bridge between two distributions using \"non-Markovian fractional Brownian motion\" would lead to \"more flexible modelling of real world variability and biological dynamics\" (lines 30-32). Could the authors provide some detail/backing for this hypothesis/claim? \n\n\nMinor comments:\n\n- Line 116: hyper-ref seems to be missing a reference to another section or appendix.\n- Lines 180-184 seem to be a repetition of lines 91-97. You could save space by just referencing the previous paragraph.\n- Line 189: \"dynamik\" -> \"dynamic\".\n- Line 192: \"the path probability for P'$\\mathbb{P}$ ...\" This is unclear or seems like a typo.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\n\n- The authors introduce a novel and principled framework for Fractional Diffusion Bridge Models (FDBMs) and provide comprehensive theoretical support for their approach.\n- The authors provide empirical support for their proposed framework through a set of experiments on synthetic (toy) data, images (unpaired data translation), and protein conformation (predicting conformational changes of proteins).\n- Moreover, the authors conduct several ablations to observe the effects of how changes in hyperparameters affect the performance of the proposed method across several scenarios.\n\n**Weaknesses:**\n\n- For the toy example in Figure 1, it would be useful to see how a (simple) baseline performs on these toy datasets. I think this would benefit in improving the overall understanding of FDBM and more clearly demonstrate the advantages of your proposed method. Moreover, this would provide a clear visual comparison with existing approaches. \n- The authors claim that \"FDBM scales robustly across high-dimensional domains ... \", but do not provide a thorough quantitative evaluation for AFHQ-512. i.e. referencing Table 3, evaluation is only provided up to AFHQ-256. It would be beneficial to include a Table 3.c, which shows results on AFHQ-512 to provide further support for the respective claim.\n- Furthermore, for the unpaired data translation task, the authors consider the task of image translation only using the \"cat\" and \"wildlife\" subsets of the AFHQ dataset. To my understanding, the AFHQ dataset also has a \"dog\" subset. It would be useful to also observe two additional scenarios for this experiment (Table 3): (1) \"dog\" -> \"wild\", \"wild\" -> \"dog\", and (2) \"dog\" -> \"cat\", \"cat\" -> \"dog\". I think the addition of these settings would strengthen the claims of this work for the unpaired data translation task and improve the robustness of these empirical results.\n- Motivation/intuition for why FBDM (compared to simpler approaches) to the application of predicting conformational changes of proteins is sparse/missing (see question below). More generally, I feel that the introduction could benefit from more contextualization and motivation to better place this paper in the field and to identify the gap this work is addressing.", "was_revised": true, "final_justification": "The authors did a good job of addressing most of my questions and comments, which include an improved motivation for the overall work and the addition of new experimental settings. They did not explicitly address my suggestions to test their method on the dhigh-imensional AFHQ-512 setting, and there is no way of confirming that the added toy example for Figure 1 makes sense. Nonetheless, their additions and responses to my concerns suffice for a score increase. I remain slightly reserved that this paper could benefit from another round of reviews, hence I remain in my position that this is a borderline paper.", "rebuttal": "We thank the reviewer for their insightful feedback. We are particularly grateful for the recognition of the novelty and theoretical depth of our proposed framework.\n\n> For the toy example in Figure 1, it would be useful to see how a (simple) baseline performs\n\nWe thank the reviewer for pointing out the missing illustration of the baseline. We have now included plots of the paths and alignments for the ABM baseline in both toy experiments. While FDBM achieves a lower Wasserstein distance than ABM on the Moons dataset, the paths and alignments appear visually indistinguishable. In contrast, on the T-shape dataset, we observe a visible advantage of FDBM in terms of alignment quality. Although we cannot share the figure during the rebuttal, we encourage the reviewer to visually compare Figure 1 of our work with Figure 2 in Somnath et al. [1]. Notably, Somnath et al. [1] visualize an average over $7$ paths for each test pair, whereas we display a single sample path for each pair.\n\n> The authors claim that \"FDBM scales robustly across high-dimensional domains … \", but do not provide a thorough quantitative evaluation for AFHQ-512\n\nWhile Table 3 provides detailed results up to AFHQ-256, we emphasize that this already constitutes a high-dimensional and complex benchmark. Specifically, the Cat $\\leftrightarrow$ Wild translation task in AFHQ-256 contains significant semantic variation and rich textural detail, even when compared to AFHQ-512 (see **Figure 2**, as well as **Figures 5 and 6** in the appendix). Our results in **Table 3 (b)** and  **Figure 3 (e,f)** show that performance stabilizes at $H = 0.4$, making it a critical configuration for analysis.\n\nWe emphasize that our evaluation for AFHQ-512 at $H = 0.4$ included complete training and **quantitative evaluation for 10 sampling runs with respective standard deviations**, within our available computational budget. Extending such detailed ablations to AFHQ-512 was not feasible under current constraints, but we believe the results provided, combined with stable trends from AFHQ-32 to AFHQ-256, offer strong evidence for the scalability of FDBM. \nWe were not able to replicate ablations for AFHQ-512, but we added an additional experiment for the base-line under best possible entropic regularization. To support the claim of scalability beyond AFHQ-256, we include below the AFHQ-512 results for $H = 0.4$, which was the worst performing stable H in AFHQ-256 (Fig 3 (c)) in comparison to the best possible baseline:\n\n|AFHQ-512|Cats $\\rightarrow$ Wild|Wild $\\rightarrow$ Cats|\n|-|-|-|\n||FID $\\downarrow$|FID $\\downarrow$|\n|SBFlow|17.79  $\\pm$ 0.66|**24.17 $\\pm$ 0.81**|\n|FDBM (H=0.4)|**14.27  $\\pm$ 0.86**|30.11 $\\pm$ 0.75|\n\nThese results are consistent with our findings on AFHQ-256, and show that FDBM maintains strong performance even as resolution increases. Visual consistency with AFHQ-256 (e.g., $H = 0.6$) is also observed, as discussed in relation to **Figures 2, 5, 6, 7, and 8**, where semantic correspondences remain stable across values of $H$.\n\nWe note that **AFHQ-512 imposes substantial computational demands**, particularly in terms of GPU memory and training runtime. This is a key reason why we restricted our analysis at this scale to representative settings. Despite these limitations, we opted to show visual results at even higher resolution than related work (e.g., [20]), which do not report either scores or images at 512 $\\times$ 512 resolution.\n\nWe hope this clarifies our design decisions and supports the robustness of the method across increasing resolution scales.\n\n> It would be useful to also observe two additional scenarios for this experiment (Table 3): (1) \"dog\" -> \"wild\", \"wild\" -> \"dog\", and (2) \"dog\" -> \"cat\", \"cat\" -> \"dog\". \n\nWe appreciate the reviewer’s suggestion to evaluate additional sub-domains in the AFHQ dataset, specifically *Dog $\\leftrightarrow$ Wild* and *Dog $\\leftrightarrow$ Cat*, to strengthen the empirical support for unpaired translation capabilities.\n\nWhile these experiments come at considerable computational cost, we allocated all remaining available compute to run ablations at **AFHQ-256 resolution** under the same settings as those used in *Cat $\\leftrightarrow$ Wild*. Due to time and resource constraints, we were unable to run evaluations at AFHQ-512 for these tasks, but ensured training and evaluation at AFHQ-256.\n\nThe results reported below showcase the robustness of FDBM across a wide range of Hurst indices $H$, and provide a comprehensive comparison to the SBFlow baseline. In all sub-domains, we observe clear improvements in FID and LPIPS, which further validates the effectiveness of FDBM in diverse unpaired translation scenarios.\n\nBelow, we list the complete quantitative results. The best scores are highlighted in **bold**.\n\n**Table 1 AFHQ-256 Dogs $\\leftrightarrow$ Wild**\n\n|Method|Dogs $\\rightarrow$ Wild||Wild $\\rightarrow$ Dogs||\n|-|-|-|-|-|\n||FID  $\\downarrow$|LPIPS  $\\downarrow$|FID  $\\downarrow$|LPIPS  $\\downarrow$|\n|SBFlow [20]|20.74 $\\pm$ 0.64|0.53 $\\pm$ 0.002|47.07 $\\pm$ 0.80|0.56 $\\pm$ 0.002|\n|FDBM (H=0.9)|20.37 $\\pm$ 0.98|**0.52 $\\pm$ 0.002**|43.11 $\\pm$ 0.68|**0.54 $\\pm$ 0.002**|\n|FDBM (H=0.8)|19.22 $\\pm$ 0.83|0.53 $\\pm$ 0.003|41.76 $\\pm$ 0.84|0.55 $\\pm$ 0.002|\n|FDBM (H=0.7)|18.11 $\\pm$ 0.75|0.53 $\\pm$ 0.002|40.08 $\\pm$ 0.73|0.56 $\\pm$ 0.002|\n|FDBM (H=0.6)|18.43 $\\pm$ 0.72|0.53 $\\pm$ 0.002|39.84 $\\pm$ 0.89|0.57 $\\pm$ 0.002|\n|FDBM (H=0.5)|**14.74 $\\pm$ 0.53**|0.55 $\\pm$ 0.002|**37.68 $\\pm$ 0.55**|0.58 $\\pm$ 0.002|\n|FDBM (H=0.4)|15.78 $\\pm$ 0.85|0.56 $\\pm$ 0.002|38.51 $\\pm$ 0.64|0.59 $\\pm$ 0.001|\n\n**Table 2 AFHQ-256  Dogs $\\leftrightarrow$ Cats**\n\n|Method|Dogs $\\rightarrow$ Cats||Cats $\\rightarrow$ Dogs||\n|-|-|-|-|-|\n|Method|FID  $\\downarrow$|LPIPS $\\downarrow$|FID  $\\downarrow$|LPIPS $\\downarrow$|\n|SBFlow [20]|**18.38 $\\pm$ 0.36**|0.56 $\\pm$ 0.002|50.08 $\\pm$ 1.38|0.56 $\\pm$ 0.002|\n|FDBM (H=0.9)|19.86 $\\pm$ 0.67|**0.55 $\\pm$ 0.002**|45.19 $\\pm$ 0.74|**0.55 $\\pm$ 0.002**|\n|FDBM (H=0.8)|20.14 $\\pm$ 0.64|0.56 $\\pm$ 0.002|45.08 $\\pm$ 0.98|**0.55 $\\pm$ 0.001**|\n|FDBM (H=0.7)|21.12 $\\pm$ 0.62|0.56 $\\pm$ 0.002|43.44 $\\pm$ 0.82|0.56 $\\pm$ 0.002|\n|FDBM (H=0.6)|22.06 $\\pm$ 0.57|0.57 $\\pm$ 0.001|41.35 $\\pm$ 0.90|0.57 $\\pm$ 0.002|\n|FDBM (H=0.5)|22.15 $\\pm$ 0.75|0.58 $\\pm$ 0.002|42.36 $\\pm$ 0.77|0.58 $\\pm$ 0.002|\n|FDBM (H=0.4)|24.79 $\\pm$ 0.68|0.59 $\\pm$ 0.002|**41.21 $\\pm$ 1.12**|0.59 $\\pm$ 0.002|\n\n> Motivation/intuition for why FBDM (compared to simpler approaches) to the application of predicting conformational changes of proteins is sparse/missing (see question below).\n\nWe thank the reviewer for raising this point, and have now incorporated the following motivation and contextualization in our introduction:\n\n*Stochastic differential equations offer a natural framework for modeling the inherent randomness and continuous-time dynamics of real-world systems. This is precisely why they serve as the backbone of state-of-the-art generative models, such as diffusion models (both latent and non-latent). Traditionally, these models assume noise driven by standard Brownian motion (BM), which is Markovian with independent increments. However, this choice is motivated by mathematical tractability and simplicity than by faithfullness and fidelity to real-world data.*\n\n*Empirical data, particularly in complex systems like proteins often exhibit long-range temporal dependencies, heavy-tailed behaviors, and intricate dynamics that are poorly captured by memoryless processes. Consequently, diffusion models driven by BM may require many iterations to approximate the target distribution, leading to slow sampling or limited expressivity. These limitations have motivated recent efforts to explore generative models with non-standard noise sources [16,17,18].*\n\n*Our work extends such line of research to bridge models, where the goal is to transform a structured, non-Gaussian prior into a complex target distribution. We specifically investigate stochastic bridges driven by fractional Brownian motion (fBM), a generalization of BM with stationary but **dependent** increments, characterized by the Hurst index $H$, which governs both roughness and long-range dependence. By using fBM as the driving process, we introduce a more expressive and flexible framework for building bridges: When $H=0.5$, our method recovers classical BM-driven bridges, while other values of $H$ flexibly allow us to model a broader range of temporal behaviors, as demonstrated in our experiments.*\n\n*In the context of protein modeling, fBM-driven diffusion processes have already proven effective due to their superdiffusive properties, which better capture long-range correlations in protein structures [19]. This has led to improvements in both sample fidelity and diversity. We build upon this foundation, proposing fBM-driven bridges as a principled extension to model protein conformational changes, as studied in our paper.*\n\nWe believe to have sufficiently addressed all the weaknesses and questions raised by the reviewer and kindly ask the reviewer to reconsider the evaluation of our work.\n\n[16] Yoon et al. Score-based Generative Models with Lévy Processes. NeurIPS 2023.\n\n[17] Nobis et al. Generative Fractional Diffusion Models. NeurIPS 2024.\n\n[18] Shariatian et al. Denoising Levy Probabilistic Models. ICLR 2024\n\n[19] Paquet et al. Annealed fractional Lévy–Itō diffusion models for protein generation. Computational and Structural Biotechnology Journal, 2024.\n\n[20] De Bortoli et al. Schrödinger bridge flow for unpaired data translation. NeurIPS 2024."}, {"review_id": "XtaEOC02wO", "reviewer": "Reviewer_v2k6", "summary": "This paper proposes fractional diffusion bridge models, a novel generative model that captures long range dependencies of the underlying stochastic process by leveraging MA-fBM. In addition to theoretical insights, the model is benc marked on protein conformation sampling and shown to perform competitively against the state-of-the-art.", "questions": "I am curious to see how the model performs in the different types of motions, namely apo & apo, apo & holo, holo & apo (different ligands), and holo & apo (same ligand), surveyed in D3PM. Providing this context will elucidate certain biases present in the model with respect to the capability to predict physically realistic conformations. Additionally, more description on the dataset splits is necessary for faithful reproduction of the model trained for this experiment.", "limitations": "yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is clearly written and the results are presented fairly intuitively. I am not very familiar with this field so I am can comment on neither the significance nor originality. \n\nThe D3PM benchmarking results are show some clear performance gains numerically, however it is not clear what these performance improvements imply in terms of the model's capability to generate meaningful and realistic conformations.", "was_revised": true, "final_justification": "I have read the authors’ rebuttal and comments to other reviewers and thank them for the detailed explanations and additional work. As I stated before, I am not very familiar with diffusion bridge models and hence cannot comment on the technical  concerns of the other reviewers. However, I do find the work to be impactful as a tool to effectively generate 3D conformations and for the rebuttal arguments to my own questions satisfactory. For these  reasons, I will raise my rating to accordingly.", "rebuttal": "We thank the reviewer for their constructing feedback. We appreciate that the reviewer highlights both the clarity of our presentation and the clear numerical performance gains of FDBM on the D3PM benchmark. \n\n>more description on the dataset splits is necessary for faithful reproduction of the model trained for this experiment.\n\nWe agree with the reviewer and now add a description of the D3PM dataset splits used in our experiments in addition to referencing the dataset description by Somnath et al. [2]. For the best comparability, we closely follow the training set-up of SBALIGN [2] with the same dataset splits. We have added *Appendix E.3 Datasets*, where we give the following detailed description of dataset preparation and split strategy: \n\n*Following the training and evaluation setup of Somnath et al. [2], we use their curated subset of the D3PM dataset, which focuses on structure pairs with Cα RMSD > 3 Å. This subset initially comprises of 2,370 ligand-free (apo) - ligand-bound (holo) pairs. To ensure high-quality alignment, Somnath et al. [2] compute the Cα RMSD between pairs of proteins' common residues superimposed using the Kabsch algorithm and retain only those examples where the computed RMSD closely matches the original D3PM value. This results in a cleaned dataset of 1,591 pairs, which is split into training, validation, and test sets of 1,291 / 150 / 150 examples, respectively. All structures are Kabsch-superimposed to remove global translational and rotational artifacts, ensuring that the model focuses solely on internal conformational changes.*\n\n>I am curious to see how the model performs in the different types of motions, namely apo & apo, apo & holo, holo & apo (different ligands), and holo & apo (same ligand), surveyed in D3PM.\n\nAs described in the SBALIGN implementation (repository provided by Somnath et al. [2]), the dataset split used in Somnath et al. [2] and our work consists exclusively of ligand-free (apo) and ligand-bound (holo) structures. Consequently, the scope of our current work is limited to apo → holo transitions. We agree that evaluating additional motion types would provide valuable insight. We explore this direction now in our future work.\n\n>however it is not clear what these performance improvements imply in terms of the model's capability to generate meaningful and realistic conformations.\n\nThe numerical improvements in RMSD and threshold accuracy (i.e., the percentage of predictions with $RMSD < \\tau$) reflect meaningful gains in biological relevance when interpreted in the context of structural biology. RMSD is a standard and widely accepted metric to assess the accuracy of predicted protein structures [2,4,5,6]. In particular, an RMSD below $2Å$ is commonly used as a threshold for correct bound structure prediction [7] and structural discernibility [2,5]. Accordingly, the proportion of predictions falling below this threshold is a direct indicator of the model's ability to generate physically realistic conformations. FDBM increases the proportion of correct and discernible predictions ( $RMSD < 2 Å$ ) from $43$% with ABM to $48$ %, while also improving the median RMSD from $2.40 Å$ to $2.12 Å$. This indicates that FDBM produces a higher fraction of near-native structures, compared to ABM and SBALIGN. To further address the reviewers concern, we added an additional metric to evaluate if FDBM is able to predict structures closer to the holo conformation than to the initial apo conformation. We computed the $\\Delta RMSD$ [6] defined via\n\n$\\Delta RMSD := RMSD(x_0,x_1)-RMSD(\\hat{x}_1,x_1)$,\n\nwhere $(x_0,x_1)$ is a pair of the D3PM test set and $\\hat{x}_1$ is predicted starting from $x_0$. Additionally, we compare our results to Sesame, a recent method based on Flow matching presented in Miñán et al. [6] that uses the exact same dataset split as Somnath et al. [2] and our work:\n\n|**D3PM Test Set**|**RMSD(Å) Median**↓|**RMSD(Å) Mean**↓|**RMSD(Å) Std**| **%RMSD(Å) < 2**↑|**%RMSD(Å) < 5**↑| **%RMSD(Å) < 10**↑|**ΔRMSD(Å) Median**↑|**ΔRMSD(Å) Mean**↑| **ΔRMSD(Å) Std**|\n|------------------------------|--------------------:|------------------:| -----------------:|-----------------:|-----------------:|------------------: |--------------------:|------------------:|-----------------:|\n|EGNN [2]|19.99|21.37|8.21|1%|1%|3%|–|–|–|\n|SBALIGN [2]|3.80|4.98|3.95|0%|69%|93%|–|–|-|\n|Sesame [5]|2.87|3.65|2.95|38%|82%|96%|2.15|3.11|4.26|\n|ABM [1] (retrained)|2.40|3.49|3.54|43%|84%|96 %|2.43|3.35|4.29|\n|FDBM(H=0.3)|2.33|3.42|3.42|43%|85%|97%|**2.52**|**3.49**|4.39|\n|FDBM(H=0.2)|**2.12**|**3.34**|3.59|**48%**|**86%**|96%|2.44|3.39|4.28|\n|FDBM(H=0.1)|2.20|3.44|3.57|46%|83%|**97%**|2.47|3.45|4.29|\n\n*Values for ABM and FDBM are averaged over 5 runs.*\n\nWe find that FDBM shows consistently better performance compared to ABM and Sesame in the rough regime ($H=0.3,0.2,0.1$), as measured by both median and mean $\\Delta RMSD$. This indicates that FDBM generated structures are closer to the holo conformations—relative to their apo starting points—than those produced by ABM or Sesame. For $H=0.2$ and $H=0.3$, FDBM matches or exceeds ABM and Sesame across all evaluated metrics.\n\n### Additional Evaluations of our FDBM\nWe would like to draw the reviewer’s attention to our rebuttals to Reviewer i6UN and Reviewer fk33 for an additional evaluation of FDBM on the the cell differentiation task introduced by Somnath et al. [2]. The dataset provided in Somnath et al. [2] consists of two snapshots: one recorded on day 2, when most cells remain undifferentiated, and another on day 4, which includes a diverse set of mature cell types. \n\nWe evaluate the performance of FDBM in the table below and compare its performance against forward-backward Schrödinger bridge models (FBSB) [12], SBALIGN [2], and ABM [1]. Consistent with our findings on protein conformational changes, FDBM shows an advantage in the rough regime ($H=0.3$ and $H=0.4$) showing better performance compared to FBSB, SBALIGN and ABM in $3$ out of $4$ metrics. The values of ABM and FDBM is averaged over $10$ training trials and $10$ sampling trials for each trained model. \n\n|**Methods**|**MMD↓**|**Wₑ↓**|**ℓ₂(PS)↓**|**RMSD↓**|\n|-|-:|-:|-:|-:|\n|FBSB [2]|1.55e-2|12.50|4.08|9.64e-1|\n|FBSB with SBALIGN [2]|**5.31e-3**|10.54|0.99|9.85e-1|\n|SBALIGN [2]|1.07e-2|11.11|1.24|9.21e-1|\n|ABM [1] (retrained)|4.10e-2|9.50|0.89|8.72e-1|\n|FDBM(H=0.3)|5.34e-2|**9.32**|0.89|**8.11e-1**|\n|FDBM(H=0.4)|4.52e-2|9.35|**0.85**|8.21e-1|\n\n\nWe hope that our response adequately addresses the questions and concerns raised by the reviewer. In light of these clarifications and additional results, we kindly invite the reviewer to raise their final rating.\n\n[1] Bortoli et al. Augmented bridge matching, 2023. \n\n[2] Somnath et al. Aligned diffusion schrödinger bridges. UAI, 2023. \n\n[4] Corso et al. DiffDock: Diffusion Steps, Twists, and Turns for Molecular Docking, ICLR 2023.\n\n[5] Miñán et al. Sesame: Opening the door to protein pockets. GEM Workshop, ICLR 2025.\n\n[6] Zhang & Geffner et al. Bending and Binding: Predicting Protein Flexibility upon Ligand Interaction using Diffusion Models. GenBio Workshop, NeurIPS 2023.\n\n[7] Chang et al. Empirical entropic contributions in computational docking: evaluation in APS reductase complexes. J Comput Chem. 2008.\n\n[12] Chen et al. Optimal Transport in Systems and Control. Annual Review of Control, Robotics, and Autonomous Systems 2021"}, {"review_id": "BTIPupojsD", "reviewer": "Reviewer_i6UN", "summary": "This work proposes a novel generative diffusion bridge model, FDBM, with generalized stochastic dynamics, driven by non-Markovian fractional Brownian motion (fBM). The framework preserves the coupling information given paired training data, and is further exteneded to the Schrodinger bridge problem to handle unpaired training data. FDBM is evaluated on both paired and unpaired data tasks, including protein docking and image translation, achieving comparable performances compared to all baselines.", "questions": "1. I have strong reservations regarding the added complexity of parameter tuning due to the introduction of two additional hyperparameters, $H$ and $K$. Could you please give a prior intuition on how hyperparameter choices (especially $H$ and $K$) may affect model performance, and further explain for the variation in optimal hyperparameters across different tasks.\n2. As noted in Weaknesses, the protein docking task is less representative. It is strongly recommended to evaluate FDBM on a molecular dynamics dataset, even a toy case like Chignolin.\n3. I wonder about the backbone model used for the protein docking task. Please make sure that it is the same as used in SBALIGN and ABM, otherwise the comparison is unfair. Meanwhile, SBALIGN, ABM and FDBM are all generative frameworks while EGNN is a GNN architecture, which should not be compared together.\n4. Could you please give some more details on pretraining and fine-tuning on AFHQ?\n5. I would like to know whether such a complex generative framework like FDBM will harm the generative efficiency of the model. Please give a comparison of the efficiency of each model on the protein docking task and image translation task.\n6. The standard deviation of RMSD should not be used as a sole metric for comparison, as both high and low values may have context-dependent justifications. Please do not bold std in Table 2.\n7. Typo: \"the the\" in line 47, 130, 179.", "limitations": "The current experiments do not illustrate the necessity of introducing more hyperparameters in FDBM, which requires a more detailed explanation.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The proposed framework demonstrates clear novelty in its formulation, supported by a rigorous theoretical foundation and well-structured and comprehensive proofs.\n- The framework is applicable to both paired data tasks and unpaired data tasks, broadening its usefulness.\n\n**Weaknesses**\n- Compared with ABM, FDBM introduces two more hyperparameters ($H$ and $K$), which greatly increases the difficulty of hyperparameter tuning.\n- The paper does not provide a prior analysis of how hyperparameter choices may affect model performance, nor does it offer a convincing explanation for the variation in optimal hyperparameters across different tasks.\n- The experiments on paired data tasks are insufficiently representative. In the protein docking task, once the protein and its binding partner are specified, the bound conformation of the complex is largely deterministic and independent of the initial configuration. As a result, the primary focus lies on accurately modeling the target marginal distribution $\\Pi_1$, while the coupling information $\\Pi_{0,1}$ is of limited relevance. Consequently, the advantages of FDBM in preserving coupling structures cannot be effectively demonstrated in this setting.", "was_revised": true, "final_justification": "My main concern was that, compared to ABM, FDBM introduces additional hyperparameters $H$ and $K$, which I believed could increase the difficulty of hyperparameter tuning. However, the authors clarified in the rebuttal that all experiments were conducted with a fixed $K=5$, and the model still achieved strong performance. I find this demonstrates that FDBM is reasonably robust to the appropriate choice of $K$. On the other hand, I agree with the authors' explanation that $H$ can be viewed as an extension of the diffusion dynamics framework. Additionally, the supplementary experiments regarding inference time further show that FDBM does not significantly impair inference efficiency, which addresses my concern. Therefore, I will raise my score to 4.", "rebuttal": "We appreciate the reviewer’s thoughtful engagement with our work and thank them for recognizing the novelty of our framework and its rigorous theoretical foundation.\n\n> the advantages of FDBM in preserving coupling structures cannot be effectively demonstrated\n\nWe would like to point out that coupling preservation is not an advantage of FDBM over ABM, as ABM preserves couplings [1]; rather, this property distinguishes both from SBALIGN. In Proposition 5, we show that ABM’s coupling preservation extends to the generalized dynamics of FDBM. To illustrate this, we replicated the toy experiment from Bortoli et al. [Figure 1, 1], where initial samples from $(-2,-2)$ are paired with $(2,2)$, and $(-2,2)$ with $(2,-2)$. We observe that SBALIGN fails to preserve this coupling. Consistent with Proposition 5, FDBM preserves the coupling while offering a wider range of trajectories: in the rough regime $(H=0.2)$, trajectories explore more of the space, whereas in the smooth regime $(H=0.9)$, nearly straight-line paths emerge. Although we cannot share the figure during the rebuttal, we have included it in the revised manuscript to illustrate Proposition 5.\n\n> It is strongly recommended to evaluate FDBM on a molecular dynamics dataset, even a toy case like Chignolin.\n\nWe thank the reviewer for the valuable suggestion to evaluate FDBM on a molecular dynamics dataset. While we agree that molecular dynamics is an interesting application domain for FDBM, conducting such experiments requires considerable setup time and computational resources. Given the limited duration of the rebuttal period, we instead opted to perform an additional experiment on the cell differentiation task introduced by Somnath et al. [2]. This task allows us to assess FDBM for cell differentiation prediction on both the distributional quality and perturbation accuracy of the generated data using distributional metrics such as Wasserstein-2 distance ($W_\\varepsilon$) [9] and kernel maximum mean discrepancy (MMD) [10], as well as the Perturbation signature $\\ell_2(\\mathrm{PS})$ [11] and RMSD. The dataset consists of two snapshots: one recorded on day 2, when most cells remain undifferentiated, and another on day 4, which includes a diverse set of mature cell types. We evaluate FDBM on this task of inferring cell differentiation in the table below and assess its performance against forward-backward Schrödinger bridge models (FBSB) [12], SBALIGN, and ABM. Consistent with our findings on protein conformational changes, FDBM shows an advantage in the rough regime ($H=0.3$ and $H=0.4$) showing superior performance compared to FBSB, SBALIGN and ABM in $3$ out of $4$ metrics. All scores are averaged over $10$ training trials and $10$ sampling trials for each trained model. \n\n|**Methods**|**MMD↓**|**Wₑ↓**|**ℓ₂(PS)↓**|**RMSD↓**|\n|-------------------|----------:|-------:|-----------:|---------:|\n|FBSB [2]|1.55e-2|12.50|4.08|9.64e-1|\n|FBSB with SBALIGN [2]|**5.31e-3**|10.54|0.99|9.85e-1|\n|SBALIGN [2]|1.07e-2|11.11|1.24|9.21e-1|\n|ABM [1] (retrained)|4.10e-2|9.50|0.89|8.72e-1|\n|FDBM(H=0.3)|5.34e-2|**9.32**|0.89|**8.11e-1**|\n|FDBM(H=0.4)|4.52e-2|9.35|**0.85**|8.21e-1|\n\n> I have strong reservations regarding the added complexity of parameter tuning due to the introduction of two additional hyperparameters,\n\nFirst, we would like to clarify that we fixed $K = 5$ throughout all experiments reported in the main paper. The choice of $K$ involves a trade-off between the quality of the fBM approximation and computational cost. Similar to Daems et al., we fix $K=5$ to only minimally increase the computational cost while having a good approximation of fBM. We now included in our Appendix a plot of the error of approximation in the different regimes of MA-fBM. For $H>0.5$ we observe a maximal $L_2$ error of 1.5e-4 and for $H\\in[0.2,0.5]$ a maximal $L_2$ error of 5e-2. Moreover, the setting $H = 0.5$ and $K = 1$ closely recovers the original model driven by Brownian motion. Rather than merely introducing additional hyperparameters, our framework extends the original model by enabling a broader class of stochastic dynamics for diffusion bridge modeling. From this perspective, we do not consider $H$ to be a hyperparameter, but rather a modeling choice—one that allows access to diffusion dynamics that were previously out of reach. We observe that FDBM performs best in the rough regime ($H < 0.5$) for both protein conformational change prediction and the cell differentiation task (see below), with $H = 0.2$ and $H = 0.3$ yielding the strongest results. We do believe that the optimal choice of $H$ depends on the specific application, as demonstrated in the task of unpaired image translation.\n\n> I wonder about the backbone model used for the protein docking task.\n\nWe use the GNN architecture from Somnath et al. [2], but—following Bortoli et al. [1]—the initial state $x_0​$ is additionally provided to the network by concatenating it with the input. Nevertheless, the GNN we use for ABM and FDBM has fewer parameters, since Somnath et al. [2] approximate two functions ($b_t$ and $\\nabla_x \\log h_t$) with a single GNN resulting in more parameters in the output layer. We would like to stress that ABM and FDBM deploy the same model architecture and summarize the number of used parameters in the following table:\n\n|**Model parameters per task (paired)**|**SBALIGN**|**ABM**|**FDBM (ours)**|\n|------------------------|-----------|-------|---------------|\n|Rotated Moons|58692|31618|31618|\n|T-Shape|19204|10754|10754|\n|Cell Differentiation|310372|177970|177970|\n|Predicting Conformations|545220|537900|537900|\n \n> while EGNN is a GNN architecture, which should not be compared together.\n\nEGNN is a widely used baseline in the literature [2,5,6], and both SBALIGN and our method incorporate a GNN backbone within the diffusion process. For this reason, we believe that a comparison with EGNN is both appropriate and informative. Additionally, we refer the reviewer to our response to Reviewer v2k6, where we included an additional baseline—Sesame [3]—which is based on flow matching.\n\n> Could you please give some more details on pretraining and fine-tuning on AFHQ?\n\nWe used the same training and sampling parameterizations for all datasets and experiments. Parameterizations for DiT-B/2 and DiT-L/2 were kept identical. See the table below for the parameterizations of all experiments. Note that for DiT architectures compute also grows with the number of tokens (i.e., patches), which grow with the image resolution.\n\n|Model|Optimizer|Learning Rate|EMA Rate|Linear Warmup|Cosine Decay|Online Finetuning|Euler-Maruyama Steps|Parameters|\n|-------|-------------------------|-------------|--------|-------------| ------------|-----------------|--------------------|-|\n|DiT-B/2|lion|0.0001|0.999|10K|90K|4K|200|130M|\n|DiT-L/2|lion|0.0001|0.999|10K|90K|4K|200|458M|\n\n> The standard deviation of RMSD should not be used as a sole metric for comparison\n\nWe thank the reviewer for pointing out this mistake and have removed the bold formatting from the standard deviation values of the RMSD.\n\n> Please give a comparison of the efficiency of each model on the protein docking task and image translation task.\n\nWe provide a complete runtime comparison in the tables below. Training times for ABM and FDBM are nearly identical and both outperform SBALIGN, which requires approximating two functions and thus involves a larger model. For sampling, ABM and FDBM again show an advantage over SBALIGN. FDBM requires, on average, only 4.22e-2 seconds more than ABM to sample a conformation over 100 Euler–Maruyama steps. This slight increase is due to simulating a higher-dimensional stochastic process. However, the effect is minor, as the dominant computational cost during sampling comes from forward passes through the GNN, which are identical for both ABM and FDBM.\n\n|**Runtime paired setting**|**SBALIGN**|**ABM**|**FDBM (ours)**|\n|---------------------------------------------------------------------------------------|--------------:|---------------:|---------------:|\n|Average training step time|0.0159±0.0075|0.01438±0.0065|0.01412±0.0063|\n|Average sampling time of one conformation over $100$ steps |0.7078±0.3409 |0.6424±0.2992 |0.6846±0.3021|\n\nThroughout all unpaired data translation experiments, we use the same model architecture for both SBFlow and FDBM, resulting in nearly identical training runtimes. The sampling algorithm during inference is the same for the paired and unpaired setting: \n\n|Runtime unpaired setting [s]| Sampling from (partially) pinned process  |Calculation of loss term|\n|-|-|-|\n|SBFLow|0.0010 $\\pm$ 0.0002|0.0132 $\\pm$ 0.0074|\n|FDBM|0.0011 $\\pm$ 0.0003|0.0132 $\\pm$ 0.0018|\n\nWe truly appreciate the reviewers detailed engagement with our work and thank the reviewer again for the valuable feedback. We hope that we have sufficiently addressed the weaknesses and questions raised by the reviewer and kindly ask the reviewer to reconsider the evaluation of our work.\n\n[1] Bortoli et al. Augmented bridge matching, 2023. \n\n[2] Somnath et al. Aligned diffusion schrödinger bridges. UAI 2023. \n\n[5] Miñán et al. Sesame: Opening the door to protein pockets. GEM Workshop, ICLR 2025.\n\n[6] Zhang et al. Bending and Binding: Predicting Protein Flexibility upon Ligand Interaction using Diffusion Models. GenBio Workshop, NeurIPS 2023.\n\n[8] Jing et al. Generative Modeling of Molecular Dynamics Trajectories, NeurIPS 2024. \n\n[9] Cuturi et al. Sinkhorn Distances: Lightspeed Computation of Optimal Transport. NeurIPS 2013.\n\n[10] Gretton et al. A Kernel TwoSample Test. JMLR 2012.\n\n[11] Bunne et al. Supervised Training of Conditional Monge Maps. NeurIPS 2022.\n\n[12] Chen et al. Optimal Transport in Systems and Control. Annual Review of Control, Robotics, and Autonomous Systems 2021"}, {"review_id": "MQOS71k3Gg", "reviewer": "Reviewer_fk33", "summary": "This paper introduces Fractional Diffusion Bridge Models (FDBM), a new class of generative stochastic bridge models that incorporate fractional Brownian motion (fBM) as a noise process. Traditional diffusion bridge models rely on Brownian motion, which is memoryless and fails to capture long-range dependencies and temporal correlations observed in real-world systems (e.g., protein conformational dynamics, visual transformations). FDBM addresses this by leveraging a Markovian approximation to fBM (MA-fBM) using a finite mixture of Ornstein–Uhlenbeck (OU) processes, which allows efficient simulation and learning while retaining non-Markovian statistical properties. The authors evaluate their method in both paired (protein conformational prediction) and unpaired (image translation) settings, achieving state-of-the-art performance compared to Brownian-based baselines across multiple metrics (RMSD, FID, LPIPS).", "questions": "- **Can you provide more intuition or visual illustrations** (e.g., in Figure 1) of how varying the Hurst index $H$ changes the nature of the generated paths? This would help readers unfamiliar with fBM better grasp its effect.\n\n\n- **Does your method generalize to discrete-time settings or other noise models** (e.g., Lévy noise, jump diffusions)? If not, what are the main challenges?\n\n- **Could you comment more on computational cost** (memory and runtime) of FDBM vs. Brownian models in high-dimensional settings, e.g., image synthesis?\n\n- Can you give more experimental details of the protein task?", "limitations": "Yes. The authors have an explicit section addressing the limitations and clearly state the potential bottlenecks (e.g., finetuning, choice of $H$, scalability). They also reflect on the non-universality of bridge reversibility for fBM, which is commendable.", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "#### **Strengths**\n\n\n* **Theoretical Soundness**: The paper rigorously derives the FDBM framework from foundational principles, including its compatibility with both bridge matching and Schrödinger bridge objectives. The augmentation trick via MA-fBM is well-motivated and mathematically justified.\n\n* **Empirical Results**: The method significantly improves over Brownian baselines in both protein and image domains. The experiments are well-chosen: synthetic datasets highlight the effect of Hurst index $H$, and real-world tasks (protein conformational change, AFHQ translation) show strong performance in practice.\n\n\n* **Clarity of Formulation**: While technically deep, the paper is careful in defining all notation and path measures. The derivations (e.g., the conditional drift for the MA-fBM bridge) are nontrivial but complete.\n\n#### **Weaknesses**\n\n* **Accessibility and Background**: The paper presumes strong familiarity with stochastic processes (e.g., fBM, OU processes, Schrödinger bridges), which may hinder accessibility for machine learning audiences. Although appendices cover more details, some higher-level intuitions could be integrated into the main text.\n\n* **Empirical Scope**: The evaluation, while thorough in its selected domains, is limited to a few types of datasets. It would be helpful to demonstrate FDBM on temporal tasks, in, say, finance or speech, to generalize further the claim about modeling long-range dependencies. Also, it makes the theory more powerful if you can make one task excel rather than working on multiple tasks but get marginal improvements.\n\n* **Hyperparameter Sensitivity**: The choice of the number of OU components $K$, reversion speeds $\\gamma_k$, and the tuning of $H$ are empirically explored, but more principled insights on selection would strengthen practical utility.\n\n* **Computational Overhead**: Although the use of MA-fBM is tractable, the augmented state dimension can become large (e.g., $K=6$ OU processes per dimension). Some discussion on scalability (e.g., for long sequences or high-dimensional states) would be useful.", "was_revised": true, "final_justification": "I appreciate the authors’ detailed rebuttal and the additional experiment on the cell differentiation task. That said, my main concerns remain. The paper is technically solid and clearly builds on a deep understanding of stochastic processes, but I found the core method difficult to follow, even after reading the responses. Key parts of the framework—like the role of the MA-fBM and how it integrates into the learning process—still lack accessible explanations, which makes the work hard to engage with for a broader ML audience.\n\nThe new temporal experiment helps but doesn’t add much in terms of empirical strength. The gains over existing baselines are small, and it's not enough to clearly justify the added complexity of the approach. I do think the idea is promising and worth pursuing, but in its current form, the paper doesn’t feel ready for publication at NeurIPS. For these reasons, I’m leaning toward a borderline reject.", "rebuttal": "We thank the reviewer for their thoughtful and thorough evaluation of our work. We appreciate that the reviewer acknowledges the *theoretical soundness*, *empirical performance* and the *clarity in our formulation*. \n\n>some higher-level intuitions could be integrated into the main text. \n\nWe kindly refer the reviewer to our response to Reviewer fb9f, where we present a revised version of our introduction. Our paper is indeed rigorously rooted in stochastic processes, which we see as a strength rather than a weakness.\n\n> It would be helpful to demonstrate FDBM on temporal tasks\n\nAs suggested, we have conducted an additional experiment on the cell differentiation task introduced by Somnath et al. [2]. This task allows us to evaluate FDBM for cell differentiation prediction on both the distributional quality and perturbation accuracy of the generated data using distributional metrics such as Wasserstein-2 distance ($W_\\varepsilon$) [9] and kernel maximum mean discrepancy (MMD) [10], as well as the Perturbation signature $\\ell_2(\\mathrm{PS})$ [11] and RMSD. The dataset provided in Somnath et al. [2] consists of two snapshots: one recorded on day 2, when most cells remain undifferentiated, and another on day 4, which includes a diverse set of mature cell types. We evaluate the performance of FDBM in the table below and compare its performance against forward-backward Schrödinger bridge models (FBSB) [12], SBALIGN [2], and ABM [1]. Consistent with our findings on protein conformational changes, FDBM shows an advantage in the rough regime ($H=0.3$ and $H=0.4$) showing better performance compared to FBSB, SBALIGN and ABM in $3$ out of $4$ metrics. All scores are averaged over $10$ training trials and $10$ sampling trials for each trained model. \n\n|**Methods**|**MMD↓**|**Wₑ↓**|**ℓ₂(PS)↓**|**RMSD↓**|\n|-------------------|----------:|-------:|-----------:|---------:|\n|FBSB [2]|1.55e-2|12.50|4.08|9.64e-1|\n|FBSB with SBALIGN [2]|**5.31e-3**|10.54|0.99|9.85e-1|\n|SBALIGN [2]|1.07e-2|11.11|1.24|9.21e-1|\n|ABM [1] (retrained)|4.10e-2|9.50|0.89|8.72e-1|\n|FDBM(H=0.3)|5.34e-2|**9.32**|0.89|**8.11e-1**|\n|FDBM(H=0.4)|4.52e-2|9.35|**0.85**|8.21e-1|\n\n>The choice of the number of OU components $K$, reversion speeds $\\gamma_k$ and the tuning of $H$ \n\nFirst of all we stress that we fix $K=5$ and $\\gamma_1,...,\\gamma_K$ throughout all experiments reported in the paper.\n- The choice of $K$ trades quality of approximation for computation cost. We fix $K=5$ to only minimally increase the computational cost while having a good approximation. We included in our Appendix a plot of the error of approximating in the different regimes of MA-fBM. For $H>0.5$ we observe a maximal $L_2$ error of 1.5e-4 and for $H\\in[0.2,0.5]$ a maximal $L_2$ error of 5e-2. \n- We follow Daems et al. [14] in the choice of $\\gamma_1,...,\\gamma_K$ as a geometrically spaced grid to minimize the $L_2$-error of approximation.\n\nWe observe that FDBM performs best in the rough regime ($H < 0.5$) for both protein conformational change prediction and the cell differentiation task, with $H = 0.2$ and $H = 0.3$ yielding the strongest results. However, we believe that the optimal choice of $H$ depends on the specific application, as demonstrated in the task of unpaired image translation.\n\n>Can you provide more intuition or visual illustrations (e.g., in Figure 1) \n\nTo better illustrate the nature of the generated paths as well as Proposition 5, we replicated the toy experiment from Bortoli et al. [Figure 1, 1], where initial samples from $(-2,-2)$ are paired with $(2,2)$, and $(-2,2)$ with $(2,-2)$. We observe that SBALIGN fails to preserve this coupling while ABM and FDBM preserves the coupling while offering in the rough regime $(H=0.2)$, trajectories that explore more of the space, whereas in the smooth regime $(H=0.9)$, nearly straight-line paths emerge. Although we cannot share the figure during the rebuttal, we included it as a figure in the revision.\n\n> Does your method generalize to discrete-time settings or other noise models (e.g., Lévy noise, jump diffusions)? \n\nInstead of sampling $t \\sim \\mathcal{U}[0, T]$ during training, one could sample from a discrete set of time points, e.g., $t \\sim \\mathcal{U}\\{0 = t_0, t_1, \\ldots, t_N = T \\}$, and use only these time steps for inference via Euler–Maruyama discretization (or any other suitable scheme). The resulting forward process $X_{t_0}, \\ldots, X_{t_N}$ would then follow the marginals of the MA-fBM-driven SDE at those discrete times. Extending Schrödinger bridge models to Lévy processes is an interesting but technically challenging endeavor, as the discontinuous nature of Lévy paths requires a generalization of the theory for continuous paths presented in Léonard [21, 22], including an extension of the Girsanov theorem.\n\n> Could you comment more on computational cost \n\nWe now provide a complete runtime comparison in the tables below. Training times for ABM and FDBM are nearly identical and both outperform SBALIGN, which requires approximating two functions and thus involves a larger model. For sampling, ABM and FDBM again show an advantage over SBALIGN. FDBM requires, on average, only 4.22e-2 seconds more than ABM to sample a conformation over 100 Euler–Maruyama steps. This slight increase is due to simulating a higher-dimensional stochastic process. However, the effect is minor, the computation during sampling is dominated by the forward passes through the GNN, which are identical for both ABM and FDBM.\n\n|**Runtime paired setting [s]**|**SBALIGN**|**ABM**|**FDBM (ours)**|\n|---------------------------------------------------------------------------------------|--------------:|---------------:|---------------:|\n|Average training step time|0.0159±0.0075|0.01438±0.0065|0.01412±0.0063|\n|Average sampling time of one conformation over $100$ steps |0.7078±0.3409 |0.6424±0.2992 |0.6846±0.3021|\n\nThroughout all unpaired data translation experiments, we use the same model architecture for both SBFlow and FDBM, resulting in nearly identical training runtimes. The sampling algorithm during inference is the same for the paired and unpaired setting.\n\n|Runtime unpaired setting [s]| Sampling from (partially) pinned process  |Calculation of loss |\n|-|-|-|\n|SBFLow|0.0010 $\\pm$ 0.0002|0.0132 $\\pm$ 0.0074|\n|FDBM|0.0011 $\\pm$ 0.0003|0.0132 $\\pm$ 0.0018|\n\n>Can you give more experimental details of the protein task?\n\nFor a description of the dataset, we refer the reviewer to our response to Reviewer v2k6. In addition, we have now included the following section on implementation details in the appendix:\n\n*The results reported in Table 2 were obtained by averaging over 5 training trials, each run for $300$ epochs, and performing one sampling trial per trained model, generating a single path over $100$ time steps. We use the AdamW optimizer with an initial learning rate of 0.001 and a training batch size of 2. During validation, inference is performed using the exponential moving average of the model parameters, which is updated at every optimization step with a decay rate of 0.9. After each epoch, we simulate trajectories on the validation set and compute the mean RMSD. The model achieving the lowest mean RMSD on the validation set is selected for final evaluation on the test set.*\n\n*We use the GNN architecture from Somnath et al. [2], but—following Bortoli et al. [1]—we additionally provide the initial state $x_0$ by concatination as input to the network. Nevertheless, the GNN we use for ABM and FDBM has fewer parameters, since Somnath et al. [2] approximate two functions ($b_t$ and $\\nabla_x \\log h_t$ via that GNN resulting in more parameters in the output layer. We would like to stress that ABM and FDBM deploy the same model architecture and summarize the number of used parameters in the following table*:\n\n|**Model parameters per task (paired)**|**SBALIGN**|**ABM**|**FDBM (ours)**|\n|------------------------|-----------|-------|---------------|\n|Rotated Moons|58692|31618|31618|\n|T-Shape|19204|10754|10754|\n|Cell Differentiation|310372|177970|177970|\n|Predicting Conformations|545220|537900|537900|\n\n**Additional empirical evaluations.**\nWe kindly refer the reviewer to our response to Reviewer v2k6, where we included the $\\Delta \\text{RMSD}$ metric to assess whether FDBM predicts structures closer to the holo conformation than the initial apo conformation in the task of protein conformational change prediction. Moreover, we included an additional baseline—Sesame [3], a recent method based on Flow matching presented in Miñán et al. [6]. We find that FDBM shows consistently better performance compared to ABM and Sesame in the rough regime ($H=0.3,0.2,0.1$), as measured by both median and mean $\\Delta RMSD$. For $H=0.2$ and $H=0.3$, FDBM matches or exceeds ABM and Sesame across all evaluated metrics.\n\nWe sincerely thank the reviewer for the valuable feedback and their detailed engagement with our work. We hope that our response adequately addresses all the questions and concerns raised by the reviewer. We kindly ask the reviewer to consider increasing their score in light of our clarifications and additional results.\n\n[1] Bortoli et al. Augmented bridge matching, 2023. \n\n[2] Somnath et al. Aligned diffusion schrödinger bridges. UAI 2023. \n\n[5] Miñán et al. Sesame: Opening the door to protein pockets. GEM Workshop, ICLR 2025.\n\n[6] Zhang et al. Bending and Binding: Predicting Protein Flexibility upon Ligand Interaction using Diffusion Models. GenBio Workshop, NeurIPS 2023.\n\n[9] Cuturi et al. Sinkhorn Distances: Lightspeed Computation of Optimal Transport. NeurIPS 2013.\n\n[10] Gretton et al. A Kernel TwoSample Test. JMLR 2012.\n\n[11] Bunne et al. Supervised Training of Conditional Monge Maps. NeurIPS 2022.\n\n[21] Christian Léonard. A survey of the schrödinger problem and some of its connections with optimal transport, 2013. \n\n[22] Christian Léonard. Girsanov theory under a finite entropy condition, 2011."}]

Author rebuttal:

Meta-review: The author proposes FDBM, a framework designed to better capture long-range dependencies and temporal correlations often observed in real-world systems—areas where existing diffusion models may fall short. FDBM achieves this by employing a Markovian approximation to fractional Brownian motion (MA-fBM), implemented as a finite mixture of Ornstein–Uhlenbeck (OU) processes. This formulation enables efficient simulation and learning while preserving essential non-Markovian statistical properties.

* Good point 
I find this idea both interesting and novel, and I vote in favor of acceptance.

* Weakness/suggestion
That said, as one reviewer noted, the paper is difficult to follow for readers outside of experts in diffusion models. I strongly encourage the authors to address this in their revision to make the presentation more accessible to a broader audience.
Decision: Accept (poster)