Underdamped Diffusion Bridges with Applications to Sampling
Authors: Denis Blessing
Julius Berner
Lorenz Richter
Gerhard Neumann
Venue: ICLR 2025 Poster (2025)
Keywords: Variational Inference
Sampling
Diffusion Models
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We provide a general framework for learning diffusion bridges, including degenerate noise and in particular underdamped versions, which are applied in the context of sampling.

Abstract: We provide a general framework for learning diffusion bridges that transport prior to target distributions. It includes existing diffusion models for generative modeling, but also underdamped versions with degenerate diffusion matrices, where the noise only acts in certain dimensions. Extending previous findings, our framework allows to rigorously show that score-matching in the underdamped case is indeed equivalent to maximizing a lower bound on the likelihood. Motivated by superior convergence properties and compatibility with sophisticated numerical integration schemes of underdamped stochastic processes, we propose *underdamped diffusion bridges*, where a general density evolution is learned rather than prescribed by a fixed noising process. We apply our method to the challenging task of sampling from unnormalized densities without access to samples from the target distribution. Across a diverse range of sampling problems, our approach demonstrates state-of-the-art performance, notably outperforming alternative methods, while requiring significantly fewer discretization steps and almost no hyperparameter tuning.

Published as a conference paper at ICLR 2025
UNDERDAMPED DIFFUSION BRIDGES WITH APPLICATIONS TO SAMPLING
Denis Blessing".. "1
Julius Berner*.2.t , Lorenz Richter"...1, Gerhard Neumann
1 Karlsruhe Institute of Technology, NVIIII,, $Zuse Institute Berlin, dida Datenschmiede GmbH, 'FZI Research Center for Information Technology
ABSTRACT
We provide general framework for learning diffusion bridges that transport prior to target distributions. includes existing diffusion models for generative model- ing, but also underdamped versions with degenerate diffusion matrices, where the noise only acts in certain dimensions. Extending previous findings, our framework allows rigorously show that score matching in the underdamped case is indeed equivalent to maximizing a lower bound on the likelihood. Motivated by superior convergence properties and compatibility with sophisticated numerical integration schemes of underdamped stochastic processes, we propose underdamped diffiusion bridges, where a general density evolution is learned rather than prescribed by a fixed noising process. We apply our method to the challenging task of sampling from unnormalized densities without access samples from the target distribu- tion. Across diverse range of sampling problems, our approach demonstrates state-of-the-an performance, notably outperforming alternative methods, while requiring significantly fewer discretization steps and no hyperparameter tuning.
INTRODUCTION
In this paper we propose general diffusion-based framework for sampling from a density
Ptarget
Ptarget
Z = Jgd
Ptarget (z) dr.
(1)
Z
where Ptarget E C(TRd RRoo) can be evaluated pointwise, but the normalization constant 2 is typi- cally intractable. This task is of great practical relevance in the natural sciences, c.g., in fields such as molecular dynamics and statistical physics (Stoltz et al., 2010; Schopmans & Friederich, 2025), but also in Bayesian statistics (Gelman et al., 2013).
Recently, multiple approaches based on diffusion processes have been proposed. where the overall idea is to learn stochastic process in such way that it transports easy prior distribution to the potentially complicated target over an artificial time. Typically, the process is defined as an ordinary Itô diffusion, in particular, demanding non-degenerate noise. In this work, aim generalize this setting to diffusion processes with degenerate noise. This is motivated by the following model from statistical physics.
Classical sampling approaches based on stochastic processes have been extensively conducted using some version of the overdamped Langevin dynamics
dx,- = V log Ptarget (X,) ds + V2dW,, Xo Pprior
whose stationary distribution is given by Ptarget (under some rather mild technical assumptions the target and on the prior Pprior). Furthermore, we can define an extended dynamics by introducing an additional variable, bringing the so-called underdamped Langevin dynamics
dX,- = Y, ds, Xo ~ Pprior:
(3a)
dY, = V log Ptarget (X,) Y,) ds + v2dw,, Yo ~o,,(0,
(3b)
"Equal contribution.
Work partially done at Caltech.

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overdamped
underdamped
uncontrolled
controlled
Figure 1: Illustration of uncontrolled (see (2) and (3)) and controlled (see (4) and (15)) diffusion processes overdamped underdamped regime, transporting the Gaussian prior distribution to target. the underdamped case, we show both the positional coordinate (left/blue) as well as the velocity (right/black). While the underdamped version enjoys better convergence guarantees, both uncontrolled diffusions only con- verge asymptotically. Learning the control,
achieve convergence in finite time.
where now the stationary distribution is given by r(z, y) := Ptarget (x) (x):Ny O. Id) (and ((z y) = Pprior (a))(yyr 0, Id) can be defined as an extended prior distribution). Intuitively, the y- y-variable can be interpreted as velocity, which is coupled to the space variable x via Hamiltonian dynamics.
While both (2) and (3) converge to the desired (extended) target distribution after infinite time, their convergence speed can be exceedingly slow, particular for multimodal targets (Eberle et al., 2019). At the same time, it has been observed numerically that the underdamped version can be significantly faster (Stoltz al., 2010). This might be attributed to the fact that the Brownian motion is only indirectly coupled to the space variable, leading to smoother paths of and lower discretization error in numerical integrators (since V log Ptarget only depends on X, but not Y). In particular, for smooth and strongly log-concave' targets, the number of steps to obtain KL. divergence be reduced from (dde²) to O(Vd/e) (Ma et al., 2021).
The idea of learned diffusion-based sampling is to reach convergence to multimodal targets after finite time. In particular, for overdamped diffusion models, the convergence rate can be shown to match the one of Langevin dynamics without the need for log-concavity assumptions as long as the learned model exhibits sufficiently small approximation error (Chen et al., 2022). In the overdamped setting, this can be readily formulated as adding a control function to the dynamics (2),
dx,, = (V log Prarget (X,) + u(X,,, s)) ds + d2d
(4)
where the task is to learn u C(R2 [0, Rd) as reach Xr Pearget (Richter Berner, 2024; Vargas et al., 2024); see Figure for an illustration. It is now natural to ask the question whether we can use the same control ideas to the (typically better behaved) underdamped dynamics (3). Motivated by this guiding question this paper includes the following:
Controlled diffusions with degenerate noise: Building on previous work based on path space measures, we generalize diffusion-basee sampling to processes with degenerate noise, in particular including controlled underdamped Langevin equations (Section 2).
Underdamped methods in generative modeling: This framework can be used to derive and analyze underdamped methods in generative modeling. In particular, we derive the ELBO and variational gap for diffusion bridges where both forward and reverse-time processes are learned.
Novel underdamped samplers: Moreover, our framework culminates in underdamped versions of existing sampling methods and in particular in the novel underdamped diffusion bridge sam- pler (Section 3). In extensive numerical experiments, we can demonstrate significantly improved performance of our method.
We provide careful ablation studies of our im- provements, including the benefits the novel integrators for controlled diffusion bridges as well training of hyperparameters (Section 4). We note that the latter eliminates the need for tuning and also significantly improves existing methods in the overdamped regime.
Numerical integrators and ablation studies:
OOr, more general, log-concave outside of region.
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1.1 RELATED WORK
Many approaches to sampling problems build an augmented target by using a sequence of den- sities bridging the prior and target distribution and defining forward and backward kernels to ap- proximately transition between the densities, often referred to as annealed importance sampling (AIS) (Neal, 2001). For instance, taking uncorrected overdamped Langevin kernels leads to Un- adjusted Langevin Annealing (ULA) (Thin et al., 2021; Wu et al., 2020). Moreover, Monte Carlo Diffussio (MCD) optimizes the extended target distribution in order to minimize the variance of the marginal likelihood estimate (Doucet et al., 2022b). Going one step further. Controlled Monte Carlo Diffusion (CMCD) (Vargas et al., 2024) proposes an objective for directly optimizing the transition kernels to match the annealed density and Sequential Controlled Langevin Diffusion (SCLD) (Chen et al., 2025) connects this attempt with resampling strategies.
Furthermore, methods relying on diffusion processes that prescribe the backward transition kernel have recently been suggested. For instance, those include the Path Integral Sampler (PIS) (Zhang & Chen, 2022; Vargas et al., 2023b; Richter, 2021). Time-Reversed Diffusion Sampler (DIS) (Berner et al., 2024). Diffusion generative flow samplers (DGFS) (Zhang et al., 2024), Denoising Diffusion Sampler (DDS) (Vargas et al., 2023a), as well as the Particle Denoising Diffusion Sampler (Phillips et al.. 2024). For those diffusion- basedd samplers, the optimal forward transition corresponds to the score of the current density, which can also be learned via its associated Fokker- Planck equation (Sun et al., 2024) or its representation via the Feynman-Kan formula (Akhound-Sadee) et al., 2024). Finally, there are methods learning both kernels separately, e.g., the (Diffusion) Bridge² Sampler (DBS) (Richter & Berner, 2024).
For some of the above methods improved convergence has been observed when using underdamped versions Hamiltonian dynamics, which can be viewed as form of momentum. In particular, ULA has been extended to Uncorrected Hamiltonian Annealing (UHA) (Geffner & Domke, 2021; Zhang et al., 2021), MCD has been extended to Langevin Diffusion Variational Inference (LDVI) (Geffner& & Domke, 2023), and the works on DDS and CMCD also proposed underdamped versions.
Our proposed framework in principle encompasses all these works special cases (see Table 2 in the appendix), noting, however, that each of the previously existing method brings some respective additional details. Moreover, we can easily derive novel algorithms using our framework, ranging from an underdamped version of DIS to an underdamped version of the Diffusion Bridge Sampler (Appendix A.9). Further, our unifying framework allows us to easily share integrators and train- ing techniques for the different methods. First, we improve tuning for all considered methods by learning hyperparameters end-to-end, overall resulting in better performance (Figure 5). Second, we advance underdamped methods with our novel integrator (Figure 4 and Figure 9). Third, we show how to scale DBS to more complex targets by using a suitable parametrization (Table 4 & Figure 11) and divergence-frre freee training objective (Proposition 2.3 vs. Proposition A.7). This makes our underdamped version of DBS a state-of-the-an -htt method across a wide range of tasks (Table 1, Figure 3, & Table 3).
2 DIFFUSION BRIDGES WITH DEGENER ATE NOISE
In this section, we lay the theoretical foundations for diffusion bridges with degenerate noise, ex- tending the frameworks suggested in Richter & Berner (2024) and Vargas et al. (2024). Relating to the example from the introduction, we note that this includes cases where the noise only appears in certain dimensions of the stochastic process and in particular includes underdamped dynamics. We refer to Appendices A. and A.2 for summary of our notation and assumptions.
The general idea of diffusion bridges to learn stochastic process that transports a given prior density to the prescribed target. This can be achieved via the concept of time-reversal (see, e.g., Figure 2). To this end, let us define the forward and reverse-time SDEs
dZ, (f + 17 u) ZZ,, 8) ds n(s) aw..
Zo ~ n.
(5)
dZ, (f + n (Z,, 8) ds + mss) āw.,
Zr
(6)
on the state space R,,, where dw, and Tw, denote forward and backward Brownian motion incre- ments (see Appendix A. for details), respectively, both living in dimension d < D. The function
We clarify the connection Schrödinger bridges and other diffusion bridges in Remark A.1.
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L(u,o) = D(F ot," ang mint o)
Figure 2: Illustration of our general framework for learning diffusion bridges with degenerate noise. Left: We consider forward and reverse-time SDEs (sec (5) and (6)), starting the (extended) prior and target and being controlled by u and respectively. Middle: We Iearn optimal controls u² and v" of the corresponding SDEs (5) and (6), respectively, by minimizing a suitable divergence D between the associated path measures pa" and Ip" the SDE trajectories (Problem 2. Right: In general, the optimal controls and we depict an alternative solution (transparent). However, every solution leads to perfect time-reversal and, in particular, represents a diffasion bridge with the correct marginals π at time = 0 and r at time & = T. We note that the trajectories can be smooth since we allow for degenerate diffusion coefficients, where the Brownian motion noise only acts in certain dimensions.
not unique
f E C(R" x tions u, v E C(RD [o, 7J. Rd) will be learned as approach the desired bridge. In our setting, the noise coefficient 7 E C((O. IR Dxd) may be degenerate in the sense that it has the shape r = (0.o)², where 0 IR D-dxd and o E C[10, 21, Rxd))) assumed to be invertible for each t E [o, Importantly, the (scaled) control functions and the (scaled) Brownian motions operate in the same dimensions. For simplicity, we assume that (f. f) with f E C(R? J0, T), Rd) and f E C(RDD x [0, T D-d), where î only depends on the last d coordinates of its D-dimensional inputs. Referring to the underdamped Langevin equation (3), we may think of Z = (X,Y)*
) is typically fixed and maps to the full space, whereas the control func-
The general idea is to learn the control functions u and such that the two processes defined in (5) and (6) are time reversals of each other. This task can be approached via measures on the space of continuous trajectories C[0,, T), PR, also called path space (see Appendix A. for details). To this end, let us denote by pu.- the measure of the forward process (5) and by n7 the measure of the backward process (6). We may consider the following optimization problem; see also Figure
Problem 2.1 (Time-reversal). Let D P P +
Rzo be divergence and let u C(R x [0, (Q, R be the set of admissible controls?. We aim to identify optimal controls u v² such that
u" u²,u² arg min D (1i (15)).
Clearly, if we can drive the divergence in (7) to zero, we have solved the time-reversal task and readily follows for the time marginals* that By = and P " л. We note that optimality in Problem 2.1 can be expressed by a local condition on the level of time marginals for any time in between prior and target.
Lemma 2.2 (Nelson's 's relation). The following statements are equivalent:
(i) Pu.. Pu.
(1)V (ii) u(:;t) - v(, =n (1)V log 1ot for all (0,T) and 1 m.
(i) u(,,1) () v(.,1) = ( log T7x for all 1 E (0, T) and Iy
Proof. The equivalence follows from the classical Nelson relation (Nelson, 1967; Anderson, 1982; Föllmer, 1986), which also holds for degenerate 77; cf. Haussmann & Pardoux (1986); Millet et al. (1989); Chen et al. (2022).
However, we cannot directly use Lemma 2.2 to approach Problem 2.1 since the marginals P. and p7 are typically intractable. Instead, to turn (7) into a feasible optimization problem, we need
We refer Appendix A.2 for assumptions on U4 We note that divergence D) is zero if and only if both its arguments coincide (in the space of probability measures P on C(O0, TJ.RD): RR); see Appendix A.1).
WWe denote the marginal of path space measure IP time E [0, 7] by P:, Similarly, we denote by Paje the conditional distribution of IP, given Pr; see Appendix A.1.
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way to explicitly compute divergences between path measures. which (in analogy likelihood ratio) typically involves the Kadon-Nikodyy derivative between those measures. A key observation is that this can indeed be achieved for forward and reverse-time processes, stated in the following proposition.
Proposition 2.3 (Likelihood of path measures). Let nj ) be the pseudoinverse of n(8) for each E [0, T). Then for -almost every Z E CI, ()PP it holds that
dru,æ
(Z))
T ||* f + 2 8) ds +
2 / II + (2)2
1
log
d (Z) = log
T(ZT)
2
L (n (Z 5) 1j (s) az, [ fn s) 77 az.
Proof. Following Vargas et al. (2024, proof of Proposition 2.2), the proof applies the Girsanov theorem to the forward and reverse-time processes; see Appendix A.5.
We refer to Proposition A.7 in the appendix for alternative version of Proposition 2.3, which, for non-degenerate noise, has been used to define previous diffusion bridge samplers (Richter & Berner, 2024). However, this version relies divergence instead of backward stochastic processes, which renders it prohibitive for high dimensions and does not guarantee an ELBO after discretization; see also Remark 3.2.
is important to highlight that the optimization task in Problem 2.1 allows for infinitely many solutions. For numerical applications may either accept this non-uniqueness (ef. Richter & Berner (2024)) or add additional constraints, such as regularizers (leading to, e.g., the so-called Schrödinger bridge (Vargas et al., 2021; De Bortoli et al., 2021)), prescribed density evolution (Vargas et al., 2024) or fixed noising process (Berner et al., 2024). Those different choices lead to different algorithms, for which we can now readily state corresponding degenerate (and thus underdamped) versions using our framework, see Appendix A.9.
Divergences and loss functions for sampling. In order to solve Problem 2.1, we need to choose a divergence D, in turn leading to a loss function L U x U + R20 via C(u, v) = D () A common choice is the Kullback-Leibler (KL) divergence, which brings the loss
dP u,
(KL(U, v) = DKL (P 2 I =En log di (Z)
While we will focus on the KL divergence in our experiments, we mention that our framework can be applied to arbitrary divergences. In particular, one can use divergences that allow for off-policy training and improved mode exploration, such as the log-variance divergence (Nüsken & Richter, 2021: Richter et al., 2020), see Remark A.2.
2.1 IMPLICATIONS FOR GENERATIVE MODELING: THE EVIDENCE LOWER BOUND
Contrary to the sampling setting described above, generative modeling typically assumes that one has access to samples X Ptarget. but cannot evaluate the (unnormalized) density. In this section we show how our general setup from the previous section can also be applied in this scenario. For instance, it readily brings an underdamped version of stochastic bridges (Chen al., 2021) and serves as theoretical foundation for underdamped diffusion models stated in Dockhorn et al. (2022).
To this end, we may approach Problem 2.1 with the forward KL divergence
dPPer log dau (Z)
DKL (p..)
Ezaeorr
For the sake of notation, we have reversed time, which can be viewed as interchanging 7 and m. Since the process corresponding to po.r starts at the target measure т, we indeed require samples from this measure to compute the divergence in (9). At the same time, looking at Proposition 2.3, we
While we optimize the measures in both arguments of the KI. divergence, the measure pu.-, corresponding the generative pracess, is the second component, which is typically referred to "forward" KT. divergeeee
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realize that the divergence cannot be computed directly, since 7 cannot be evaluated. A workaround to instead consider an evidence lower bound (ELBO) (or, equivalently, a lower bound on the log- log-likelhhood. In our setting, we have the following decomposition.
Lemma 2.4 (ELBO for generative modeling). It holds that
E Ezo-r llgg Po (Zo)) = (P"M + Bzo [log T (Zo)] - DKL (Pu
(10)
evidence / log-likelhhood
variational gap
ELBO
where ((,,1) u(;,() = ( log !
Proof. This follows from Lemma 2.2 and the chain rule for KL. divergences; see Appendix A.5.
Crucially, observe that the ELBO in Lemma 2.4 does not depend on the target anymore the dependency cancels between the two terms (cf. Proposition 2.3). Moreover, the variational gap is zero if and only if ü almost everywhere, i.e., the path measures time-reversals conditioned on the same terminal condition due to Lemma 2.2. The ELBO is maximized when additionally 1P equals the target measure т, i.c., if and only if we found minimizer (u", v²) of Problem 2.1. In consequence, it provides viable objective to learn stochastic bridges in an underdamped setting (or, more generally, with degenerate noise coefficients 17) using samples from the target distribution т.
We note that for non- non-degenerate coefficients 17, the ELBO from Lemma 2.4 has already been derived in Chen et al. (2021); see also Richter & Berner (2024); Vargas et al. (2024). For diffusion models, i.e., 0 and f such that 150.T this ELBO reduces to the one derived by Berner et al. (2024); particular, has been shown that maximizing the ELBO is equivalent to minimizing the denoising score matching objective (with specific weighting of noise scales) typically used in practice.
Huang et al. (2021).
For general forward and backward processes, allowing for degenerate noise, stated in (5) and (6), the derivation the ELBO is less explored. For (underdamped) diffusion models with degenerate 7, a corresponding (hybrid) score matching loss has been suggested and connected to likelihood optimization by Dockhorn et al. (2022, Appendix B.3). In the following proposition, we show that this also follows special case from Lemma 2.4.
Proposition 2.5 (Underdamped score matching maximizes the likelihood). For the ELBO defined in (10) (setting v- 0) it holds
ELBO(u) = - s (..
[Ilu(Ze,)) +n T( log 50. (Z.lZo))l + const.,
where the constant does not depend on
Proof. Following Huang et al. (2021, Appendix A), the proof combines Proposition 2.3 with Stokes' theorem; see Appendix A.3. Note that our notation u learns the negative and scaled score.
UNDERDAMPED DIFFUSION BRIDGES
order to approach Problem 2.1 and minimize divergences (such as the KL divergence) practice, we need to numerically approximate the Radon-Nikodys derivative in Proposition 2.3. Analogously to Vargas et al. (2024, Proposition E. I.,,, we can discretize the appearing integrals to show that
(Zo) I - Po+ 1/r (2n+1 2n.)
dP T
Z ~
(11)
p0,7
T (2x) NN-1 Pa n+ (Zn) 22+1)
where the expressions for the forward and backward transition kernels ÿ7 and T depend on the choice of the integrator for Z.
Since we have degenerate diffusion matrices, the backward kernel 5 can exhibit vanishing val- ues, which requires careful choice of the integrators for Z. In particular, naively using an Euler- Maruyama scheme 2012; Doucet et al., 2022b); see also Figure 4.
integrator is typically not well-suited (Leimkuhler & Reich, 2004; Neal,
We therefore consider alternative integration methods, specifically splitting schemes (Bou-Rabe & Owhadi, 2010; Melchionna, 2007), which divide the SDE into simpler parts that can be integrated individually before combining them. Such methods are particularly useful when certain parts be

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solved exactly. To formalize splitting schemes, leverage the Fokker-Planck operator framework, proposing a decomposition the generator for diffusion processes of the form (5).
We can define L via the (kinetic) Bokker-Plancee equation
d.p = Lp with Lp = V (If + mu)p) + (e(mmV>p))
(12)
governing the evolution of the density t(-,t) = Pit of the solution to the SDE in (5). In order to approximate the generator L. we want assume suitable structure for and 7, such that we decompose L into simpler pieces. For this, we come back to the setting of the underdamped Langevin equation stated in the introduction in equation (3), We can readily see that its controlled counterpart can be incorporated in the framework presented in Section 2 by making the choices
D 2dl, = (X,Y) and
f(z, 8) (v. 8) - 300 po)(()) 7= (0,0)
(13)
in (5) and (6), where 0 E Raddd and f E C(R2 x [0, is chosen appropriately. Follow- ing Monmarché (2021); Geffner & Domke (2023) we split the generator as LA + LB Lo (sometimes referred to as free transport, acceleration, and damping) with
LAP = y VaP, Lap = -- VuP, Lop = V. (gp) + Tr(ooo To(ooTV))))
(14)
where g(2, 8) == -0 T (s)y + o(s)u((, y. s), resulting in
dX, dY.
0
ds f()
ds +
[-3300 (s)Y, + ou(Z,,8)) ds (s)āw,
(15)
where use a standard normal for the last d components of the initial and terminal distributions following Geffner & Domke (2023), i.e.,
T (z, v) Pprior (v) O, Id) and T(., v) Pearget (r:) 0, Id).
(16)
According to the Trotter theorem (Trotter, 1959) and the Strang splitting formula (Strang, 1968), the time evolution of the system can be approximated as:
e to = jefad cCu ( + +ONA
(17)
where a finite number of time steps of length A approximates the system dynamics. For higher accuracy, symmetric splitting can be used:
( + ta+ se [efo e Co c + O(NA3).
(18)
which reduces the approximation error (Yoshida, 1990). The optimal composition of terms is generally problem-dependeet and has been extensively studied for uncontrolled Langevin dynam- ics (Monmarché, 2021). For the controlled setting, prior works often use the OBAB ordering (Geffner & Domke, 2023; Doucet et al., 2022a). In this work, we additionally consider OBABO and BAOAB, which show improved performance (cf. Section 4).
Further details on the integrators for forward and backward kernels p and P corresponding to these splitting schemes can be found in Appendix A.8. We refer to Algorithm in the appendix for an overview our method and to Appendix A.10 for further details. A few remarks are in order (see also Remark A.3 for note higher-order Langevin equations).
Remark 3.1 (Mass matrix). Previous works, such as Geffner & Domke (2021) and Doucet et al. (2022b), consider incorporating a mass matrix M E C(OO, T) Rdxd) into the SDE formulation in (15) and terminal conditions. For simplicity, section. However, additional details on its inclusion and effects can be found in Appendix A.7. Furthermore, we conduct experiments where we learn the mass matrix, as discussed in Section 4.
have omitted this consideration in the current
Remark 3.2 (Discrete Andon-Nikodym derivative). We note that our discretization of the Radon- Nikodym derivative in (11) corresponds to a (discrete-time) Radon-Nikodym derivative between the joint distributions of the discretized forward and backward processes. In particular, we can analogously define KL divergence which allows us to obtain (guaranteed) lower bound for the
WWe denote by Tr the trace and by V the derivative operator w.r.t. spatial variahle 2; see Appendix A.I.

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Table 1: Results for benchmark problems of various dimensions d, averaged across four runs. Evaluation criteria include
errors for estimating the log-normalizing constant, A log Z. effective sample size ESS, Sinkhorn distance W,, and a lower bound (LB) on log 2; see Appendix 1014 for details on the metries. The best results are highlighted in bold. Arrows (1. +) indicate whether higher or lower values are preferable. Blue shading indicates that the method uses the underdamped Langevin equation.
Funnel (d = 10)
Many Well (d 5U)
LGCP (d 1600)
Method
A log Z
ESS
W2 +
2
ESS
log 2 (LB) 1
ESS
0. 0.14022..00 000
169.859go.11 1as
0.0101.0000
0.19900.0
482.024go oce
0.020
ULA
0.1 20000
20.016
159.212:0 ows
0.000
0.418
484.087 063
0. 0.030g0:004
0.1 04473
0,206=0.026
164.967=0.a54
0.005+0.002
0.737=0.00
183, 113,137=00.0
0.031=0.004
MCD
0.088+0.co8 0.375+0,016
144. 104.753=0.115
0.005+0..00
0.866=0.012
18 18
0.032-0.006
CMCD
0.0233
104.644:9.74
0.004::.oom
a.859
a83.875
0.032
0,26810.210
1.3691
148.990.11.33
0.008 19.00a
0.585
483.535 1n.22a
0.028_.o.om
0.047 0.047 co.coa
0.498-1o.0a1
107.458_..aa0 8.6
0.0061.0000
1.7082
405.680.. 4.019
0.015
DIS
con
0.5501..009
114.580 457
0.0051100000
0.85620.003
diverged
diverged
+0.00s
0.603=0.014
102. 658-0.666
0.005=0.001
0.887=00.004
186,376=11.022
0.032=0.002
DBS
0. 0.010=0.001
0.779=0.000
101. 101.118==..4 425
0.005+0.000
0. 0.898=0.00g o02
197 177.5511=o..s 188
0.171=0.017
log- normalizzation constant log Z in discrete time. On the other hand, this is not the case we discretize the divergence-ber basee Radon-Nikodye derivative in Proposition A.7 done in previous work (Berner et al., 2024; Richter & Berner, 2024). Moreover, we can still optimize the divergences between the corresponding discrete path measures as presented in (8) and Appendix 1.10.5. Finally, we note that the discretized Radon- Nadon-Nikodym derivative does not depend on f for the integrators considered in Appendix A.8. We thus choose f to have a good initialization for the process Z, see Appendix A.
Remark 3.3 (Properties of the score). Since the target density Ptarget in (16) only appears in the coordinates where 17 vanishes, Nelson's 's identity in Lemma 2.2 shows that
u (ar, y, 7) v² (2, = a T (y; Id).
(19)
i.e, the optimal controls =" and e- do not depend on the score of the target distribution, Va log Ptarget: at terminal time T, as in the case of corresponding overdamped versions. This can lead to numerical benefits in cases where this score would attain large values, e.g., when Ptarget is essentially supported on a lower dimensional manifold (Dockhorn al., 2022; Chen et al., 2022).
4 NUMERICAL EXPERIMENTS
In this section, we present a comparative analysis of underdamped approaches against their over- damped counterparts. We consider five diffusion-based sampling methods, specifically, Unad- justed Langevin Annealing (ULA) (Thin al., 2021; Geffner & Domke, 2021), Monte Carlo Diffu- sions (MCD) (Doucet et al., 2022b; Geffner & Domke, 2023), Controlled Monte Carlo Diffusionss (CMCD) (Vargas et al., 2024). ReverRee Diffusion Sampler (DIS)' (Berner al., 2024), and Diffusiio Bridge Sampler (DBS) (Richter & Berner, 2024). We stress that the underdamped versions of DIS and DBS have not been considered before.
To ensure fair comparison, all experiments are conducted under identical settings. Our evaluation methodology adheres to the protocol suggested in Blessing et al. (2024). For a comprehensive overview of the experimental setup and additional details, we refer to Appendix A.10. Moreover, we provide further numerical results in Appendix 10.5, including the comparison to competing state-of -he methods. The code is publicly available..
4.1 BENCHMARK PROBLEMS
We evaluate the different methods on various real-world and synthetic benchmark examples.
It is worth noting that we do not separately consider the Denoising Diffusion Sampler (DDS) ( (Vargas et al., 2023a), as it can be viewed as special case of DIS (see Appendix A. in Berner et al. (2024)). https:: / /i thub. com/ i 981 ess / /n///nndrrddr Udmpedda fusi oneridges
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Punnel (4
Crodit 25)
Seads (4 26)
Canoer (4 31)
07
30
125
38
Drowniam (d 323
lonosplazee (d) 25)
Ddany Well 511)
Sonar (d 61)
11.0
007
100
ULA
MCD
CMCD
131
DBS
OD
Figure 3: Effective sample size (ESS) for real-world benchmark problems of various dimensions d, averaged across four seeds. Here, N refers number discretization steps. Dashed and solid lines indicate usage of the overdamped (OD) and underdamped (UD) Langevin dynamics, respectively.
EM (OD)
EM (UD)
OBAB
BAOAB
OBABO
wallookk diee (00301
32
Figure 4: Effective sample size (ESS) and wallelock time (in seconds) of the diffuusin bridge sampler (DBS) for different integration schemes, averaged across multiple benchmark problems and four seeds. Integration schemes include Salcr-Maruyama (EM) for over- (OD) and underdamped (UD) Langevin dynamics and various splitting schemes (OBAB, BAOAB, OBABO).
Real-world benchmark problems. We consider seven real-world benchmark problems: Four Bayesian inference tasks, namely Credit (d = 25). Cancer (d = 31), Ionosphere (d = 35), and Somar (d = 61). Additionally, we choose Seeds (d 26) and Brownian (d 32), where the goal is to perform inference over the parameters of a random effect regression model, and the time discretization of Brownian motion, respectively. Lastly, consider LGCP (d 1600), high- dimensional Log Gaussian Cox process (Møller et al., 1998).
Synthetic benchmark problems. We consider two synthetic benchmark problems in this work: The challenging Funnel distribution (d = 10) introduced by Neal (2003), whose shape resembles funnel, where one part is tight and highly concentrated, while the other is spread out over wide region. Moreover, we choose the Many Well (d = 50) target, highly multi-modal distribution with 25 32 modes.
Mall
4.2 RESULTS
Underdamped vs. overdamped. Our analysis of both real-world and synthetic benchmark prob- lems reveals consistent improvements when using underdamped Langevin dynamics compared its overdamped counterpart, as illustrated in Table I and Figure 3. The underdamped diffusion bridge sampler (DBS) demonstrates particularly impressive performance, consistently outperforming alter- native methods. Remarkably, competing methods that utilize significantly more steps.
with few N 8 discretization steps, often surpasses
Numerical integration schemes. We further examine various numerical schemes for the diffusion bridge sampler (DBS) introduced in Section 3. Results and a discussion for other methods can be found in Appendix A. 10.5. To provide concise overview, we present the average effective sample

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0
31
M
Figure 5: Effective sample size (ESS) of the underdamped diffusion bridge sampler (DBS) for various com- binations of learned parameters, averaged across multiple benchmark problems and four seeds using N 64 discretization steps. Hyperparameters include mass matrix M, diffusion matrix a, terminal time T. and ex- tended prior distribution See Figure for the results with discretization steps.
size (ESS) and wallelock time across all tasks, excluding LGCP, in Figure 4. Detailed results for individual benchmarks can be found in Appendix 1.10.5. While is known that classical Euler methods are not well-suited for underdamped dynamics (Leimkuhler & Reich, 2004), our findings indicate that both OBAB and BAOAB schemes offer significant improvements without incurring additional computational costs. The OBABO scheme yields the best results overall, albeit at the expense of increased computational demands due to the need for double evaluation of the control per discretization step. However, it is worth noting that in many real-world applications, target evaluations often constitute the primary computational bottleneck. In such scenarios, OBABO may be the preferred choice despite its higher computational requirements.
End-to-end hyperparameter learning. Finally, we examine the impact of end-to-end learning of various hyperparameters on the performance of the underdamped diffusion bridge sampler. Our investigation focuses on optimizing the (diagonal) mass matrix M (cf. Appendix A.7), diffusion matrix o, terminal time T, and prior distribution æ. Figures 5 and 10 illustrate the effective sample size, averaged across all tasks (excluding LGCP) for N 64 and N 8 diffusion steps, respec- tively. The results reveal that learning these parameters, particularly the terminal time and prior distribution, leads to substantial performance gains. We note that this feature improves the method's usability and accessibility by minimizing or eliminating the need for manual hyperparameter tuning.
CONCLUSION AND OUTLOOK
In this work we have formulated general framework for diffusion bridges including degenerate stochastic processes. In particular, we propose the novel underdamped diffusion bridge sampler, which achieves state-of-the-aat results on multiple sampling tasks without hyperparameter tuning and only few discretization steps. We provide careful ablation studies showing that our improve- ments are due to the combination of underdamped dynamics, our novel numerical integrators, learn- ing both the forward and backward processes as well as end-to-end learned hyperparameters. Our results also suggest to extend the method by Chen et al. (2021) and benchmark underdamped diffu- sion bridges for generative modeling using the ELBO derived in Lemma 2.4.
Finally, our favorable findings encourage further investigation of the thcoretical convergence rate of underdamped diffusion samplers. Similar to what has already been observed in generative model- ing by Dockhorn et al. (2022), we find significant and consistent improvements over overdamped versions, in particular also for high-dimensional targets with only few discretization steps N. However, previous results showed that (for the case = 0), the improved convergence rates of underdamped Langevin dynamics do not carry over the learned setting, since (different from the score V log Ptarget in Langevin dynamics) the control u depends not only on the smooth process X, but also on Y (Chen al., 2022). Specifically, it can be shown that small KL divergence between the path measures generally requires the step size to scale least linearly in (instead of vū). While the tightness of our lower bounds on log Z corresponds to such KL divergences, we believe that the results can still can be reconciled with our empirical findings due to the following reasons: (1) our samplers are initialized Langevin dynamics (see Appendix A. 10) such that theoretical ben- efits of the underdamped case hold at least initially (2) the learning problem becomes numerically better behaved (see (19)), leading to better approximation of the optimal parameters, (3) learning both u and v as well as the prior म, diffusion coefficient a, and terminal time T (see Figure 5) can reduce the discretization error.
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ACKNOWLEDGEMENTS
J.B. acknowledges support from the Wally Baer and Jeri Weiss Postdoctoral Fellowship. The re- search of L.R. was partially funded by Deutsche Forschungsgemminschaatt (DFG) through the grant CRC 1114 "Scaling Cascades in Complex Systems" (project A05, project number 235221301), D.B. acknowledges support by funding from the pilot program Core Informatics of the Helmholtz Association (HGF) and the state of Baden- Bedenn-Württennee Württemberg through bwHPC, well the HoreKa supercomputer funded by the Ministry of Science, Research and the Arts Baden-W Württemberg and by the German Federal Ministry of Education and Research.
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A
APPENDIX
CONTENTS
A.I Notation
16
A.2 Assumptions
17
A.3 Further remarks
18
A.4
Auxiliary results
19
A.5 Proofs
20
A.6
Additional statements on diffusion models
A.7
Including mass matrix..
22
A.8
Numerical discretization schemes
22
A.8.1
Euler-Maruyama
23
A.8.2 OBAB
23
A.8.3
BАОAB
24
A.8.4
OBABO
24
A.9
Underdamped version of previous diffusion -baeed sampling methods
25
A.10
Further computational details
26
A.10.1
Algorithm.
26
A.10.2 SDE preconditioning
26
A.1..3 Experimental setup
28
A.10.4 Evaluation criteria
A.10.5 Further experiments and comparisons
30
A.1 NOTATION
We denote by Tr(E) and Σ+ the trace and the (hoore-eerrose) pseudoinverse of a real-valued matrix E, by || the Euclidean norm of vector μ., and by μr He the Euclidean inner product between vectors and (12.
For function p E C(RD T), R), depending the variables (x. y) Rd RD-d ~ RD and & E T), we denote by 0.p it partial derivative w.r.t. the time coordinate and by VaP and vyP its gradients w.r.t. the spatial variables z and y, respectively. Here, C(A, B) denotes the set of all continuous functions mapping from the set A to the set B. Moreover, we denote by
V AP WP
p
(20)
the gradient w.r.t. both spatial variables z = (x, w). We analogously denote by V the Hessian of
p w.r.t. the spatial variables. Similarly, define V f
ED1 fs. to be the divergence of
(dme-depenndett vector field f - (f:) C(R" [0, RD) w.r.t. the spatial variables.
We denote by W(, Σ) multivariate normal distribution with mean E Rd and (positive semi- definite matrix) covariance matrix Σ ddxd and write AZ; p. Σ) for the evaluation of its density (w.rtt the Lebesgue measure) at E Rdd, Moreover, we denote by Unif([0, TJ) the uniform distri- bution on j0, T). For Rd random variable with law IP and function f E (Rd R). we denote by
Ex~w [S(X] = f saw
(21)
the expected value of the random variable f(X).
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For suitable continuous stochastic processes Z (Zi)ee(a.7] and Y (Yi)eeo.T). we define for- ward and backward Itō integrals via the limits
'. x. dy, lim 20000
Σ
X (Vezts - Yez));
(22)
1=0
7 X. dy,
Kn Xtg
lim 200
Σ
(Yi7 - Yo;),
(23)
(=U
where 1 < 18 <... < 2
7 is an increasing sequence of subdivisions of [t. 7] with mesh size
tending to zero; see Vargas et al. (2024) for details. The relation between forward and backward integrals given in Lemma A.6.
We denote by P the set of probability measures on C((O, 71, P), equipped with the Borel o-field associated with the topology of uniform convergence on compact sets. For suitable vector fields u, and distributions g, we denote by Iu." E P and Pu, E P the forward and reverse-time path measures, i.c., the laws or pushforwards on C((0, 7], P), of the solutions Z = (Z))r to the SDEs
Z, Zo + [( (f nu+
nu) (Z., s) + [ n(s) ((s)...
Zo n,
(24)
2= = ZT f: (++ (Z,, 8) J. n())
ZT ~T
(25)
respectively. In the above. W denotes a standard d-dimensional Brownian motion satisfying the usual conditions, see, e.g.. Kunita (2019). Note that consider degenerate diffusion coefficients of the form n= o)) .nMoreover, we assume that the first components of f (f. T. namely f. satisfies Vz.(2z -) for every = (x, y) E RD, i.e., f only depends on y but not on x. Finally, denote the marginal of a path space measure P time t E [0, T] by Pr, which can be interpreted as the pushforward under the evaluation Z + Ze- Moreover, we denote by Paje the conditional distribution of P, given P..
A.2 ASSUMPTIONS
Throughout the paper, we assume that all vector fields are smooth, i.e., for a vector field g it holds g C [0, )d), and satisfy global Lipschitz condition (uniformly in time), i.e., there exists constant C such that for all 21, 22 E IR' and t E [O,T] it holds that
(19(21,)) L) s( OII S C||=1 =22l-
(26)
These assumptions also define the set of admissible controls U C (0,7),
Moreover, we assume that the diffusion coefficients appearing in the dimensions with the control, a, are invertible for all t [0, and satisfy that C ((0, 7]. Rdxd). RR, Our continuity assump- tions on the SDE coefficient functions and the global Lipschitz condition in (26) guarantee strong solutions with pathwise uniqueness (see, e.g., Le Gall (2016, Section 8.2)) and are sufficient for Gir- sanov's ' theorem in Theorem A.4 to hold (see, e.g., Delyon & Hu (2006)). Moreover, our conditions allow the definition of the forward and backward Itô integrals via limits of time discretizations as in (22) and (23) that are independent of the specific sequence of refinements (Vargas et al., 2024).
Finally, we assume that all SDEs admit densities of their time marginals (w.r.t the Lebesgue mea- sure) that are sufficiently smooth such that we have strong solutions to the corresponding Fokker-
SSfficient conditions for the existence of densities can found Millet al. (1989, Proposition 4.1) and Haussmann & Pardoux (1986, Theorem 3.1). For imeeinddpendent SDE cocfficient functions, a result by Kolmogoroff (1931) guarantees that the Fokker-Planck equation is satisfied if the density is in CR2 [o. T], IR; see also Pavliotis (2014, Proposition 3.8). and Schilling & Partzsch (2014, 19.6 Proposi- tion). However, we note that popular results by Friedman (1964, Section 1.6) (see also Friedman (1975, Section 5) and Durrett (1984, Section 9.7)) for showing existence and uniqueness of solutions to Fokker- Planckk equa- tions require uniform ellipticity assumptions, which are not satisfied for our degenerate diffusion coefficients. We refer to Bogachey et al. (2022, Sections 6.i( and 9.8 9.8()-(iii) for existence and uniqueness in the degen- erate case and note that we only make use of the Fokker- Planck equation for motivating our splitting schemes in Section 3.
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Planck equations. The existence of continuously differentiable densities and our assumptions on the SDE coefficient functions are sufficient for Nelson's 's relation in Lemma 2.2 to hold; see, e.g. Millet et al. (1989). While use the above assumptions to simplify the presentation, note they can be significantly relaxed.
A.3 FURTHER REMARKS
Remark A.1 (Stochastic bridges and bridge sampling). By stochastic bridge or diffusion bridge (also referred to as general bridge by Richter & Berner (2024)), we refer to an SDE that satisfies the respectively. For given diffusion coefficient
marginals Porior and Ptarget at times = 0 and of the SDE, there exist infinitely many drifts satisfying these constraints. In particular, for every sufficiently regular density evolution between the prior and target, we can find drift (given by a unique gradient field) that establishes a corresponding stochastic bridge; see, e.g., Vargas et al. (2024, Proposition 3.4) and Neklyudov et al. (2023, Appendix B.3).
However, any stochastic bridge solves our problem of sampling from Pearget and the non-uniqueness can even lead to better performance in gradient- based optimization (Sun et al., 2024; Blessing al., 2024). Other previous methods have obtained unique objectives by prescribing the density evolution, e.g., as diffusion process in DIS (Berner et al., 2024) or geometric annealing between prior and target in CMCD (Vargas et al., 2024).
Another popular approach for obtaining uniqueness consists of minimizing the distance to a ref- erence process (additionally to satisfying the marginals). In case the distance is measured via Kullback- divergence between the path measures of the bridge and reference process, this setting is often referred to (dynamical) Schrödinger bridge problem. In the context of samplers, reference processes have been chosen as scaled Brownian motions in PIS (Zhang & Chen, 2022) and ergodic processes in DDS (Vargas et al., 2023a); see also Richter & Berner (2024) for an overview.
A special case of such a Schrödinger bridge problem is given if the marginals Pprior and Ptarget are Dirac measures. Sampling from the solution to such problem is equivalent sampling from the reference SDE conditioned on the start and end point at the times t = 0 and t = T (specified by the Dirac measures). For instance, if the reference measure is a Brownian motion, solutions are commonly referred to as Brownian bridges. As special cases of our considered bridges, solutions to such problems are also sometimes called diffussion bridges and we refer to Schauer et al. (2013); Heng et al. (2021) for further details and numerical approaches. However, our sampling problem is in some form orthogonal to such tasks: in case of a Dirac target distribution, sampling is trivial and one is interested in the conditional trajectories. For the sampling problem, the trajectories are not (directly) relevant and one is interested in samples from a general target distribution.
Remark A.2 (Log-variance loss). As an alternative to the KL divergence in (8), we can consider the log-variance (LV) loss defined as
dus
( v) DIV (PuT (o.T) Var z-pw. log
(27)
d
where the expectation taken with respect path space measure corresponding to forward process of the form (5), but with the control replaced by arbitrary control w E U. This allows for off-policy training and avoids the need to differentiate through the simulation of the SDE. Moreover, the estimator achieves zero variance at the optimum (u", v²), see Richter et al. (2020); Nüsken & Richter (2021); Richter& & Berner (2024).
Remark A.3 (Higher-order Langevin equations). We note that our general framework from Sec- tion 2 can readily be used for higher-order dynamics and in particular higher-order Langevin equa- tions, where next to a position and velocity variable one considers acceleration. & Liu (2024), corresponding trajectories become smoother the higher the order, which can lead to improved performance of (uncontrolled) Langevin dynamics. Also, Mou et al. (2021) observed improved convergence of third-order Langevin dynamics for convex potentials. We leave related extensions to diffusion bridges for future work.
argued by Shi
In the context of generative modeling, also more general settings, referred to as mean-field games or gen- eralized Schrödinger hridges, have been explored; see, e.g., Liu et al. (2022); Koshizuka & Sato (2023); Liu al. (2024).
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A.4 AUXILIARY RESULTS
-almost Theorem A.4 (Girsanov theorem). For -lmm-a every Z E C( it holds that
dl
(2) = -0=- [/'
log dm
2 |lu w|| + (17 + w) (u - w) (Z,, ds S
(28a)
1 //' (Iln* ff w || f + ull?) (Z 8) ds + S
(28b)
2
where
- (u- (u- (8) dz.
=
(29)
In particular, for Z pu.. we obtain that
dP.5 0, 1 म
- (Z) 20 ||u (2), + / (u wZ) (w- ãB.
log
(30)
Proof. See Särkkä & Sottinen (2007); Üstünel & Zakai (2013); Chen al. (2022).
Theorem A.5 (Reverse-time Girsanov theorem). For Plms-almost -lmost every Z E C((O,T),R") it holds
that
pu.5 Pw,
d'u.. dPw (Z)
f (u-w) ()) ())) (s)dz
log
(Z) = log
(31)
(u w) (z. n Jz.
(32)
Proof. Using Theorem A.4 and the definitions in (22) and (23), we observe that d0%."
diw. (7) equals
the Radon-Nikodyn derivative between the path spaces measures corresponding to forward SDEs as in (5) with initial conditions and all functions f, u, w, and n) reversed in time, evaluated at t + -t- We can now substitute t + T - t to proof the claim; see also Vargas et al. (2024, Proof of Proposition 2.2).
Lemma A.6 (Conversion formula), For ~ pw and suitable E C(RP x (0,T),RP) it holds that
S (1(2,))) ãz, = { o(K,,)) dz, + f. V V (m s), ds.
(33)
Proof. Similar to the conversion formula Vargas et al. (2024, Remark 3), the result follows from combining (22) and (23). First, rewrite the problem by observing that
/ g(Zz. (.. .Z. [ g(Z,,)) dz, / 8) -JW, 1 (22,8)-JW.
where g = =n Then we can compute
8(2,,8) TW. = lim Σ
T(Zng +ZZ (1)) (Wi,, - Wey) Way)- 3(Zn,)) dw.
1=0
=2 g(Z.. s) dw. - ö(Z,, 8) dw.,
where o denotes Stratonovich integration. The result now follows from the relationship between Itō and Stratonovich stochastic integrals, i.c.,
J. s) J ()),,,) dw, + 2
1 f/ V (z.)) s) ds.
(34)
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see, e.g., Kloeden & Platen (1992, Section 4.9).
A.5 PROOFS
Proof of Proposition 2.3. The proof follows the one by Vargas et al. (2024, proof of Proposition 2.2).
Using disintegration (Léonard, 2014), we first observe that d.
r(Zr) dww. (2)
for w -1 f.
a| 7(Z))
Thus, it holds that
dI dIPw,
dlp d
(Zoo
log
(Z)
log
(Z)
log
log
(35)
diPu,
d Pu,7
T(ZT)
The result now follows by applying the Girsanov theorem; see Theorem A.4 and Theorem A.5. 1T
Proof of Lemma 2.4. Using Lemma 2.2 and the chain rule for the KL divergence, we observe that
DKL (P"TT DKL (1 DKL 1ipiiiPp + (r
(36)
where F I We that the Girsanov theorem (see Theorem A.4) implies that the variational gap can equivalently be written as
T /"'
DKL (P (P"TTIP"" = =E Foo
(o(Z., u(Z): +1 log (Z.) ds
J
also Vargas et al. (2024, Appendix C).
Proof of Proposition 2.5. The proof extends the by Huang al. (2021, Appendix A), Berner et al. (2024, Lemma A), and (Vargas et al., 2024, Appendix C.2) to the case of degenerate diffu- sion coefficients 17. Using Proposition A.7 and a Monte Carlo approximation, we first observe that, for the case u = 0, the ELBO can be represented as
[uul V (mj + n) (Z 5) ds 1/-
ELBO E. Z- log (ZT)
(37)
[Glluje- v -mn) (2,, const,
-T a0 Un.
(38)
where the last expression can be viewed as an extension of implicit score matching (Hyvärinen & Dayan, 2005) to degenerate 77.
Completing the square and using the tower property (37), remains to show that
Eir (Z.) Zo] = - ().,))))))
(39)
for fixed s E [0, where we used the abbreviations
Vp(=)
p(=) % (= Zo) and r(2) = u(2,s) (nT(() (s) ()(s)u(2,s))
(40)
p(2)
Under suitable assumptions, the statement in (39) follows from the computation
E(r(Z,) [ r(2)p(2)
(2, / V (p)
(41)
- IV (71) 3))
(42)
where we used identities for divergences and Stokes' theorem.
CConsidering (kinetic) Fokker-Planck equation (12), Lebesgue measure an invariant measure of the SDE in (5) with control vr -nn f if and only f merely depends on the last d coordinates.
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A.6 ADDITIONAL STATEMENTS ON DIFFUSION MODELS
The following proposition is an alternative version of Proposition 2.3, which, instead of backward integrations, depends on the divergence operation and does not rely on computing the pseudoinverse of 1).
Proposition A.7 (Radon- Nikodym derivative). For a process Z ~ Pu: as defined in (5) holds
dPu.
(Z) log (Zo)
T(Zx) + 1/² (u--v)
π7
[T]
v) (w
u u+
v V. + no) (Zg, 8) ds
log dlPer
2
L (u- u) 8) aW..
Proof.
This follows from combining Proposition 2.3 with Lemma A.6. In particular, note that for Z Pus it holds that
dlpux
(Zo) 1 { 11(1 + w)l 8)ds + 1 S/" 11 f ())? ds 2
log
(2) log
2 2
dP
T(Zr)
+ /. (n* f + (),, ()..s)).7) 17 (s) az) - 1-' (n1 ( (2). n7 (s) (s)
log (7z) 1 T
2 || f + 3 8) ds + 1
2 L
Ir I
11( w) (Z,, ds
7(ZT)
(u - v) 8) n (s) (oydzz. - /-" V. V (m no) 8) ss ds
log (Zo)
!" (u-v)- (w- u +
T
v (m + +wo) (Z." 8)
(ZT)
2
+ [/ (u (Zs 8) aw.
0 Idaxd Together with our assumption that the first component f of
We note that 1p1=
f = (). only depends on the last coordinates, we obtain that V (177 f) = V ff which proves the claim.
O
Remark A.8 (PDE perspective). Similar Berner et al. (2024); Sun al. (2024), we can also derive
the expression in Proposition A.7 using the underlying PDEs. To this end, we recall that the density
p of the solution Z to the SDE in stated (5) is governed by the (kinetic) Fokker-Planch equation in (12). Using the Hopf-Cole transformation V == logp, we get the Bamillon-lacobi-Bellee equation
a.V = - div(f + mm) VV - (f + mu) + 2 I TVV + Tr(m Tr(mmTV'V V²V).
Moreover, Itô's formula implies that
0) = S'' (0.V Trrn1 Tr(mmTmmvV + (f mw) VV) (Zs: ss
V(Zr, -
VV VV () n7()) dW.,
where 7 ~~ -u. Following the same computations in Proposition 3.1 in Sun et al. (2024) and minimizing the squared residual of the above Itô formula, we obtain the loss
(
27 )
dI
_ (u, v) E Eun
(2))
where the Radon-Nikodym derivative is equivalent to the one given in Proposition A.7.
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A.7 INCLUDING MASS MATRIX
In Section 3, we omitted the mass matrix M for simplicity. Here, we give further details on the SDEs when the mass matrix is incorporated. It can be incorporated the framework presented in Section 2 by making the choices D = 2d, Z (X,Y) Y and
F(.,y.s) = (v. flz y. - -00 (s)u)²
(43)
in (5) and (6), where Oa Raxd f C(R* x [). is chosen appropriately and o, M ( T] Rd For the terminal conditions, the standard normal for the last d components of the initial and terminal distributions is replaced by Gaussian whose covariance matrix given by the mass, i.e.
(x, v) = Pprior (w: 0, M) and T(.,V) Pearget (r))
(44)
We, therefore, get the forward and reverse-time processes dX. M- ds,
Xo Ppriors (45a)
dY, (f( 100 aM ((s.s) ds (2 aw,, Yo (O,
(45b)
and
dx, = M-1yy, ds,
Xy ~ Ptarget,
(46a)
dY, = ()) -0:00 -o0 + 2 ds + (1) āw., Yr ~ M).
(46b)
In a similar spirit to the diffusion matrix a, one can also learn the mass matrix. However, our experiments (Section 4) showed little improvements when doing so.
A.8 NUMERICAL DISCRETIZATION SCHEMES
In this section we provide details on the numerical integration schemes discussed in this work, called OBAB, BAOAB, and OBABO. In particular, we derive the transition kernels p and for computing the discrete-time approximation of the Radon-Nikodym derivative as
dP
π (Zo) II Po+ 1/ [2n+1] 2n)) T T(2x)
d (Z)
(47)
) I Pa. m+ 2n Zn+z)
We note that such splitting schemes well-studied in the uncontrolled setting, see, e.g., Section in Leimkuhler & Matthews (2015) or Section 2.2.3.2 in Stoltz et al. (2010). The controlled setting, and in particular the approximation of the Radon-Nikodyn derivative between path space measures, has to the best of our knowledge only been considered for OBAB yet Geffner & Domke (2023);
Doucet al. (2022b).
For convenience, let us recall the following splitting for the forward SDE that is used throughout this section, i.e.,
dx2 dY.
0 ds
f(X..
s)
ds +
(-3o0 (s) + ou(Z,, 8)) ds (s)TW.
(48)
and let us use the following splitting for the reverse SDE
dx."
0 [- (o))oo ov(Zs, 8)) ds + a(s)TW.
0 ds
+ f(X..s) ds
(49)
dY.
Here, we use arrows to indicate whether the corresponding splitting belongs to the generative or inference SDE. To simplify the notation, we define on := o(n)), f. = ff (XXssmA)) Pu+ = Pnt1l (Zn+ Zn)) and analogously for the backward transition Pant11-
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A.8.1 MARUYMA
We follow Geffner & Domke (2023) and leverage a semi-implicit Fuler-Maruyama (EM) scheme, where the velocity update is computed first and then used move the position, i.e.,
Pu+1 (1 oo A) + (an((Zn frA+ on v VDEn.
(50)
Rx+1 =x, + -
(51)
with En ~ 10, and step size A > O. We further define 2 2-+1= +( Pn+1) = (8n, + Pn+1A, A, P which helps to simplify the discrete-time approximation of the Radon-Nikodym derivative as shown the following. We obtain the forward transition density
F1 -N (Puta (1(1 _ano A) an (2, AA)A + Ta on A)
x 18 Pnt1) (Z=+1),
where 5 is the Dirac delta distribution. Integrating the reverse process (49) using the semi-implicit EM scheme, we obtain
X, Xat1 A
(52)
Y, P1 + 20+10 n+I A) - 1)))) -Fut1A + tn+11 V
(53)
with corresponding transition density
Pn|n+1 =N (fn (1+2(11 + 200+10 ont1v(22tz) (n (n+ (1+))))))+)))))) Fut
x 2- 11) (Rn, Pnt1),
(54)
resulting in the following ratio between forward and backward transitions
N (En+ilEnnt fo A) + wn nA)a + Tn
Pn+1|n
Polnt1
N (Palaatit + fo ant (2 (n + 1)4)A Jatz A ont1o
(55)
as the ratio between the two Dirac delta distribution cancel.
A.8.2 OBAB
Composing the splitting terms ŌBAB yields the integrator
y n = P1(1 - _ n A) o u 2n, nd)A + on Asn, E. ~ ~0,
(56a)
D" = =Y, J
Xn+1 -xn + Y
+
(56b)
Ynt1 = =P" + Int1
with Znt1 4 2(8m, P.). The resulting forward transition is given by
Patlin 6 .2:) (Zn+1) P.
Jono A) + (,)
The inference SDE, i.c., ÖBAB, is integrated as
D" = Patl - Jn+1 A 2
8, = 8nt1 - 4 A
Φ-1
(57)
V, P." -
Y, p fo, A) (Z nd)A V. En N,,(1,))
(58)
with (X,, f/) - (Znt1): giving the backward transition
Po|nt1 = 6, i(Zm,.)) (8n, PON (Fn P (1 fono A) on, n4),
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This results in the following ratio between forward and backward transitions
N (1 [ + on un (Zn nA)A, on
Pa+1/n
(59)
Pula+1
N
Y (1 200TTAA , Zn n 0
A.8.3 BAOAB
Composing the splitting terms BAÖAB yields the integrator
Y = Pn fn
1
(60)
X -
P" = -Y,(1 0
n A) + to F
+
(61)
Rat1 82, + +8
$2
(62)
Pn+i = =P, + f A
with En (0, I). (&, vn $ 1 (Zn), and Zn41 =X, Hence, we obtain the forward
transition density
Pn+1/n =6 22 8e") ( (Zx+1) x 6 (Zn) (x (K Pn)
(63)
xN 82" P1( fo A) (n
For BÄÖAB we obtain
F 2 = P An41 - fa+13 2
:-1
(64)
X, 8nt1 8"
P = Pn(11 (1
fon (1+ A) - -nne( "" Rn)A + on VAn
(65)
Xn X, -P'S
x-1
(66)
Ý, = V - fn -
with (8, )") 17 (2n-1 and Zn . 18, (.). Moreover, have
Poln+1 =6 5, (8,, 82) (Zn) 60.¹ (Zm41) (8 Pn)
(67)
xN (%% 19"(( T T A) an ",,,"""""""" (.mm))A,,,.. oT)
We therefore obtain the following ratio between forward and backward transitions as
Pa+ljn No Y 11 - , A) + or u( o
(68)
Pn|n+11
N 8; P (1+ J,, A) o, o, 8" AA).oonO,
A.8.4 OBABO
Composing the splitting terms ŌBABŌ yields the integrator
P = Pm(1 to T n A) + a Z.,, nA) + on (())
(69)
Y P + F
8at1 = 8n + P"
(70)
4rd = 5
L P = Y,"" (1 fo A) + t (n (n+1)A)) + / Ae(2))
(71)
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with 5%2 c(2) is given by
,n (0, and (Xn+1; PT")) $(8n, Pr). The resulting forward transition density
Pn =N ( (n P"" (1 fon0 A) + on V,"" . + 1) 20n0
x 6 82. (8x+1,
(72)
x Po m
Pn on A) + on un nd) 2
2. 200
The inference SDE, i... ÖBABO, is integrated as
T" =Pati(1 tono A) o, on (n +1A) + on /220
(73)
a
D" = a
=p"" - -Int1 A
n
2
8, =8a+1 -8n A Φ-1
(74)
P = P" - -nn
Pn Pa(1 + 1on n A) (n P,, (n+ 1) + on / A
/
A
(75)
with (8n, Yn) x0 1 18nn1, Px"") and gives the following backward transition densties
Pn|n+1 =N (Pn P/(1 +
1 i ono A) vn pn,(n }) 2
ó (8n4 (X pn))
(76)
N ( Pnti(1+ + tono, A) o, on (n + 1)A) fono A)
resulting in the following ratio between forward and backward transitions
N (Fnt1)
8 a (1 tono () + on (( &n+1, V."", (n + 1) 4 2 Io
Patlin
Pn n+1
N
f p Pn+1(1 (1 Io A) on ,n (Zn41 (n + 1)2) 4 IIT
Po 8a fn A) + or u 2n, nA) 3, 2 TTA
N (1+ to A) on (Kn,, P (n 1)A A AT)
A.9 UNDERDAMPED VERSION OF PREVIOUS DIFFUSION- BASED SAMPLING METHODS
In this section outline how our framework in Section 2 includes previous diffusion- -based sam- pling methods. First, note that setting the drift f and controls and in (15) specific values recovers underdamped methods of ULA, MCD, and CMCD, see Table 2. Moreover, we can also introduce reference processes with controls ū and ō that satisfy
d 10.
1,
(77)
where ãे and F known reference distributions. In other words, this assumes having knowledge of perfect time-reversal for specific controls ã, T and marginals म, 7 (cf. Section 3.3 in Richter & Berner (2024)). We remark that these processes take a role similar the Brownian motion used in
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the proof of Proposition 2.3. In particular, by applying Proposition 2.3 twice, obtain that
d.
dlPu,
d
log
(Z) log
(Z)
log
dieE, (Z)
diur
dicr
.. ((oo)
log T(ZT)
log
(())
F(Zr)
1 T
2 J ((v 0) (2n²* f 2 + - (u (2m f + + ()) (Zs, s, s) ds
(u- (Z,, 8) 11 (s) az - [f (v- 8) (z)
Several previous methods, such as versions of PIS and DDS, can be recovered by fixing e and using the choices ह as well ãे n, which significantly simplifies the above expression (Vargas et al., 2024; Richter & Berner, 2024).
A.10 FURTHER COMPUTATIONAL DETAILS
In this section we provide additional computational details.
A. 10.1
ALGORITIM
In Algorithm 1 display the training process of our underdamped diffusion bridge sampler.
A.10.2 SDE PRECONDITIONING
Here, we introduce a preconditioning technique that modifies the SDEs to improve numerical be- havior while preserving the optimal control for both overdamped and underdamped diffusions.
Overdamped diffusions.
Consider the controlled forward and backward pair of SDEs given by
dx, (f + o s) + (s) aw.,
Xo Pprior:
(78)
dX, = (J ( (X., ds ds + (s))s(sw
X7 Purget,
(79)
with forward and backward Brownian motion dw, and dW. f C(R" T) Rd), dif- fusion coefficient a E T), control functions u, V E C(R' x I), and path measures Pa.pe and Iou.PT corresponding to (78) and (79), respectively, with Po = Pprior and Pr = Purget- Note that for denoising diffusion sampler such as DIS, we h2ve f(x.-) 200 x and v = 0, i.e., an Ornstein-Uhlenberr (OU) process. While such a choice for ensures for a suitable a that Po se 1), may lead to a poor initialization for the path measure Pu.co Similarly, for DBS it helps significantly to choose f = v log Purget or f V where see Appendix A. to obtain good initialization for Nu How- ever, this may again lead to poor initialization for Pu.er. We can alleviate these problems by preconditioning of the SDEs: For DIS, we set ou ou - 2f with fxx = 200 T x, resulting in the forward SDE
((x, 8) x Pprior
1-8(s) (x
(x); (x)))(e)) (()))ee ((=)
dX,, (-f+ (X,, ss + (s)
Xo ~ Pprior
(80)
Note that when initializing ūt O, we have Tã. ~o 1) for all t E [0.T] which may lead to more stable training. Similarly, for ULA, MCD, and DBS, we set ou backward SDE
2 to obtain the
dx. --f (X,, ds o(s) dw
Xx Pragett
(81)
where V log y for ULA and MCD. Moreover, DBS uses preconditioning for all choices of f discussed in Appendix A.10.5. The impact of preconditioning is illustrated in Figure 6. A few remarks are order.
Remark A.9 (Preconditioning for CMCD). The controlled Monte Carlo diffusion sampler (CMCD, Vargas al. (2024)) prescribes path of densities (v(,, ()) and uses f 200 V log v.
12Please note that the sign differs from most existing literature the process initialized at the target starts t T instead of f
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DIS (No preonditioning)
DIS (With preconditioning)
DBS (No preconitiooing)
DBS (With preconditioning)
DBS-UD (No prcouditionig))
DBS-UD (With preconittining)
Figure 6: Illustration of the effect preconditionig as explained Appendix A. 1.1 at initialization. with u, = The red trajectories start from the prior distribution (uni-modal Gaussian distribution) and are integrated forward in time (from left to right), whereas the blue trajectories start at the target (bi-modal distribution) and move backward in time (from right left). The non-transparent trajectories correspond to the SDE that affected by preconditioning.
Therefore, Nelsons relation (see Lemma 2.2) yields u - v o V log resulting in the pair of SDEs
dX, = (2 v log + +ou (X, 3) ds dW)
Xo ~ Ppriors
(82)
dX, (- (-200 V + u)) (X, ds o(s) aw,,
Xy Parget
(83)
by replacing v = a V log As a consequence, CMCD does not require preconditioning as the sign flip of f naturally occurs due to Nelsons relation.
Remark A.10 (Preconditioning for DIS). The time-reversed diffusion sampler (DIS, Berner et al. (2024)) and related methods (Zhang & Chen, 2022; Vargas et al., 2023a) incorporate V log Peaget (or v vlogv) into the SDE by treating it as part of the control, i.c.
u(x, 8) = in u1 s) + (2(5) V log (z))
(84)
However, we found that this approach can lead to worse numerical behavior. As a result, we opted use this form of preconditioning.
Underdamped diffusions. chosen in (13), i.e., We consider the pair of SDEs given by (5) and (6) where the drift is
f(r,y.s) y. F(x, - To)y)
(85)
In a similar fashion to the overdamped case, we can precondition the underdamped SDEs: Assume for now that f. u 0. Then, (85) results in an OU process for Y which is desirable as both initial and terminal density of the velocity are Gaussian, see (16). However, the sign flip when performing the timerreversal may again lead to a bad initialization for the path measure p=,,, We therefore transform the control in the backward process (6) n noo y, +) + aa y, resulting in the
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Algorithm Training of underdamped diffusion bridge sampler
Require:
D See Appendix A.10 for details
model: neural networks uæ, V, with initial parameters ((0), ,
fixed hyperparameters: number of gradient steps K, number of discretization steps N, batch size
optimizer method step, integrator method integrate
learned hyperparameters: prior distribution Pycior NN(,, (fag(softplussmn2))) diffusion and
mass matrices diag(softplus(ne)) and M diag(sotphus(mm))) and terminal time T
N so7t with initial parameters T)μ "E No .1M 11a
(0) (0) (0) (0) (0)
mTo O {0(00 (0) ?i 1 ; Mid (10)
}. Pprior @ (0, M), 7= Pearget @ 00 M)
for + 0,
do
for + 1 m do
b Approximate cost (batched in practice) b See (16)
Z ~ π
md.00 log r (Zo)
for ni O.....
do
Zu41 integrateeee ((4)
See Appendix A.8 b Sec (11)
mddn+1 + mdi, + log Pn+1 (2n+1 [Zn) - (el))-log.. logpnn+ (2n+1)
n + - log 7 x(2x)
++ - m
Compute loss Gradient descent
(k+11) step (0 VoL
return optimized parameters (kk)
backward SDE
dY, (j (f+nØ) (Y,, ().,s) ds (s)) (s)Y, (
n(s)d,, Y (d0,
(86)
The impact of preconditioning for the underdamped setting is illustrated in Figure 6.
1.1..3 EXPERIMENTAL SETUP
Here, we provide further details on our experimental setup. Moreover, we provide an algorithmic description of the training of an underdamped diffusion sampler in Algorithm 1.
General setting. All experiments are conducted using the Jax library (Bradbury et al., 2021). Our default experimental setup, unless specified otherwise, is as follows: We use batch size of 2000 (halved if memory-constrained)) and train for 1 140k gradient steps to ensure approximate conver- gence. We the Adam optimizer (Kingma & Ba, 2015), gradient clipping with value of 1, and a learning rate scheduler that starts at x 10 10-3 and uses a cosine decay starting at 60k gra- dient steps. We utilized 128 discretization steps and the EM and OBABO schemes to integrate the overdamped and underdamped Langevin equations, respectively. The control functions ug and vy with parameters 0 and 7, respectively, were parameterized two-layer neural networks with 128 neurons. Unlike Zhang & Chen (2022), we did not include the score of the target density as part of the parameterized control functions úy and Uny Inspired by Nichol & Dhariwal (2021), we applied
square scheduler for the discretization step size, i.c., A cos? (3 // at step preconditioning explained in
n, where : j0, oc) (0, oo) is leamned. Furthermore, Appendix 10.. The diffusion matrix a and the mass matrix M were parameterized as diagonal matrices, and we learned the parameters μ and E for the prior distribution Pprior N E), with Σ also set as a diagonal matrix. We enforced non-negativity of a and made a, M, and Σ positive semidefinite via an element- softplus transformation.
For the methods that use geometric annealing (see Table 2), that is, v(x, s) x Pprior (x)) (x), where 3: [0, 21 + [0, 1] is monotonically increasing function satisfying 3(0) = 0 and ((T)) ) = 1, additionally learn the annealing schedule 8. Similar to prior works (Doucet al., 2022b), parameterize an increasing sequence of N steps using unconstrained parameters b(s). We map these our annealing schedule with
1-3(s) (x=(()()))))))))
B(mA)= Enso (b())))s(((11A)) MN In soltssoslbbuullll. (bnoA))
(87)
where softplus ensures non-negativ Further, we fix (0) 0 and ( 1, ensuring 3(n' < 3(nA) when n'' I n. We initialized such that 3 is linear interpolation between 0 and I.
28

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Published as a conference paper at ICLR 2025
Table 2: Comparison of different diffusion-based sampling methods based on f, v as defined in the text.
Method
ULA
ad
V= log
0
MCD
od V= log
0
Iearned
CMCD
o00 Vz log
learned
log u
DIS
-ao Vx log Pprior
learned
0
DBS
arbitrary learned
learned
Moreover, we initialized o = E Id and μ = 0 for all experiments. In the case of 0.1, while for the remaining benchmark
the Brownian, LGCP, and ManyWell tasks, problems, we chose a = 0.01 to avoid numerical instabilities encountered with a = 0.1.
Evaluation protocol and model selection. We follow the evaluation protocol of prior work (Blessing et al., 2024) and evaluate all performance criteria 100 times during training, using 2000 samples for each evaluation. To smooth out short-term fluctuations and obtain more robust results within single run, we apply a running average with window of 5 evaluations. We conduct each experiment using four different random seeds and average the best results of each run.
Benchmark problem details. All benchmark problems, with the exception of ManyWell, were taken from the benchmark suite of Blessing et (2024). In their work, the authors used uninfor- mative prior for the parameters in the Bayesian logistic regression models for the Credit and Cancer tasks, which frequently caused numerical instabilities. To maintain the challenge of the tasks while ensuring stability, we opted for Gaussian prior with zero mean and variance of 100. For more detailed descriptions of the tasks, we refer readers to Blessing et (2024).
The Many Well target involves a d-dimensional double well potential, corresponding to the (unnor-
malized) density
1
Purret (x) exp
(
8)2
2
\=
with m E representing the number of combined double wells (resulting in 2m modes), and a (0, x)) (see also Wu al. (2020)). In our experiments, set 50, m = 5 and 8 = 2. Since Ptarget factorizes across dimensions, we can compute reference solution for log Z via numerical integration, as described in Midgley al. (2023).
separation parameter
10.0.4 EVALUATION CRITERIA
Here, we provide further information on how our evaluation criteria are computed. To evaluate our metrics, we consider 2 10 samples (z())
Effective sample size (ESS). We compute the (normalized) ESS as
(E w(())
ESS
(88)
n Σ (wi(t))²
where (w))) are the unnormalized importance weights of the samples (Z()) in path space given
D0,
(1) = Z (Z(1).
(89)
Note that the dependence Z vanishes when replacing a.
di
with the expression in Proposi-
tion 2.3.
29

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Table 3: Results for lower bounds log 2 for various real-world benchmark problems. Higher values indicate better performance. The best results are highlighted bold. Blue shading indicates that the method uses underdamped Langevin dynamics. Red shading indicate competing state-of-the-an methods.
Method
Credit
Seeds
Cancer
Brownian
lonosphere
Sonar
188552442..44
=0.00
63 4 184
1.081=0.000 004
-11.67332..00
108. 595 000
DBS
-585. 11 112.0.00
-73. 42320.001
77 88188811000 01.014
1 136 001
-111. 111.630=o.0m 001
-108. 158=0.004
GMMVI
585.098--0 one
-73. 773.415--- 0.000
-7..988-0000 004
1,092 ons
-111 -111.82++..0
108. 726++0007
SMC
- 698, 4.14
-74.999eo.10
194.059=o.az
-1.874 oz2
-114.751
11113356
CRAFT
-73.793 (5,77933..00
99.773 ouT
0.886.20.0 ons
-112.386
-155 1
FAB
585. 1.00
-73.418 10.022
78.2277 805
10000
-111.678
-18.5000 1o.c0
Sinkhorn distance. We estimate the Sinkhorn distance W (Cuturi, 2013), i.e., an entropy regu- larized optimal transport distance between set of samples from the model and target using the Jax ott library (Cuturi et al., 2022).
nog-aionn constant. For the computation of the log- normalizing constant log Z in the gen- eral diffusion bridge setting, we note that for any E lI it holds that
E
Ez-pur
(90)
dlPu.
Together with Proposition 2.3, this shows that
log Z E zu
dP log dP u dP
TT
(Z)
F(ZT) π (Zo)
(91)
where 7(Zr)= = Ptarger (X2 (W Id) and P.p0 denotes the path space measure of the process Z with initial condition Zo = 2o E 2d (analogously for P see e.g. Léonard (2013).
If a u² and u = v², the expression in the expectation is almost surely constant, which implies
F(ZT)
log 2 = log d..... 10
(2) π (Zo)
(92)
If we only have approximations of u² and v², Jensen's 's inequality shows that the right-hand side in (92) yields a lower bound to log Z. For other methods, the log-normalizing constants can be computed analogously, by replacing u, v accordingly, see e.g. Berner et al. (2024) for DIS. Our experiments use the lower bound as an estimator for log 2 when labeled with "LB".
A.5 FURTHER EXPERIMENTS AND COMPARISONS
Comparison with competing methods. extend evaluation by comparing DBS against sev- eral state-of-the-ant techniques, including Gaussian Mixture Model Variational Inference (GMMVI) (Arenz al., 2022), Sequential Monte Carlo (SMC) (Del Moral al., 2006), Continual Repeated Annealed Flow Transport (CRAFT) (Arbel et al., 2021; Matthews et al., 2022), and Flow Annealed Importance Sampling Bootstrap (FAB) (Midgley et al., 2023). The results, presented Table 3, are primarily drawn from Blessing et al. (2024), where hyperparameters were carefully optimized. Since our experimental setup differs for the Credit and Cancer tasks (detailed in Section A.10), we adhered to the tuning recommendations provided by Blessing et al. (2024). Across most tasks, we observe that the underdamped variants of DBS and CMCD consistently yield similar or tighter bounds log Z compared to the competing methods, without the necessity for hyperparameter tuning. Notably, the underdamped version of DBS consistently performs well across all tasks and demonstrates robustness, evidenced by the low variance between different random seeds.
Choice of integrator. To complement the results from Section 4, we conducted an ablation study evaluating the performance and runtime of different integrators for ULA, MCD, CMCD, and DIS. The results are presented in Figures 7 and 8. Consistent with previous findings, the OBABO inte- grator delivers the best overall performance, with the exception of ULA. We hypothesize that in the
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UTA
MCD
0.2
1
1888
RSSS
0.2
N
N
CMCD
DIS
0.6
0.6
0.4
138-
0.4
0.2
0.2
16
12
64
128
16
32
128
EM (OD)
EM (UD)
OBAB
BAOAB
OBABO
Figure 7: Effective sample size (ESS) for various methods (ULA, MCD, CMCD, DIS) and different integration schemes, averaged across multiple benchmark problems and four seeds. Integration schemes include Buler- Maruyama (EM) for over- (OD) and underdamped (OD) Langevin dynamies and various splitting schemes
(OBAB, BAOAB, OBABO).
case of ULA, simulating the controlled part (O) twice offers little advantage, as both the forward and backward processes are uncontrolled for this method.
Caslit
0.8
0.6
0.5
135
0.11
1355
135
0.4
0.2
0-
wailellods time
104
wallelock time
10
wallekock time
101
Caunee (4 - 21)
Insooplace (6 55)
Sazuet (5) 61)
0.5
00.5
0.50
138
138
0.4
0.2
0.1
0.2
walleleok
wallebok
wallokoc time
RM (OD)
FM (UD)
OBAB
BAOAB
OBABO
Figure 8: Effective sample size (ESS) over wallelock time diffusion bridge sampler (DBS) with 128 diffusion steps for different integration schemes, multiple benchmark problems, and four seeds. Integration schemes include Euler-Maruyane (EM) for over- (OD) and underdamped (UD) Langevin dynamics and various splitting schemes (OBAB, BAOAB, OBABO).
Additional results for DBS. We present further details regarding the results discussed in Sec- tion 4. Specifically, provide breakdown of the performance of different integration schemes
31

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Published as a conference paper at ICLR 2025
Place: 14 18)
Crddit (dl- 25)
Needa pd
Cheam (f-
FLB
Diver (..
115
044
142-
RM EM (UD)
OBAB
BAOAB
OBARO
Figure 9: Effective sample size (ESS) of the diffusion bridge sampler (DBS) for different integration schemes, multiple benchmark problems, and four seeds. Integration schemes include Guler-Mauuyamee (EM) for (OD) underdamped (UD) Langevin dynamics and various splitting schemes (OBAB, BAOAB, OBABO).
0.4
ESS
M
T
M
M
M
M
M
M
M
T
17
Figure 10: Effective sample size (ESS) of the underdamped diffusion bridge sampler (DBS) for various com- binations of learned parameters, averaged across multiple benchmark problems and four seeds with N discretization steps. Parameters include matrix M, diffusion matrix a, terminal time T, and extended prior distribution æ.
across all tasks in Figure 9 (ESS values) and Table 5 (log Z (LB) values). Overall, observe a notable improvement in performance with (symmetric) splitting schemes compared to Euler- Maruyama discretization. However, the number of discretization steps increases, the performance differences between OBAB, BAOAB, and OBABO become less pronounced. Interestingly. OBABO tends yield substantial performance gains when the number of discretization steps is small. Fur- thermore, we examine the impact of parameter learning for N = 8 discretization steps, with the results shown in Figure 10. Surprisingly, while learning either the terminal time T the parame- ters of the prior distribution yields modest improvements, learning both leads to a remarkable 5x performance increase.
Choice drift for DBS. The drift term f in the diffusion bridge sampler (DBS) can be freely chosen. To explore the impact of different drift choices, we conducted ablation study. We tested several options: no drift, V, log Pprior: V. log Purren and a geometric annealing path, represented by V. log where s(x,s) Pprior (1-210)) (2))) (x). We also tested using learned function for presented in Table 4 and Figure 11. The findings suggest that the most consistent performance is achieved when using the learned geometric annealing path as the drift f. Interestingly, using the score of the target distribution, Vz log Phargtt resulted in worse performance compared to no drift for overdamped DBS and only marginal improvements for underdamped DBS.
3. The results of these experiments
32

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wallebok [s010
0.6
-1-
138
0.4
0.2
0-
V. Valeg
loge (burnd))
Vi Aig
Fo Pe
loge (sarned)
overdamped
underdamped
Figure 11: Effective sample size (ESS) and wallclock time for various drifts J of the underdamped and overdamped diffusion bridge sampler, averaged across multiple benchmark problems and four seeds. Here, 1-8(()
((a) (z), where "learned" indicates that 8 is learned (end-to-end). v(o x Pyrior (x)
Table 4: Lower bounds on log 3 for different drift function f for DBS on various benchmark problems. Higher values indicate better performance. The best results are highlighted in bold. Here, M(z, s) oc 1==(0> ((+)
Perior (x)p (x), where "Icamed" indicates that 8 is learned (end-to-end). Blue shading indicates that the method uses underdamped Langevin dynamics.
Funnel
Credit
Seeds
Cancer
Beowraion
lonosphere
Mall
Senar
0.212:
8L.
111.778
38 o000
108 905..
-0.165
566.
778 sossce
-l. 807 s4.
-111.546
42 42/77
1000 110.715--a
V. log
-0.216
565 17Bessss
778 388.88
8t. st2
0.787
-111.44ls
12.772
108 503
0.912
V. Pog
-0. 18l2s
-085 85722
-75. 207
122 10tg
0.501
970kg one
N/A
100 163
0.09619.004 o04
20.82 565
70.445.0.ce
81. 25015.0
1.061i.ce
111 826.1
42 78222.300
109 264 623
V. y
0,183in
499006411435
73.442.0.com
83,8811.000
1000uacca
131,678.a
42
108 616.
-0.10eemee
127
-8.482244.0
1.106
-111,661
42. 436
V. log
-0.175a
566. 106
778. 33.458
75. 6.0.8.0
1.074
-131.678e
42 700
108 503
(lamad))
-0, 102ga
-885
112ga
-75 42/4222
77. 77.886:23
1.157
-111 6386
12 7033g
-108 454ggn
Table 5: Results for lower bounds log 2 for various benchmark problems, integration methods, and dis- cretization steps for DBS. Higher values indicate better performance. The best results are highlighted bold. Blue shading indicates that the method uses underdamped Langevin dynamics.
Integrator
Punnel of 100
Credit (dl - 20x
Seeds tsl 20
Cancer td
Brommian 6d 22) Ionoppere (sl 35) ManyWell (d) 500
Senar Gf
N
(M
-385, W0
-73.613en
-80.900gg 183
D. 0.196aa sta
-111 11118888 so.
(3:082es om
-100 01
EM (M())))
-0/72bsno
-100 Venuas
-13 222
-96.286
-3 L5bat
-123 12
22,02
-155.001 801
OBAB
-585, 165
773
79 167
0.90
837
OBABO
-895, 1790noes
733 580
8188
82b.a eea
035
EM (0D)
0.46 wuka
G85 (85.7922 Luuu
75,061 evea
81,025 1:34
108,002 woee
EM (Lib)
-1 Sddeena son
-187, 020enass
-73/782ennne
-83 (0M5 ums
0.421
-123
128
-137 801
OBAB
-885, 163 nuses
-73,520
-96.188...72
0.94
-111.75
108. suate
B4GMB
420
73.615
111.720.x
42.548
754
OBABO
585. awos
73.401.:
78.
18.404.1.00
111
FM (CR))
452gmvo
-585 911
-23 oer
-811007
8.89 aw
730g
-108 7M8
EM (UD)
0.425.0vxx
585.00 1es.
75,027 evea
86,307 ue.
111.9780 wo
42.302.1.0
100 100.37% soos
OBAB
585 126: ooa
73, 663
77 22.2212 mo
1.02 an
111 650
100 6122 on
RAQAНI
-0317
-585, 1i7on
-73. 003-163:2 (63
305
-111 677
61/661
106 6030
OBABO
048
42.729..u
601
507
FM (18D)
-305 201
-73 ton
-78-78774
0772
837
03/661
-100 1ilkaa
OBAB
-185. 115mnome
-73. 6il
-77 44
10822 so
111 d52 oo
100 aa
BAGAB
0.006 oua
185. 1885.115.en.
73,641 4om.
77.550
Lobliwee
111.6500
OBABO
16420-00s
548 113c2
77 131
77.
111 648
73000
108 501
128
-83.905-4.13
-111477
(UD)
0240.000
78.7141..777
111.750
0BAB
585
118:
18.055
1.11
131
13.7311100
BAOAB
-0.159--..
-305
112
-73.
-73.125.eve:
-17.87444000
Li11
-111.557 so
-108 457 c..04
OBABO
0.103
585.
0126:112::00
73.423
77.881isxu1
111.636 ox02
42.763.u.ss
118.
33


Reviews:
[{"review_id": "qb0cSlfmQy", "reviewer": "Reviewer_nE8S", "summary": "The authors explore training diffusion bridges with the following focus:\n- investigating the case where the reference diffusion process has degenerate noise\n- explores the under-damped Langevin case\n- numerical experiments", "questions": "See weaknesses\n\n> learn the control functions u and v such that the two processes defined in (5) and (6) are time reversals with respect to each other.\nQuotes from page 3.\n\nIf the marginals are fixed and preserved and the forward/ backware are time reversals of each other, then this would be the Schrodinger bridge?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "The authors explore underdamped diffusions for sampling and appear to achieve record performance in Table 1, Figure 3 etc. compared to the baselines considered for the datasets considered, including some medium dimensional data d>1000.\n\nThe authors explore a variety of known splitting based sampling schemes to improve performance.  This appears under-explored in the literature of diffusion models.\n\nThe authors claim their method enables end-to-end hyper-parameter tuning. This appears to be a novel and valuable contribution though I do not understand how this works and it does not appear well explained in the paper (as detailed in weaknesses below).", "weaknesses": "It is not clear to me what the contributions are in this submission nor the significance. \n\nIt appears most of the propositions/ theorems are from other works. The authors claim that the novelty in this work is extending these to the degenerate noise case. It is not clear to me why the degenerate noise case is useful or important (bullet 1 of contributions).\n\n>Even though for (underdamped) diffusion models with degenerate η, a corresponding (hybrid) score matching loss has been suggested by Dockhorn et al. (2021), it has so far been unclear whether it can also be connected to an ELBO, i.e., to likelihood optimization.\n\nLikelihood based training for underdamped diffusion models has indeed been investigated in Dockhorn et al. (2021), see e.g. sections\n\"B.3 CLD OBJECTIVE\"; \"F.1.2 MAXIMUM LIKELIHOOD TRAINING\".  (bullet 2 of contributions).\n\n> Novel underdamped samplers:\nSplitting based samplers  have been used before, as cited by the authors e.g. Monmarché · 2021. It is claimed by the authros (bullet 3 of contributions) that these samplers are novel. Is the novelty in using these for sampling with diffusions? I appreciate these splitting methods have been used before with damped diffusions and for diffusion models, but the full design space has not been explored.\n\nThe promise of end-to-end training is very attractive (bullet 4 of contributions). How is this achieved in practice? It appears this is only discussed in the numerical experiments yet is one of the main selling points of this work. Can the authors elaborate more on this? It appears from Figure 5 that learning hyper parameters is key to good performance.\n\nThe simulation based loss considered appears to be the same as that proposed in \"Guided proposals for simulating\nmulti-dimensional diffusion bridges\", [1] for forward KL eqn (9) and (11). If I understand correctly, this requires simulating the whole diffusion trajectory under some discretisation to learn the drift and has been considered by [1] for simple drift terms. The approach by [2] requires simulating in the nonlinear diffusion case up to time t and not the whole trajectory (simulation free for affine SDE); and uses a score matching loss (reverse KL) to train a neural diffusion bridge and then to sample diffusion bridges for a range of nonlinear reference diffusions.\n\nThe works [1,2] have not been discussed and appear quite relevant. \n\n[1] Guided proposals for simulating multi-dimensional diffusion bridges SCHAUER et al 2015 \\\n[2] Simulating Diffusion Bridges with Score Matching, Heng et al 2021", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "R1iCwgGNvW", "reviewer": "Reviewer_5Kic", "summary": "This paper generalizes the framework for learning diffusion bridges that transport prior to target distributions, as well as proposing a new method called underdamped diffusion bridges.", "questions": "See weakness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-organized. The explanation is clear and intuitive. Overall, it is easy to follow the logic and flow of the paper.\n2. The detailed proofs, experiment details, as well as the code, are explicitly provided.", "weaknesses": "1. The notation is a little confusing. Maybe you should provide an explanation about the notation as it shows up in the main text firstly.\n2. The theoretical benefits of underdamped diffusion bridges are lacked. Could you provide some explanations on its good performance on theory side?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4Z2KNrQ1T6", "reviewer": "Reviewer_Qe9N", "summary": "This work deals with the problem of sampling from a target distribution for which we can evaluate the unnormalized density, which is a common and general problem in statistics and machine learning.\nThe considered setting is a dynamic one, with the objective of learning a SDE which, over a finite time interval, produces terminal samples approximately distributed according to the target distribution.\n\nThe authors aim to advance the field by delivering the following contributions:\n1. extending and unifying previous frameworks, including diffusion processes whose diffusion coefficient can be zero across some dimensions / times\n2. propose the use of underdamped dynamics, on an extended space, for sampling\n3. propose the use of numerical integrators appropriate for the setting of point 2\n4. complement the theoretical developments with numerical experiments illustrating the practical relevance of 2. and 3.", "questions": "1. Is it correct to say that without loss of generality f(x, t) could be absorbed into u(x, t), and thus v(x, t), in the sense that (5) and (6) are learnt so that (6) is the time-reversal of (5) (and viceversa)? I found unclear why f(x, t) is needed at all in the framework considered in this work, given also that it complicates some derivations. Is it there only to connect with prior works?\n\n2. Does the divergence-based RN derivative formulation of Prop. A. 4. offer any advantage over the result of Proposition 2.3?\n\n3. Equation (24) is not the linear growth-condition, but the globally Lipschitz condition (uniformly in time), that is used to establish uniqueness of the (strong) SDE solution. The growth-condition is missing, and also some condition is required on the diffusion coefficient even when state-independent as here.\n\n4. A large number of conditions are reported at the beginning of pag 15., what condition implies what? Specifically, it is unclear to me how they suffices for:\n- Girsanov: which requires some integrability condition\n- Existance and uniqueness of solution to Fokker-Planck equation (PDE for marginal density evolution), which, from what I understand, is an assumption in [1], and justified later derivations of the same result not relying on that assumption, which at the time was very difficult to establish based on verifiable conditions (it is possible that more recent developments changed the situation, if so the authors can point this out)\n\n5. I did not understand in the proof of Lemma A.3. what yields the equation following \"First, we rewrite the problem by observing that\", in particular the last two terms (the two stochastic integrals, in different time directions)\n\n6. The authors could expand on Remark 3.3.: I quickly glanced at the references but did not spot the relevant parts.\nIs it the case that the drift of the process matching a terminal distribution living on a manifold remains bounded as the process approaches terminal time?\n\n7. What role do the distribution for the augmented space in (16) play? (On a side note: the construction on the enlarged space could be given more preeminence)\n\n[1] Anderson - Reverse-time diffusion equation models", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, with some limitations (see weakness section below), I found the reading enjoyable and the goal of providing an unified and clear framework is commendable.\n\nThis work covers considerable ground, briefly discussing the context where samples (instead of the density) are available for the target distribution, introducing numerical integrators appropriate for the considered setting, and carrying out numerical experiments to complement the theoretical framework.\n\nThe rating system do not allow me to be so granular, but I find this contribution to be closer to 7 than to 6.\nI am open to revise my rating in view of the future developments during the rebuttal.", "weaknesses": "> In this section, we lay the theoretical foundations for diffusion bridges with degenerate noise, extending the frameworks suggested in Richter & Berner (2024) and Vargas et al. (2024).\n\nPart of the contributions of this work consists of extending previous results to the case where the diffusion coefficient can be zero for some dimensions. I have two remarks on this point.\n\nFirstly, it is not completely clear what specific prior results actually require to be extended, in the sense that prior derivations leveraged results and / or proof techniques that would not hold in this newly considered setting.\n\nSecondly, it is a bit unclear to what level of rigorousness (specific points are raised in the following, see questions 3,4) are the results established.\n\nIn general, if I understood correctly, and with the exception of Proposition 2.5, the results are in line with the non-degenerate case (same form as prior results).\nSo a significant contribution would, on a solid theoretical level, establish the desired results in the more general setting here considered.\n\nAt this stage, it is unclear to this reviewer how significant the contribution 1. really is (in the sense: how non-trivial, and how rigorous).\n\n> Our proposed framework encompasses all these works as special cases (Tab. 2).\n\nOn a side, I find this claim not completely accurate, in the sense that the mentioned works often contains additional considerations / objectives not present in this reviewed paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uaHKTbnMMz", "reviewer": "Reviewer_xTDP", "summary": "The paper introduces underdamped controlled SDEs for sampling from an unnormalized target density. This approach can be seen as a generalization of earlier sampling algorithms that utilize over-damped dynamics or critically damped Langevin dynamics in diffusion generative modeling. The authors demonstrate that under-damped dynamics achieve superior performance in sampling tasks.", "questions": "To simulate Hamiltonian dynamics, the authors suggest approximation techniques such as OBAB, BAOAB, and OBABO. However, it seems that these approximations may introduce additional costs during training and inference. Could this potentially negate the advantages that the authors claim in formulating the task using underdamped dynamics, such as the capability to minimize the discretization step due to the presence of degenerate noise?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The paper is well-motivated and clearly written, with the derivations appearing to be correct. \n* Extending previous diffusion-based sampling algorithms into under-damped dynamic is novel. \n* The proposed algorithm demonstrates superior performance in sampling tasks compared to recent methods.", "weaknesses": "* I’m not an expert in Hamiltonian systems, but many of the paper’s derivations are heavily based on the theories developed by Vargas et al 2024 (https://arxiv.org/pdf/2307.01050). While the formulation of the sampling problem using under-damped dynamics demonstrates some performance improvement, I’m uncertain about the theoretical significance of this extension beyond its practical benefits. \n* Is there a substantial challenge in formulating the task with under-damped dynamics that has not been addressed in prior works? In my view, the primary difficulty lies in sampling from the underdamped dynamics; however, the approximation techniques have already been established in earlier research.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mqwNqiyNRf", "reviewer": "Reviewer_znkS", "summary": "This paper successfully integrates the theories of diffusion-based models spanning several existing studies, providing greater flexibility in sampling computational methods. As an application, the authors propose a new sampling method using underdamped Langevin dynamics and validate its effectiveness through numerical experiments.", "questions": "1. Under Lemma 2.4, it is stated that \"the ELBO in Lemma 2.4 does not depend on the target $\\tau$\". However, the current form of the ELBO is $\\mathbb{E}\\_{Z_0 \\sim \\tau}\\left[\\log \\tau\\left(Z_0\\right)\\right]-D_{\\mathrm{KL}}\\left(\\overrightarrow{\\mathbb{P}}^{v, \\tau} \\mid \\overleftarrow{\\mathbb{P}}^{u, \\pi} \\right)$, which depends on $\\tau$. Could you clarify this point?", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 2, "contribution": 3, "strengths": "### Originality\nThe idea of considering underdamped Langevin dynamics as a diffusion bridge with degenerate noise allows many existing methods related to diffusion-based sampling to be handled in a unified manner. This demonstrates the originality of the authors' idea.\n### Quality\nThe existing research is well-organized. Additionally, the effectiveness of the proposed method is robustly verified through ablation studies in the numerical experiments.\n### Clarity\nIn Figure 5, the effective hyperparameters are clearly indicated.\n### Significance\nThe finding that accelerated dynamics with efficient sampling can produce even better sampling methods when combined with score-matching techniques\" is considered a significant and fundamental result.", "weaknesses": "### Theory Part\nIn Sections 2 and 3, the theory and algorithms of diffusion bridges are constructed under degenerate noise. However, I believe that the non-trivial impact of the degeneracy on the theory is minimal. In this sense, the novelty of this theory may be somewhat limited.\n### Numerical Experiment\nThe effectiveness of the proposed method in sampling is well-validated. However, the algorithm for generative modeling mentioned in Subsection 2.1 lacks experimental validation, leaving the effectiveness of the proposed method in generative modeling unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)