Conditioning Diffusions Using Malliavin Calculus
Authors: Jakiw Pidstrigach
Elizabeth Louise Baker
Carles Domingo-Enrich
George Deligiannidis
Nikolas Nüsken
Venue: ICML 2025 poster (2025)
Keywords: Bridges
Optimal control
stochastic differential equations
diffusion processes
Malliavin calculus
conditioning
Primary area: probabilistic_methods->monte_carlo_and_sampling_methods
TL;DR: We propose a Malliavin-calculus-based framework for steering diffusions under singular rewards (e.g.\ diffusion bridges), enabling stable training and outperforming existing gradient-based methods that fail when the reward is non-differentiable.

Abstract: In generative modelling and stochastic optimal control, a central computational task is to modify a reference diffusion process to maximise a given terminal-time reward. 
Most existing methods require this reward to be differentiable, using gradients to steer the diffusion towards favourable outcomes. 
However, in many practical settings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero otherwise.
We introduce a novel framework, based on Malliavin calculus and centred around a generalisation of the Tweedie score formula to nonlinear stochastic differential equations, that enables the development of methods robust to such singularities.
This allows our approach to handle a broad range of applications, like diffusion bridges, or adding conditional controls to an already trained diffusion model.
We demonstrate that our approach offers stable and reliable training, outperforming existing techniques. 
As a byproduct, we also introduce a novel score matching objective. Our loss functions are formulated such that they could readily be extended to manifold-valued and infinite dimensional diffusions.

Conditioning Diffusions Using Malliavin Calculus
Jakiw Pidstrigach"
Elizabeth L. Baker Carles Domingo-Enrichi George Deligiannidis
Nikolas Nüsken *4
Abstract
In generative modelling and stochastic optimal control, central computational task is to modify a reference diffusion process to maximise a given terminal-time -iime reward. Most existing methods re- quire this reward to be differentiable, using gradi- ents to steer the diffusion towards favourable out- comes. However, in many practical setrings, like diffusion bridges, the reward is singular, taking an infinite value if the target is hit and zero other- wise. We introduce a novel framework, based on Malliavin calculus and centred around a generali- sation of the Tweedie score formula to nonlinear stochastic differential equations, that enables the development of methods robust to such singulari- ties. This allows approach handle broad range of applications, like diffusion bridges, or adding conditional controls to an already trained diffusion model. We demonstrate that our ap- proach offers stable and reliable training, outper- forming existing techniques. a byproduct, we also introduce a novel score matching objective. Our loss functions formulated such that they could readily be extended to manifold-valoe valued and infinite dimensional diffusions.
0.
0.0
03
-23
-23
-14
-11
(a)
(b)
Figure 1: The figure depicts particle trajectories in dou- ble well potential, with the background color indicating potential intensity. The potential has two metastable states at æ = 1 and x = -. In (a) we observe that under the diffusion dynamics, particles initialised at typically remain confined to their well, rarely crossing the barrier to 1. On the right, in (b), the diffusion is conditioned on the rare event of transitioning between the two metastable states.
and generative modelling (Dhariwal & Nichol, 2021; Ho & Salimans, 2021), including guidance for generative models (Zhang et al., 2023; Denker et al., 2024).
1. Introduction
Reference System. To demonstrate some of the core ideas, us consider a diffusion process of the form
Simulating conditioned diffusions is central computational task in many applications, ranging from molecular dynamics and physical chemistry (Dellago et al.. 2002; Bolhuis et al., 2002; Vanden-Eijnden et al., 2010) and genetics (Wang et al., 2011) 1) to finance, econometrics (Bladt & Sørensen, 2014; Elerian al., 2001; Durham Gallant, 2002), evolutionary biology (Arnaudon al., 2023; 2017: Baker et al., 2024),
dX, = b(X,) dt + dB,,
(1)
where the drift vector field b is given-either from learned generative model or from physically financially moti- vated considerations. Samples from (1) can be obtained by straightforward numerical simulation.
Guidance and Control. In many applications, desir- able to condition the reference process (1). For instance, in generative modeling, large pretrained model may not offer the necessary task-specific controls. One may then seek to impose such controls post hoc-foo example, to generate images matching a desired edge map or human pose, or to sample proteins with properties tailored to a particular application. Similarly, when the reference process mod- els weather dynamics, interest may lie in rare or extreme
Equual contribution 'Department Statistics, University Ox- ford, UK 'Department of Computer Science, University of Copen- hagen, Denmark 2 "Microsoft Research New England, Cambridge, USA 'Department of Mathematics, King's College, London, UK. Correspondence to: Jakiw Pidstrigach <mail@jakive jamail@jakiw.com.. Bliza- beth L. Baker <eloba@ dtt.
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Conditioning Diffusions Using Malliavin Calculus
scenarios rather than typical ones.
valid provided f and g vanish sufficiently fast at infinity, of- fers blueprint for handling singularities: the left-hand side can be given rigorous meaning even if g lacks differentia- bility, as long as f is smooth. Numerically, large (exploding) gradients can be avoided by shifting differentiation onto the factor with better properties.
These objectives can be for- malised within the framework of Bayesian inverse problems (Stuart, 2010). To this end, let G be an observation operator and Y the corresponding observation at terminal time:
Bayesian Inverse Problems.
Y ((T)).
Outline and Contributions. Guided by (5), we develop loss functions L(u) whose unique minimisers uz(2; y) im- plement the conditioning of (1) via the controlled diffusion in (3), for any condition Y y at once. Towards this goal,
The goal is to sample (Xt)oSigT given yy=- y. This setting encompasses all previous examples; for instance, G may extract edge or pose information from an image. Our for- mulation allows G be a full or partial (potentially noisy) observation of Xr. Note that G may be stochastie; for example, G(Xr) = XT + E, where & is random variable.
we recall the well-known (Eberle, 2015; Denker et al., 2024; Shi et al., 2024; Du et al., 2024) connection of diffusion bridges to Doob's h-traasform in Section 1, in particular the relevance of conditional score functions,
Stochastic Optimal Control and Conditioning Diffusiios, In order to condition on Y, we add a control term uy to the reference process (I):
dX, = b(x,) dt + u. (X,) dt + dB,,
(3)
lift (5) to integration by parts on the space of trajecto- ries; the role of the Lebesgue measure dr is replaced by the law of (Xe)oSIST, and the ordinary derivative 02 is replaced by the Malliavin derivative (Nualart, 2006). Based on this, we derive novel formula for conditional scores that generalises Tweedie's formula (Efron, 2011) for denoising score matching (Vincent, 2011),
The optimal control will have the property that each sample (X1)oStST from (3) is sample from the distribution of (1), conditioned on Y y-
Likelihoods and Rewards with Gradients. The function G induces a likelihood or reward
g(,;y) == p(Y = y =
(4)
Current methods, framed in terms of stochastic optimal con- trol (Domingo-Enrice Enrich, 2024; Zhang & Chen, 2022; Berner et al., 2024), critically rely on gradient information Vg or V Vogg g to guide the process and learn Uy-
discuss implementation details and showcase numeri- cal performance in Section 4.
Furthermore, we recover existing methods for stochastic optimal control from our framework, see Section 3,2.
Singular Rewards. In many settings there is no gradient information available for g- One case where this happens is if G is an indicator function or discontinuous, as would be the case for classification. But even if G is smooth but deterministic, the induced likelihood is often singular if one does not add artificial noise to the observations. Even in the seemingly straightforward case of G = Id, the reward would be given by a Dirac delta distribution g( = (ZT). which is not even continuous. The case of G Id conditions diffusions to end at specific state and is known under the term disfusion bridges (see Figure I(a)). Due to g being a Dirac it is in some sense the most challenging setting and will be a guiding problem for us in the development of the methodology.
2. Theoretical Background and Main Result
Theorem 2.1 below serves as the mathematical foundation of our methodology. Before presenting it, we introduce the necessary notation and assumptions.
Throughout, we consider diffusion processes of the form
dX, = be(( dt + (X,) dB,
Xo = Io. (6)
where 6: 10, R. + R is a smooth drift of at most linear growth, 2u R" is fixed initial state, B, is standard Brownian motion, and a [0, T] x R - Bann specifies the volatility, assumed to be symmetric, strictly positive definite and bounded, with bounded inverse. A key component is the Jacobian Jı,s associated to (6), which is a matrix-values stochastic process satisfying
Integration by Parts on Path Space. In this paper, we circumvent these issues altogether by constructing numeri- cal methods that remain unaffected by singularities in the likelihood g. The key idea is best introduced via an analogy: replace, for the moment, the trajectory space associated with (1) by the real line. The classical cinggration-by-part identity
dJe. Vb, (X.) dt Vo, dB, (7)
with initial condition J/1s = 1d, for fixed s E [0, In- tuitively, Jej. measures the sensitivity of (6) at time with respect to a small perturbation at an earlier time s < t. More precisely, represents the derivative process, i.e.
fl) =- g(x)0,f(x) dr, (5)
2

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Conditioning Diffusions Using Malliavin Calculus
Jej. Vx, X (Williams & Rogers, 1979, Chapter V.13). The process Jtjs plays central role in adjoint methods for gradient computation (Li et al., 2020) and stochastic optimal control Gommingo-Enriche et al., 2024b), and we highlight the fact that simulating the full matrix-valued evolution is typically unnecessary as only matrix- vector products are re- quired (see Appendix A). Our main theoretical contribution can now be stated as follows.
are gradient free. On the technical technical level this corre- sponds to the conditional expectation being the minimiser of the L loss, see (17).
In the remainder of this section we prove Theorem 2.1 and illustrate its connection to integration by parts, as hinted at in (5). Implementation details are deferred to Section 4.
Algorithm 1 BBEL Training Step
Theorem 2.1. Let a: : [0. -+ Fax= be a matrix-valued
Require: a [0, 1] - R initial condition zo. batch size N, current drift approximation u², time grid {to, tt..t.t..
differentiable function such that ATI. oer 0gg is invert- ible for all E (0, Define the score process
S, 5 - I of dB,
1: for i = 1 to N do
(8)
2: Sample sample path X with corresponding Brown- ian motion path B from the SDE (6).
as well as the loss functional
3: Sample an observation Y = G(T)) from (2).
4: Compute the Monte Carlo estimator for S, using (8) (for details see Algorithm 2) along the path (X, B).
!'- [lu((Xi;Y) - S.lll ds
C(w) EE
(9)
Calculate the single-path loss
where the expectation is taken with respect to the injected noise (Be) driving (6), (7) and (8), as well as poten- tial noise in G in (2). Then L admits a unique minimiser u², and for any y E Rd. the law of
M Σ
4())
(x,
6: end for Sum for the full-batch loss
dx2,- = (x() dt + (X())) (X5.)) dt + +x,(X,) dB.
(10)
N
C (0) ((0).
coincides with the conditional law of (6), given Y y-
Based on Theorem 2.1, we propose (i) estimate the pectation in (9) using Monte Carlo, (ii) parameterise the drift (including the conditioning) by a neural network, and (iii) learn the parameters of ue through gradient-descent type updates. We have formalised the resulting training procedure in Algorithm 1, and, given the close relationship (BEL)
8: Take a gradient step on L (0) with your favourite opti- miser.
2.1. Doob's h-transform and Conditional Scores
between Theorem 2.1 and the Bismuth-Elworthy formula from Malliavin calculus (see Section 2.2), refer to these methods as BEL-algorithms. Importantly, the score process (8) can be simulated efficiently by directly updating J.T (X,,) T)-1 dBB along (7), without simulating the full dynamics of J (see Appendix A for details). The main the matrix-valued fune- tion [0, T] - Ra based on variance and stability considerations we give guidance on its choice in Section 3. Furthermore, specific choices of allow us to connect existing methods to the framework of Theorem 2.1; those are explained in Sections 3.2.1 and 3.2.2.
As a first step towards Theorem 2.1, we recall the fact that the optimal drift in (10) can be expressed in terms of condi- tional scores (Rogers & Williams, 2000, p. 83);
Proposition 2.3 (Doob's h--ransform).. For E Rd, let
ui (z; v) = ox(x)), (z) T V. logg(Y = = / X, = x):
(11)
hyperparameter in Algorithm
where p(Y = y X, 2)) is the probability Yf y given
X, x. Then the corresponding controlled diffusiion (10) reproduces the conditional law of (6), given Y == y.
Similar closed-form formulations are available in the context of optimal control (Nüsken & Richter, 2021, Section 2.2) and time reversals (Boffi & Vanden-Eijnnen, 2024; Song et al., 2021b; Ho et al., 2020). Note that for t T, the conditional score coincides with the gradient of the log- reward, V, log p(Y = y Xx x) = V. log g(2; y). As a consequence, if g is singular, (11) becomes singular (and possibly numerically unstable) as t - T. We deal with this issue in the next subsection.
Remark 2.2 (Amortisation). After training according to Algorithm 1, the learned control un uz(:; y) performs condi- tioning of (1) for arbitrary y. without the need for retraining. On a related note, the constructions in (8) and (9) as well as in Algorithm 1 depend on the observation operator G only implicitly through the samples Y = G(Xr) and do not access or require any particular structure; in particular, they
3

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Conditioning Diffusions Using Malliavin Calculus
2.2. Malliavin Calculus and Integration by Parts
place, the analogue of (5) on Wiener space becomes
While Proposition 2.3 in principle identifies the desired control vector field, the right-hand side of (11) will be un- available in all but simple toy examples. The following result provides formula that is amenable to Monte Carlo simulation:
[[/
9 S. f dB;
E
Dig ft
E
(15)
for appropriate choices of g and the stochastic process fe- Using (13), we write
Proposition 2.4 (Generalised Tweedie formula). The con- ditional score is given by the conditional expectation of the score process,
V z logrr 1x,=))= V.Bliar (XT) X, x] (ri(Xx= x,=x)
and, using the chain rule, we further obtain
Ve log p(Y = X x)
(12)
= =E St X, = 2, x, yl.
V V2 (XT) X =r]
= = my dgr (XT) Xx,
for all t [0, T) and x E RR E Rdd
Remark 2.5. In the case when (6) is linear, Proposition 2.4 simplifies to the celebrated Tweedie formula (Efron, 2011), given, e.g., by
Here, Vx Jr gradient Vx, 8gy is understood in the sense of distributions (Duistermaat et al., 2010, Chapter 4). To complete the proof, it remains to eliminate the derivative on 6zz via the inte- gration by parts formula (15)-the the left-hand side of (15) precisely explains the appearance of the stochastic integral in the score process (8). For this, we need to convert the conventional gradient Vzr into the Malliavin derivative Dr. This is achieved using the matrix-valued function a, as detailed in Appendix D.1. In fact, there are infinitely many ways to carry out this conversion, each determined by a particular choice of a. We note that the proof strat- egy, particularly the conversion of Vær into Dr and the use of integration by parts, closely resembles the proof of the Floortyyl (BEL) formula from Malliavin calcu- lus (Bismut, 1984; Elworthy & Li, 1994), which inspired the name of Algorithm 1.
as explained Section 2, and the
V Ver p = =T) = (E[X, X7 x7] - TT)-
Similarly, the denoising score matching objective for dif- fusion models (Song et al., 2021a) can be seen as an in- stance of (9) for a specific choice of a. Consequently, (12) is a generalisation nonlinear SDEs, also allowing us to derive novel regression targets for diffusion models; see Appendix B.
Proof sketch (Proposition 2.4). See Appendix D.1 for full details. Without loss of generality, we may consider the dif- fusion bridge setting (i.e., G(Xr) = Xr), see Lemma D.2. To show (12), we rely on two main ideas:
Firstly, to express the transition probability in terms of expectation, we use the fact that whenever a random vari- able X admits a smooth Lebesgue density px, we have the representation
2.3. Amortised Bridges and Forward-KL.
While Propositions 2.3 and 2.4 together provide an expres- sion for the desired control vector field ut. the left-hand side of (12) requires access to the targeted distribution of (6), conditioned on X, = x and Y = y; the construction might thus appear circular. Fortunately, taking expectations over X, and Y in (12) not only recovers the straightforwarddto- simulate reference diffusion (6) as a mixture of its bridges, but also allows us to amortise the learning procedure, infer- ring all the bridges simultaneously.
Px (x) = E[0,
(13)
see Duistermaat et al. (2010) for a rigorous general account or Watanabe (1987, Section 2.1) for the statement in the context of Malliavin calculus.
Secondly, as outlined in the introduction, we elevate the antegration-by-part formula (5) to Wiener space, i.e., to the space of sample paths of Brownian motion (Bo)ssgg- To this end, we introduce the Malliavin derivative D., which represents differentiation with respect to the infinitesimal noise increment d B,: for a functional g that depends on the realisation of (Bi)usigT, the Malliavin derivative is given by
The following proof of our main result reflects this idea:
Proof of Theorem 2.1. The minimiser of the iqaat-squares "local-in-time" loss
Clocal (w) = E [X,, Y) -s
(16)
0g Dig adB."
(14)
is given by the conditional expectation of S with respect X, and Y, i.e.
A rigorous treatment of (14) requires the framework of calculus on Wiener space; see Nualart (2006). With (14) in
(x, = E(S, X, = x, Y = w).
(17)
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Conditioning Diffusions Using Malliavin Calculus
Combining Proposition 2.3 and Proposition 2.4, this implies that the minimiser in is equal to the optimal control term. By exchanging the expectation with the integral in (9), we get that ue is equal to the optimal control V x, poy Y X2), for each & E [0,7].
1.5
1.0
8
Remark 2.6 (KI. interpretation). The time-integrated lo- cal loss can be interpreted terms of the amortized KI- divergence between measures on path space.
0.0
0.0
0.2
0.4
0.6
0.8
1.0
$
S. Clocul(u)) dt E (IP PP)] +
(18)
Figure 2: The variance-optimal weighting o', (see Lemma 3.1) for the simplified setting in which X₃ is conditioned Brownian motion.
with a constant C > 0 that does not depend the expectation is taken with respect to Y and 1 refers to the distribution of trajectories induced by the diffusion conditioned on Y = The measure IP" refers to the distri- bution associated to the current control u. The forward-Ko -L divergence in (18) mode-covering (Naesseth et al., 2020), which is desirable property for conditional generation and rare event simulation. Furthermore, the KT-rrepresentation (18) allows to connect the BEL-framework from Algo- rithm to the previous works by Heng et al. (2022) and Baker et al. (2025). However, the former requires addi- tional numerical overhead in learning backward processes, whereas the latter relies on additional simulations for a Feynman- Kac type construction. The accuracy of these approaches is evaluated experimentally in Section 4. Fi- approach from
Here,
Since the algorithm works by generating independent sam- plcs from S, and then regressing against those, a lower variance of S, is expected to lower the variance of the gra- dients. In the following result we compute and optimise the variance in simplified setting, providing guidance for the choice of a::
Lemma 3.1. Set = =, 1, 0 and a = 1, ie.. X₂ is a onedimmesioonal Brownian motion conditioned on X, = xn and X1 = I1. Then the variance of the Monte- Carlo estimator
V logPno(B1 = Bo zo) 2 S "
nally, we remark that (18) distinguishes the work by Du et al. (2024), who use forward-KL, to- gether with subsequent (Gaussian) approximations. For background on the "measures on path space" perspective, see Nüsken & Richter (2021), and for a proof of (18) see Appendix D.2.
is given by
01 ao (
(a') (- (a1 (a1 oz)2 (01=
dt dz). (19) (1 (1-13 1)2
assuming that 91-a= + 0 s - and letting d 3 11118 No- v/1- The variance is minimised for the choice
3. On the Choice of a
-1 (1-((1)))(1++55)
a = (1-
(20)
Theorem 2.1 provides whole class of loss functions for dif- fusion bridge simulation: each choice of a yields a slightly different approach to approximating the drift (11), provid- ing considerable flexibility. We now exploit this flexibility for two purposes: in Section 3.1, we derive an optimal choice of under simplified setting, and in Section 3.2, connect our bridge losses to existing algorithms used in stochastic optimal control and diffusion bridge simulation.
The proof of Lemma 3.1 can be found in Appendix E, and Figure shows the derivative of of the optimal variance- reducing choice in (20). Interpreted as weights in the score process (8), we that (20) weights the initial increments of the Brownian motion relatively highly in comparison to increments closer to the terminal time. The choice of a will further be explored in Section 4.
3.1. Optimal for Reduced Variance
3.2. Connections to other Algorithms
Each choice of in Theorem 2.1 implies different regres- sion target for the neural network. While all of these targets have the same mean, namely the diffusion bridge drift in (11), they differ in variance:
3.2.1. 1. REPARAMETRISATION
Inspired by the reparametrisation trick in variational infer- ence (Kingma et al., 2019, Section 2.4), Domingo-Enin Ennich et al. (2024b) derived a novel methodology for stochastic optimal control. A variant of it id-adepeed to our setting can be recovered from Theorem 2.1 by a specific choice of
ATie J."
Var(S.) = Var
5

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Conditioning Diffusions Using Malliavin Calculus
Lemma 3.2. Let M [0, T] + Raxn be a matrix-valued differentiable finction such that Mo Td and Mr O. Then the score process from (8) has the equivalent represen- tation
regression. In Heng al. (2022), the authors applied similar Gaussian approximation the time-reversal of diffusion to learn bridges of the time-reversed process.
However, this approximation is performed in the density domain P X2), and its impact on the accuracy of V log Pt+ (Xe4)e(Xx 8t Xx) is unclear. By the generalised Tweedie formula (see Proposition 2.4), we obtain the ex- plicit relation:
S. S. f. (M.VOT (X) dB,. (21)
Proof. The correspondence between (8) is via the choice ag = J4. Mc; see Appendix D.3.
V log P1+ (X Xx,) ESS, Xx, Xe+öol:
(25)
The significance of Lemma 3.2 is that (i) it extends the reparameterisation trick to the singular reward setting and (ii) it gives conceptual insights into the choices of a and Mj in the respective methods. For further intuition into the role of M, we refer to Domingo-Enrich et al. (2024b) and Appendix D.3.
where S, is defined in (8) with terminal time T = t + St. The Gaussian approximation can now be directly related to a discretisation scheme for S, in (25). We prove this in the case of = 1:
Lemma 3.4. Approximating p(X | Xi) by Gaussian (24a) and regressing against its score is equivalent to choos- ing a² = 16, and approximating the stochastic integral in S. (8) as
3.2.2. GAUSSIAN APPROXIMATIONS
Another method to learn V logp(Y y X, = xz), identi- fied as the conditional drift in Proposition 2.4, is to approxi- mate the transition densities as Gaussian and regress against their score. We now derive methods based on Gaussian ap- proximation and show how they can be reformulated as BEL algorithms for specific choices of a. Moreover, the BEL algorithm provides deeper insight into the approximation error introduced by the Gaussian assumption.
L+8
Ja = Je+51je - Bt),
(26)
and furthermore approximating J2t81e by an Euler- Maryuama step on (7):
Je+54u Re ld (d
(27)
The proofs for this section can be found in Appendix D.4.
Rather than regressing directly against the target, we first observe that one can regress against the optimal drift for
3.3. Summary of Choices
small timestep:
Based on the above discussion we will now summarise some choices for the function 0e in Theorem 2.1. Each choice leads to a different loss function, and we compare the different choices empirically in Section 4.
Lemma 3.3. For any 2
CGA (u, s,t)
(22)
= =X(llu(X,s, Vx. (X x.)]
BEL optimal: The first choice is setting ag in Lemma 3.1. Although this only gives optimal variance in the specific case of Brownian motion, it still may do well in other problem settings, especially those similar to Brownian motions.
is minimised by the diffusion bridge term uz (Ts: v), see (11).
Thus, optimising CGA for each s yields the diffusion bridge drift. To achieve this, one must select a 2 for each After discretising the time domain into {lo, 41 = to + St.....ty tx = to + N8t). suitable loss function is:
a2 such that o² [ss)+ In this case, ar 0, As. This means are only using local information to approximate the score.
BEL first: Based on Lemma 3.3
N-1
BEL average: Another choice is setting a, = This leads to the traditional Bismut-Elwortd formula (Bismut, 1984; Elworthy & Li, 1994).
CGA = Σ
CGA (u, t, + 6t).
(23)
For small 8t, the transition density can be approximated by Gaussian, for example, via Euler-Maruyama step:
BEL last: This algorithm uses a' 1r--a This is similar to the stochastic optimal control setting, where the gradient of the target function is propagated back through the whole trajectory.
p(X++51 X₂) = N (X, + St by (X,), ) 5t a (X))), (24a)
a. (X.) = or (x, (X.)
(24b)
Reparametrisation: Corresponding to the discussion on the reparametrisation trick in Lemma 3.2, we set ag = Jaj with M, = T-s ld.
This
provides an
explicit expression for Xt), which can be used for
Xx, log (x+55ee(Xeess (Xesst
6

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Conditioning Diffusions Using Malliavin Calculus
2.0
1.0
10
50
00
-10
-10
-15
50 18 -20
-20
(a)
(b)
(b)
Figure 3: In (a), we plot the double-well potential (31) for In (b), we sample 1,000 paths from the uncondi- tioned SDE (30) and observe that they remain confined to single potential minimum, failing transition between them. This highlights the inherent difficulty of the problem.
1.5
1.5
65
0..
-10
-15
50 88 -20
4. Experiments
0.4
(c)
(d)
In this section, do empirical evaluation of our meth- ods. Full experimental details are provided in Appendix C. In Section 4.1 and Section 4.2 we study the case of diffusion bridges, i.e. Y Xy,, or a Dirac-delta reward function g, since this can be seen as the most challenging setup: In Section 4.1, we conduct experiments where the true transi- tion densities are available for evaluation. In Section 4.2, demonstrate our method on bridges of stochastic shape processes, which have applications in biology. We also com- pare to related methods (Heng et al., 2022) and (Baker et al., 2025) and show favourable results. In Section 4.3 we apply our methodology to diffusion models or equivalently, flow matching algorithms.
Figure 4: Panel (a) shows ground truth paths from (30), conditioned on transitioning from state 1 to state -1. Panels (b), (c) and (d) present the paths generated by the BEL-first, BEL-llst and BEL-oottial respectively.
the first coordinate, we maintain an approximately constant event probability whilst varying the problem dimensionality.
4. 1.1.2 BROWNIAN MOTION
The first experiment we consider involves conditioning Brownian motion:
4.1. Controlled Environment Experiments
dX, dB,.
(28)
In this subsection, we introduce two experiments where the true diffusion bridge drift can be computed, allowing us to evaluate our methods against the ground truth. We explain the experiment setup in Appendix C.1.I.
For this SDE, the diffusion bridge drift (11) has closed- form solution given by:
4.1.1. METHODOLOGY
loggij((X: = X,= xx) 1
(29)
For the experiments in Section 4.1.2 and Section 4.1.3, we consider one-dimensional SDEs, simulated indepen- dently across all dimensions. Trajectories are conditioned to start at finit = (1, 1,...,1) 1) E RD and end at yinnll = (-1, I-1,,,...,11 1) RD, modifying the first coordinate.
This allows us to compare our methods against the true bridges. The results are presented Table 4. We observe that the reparametrisation trick algorithm performs well for the one-dimensional Brownian motion, while the BEL average achieves the best performance in 10 dimensions. For an explanation of the metrics see Appendix C.1.2
This setup isolates the effect of the event from the di- mensionality of the state space. Conditioning all coordinates would cause the probability of the transition event wvent-being the product of independent dieedimnsio transition prob- abilities to scale as p". where is the probability of the event one dimension. contrast, by constraining only
4.1.3. DOUBLE- WELL
In this experiment, we consider the double-wel problem as described by Nüsken Richter (2021). This model is given
7

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Conditioning Diffusions Using Malliavin Calculus
by the SDE:
dx, VU,(X,)dI + d (x) = -
(30)
(31)
For = 5, we visualize the potential in Figure 3(a). The potential exhibits two minima correspond metastable states. Since the drift term in (30) drives trajectories toward the minima of Us, transitions between = -1 and z = are rare events, with their probability decreasing exponentially as the potential barrier height increases (Kramers, 1940; Berglund, 2013). We illustrate this by plotting 1000 sample paths the uncon- ditioned process in Figure 3(b). none of which crosses the barrier.
-11
1 and x I, which
-11
-15
-2
-13
(a)
(b)
To obtain ground truth for this example, we numerically estimate f(x) = (X] = -1 X, I) and compute its logarithmic gradient. We then compare the paths of the process under the true drift with our estimates Figure 4. higher-
-10
-11
-15
-30
-55
50
85
12
15 -35
-10
-23
02
03
10 35
(c)
(d)
Additionally, we extend the experiment dimensional settings described in Section 4.1.1. The first and second marginals of 10 dimensional process are shown in Figure 5.
Figure 5: The first and second coordinates of a 10- dimensional process sampled from the SDE in (30) are shown in (a). The process is conditioned on transitioning from 1 to -1 in the first coordinate while remaining at 1 in all others. Panels (b), (c), and (d) depict the paths gen- erated by the BEL-first, BEL-average, and BEL-optimal a algorithms, respectively. All plots inclucde underlying con- tour lines representing the level sets of the potential.
Finally, we quantitatively compare the performance of differ- ent algorithms in Table 7. We observe that BEL Last and the Reparameterisation Trick exhibit the lowest performance, aligning with the empirical experience of the authors. For explanation of the metries, see Appendix C.1.2. In Figure also provide plots for the reparametrisstion trick.
4.2. Shape Processes
that nearby points remain highly correlated. This property makes learning bridges particularly challenging. An ex- ample trajectory of the unconditioned process is shown in Appendix C.2.
We demonstrate methodology on stochastic shape pro- cesses (Araaudn et al., 2023), which used in computa- tional anatomy to model morphological changes in human organs due to disease (Arnaudon et al., 2017). In evolu- tionary biology, they help analyse morphometric changes in species, such as how butterfly wing shapes evolve along phylogenetic trees (Baker al., 2024).
We apply our method, BEL average (i.e. f), to learn bridges of the shape SDE, conditioning the process on a cir- cle of radius 1.5, We compare our approach to the methods of Heng et al. (2022) and Baker al. (2025). The latter uses "adjoint" processes derived from Feynman- Kac represen- tation for the conditional expectation in (11), while Heng et al. (2022) estimate the time-reversed bridge process by regressing against the time-reversed diffusion bridge drift for small time steps, following Lemma 3.3. However, due to the time-reversal their drift term involves the gradient (X,, = It. X, = xx), where differentiation taken with respect the final time point rather than the initial one.
Following the setup in (Sommer et al., 2021), consider shape represented by zo 22N where N points discretise a two-dimensional shape. Let [ C RR² be equidistant points, and (B/ 1 be independent Brownian motions in R The SDE governing the shape evolution is given by
}
M
dxf
Σ
- dB),
1sis N,
(32)
1=1
2
Our findings are given in Table 1 and show that BEL average outperforms competing methods. We plot trajectory from the conditioned process learned via BEL average, alongside trajectory from the unconditioned process in Appendix C.2. The next best method is Heng al. (2022), which can be
k(z, v)
(33)
8
where la is a Gaussian kernel with parameters K. 3 E R. For each the map Zo + 2, is a diffeomorphism, ensuring
8

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Conditioning Diffusions Using Malliavin Calculus
Method
Dist
BEL average 0.085
Time reversal 0.090
Adjoint paths 0.498
Untrained 1.396
Table I: A comparison between different methods for learn ing bridges for shape processes. We see that our proposed method BEL average outperforms other existing methods.
(a)
Figure 6: In panel (a), the top-left image shows the ground truth used for conditioning. The remaining 15 images are samples generated by the diffusion model conditioned on the upper-left quarter of the ground truth image. Panel (b) displays only the conditioning inputs: again, the top- left image is the ground truth, while the others show the corresponding conditioned quarters used for generation.
interpreted as first applying BEL to the time-reversed pro- cess, approximating the transition densities via Gaussians ((24a)), and using fixed start point instead of amortisation. However, since their approach involves time-reversal, they do not take gradients of the drift or diffusion terms. BEL av- erage not only uses this gradient information, but non-local information about the score.
4.3. Image Experiments
We train a diffusion model using a flow matching loss with a U-Net architecture on the MNISS dataset (Xiao et al., 2017).
Remark 4.1. The model is trained deterministically using the flow matching loss. We then apply the memoryless schedule from Domingo-Enrien Enrich et al. (2024a) to reinterpret the trained model an equivalent SDE (or diffusion model).
5. Conclusion
In this work, we introduced novel class of loss functions to estimate the control of conditioned diffusion process. When applied to diffusion models, we discover novel meth- ods to add controls to an already trained diffusion model. As a byproduct, we also find novel denoising score matching objectives, which can be used for time reversal of SDEs, or to train a diffusion model. Our approach leverages Malliavin calculus and integration by parts to handle singular losses, enabling more stable and accurate estimation. Thanks to its generality, the framework accommodates a wide range of conditioning cypes-continuous, discrete, noisy, or noise- less-and and could be extended to manifold-valuee and settings in future works.
Next, condition the resulting SDE to produce images with specified lefer-left corner. Importantly, this condi- tioning does not require adding artificial noise to the obser- vation. More specifically, the law of XT is by design the data distribution. We condition on Y = ((Xr) where G selects the upper left corner of the image. Once trained we can therefore choose an arbitrary upper left corner y and sample from images matching this, using the learned addi- tional drift u u(X,, y). Note only need train once and then can sample using any corner. We demonstrate the result by conditioning on the upper-left corner of a jacket image in Figure 6.
A Acknowledgments
Remark 4.2. When both the forward and reverse dynamics of an SDE are known, one can construct bridges, as shown in Heng et al. (2022). Diffusion models form a special case of this setting: not only are both directions available, but the reverse dynamics are particularly easy simulate. This property has been leveraged in Denker et al. (2024) to simplify the methodology of Heng et al. (2022). However, both approaches implicitly assume that the learned score is exact. In contrast, our method makes no such assumption. Since we do not require access to the reverse dynamics, we directly learn the correct conditioning for the learned score-eee even when it is inexact.
EB is supported by the Novo Nordisk Foundation grant NNFI 8OC00500000 a research grant (VII40582) from VII. LUM FONDEN, and UCPH Data+ Strategy 2023 funds for interdisciplinary research. GD and JP were supported by the Engineering and Physical Sciences Research Council [grant number EP/YO18273/1].
Impact Statement
This work has connections to diffusion models, and could used in that setting. As such, it shares the societal and ethical implications of generative modelling.

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Conditioning Diffusions Using Malliavin Calculus
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Conditioning Diffusions Using Malliavin Calculus
A. Adjoint SDE for Calculating S
Algorithm 2 Adjoint SDE Method for Calculating S²
Require: Simulation path {Xt), Brownian increments {6Br), where E (0, ..;T. T).
Initialise ST
2: for tort= T 5t to 0 (backwards in steps of öt) do
3: Calculate Dj via (36).
4: Update S, S1+8t + Dr.
5: end for
6: Compute score process S, for s E {0, 8t,..,...,777 T via (37).
We employ an adjoint stochastic differential equation (SDE) method, inspired by adjoint ordinary differential equations (ODEs) (Kidger, 2022; Pontryagin, 2018; Chen et al., 2018), to compute Sg. Specifically, we define the auxiliary variable S, as:
8, : f. (TT() dB,
(34)
which corresponds to the score process equation (8) when = disregarding the normalisation factor We assume access to the following data obtained from discretised Euler-Maruyama simulation of the SDE (6):
Xo, X Xx: The states of the discretised SDE.
SB, . B oo: The Brownian increments used in the simulation, where X = X. (o
Note that ST 0. We can recursively compute S, from S.48t using:
S. f J.
(X.) (x) 144501.54a + fatth JeJo (
(X.)
making use of the semigroup property Jejs J1lot6edo+881e Approximating the integral term, obtain
S, =c J S.t5t + J. (a,)) (X.)8B,.
(35)
where 7 can be any number in J E [s, + 5t). For T = we have Japs Id. The term Js+81Jo can be approximated using an Euler-Maryuama step on (7), leading to Jot5 2 Id + SlV,, (X,) + (X.(X.)....
Remark A.I. Here, the term (X,, 5B, is meant the derivative with respect of the map
x x RER + oo()) e R".
This allows to compute the difference term:
D. 5,- - SS+st = 1.-0 2--5t
JT.(o((())))- dB, = (8tVb(X.) To...))5B.)....a (x.))'(X,) (36)
Crucially, note that these Jacobian-vector products are efficiently computed using reverse-mode autodifferentiation, avoiding explicit Jacobian computation.
The algorithm proceeds by initialising S, = 0 and iteratively computing S, backward in time for t E (T - ot....,88 8t,0 0}. Finally, S, is obtained as
T St
S, R A
Σ
afDr
(37)
restoring the normalisation that was dropped in (34).
13

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Conditioning Diffusions Using Malliavin Calculus
B. Denoising Score Matching and Tweedies Formula
Here we show how our results can applied to derive a formula for the score log diffusion process. This could be done by applying Proposition 2.4 a trivial conditioning Y and treating the reverse dynamics of the SDE (6). However, is instructive rederive the analogous equation in this simplified setting, which we do in this section. We will that it generalises Tweedie's 's formula.
Assume we have an SDE
dx, = bx dt + oz(X))
(38)
and denote by Pr the density of X,, Then we have the following representation and score matching objective: Lemma B.1. The score can be represented as
V Vlog E [% (os (ox( dB, X1
Here a', is function which satisfies So d' ds = 1, and Jıja is the Jacobian which follows the flow (7).
In particular, for any such a, the loss
C(u)= = E [lue(X,) - / (Ta d.d..ll
(39)
has unique minimiser given by ux(x) V V.
Remark B.2. The score matching loss (39) is generalisation of (42) below to nonlinear SDEs. Note that in contrast to (42), the construction (39) straightforwardly generalises to Riemannian manifolds by replacing the squared norm and the Brownian motion by their Riemannian counterparts (Hsu, 2002), and noting that Jels maps between the tangent spaces Tx, .M and Tx.M in such a way that the stochastic integral is well defined. The loss (39) also generalises to infinite dimensions, replacing the squared norm by a Hilbert space norm and the Brownian motion by an appropriate Wiener process. The relationship between the Wiener process and the chosen norm however more intricate. In the diffusion model case (the linear case), they have been worked out in Pidstrigach al. (2024, Section 2.4, Section 6).
Proof. By the chain rule for Malliavin derivatives can write
(x,(( Vx((x))),
where
J/ Vx
the Jacobian. This implies that
Vi((X,) (x,)
4/s
assuming that o, is invertible. As this relationship holds for all s we can integrate over & to get
Tx(())= D J a' ds
making use of the fact that a', integrates to 1. Taking expectations and using Malliavin integration by parts (see (15)), we arrive at
E V E [[ D.P((X))ss(X)) "1 a ds -B [x(x)) L. (a.(X.)) 1/o og dB,
Based the framework from Watanabe (1987), apply this to yp dn the Dirac delta centred at vr E R", and get
V (x)) = Vz log / (2) p((x) Vz (2) p((x) (.B(
1
B E (x.) ( (x,( AB, =B [ ((x(,(()))))1)) d'd X =
(x)
Since the conditional expectation is the minimiser of the 2 distance, (39) follows.
14

---

Conditioning Diffusions Using Malliavin Calculus
Lemma B.3, Assume that the forward SDE (38) is Ornstein-Uhleebbek process
dx, 2 dt + dB,,
(40)
Then for the choice a', = we get Tweedie's s formula:
X. V log - (x e/2 X4, =
1
(41)
and, in particular, the loss (39) simplifies the well-known denoising score matching loss:
((u) E u(Xi) - 1 T (X+ - 112
(42)
Proof. We have that
J1ss el(ts))/2dd
Therefore,
V log = E [ (o(x)) dBd Xt
% E d₂ dB,
EE
x. =
If we choose a, ex-t, then
V loggr(xx)
=
1
E
2
dB, X,
(43)
1 e
However, the solution of (40) is given by
X, e + f
- dBr.
Plugging this into (43), we get
V log Pe())
p-- B[X, 4/2 X, =x]
Now, (42) follows again since the conditional expectation is the minimiser of the L distance.
C. Experimental Details
C.1. Controlled Environment Experiments
C.I.I. EXPERIMENT SETUP
We discretised the time domain [0, 1] into 200 equivariant grid points for the simulation and used Euler-Maryuama scheme to simulate paths of the unconditioned SDE (6). We approximated S, (8) by starting the final increment of d., and then using efficient adjoint method for SDEs propagate the derivative information backwards, The full Jacobian matrix Jaj is never calculated.
For the algorithms BEL first and BEL last we used 81 200 (see Section 3.3).
We used a batch size of 2048 and iterated through 20 000 batches. We used the Adam optimizer and a neural network architecture which is loosely inspired by UNets. It projects the input data up to 256 dimensions and then has fully connected layers of size [256, 128, 64, 32, 64, 128, 256] with skip connections. The last layer is then fully connected layer to the output dimension.
15

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Conditioning Diffusions Using Malliavin Calculus
15
15
1.0
0.5
-2.0
-1.5
1.0
(b)
Figure 7: We plot the results from the double well experiments for the reparametrisation trick corresponding to (a) Figure the double well experiment and (b) Figure 5, the ten dimensional double well experiment.
1.1.2. METRICS
Both metrics compare the simulated paths with paths from the ground truth when started in Pini and conditioned on landing in Ytinal (see Section 4.1.I), if the algorithms can approximate rare events even though they are trained on an amortised objective. We used 15 000 simulations of the trained models to calculate the metrics.
Dist. This metric calculates the average Euclidean distance of the final state of the path to the conditioned Vrinal
MV. This metric calculates the coordinate-wise mean and variance along the paths of the SDE. It then compares those to the coordinate-wise mean and variance of paths m ygen simulated with the ground truth drift, and calculates
200
MV
1
m
mil² + 11/ 31/2112
(44)
200
where my and i; are the variance vectors at time normal distributions.
The form of (44) is inspired by the Wasserstein-2 distance of two
200
C.1.3. RESULTS
Here provide the tables with the results for our controlled environment experiments for Brownian motion Table 4.
We also provide figures for the reparametrisation trick for the double well experiments in Figure 7.
C.2. Shape Processes
For the kernel parameters in the SDE (32) associated parameters in Yang et al. (2025) to train the model with the Adam optimiser. We discretise the time domain [0, 1] into 100 equivariant grid points and use the Fuler-Maruyama scheme to simulate paths of the SDEs. For each method, we train on a total of 102.400 trajectories with batch size of 128, We compare to the time-reversal method (Heng et al., 2022) and the adjoint method (Baker et al., 2025) using the code provided by Baker et al. (2025).
= 0.1 and B 1.0. For all methods
the neural network and
To evaluate performance, we use the mean pointwise distance between the target shape (vi ) N and the final points of M
16

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Conditioning Diffusions Using Malliavin Calculus
Table 2: Dimension
Loss
MV
Dist
BEL average
7.6 x 10
3.2 x 10
BEL first
8.6 10
2.1 10
BEL last
x x 10-1
5.6 10
BEL optimal
1.1 10-1
3 10-3
Reparametriiation Trick
7.6 x 10
2.2 10-3
Table 3: Dimension 10
Loss
MV
Dist
BEL average
4.5 x 107
2.3 x 10 2
BEL first
4.5 x 10
7.1 X 1
BEL last
4.8 x 10
1.9 X 10
BEL optimal
4.5 10
10
Reparametrization Trick
4.5 10
9.4 1
Table 4: Performance of various algorithms for conditioning a Brownian motion (see Section 4.1.2).
Table 5: Dimension 1
Loss
MV
Dist
BEL average
8.1 x 10-1
1.6 X 10 1
BEL first
2.9 x 10-1
8.4 X 10 2
BEL last
1.1 x 10
1.1 x 10
BEL optimal
8.3 X 10-1
1.9 X 10-1
Reparametrization Trick
9.2 x 10-1
2.1 10-1
Table 6: Dimension 10
Loss
MV
Dist
BEL average
3.4 10 1
3.0 10
BEL first
2.1 10
3.3 10 1
BEI. last
5.9 x 10
1.1 x 10
BEL optimal
3.1 x 10
3.0 x 10 1
Reparametrization Trick
5.5 X 10
5.2 x 10
Table 7: Performance of our proposed algorithms for conditioning the double well SDE (see Section 4.1.3). The best- performing algorithm metrics are marked in red.
17

---

Conditioning Diffusions Using Malliavin Calculus
1.5
1,0
-1.0
-1.
-15
-10
-05
0.0
0.5
1.0
1.5
(a)
(b)
Figure 8: We plot one trajectory from the unconditioned stochastic shape process in (a), started at circle radius and one trajectory from the process conditioned via the proposed method BEL. average to end a circle of radius 1.5 in (b).
sampled bridges {ar) {
M
N
(45)
MN
==
We compute this metric over M 512 trajectories with N 50 points and report results Table
D. Proofs
D.1. Proof of Proposition 2.4
We start by proving the following preliminary result, which is slight generalisation of the Bismuth- Eesmuth-Elloorty-Li. formula (Bismut, 1984; Elworthy & Li, 1994). Results in similar spirit (although for transition densities instead of scores) can be found in Milstein et al. (2004).
Theorem D.1. Let a: [0, T] + gaxx be a matrix-valued differentiable function such that Ari = C'r ax is invertible for all 8 E T). For >> s we have the representation formula
=1 10=EB(0o ((XT))
[[ (or( T( Td, X, = xe ATTs
Vx, B(((T) X. =x,]= E
(46)
Proof. Case Y Xx (or G Id). For any s < T it holds that
Dxp(Xr)) V D.X, VIX(K)) by the chain rule of Malliavin calculus (Nualart, 2006, Chapter 2). Therefore,
v((XT) = (=p((r)os(Xx)
(47)
and
x Vep(x))Jro0 = (X.) = (x.(
Since (47) holds for any 8, we can integrate along a';
- 1 "Sipoooo( -0)
vx(((F) -
18

---

Conditioning Diffusions Using Malliavin Calculus
We now apply the Malliavin integration by parts (15) obtain
J. T Dxsp(Xx)oo((x)))++111l0
E V
E
E (((x)) S'" Jsjoa')) dB, (ar (ur-au)-!
In order to prove Proposition 2.4 (and later Lemma 3.3), we now prove that the score is a martingale, when conditioned on the observation Y:
Lemma D.2. Let P then we have that
Vx, pY
Xx) E Pog xx) Y, X2).
(48)
Proof. We have
V log pY (x)
Vz, / P(Y xx==r))p(x===== oт X,)dær
p(Y Xx)
P(Y Xx,) Uze / p(Y =x==zr)p(rr> = ør X. = xx)p(X, = Xx)dd,drr
Uze / p(Y XX x xr Xx = xx)p((x = =, Xx) x, logp(X, = as Xx)ddaddrT
p(Y X,)
VVVx, x, log Pole Xx) Xt, Y].
Proposition 2.4 can now be proved using yp 83 in Theorem D.I. Intuitively, this corresponds to approximating the Dirac delta distribution by sequence of peaked Gaussians fnx and then taking the limit. On technical level, extending Theorem D.I is supported by the framework of Watanabe distributions, see Watanabe (1987, Section 2).
Proof of Proposition 2.4. Recall that the main ideas of the proof have already been outlined in the main text. Applying Theorem D.I with 9 6my, for fixed æT E 12, leads
V VogPPr)((XT = IT X, xx)
Vz, E Beo (XT) Xg = xz]
(XT xT X. = Ix)
dur (XT) '' (((xx)))'depac)) dB, Xg Eg
A AT
E
Pr Pri (XT FT
Xs F.)
dB, X. [" (or(X.)) TdB
E
v,, X7
ATI.
The final result of Proposition 2.4 follows transposing a and transposing the last equation; this is only a cosmetic change so that the result is more in line with algorithmic implementations.
Now, using Lemma D.2 for t T we get that
V log po = X, x.)
E x, log PTIs (Xr Xx)) Y,X,]
E
dB, Xx, Xr |r
(xx( -1) TdB, dB, Y.X,
19

---

Conditioning Diffusions Using Malliavin Calculus
D.2. Equivalence KL-Loss
Lemma D.3. Letting C. (ue) be defined as in (16). Then can be interpreted in terms of an amortized Kellback-Leibblee
divergence between measures on path space as follows:
' fiocal(ue) di [
P")]
Proof. We have that
Ct Loccal(u (ue) == (x)lu(X,) Spll Y - =y]
= Blllu(X,) - (S: - -E[S, X,, Y] + E[s, x, 1 Y yl
E[]lu(X,)) - E[S, X,, y]1 Y = + B||| B[S, x,, YIl |Y
[X]) V log = 2 Y vl [. [Ss X 1
|Y= y)
where we used that [ES, X,, Xr] is an 1.2 orthogonal projection of S, onto the set of a X7) measurable random variables which u(X,) is an element of. Integrating over this we obtain
L C KL(P" P*) + / S; - EiS, Xr)]1 XT = zrldf,
where the second term is independent of u. Hence, assuming that the second term is integrable, the result follows by taking expectations on both sides. However, note that they have different finite-sample properties.
D.3. Reparametrisation Method
The pathwise reparameterisation trick has been introduced by Domingo-Enn Enrich al. (2024b) in order to derive the stochastic optimal control matching loss. It is the following result:
Lemma D.4 (DDomingo-Enrich et al., 2024b), Prop. 1). Let Xx (X;) be the solution of the SDEAX; b(XF di + (t) d.B with initial condition X = Assume that f : Rd [0, TR and Rdd - R are differentiable. For each funccttoo
t E [0, T). let M, [] + R ddxd be an arbitrary continuously differentiable function matrix-valued function such that Mt(t) Id. We have that
E exp (-
1-" s(X; 1(Xg, sds 9
E (
1'" M.vaJ(XE 8) MrV I' (M....(XE)))) MD) -
(49)
- (-x.s---
x cxp
-(
fxx
Looking at the proof of this result in Domingo-Enrich et al. (2024b, Subsec. C.2), we observe that when g is not differentiable, if we impose that Mr = 0, then (49) still holds. If we additionally set f = 0 and b lime-independent, we obtain
V V exp ( ((xx)) EE [[. (M.V26(x MM)(0 exp ( 9(x7))
(50)
Next, we prove that relying on our approach, we can recover and generalise this result to compute v.E(((({t) for yp that are not necessarily strictly positive or differentiable:
Proof. We apply Theorem D.I for a, = J. M, and observe that
T, Jojoa " (Jsjaaa))) - "J'poa = o, ' M - --V6(XX))5 = 'M' Vb(X))M.).
20

---

Conditioning Diffusions Using Malliavin Calculus
Therefore
[(v=)(X7)) = E ((Xx)) -- T ('o'"J))'') dBs (ar)-
= E p(xT)) / (M, Vb(X,) dB, (ar- - (00-
Since M, is chosen in such way that Mo ld and Mr O, we have ar ap -dd, and the result follows.
D.4. Gaussian Approximations
Now use Lemma D.2 show the equivalence of the losses:
Lemma D.5. (Lemmaa 3.3) It holds that
(X Vx, logp(Y x,) = E|luu Vx, logPa/((X, [.)
(51)
Proof. We have that
ue (X, Vx, LY x,)I ([lluu(X, B[Vx, x, log X,) X,, Y]1
by Lemma D.2. Since the conditional expectation E[- X,, YJ is a orthogonal projection onto the subspace (X,,, Y)- measurable random variables in I,2, and st,, X,, Y) is an element of that subspace, we have that
Bl]lur( Xe, P) Vx, logp(X, 1=
Fl (X,, Y) [ x, log (X, X1) X,, X,yY]1+++ B x, log (X, X,) E x, logPaj((X, Xx) Xx, Y]I (X Y) V x, log P X,) X,, Y]]] [B[]] x, Pog (x.)1²]
- [ log log X₂) (x,Y)1
This proves the statement.
Finally, we prove
Lemma D.6 (Lemma 3.4). Approximating p(Xt-6t X1) by a Gaussian (24a) and regressing against its score is equivalent to choosing o' = 1 [1,+61]] and approximating the stochastic integral S (8) as
6+
JandBB zo J1488e(B148: B.),
(52)
and farthermre approximating Je48 by an Euler-Maryuama step (7):
Je48t RJ ld (btV(tt
(53)
Proof. Since approximate
p(Xe45l (x)) R N(X, 68b,(
we have that
1 (Id St X.)) B.). (54)
V Vlogp(Xr488 X,) (Id + (.) (X, 8(b(xx)))
51
Since 04445 0, plugging the normalization a145c-ar are the same.
into S, (8) shows that the two expressions (8) and (54)
21

---

Conditioning Diffusions Using Malliavin Calculus
E. Gaussian Analysis of the Variance
In this Section, we prove Lemma 3.1. First we determine the variance the Monte Carlo estimator: Lemma E.1. Ser = 1, n = 1, b= = 0 and o = 1, i.e, X₃ is one-dimensional Brownian motion conditioned on X, = Iy and 11 = T1. Then the variance of the Monte-Carlo estimator
v logPijo(B = Bo xo) 2 L/ d
given by
-
dgg (Lo (at) dt + /'² [a) -x)²
dt + (x1 =009)
(55)
01
(1 (1-17 1)2
assuming that 0z VI-s
as
+
Proof. Without loss of generality, we assume that a integrates to and we the notation 20 and y øT. We need the following preliminary facts,
E EBB, B1 y, Bu = x] = E(W, sW1 + (1 - s). + -syl = (1 - s) + sy
x: B.
E logp(B; B1 B.) B1 = Bg x]
E
B1 = y. Bo (y - x),
-
where W. is standard (unconditioned) Brownian motion. The expectation of the Monte Carlo estimator can then be computed as
E
0, '='d B1 = y. Bo = x = E
a,dB, B1 = yy =
E
a ddW, v logp((B B.)ds) B y, Bo
E
a V logg(BB = x B.)ds
Bi = y. Bo = x
E
a. logg Bi = x Ba)ds
[ (y r)a,ds (g (x)x.. x).
For the square of the estimator obtain
E (T. -.- oo-'
27 ) [1, d'W,,, + BB, logp(B: = ==1B.)ds) B.)dss) ²
=E E
|B =
E [1/ ".d.., )9 +E S a.dW, / al a'ly- x)ds
+E [1%² /
B
27
ds
1
L Pss + +
(
y B
27
a
1 1- d8
Moreover,
(/
/ y - B ds
/ ag 1- 0, ds dt
B. - B,
E
dg 8
E
S.') ((' (otat, min(s, (8,)) 1) st
a' d.a,E(B, BB )
ds dt
(56)
(1 (1= -)
- (1 -
22

---

(8.)² s² ds, Conditioning Diffusions Using Malliavin Calculus
The first term in (56) can simplified,
So' L. a'ol 3 1) -st st 5)(1 -1)
min(s,
J. L 0g (1 8)(1 - -)
st
do + S/¹ S. a'al (1 (1= - -t)
t- st
ds dl
8(1 - 1)
ds + f J. a'al (1- s) 1)
1(1 - 8)
ds dt
s) - ()
8
ds dl S.¹ J. dgal (1 (1-1) dodd
t
dodl
8)
ag (1 dfd'ds+ + S oi a1 L. a,dect
s
t -
s
ag
(a1 -a,) ds
s)
2
u(s (s))
ds,
--s s
where we have defined u(s) = _ a²)². Now we see that
1
+ [/ (T-3)
)
2
u(s) ds -2 u(s)
[2(((
u(s)ds ds
0 8
1 8 0
1
=
(1 8)2 u()) ds
/ (01 ),))
ds,
(1 (1-s) 8)2
where we have used the fact that 01 - 0g = - 5) by assumption. The second term in (56) simplifies as follows:
1. a. -v1
- B] - 8]
ds = S/' //' "'a/ol(x m) - sx) y) - t) ds dt (1-s)(1
(1 s)=s)(1= t)
(x aa--vl y)² ((') 1/ a'a' ds dt (z -y)
so that collecting all the terms yields the claimed result.
In the following determine the optimal choice of a in the simplified setting:
Lemma E.2. Assume the setting from Lemma 3.1. Then the optimal a has derivative
d² = (2 (1+vB)) (1 s){((1+v3),
Proof. We assume without loss of generality that a1 = 1 and an = 0. We need to optimise the following term,
L. (() ds S/' (a1
a,) -ag) ds L/' (Bi_s'²+
(81-0)² ds -- =-f" (1 (1-s)² s)z
=%/' (0.)) +
(1- s)²
where we have set Bg = -0--- We define
F(B) [( (B.)2 +
(B.)² s2 ds,
since the last term in (19) is independent of 3,. Assuming that a is a minimizer of F, follows that d F(3 + esp) = 0 for any yp such that the a defined through | + exp satisfies the assumptions of Lemma 3.1. In particular, this condition has be satisfied for any weakly differentiable to such that 4P0 = 1 = 1. Given that, we calculate
d de S.'' ((8'+E0)) (Bs + exp) 82
ds = 2 S/' (0' (8,' + +sp') + ((B, + tep) ds.
82
23

---

Conditioning Diffusions Using Malliavin Calculus
Therefore, arrive
0 -
4 33"
43, s2
ds
[-
-pB_" +
ds.
82
Since this equation needs to hold for any continuous yp with 0-boundary conditions it follows that
Bg g2
3
Solving this with the boundary conditions Pg and 31 = 1 leads to
B, = (1+v3))
B' [ (1 (1+ v5) 2 -2(
as claimed.
F. Related Work
Numerous algorithms exist for bridge simulation. For example, Markov chain Monte Carlo-based methods (Delyon Hu, 2006; Bayer & Schoenmakers, 2013; Bladt & Sørensen, 2014; Schauer al., 2017) and more recently, machine learning-based works (Heng et al., 2022; Du et al., 2024; Baker et al., 2025; Zhao et al., 2024).
In the context of diffusion models, the most relevant approaches are Zhang et al. (2023); Denker et al. (2024). These methods rely on a fundamental equality in their derivation:
V x,
= Xx, logp(X.|X7)) + Vx, logp(X.) R Vx, logp(X.|Xx)) + (X.)..
This formulation allows the first term on the right to be computed analytically, while the second term relies on the pre-truee pre-trained score network from the diffusion process. Although this approach offers computational efficiency, it assumes that the score network accurately approximates V log (X,, the gradient of the distribution from which X, is sampled. This assumption frequently fails in practice and may become increasingly invalid after network fine-tuning or other modifications. Our approach avoids this limitation by directly learning optimal control for the current drift estimate sy (X,), regardless of its origin.
Guidance methods share similar objectives of introducing post-hoe control (Chung al., 2022; Song et al., 2023; Boys et al., 2023). However, these techniques rely on cheap approximations of the optimal control V log Pr [Xt), producing samples that deviate from the true posterior or conditional distribution. Instead, they generate samples resembling high- high-likelihood draws from the prior distribution. While adequate for many applications, these methods cannot provide authentic conditional samples from the Bayesian posterior when such precision required.
There are also two concurrent works on Malliavin calculus and diffusion models. However, they differ in their goals, techniques and results. In Mirafzali et al. (2025) the authors establish different Malliavin based score formula. In contrast to Lemma B. this construction relies on second-order Malliavin derivatives (Mirafzali et al., 2025, Theorem 3.9) and the inverse of the Malliavin matrix. In Greco (2025) the author uses Malliavin calculus to study diffusion models in infinite dimensions. This can be seen as giving justification for Tweedie's 's formula (see 2.5) in the infinite dimensional case, extending the results from Pidstrigach et al. (2024).
24


Reviews:
[{"review_id": "ezURY2rCNl", "reviewer": "Reviewer_vRjP", "summary": "This paper introduces a novel approach to conditioning diffusion processes using Malliavin calculus, enabling stable training of score-based diffusion bridges. By replacing the ordinary derivative by Malliavin derivative, their framework unifies and extends existing diffusion bridge methods. Through controlled experiments on Brownian motion bridges and double-well potentials, the proposed method performs better than the baselines in accurately modeling conditioned diffusions.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review! We're glad you see our method as unifying and extending diffusion bridge approaches. Below, we address your questions and concerns.\n\n***For theorem 2.1 as $\\sigma(X_t)$ becomes singular, wouldn't you have stability issues with its inverse? & The method assumes that the matrix A and diffusion coefficient $\\sigma(X_t)$ are always invertible...***\n\nThanks for the question!\n\nFor $A$: Since $\\alpha$ is user-chosen, it can always be selected to make $A$ invertible. All natural choices for $\\alpha$ do so.\n\nFor $\\sigma$: It is helpful to discuss two slightly different cases: \n\n1) If $\\sigma$ is small but positive definite, stability can be partly managed via small step sizes or adjusting $\\alpha$. However, small noise makes the bridge problem harder in general (regardless of the method), as the dynamics become rigid; for $\\sigma = 0$, the problem is ill-defined.\n\n2) The diffusion coefficient is sometimes genuinely rank deficient, for example in the underdamped Langevin dynamics,\n$\\mathrm{d}q_t = p_t \\, \\mathrm{d}t,$\n    $\\mathrm{d} p_t = - \\nabla V(q_t) \\, \\mathrm{d}t - \\gamma p_t \\, \\mathrm{d}t + \\sqrt{2\\gamma} \\mathrm{d}W_t,$\nof great interest in molecular dynamics (here the diffusion matrix would be  \\begin{pmatrix} 0 & 0 \\\\\\\\ 0 & \\sqrt{2 \\gamma}\n\\end{pmatrix} with noise only acting on the $p$-variable). In this case (and in similar other cases), it is possible to extend Theorem 2.1 using pseudo-inverses, and the proposed methodology generalises straightforwardly. On a technical level, this extension requires \"hypoellipticity\" which guarantees smooth transition probabilities of the underlying dynamics. Since underdamped Langevin dynamics is important for applications, we suggest adding a short explanation in the appendix. \n\n***There seems to be a implicit circular dependency...***\n\nWe are a bit unsure exactly which proof you mean, but we assume you mean in Theorem 2.1. To clarify, the proof steps are as follows:\n\n1. By Doob’s $h$-transform (Proposition 2.2) for eq (8) to coincide with the conditional law, we need to show that $u^*(x; x_T) = \\nabla \\log p_{T\\mid t}(X_T = x_T \\mid X_t = x)$. \n2. By Proposition 2.3, we know $\\nabla \\log p_{T\\mid t}(X_T = x_T \\mid X_t = x) = \\mathbb{E}[\\mathcal{S}_t \\mid X_t=x, X_T=x_T]$.\n3. In the proof of Theorem 2.1 we show that the minimiser of $\\mathcal{L}(u)$ is given by $\\mathbb{E}[\\mathcal{S}_t \\mid X_t=x, X_T=x_T]$. Since this is equal to the term arising from Doob’s $h$-transform, eq (8) coincides with the SDE under the conditional law. \n\nIn particular, in equations (14) and (15), $\\mathcal{S}_t$ is already fixed and does not depend on $u$, hence there is no circularity.\n\nIf this is what you mean, we’d be glad to include the above steps in proof of 2.1 making it more explicit! \n\n\n***The quality of the writing still needs to be improved...***\n\nThanks for pointing this out! We will fix this immediately.\n\n***Lack of real world experiments conducted leaving applicability beyond theory uncertain.***\nWe based our experiment setup on papers in the literature that also concentrate on similar problems [1, 2, 3, 4, 5]. Indeed, the double well experiment is a challenging problem in conditioning due to the high energy barrier between wells and therefore the transitioning events are very rare. However, we agree it is interesting to consider image generation, and have since experimented with this! \n\nWe condition a pretrained model to only generate images matching with a given top left corner. Please see our answer to Reviewer q3Kt (Section q3Kt.1) for details, and see the samples linked here: https://postimg.cc/bsbNgMY2.\n\nWe hope we've conveyed the method's applicability and flexibility! We'll include this experiment to show it scales to high-dimensional SDEs and has practical relevance beyond the theory.\n\n[1] https://arxiv.org/abs/2111.07243\\\n\n[2] https://arxiv.org/abs/2407.15455\\\n\n[3] https://link.springer.com/article/10.1007/s42985-021-00102-x\\\n\n[4] https://arxiv.org/pdf/2312.02027v4 \\\n\n[5] https://projecteuclid.org/journals/bernoulli/volume-23/issue-4A/Guided-proposals-for-simulating-multi-dimensional-diffusion-bridges/10.3150/16-BEJ833.full.\n\n***Figure 1 hardly convey the message...start by introducing a motivating example...***\n\n\nWe agree we should include more information in this first image, and update it as follows: https://postimg.cc/NK7gq2JB.\nWe hope you agree that this better illustrates the motivation and challenge of the paper, and it also gives us the opportunity of referring to it in the introduction when explaining the reward (and its potential singularity).\n\n**...do you think your method will be a good fit for high-dimensional task like image generation?**\n\nYes, as we have seen from our experiment on images it does scale to high-dimensional tasks! The Jacobian calculation can be done very efficiently using vector-matrix products; for more details on this and on computing $\\mathcal{S}$ please see our answer to Reviewer 4Bsi (Section 4Bsi.3).", "claims_and_evidence": "Their main claim is this novel formula for conditional scores in denoising score matching in Theorem 2.1. \nTo the best of my knowledge, the formulation of replacing ordinary derivatives with Malliavin derivatives is new and the authors provide detailed proof for their main theorem upfront.", "methods_and_evaluation_criteria": "The proposed method is valid for learning the diffusion bridge problem. The evaluations are toy experiments based.", "theoretical_claims": "For theorem 2.1 as \\sigma(X_t) becomes singular, wouldn't you have stability issues with its inverse? \n\nThere seems to be a implicit circular dependency between S_t and u_t in your proof for showing the u_t is the unique minimizer, could you make a concise proof in your Appendix to show this is not the case?", "experimental_designs_or_analyses": "The experiments are done with two toy experiments set up, given the main contribution of the paper is the theorem, the experimental designs might be acceptable.", "relation_to_broader_scientific_literature": "This paper builds on prior work in diffusion bridge learning, score-based generative modeling, and stochastic control, extending methods like Doob's h-transform by using Malliavin calculus for stable conditional score estimation.", "essential_references_not_discussed": "To my best knowledge, close related works are mentioned in the paper.", "other_strengths_and_weaknesses": "**Strengths**\n\nExperiments on double-well potential and Brownian motion bridges showcase the effectiveness of the method in capturing rare event transitions.\n\nThe derivation of a generalized Tweedie’s formula and path-space integration by parts provides a unifying theoretical framework.\n\n\n**Weaknesses**\n\nThe method assumes that the matrix A and diffusion coefficient \\sigma(x_t) are always invertible, which may not hold in degenerate or low-noise regimes.\n\nThe quality of the writing sill need to be improved i.e. some of the figures are mis-referenced.\n\nLack of real world experiments conducted leaving applicability beyond theory uncertain.", "other_comments_or_suggestions": "In section 4.1.3, some of the references to Figure 5 could be Figure 3? \n\nFigure 1 hardly convey the message of the paper, try either include conclusion in the comments or other examples. \n\nI would start by introducing a motivating example as mentioned  by you, \"(approximating the Dirac measure by highly peaked Gaussians) often face numerical instability.\"  Then showcase that your method is able to solve the instability problem.", "questions_for_authors": "Some of the questions, please check the theorems section and weakness section of the review. \n\nState-of-art  diffusion based models/flow matching models all have the assumption about the highly peaked Gaussians near the t equals to 0, your method seems to circumvent this problem.  With Jacobian calculation and matrix inverse in calculating S, do you think your method  will be a good fit for high-dimensional task like image generation? \n\n\nSome of the questions, please check the theorems section and weaknesses section of the review.\nState-of-the-art diffusion-based models and flow matching models assume highly peaked Gaussians near t=0, but your method appears to circumvent this issue. With Jacobian calculations and matrix inversions involved in computing S, do you think your method is well-suited for high-dimensional tasks like image generation?", "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": "0dqUiXgxfb", "reviewer": "Reviewer_WkxC", "summary": "The work develops a technique for learning the control process that\nconditions a diffusion on the terminal state given the initial state.\nA key advantage of their method is that it is robust to singular\nrewards. They test their algorithm on multi-well toy experiments,\nand show that their algorithm successfully allows conditioning on\ntransitions between the wells. They also test on a Shape process\nand obtain better performance than previous methods.\n\n## Update after rebuttal\n\nThe rebuttal was good, and I appreciate the additional experiment, but it still seems fairly small scale to me so was not enough to convince me to increase my score, as the extent of the importance of the results in the paper is not conveyed to me. However, I am positive about this paper and maintain my vote leaning toward accept.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We really appreciate your positive review and that you think our paper is well written!\n\n***One weakness is that the experiments are fairly toy***\n\nWe based our experiments on setups from the most closely connected literature and are glad you found them more substantial than in similar papers! \n\nWe think the included experiments are good tests for our method since \n1) In the double well experiment, there is a high energy barrier between transitioning between wells, meaning it happens very rarely. Therefore, conditioning to go from one well to another is challenging.\n2) In the shape models, points close to one another are very correlated making transitioning to another given shape challenging.\n\nHowever, we agree these are fairly toy. In order to resolve this we have now conducted an experiment on images, which we hope will convince you of the applicability of our method to numerous situations, including high dimensions!\n\nWe use the following setup:\n\n1. As the base SDE we take a pretrained diffusion model SDE trained to go from the noise distribution to the data distribution (for this we have used Fashion MNIST).\n\n2. We condition the SDE on the upper left corner of the image to take a given specific value. Practically, this corresponds to the task of completing partial images.\n\n3. We link our results here: https://postimg.cc/bsbNgMY2 \\\nThe true \"left corner\" is from the first image in the top left corner. We see all the samples match well, generating only images with the same left upper corner.\n\nStrictly speaking, this experiment goes slightly beyond the considered set-up, as conditioning on a part of the image translates into \"pinning down\" only some of the coordinates of the endpoint. Indeed the proposed method is very flexible, and applies to conditioning on any observation $Y= G(X_T)$, with no assumptions on the function $G$ (e.g. $G$ need not be differentiable and $Y$ could be discrete or continuous). For more details, please see our answer to Reviewer q3Kt (see Section q3Kt.1). We realise this is not so clear in the paper right now and will include this discussion!\n\nWe hope we have managed to address your concern.\n\n\n***The order of the numbers of some Figures/Tables is strange.***\nThanks for pointing this out! We've updated this now so that the tables and figures are named in order of appearance. \n\n\n***What is the computation time like? Is this method practical?***\n\nTraining the conditioning drift for the image task took us about 30-60 minutes on a high-end GPU, so yes, this method can potentially be applied in large-scale settings. Note that after this training time one can condition on *any* upper left corner (for example jackets, shoes, etc). Please do let us know if you want anymore information on this!\n\nFor a more detailed complexity analysis of the algorithm (including the number of forward and backward passes) please refer to our answer to Reviewer 4Bsi (Section 4Bsi.3).\n\nThanks again for your review. We hope we have managed to resolve your concerns by including the extra experiment and an analysis on the complexity of the method.", "claims_and_evidence": "Yes, the evidence is sufficient for the claims.", "methods_and_evaluation_criteria": "This seems primarily a theoretical paper, so the experiments\nseem appropriate. The practical usefulness of the approach\nremains a bit unclear as the experiments do not seem to be\nreal applications.", "theoretical_claims": "I did not check the proofs thoroughly.", "experimental_designs_or_analyses": "I did not check them in too much detail. The plots of the\ndiffusion paths show that their method can condition on the transitions.\nMoreover, the metrics in the tables also show improvement (they considered\nmetrics of distance to the target location as well as a kind of\nerror to the ground truth drift).", "relation_to_broader_scientific_literature": "Adequately discussed.", "essential_references_not_discussed": "None that I can think of.", "other_strengths_and_weaknesses": "The paper is well written.\n\nOne weakness is that the experiments are fairly toy, though checking\nother related published works, it seems this is common in this field,\nand the experiments may be more substantial than usual in similar papers.", "other_comments_or_suggestions": "The order of the numbers of some Figures/Tables is strange.\nFor example, Table 1 is introduced last in the main paper, while other\ntables like Table 4 are introduced earlier.", "questions_for_authors": "What is the computation time like?\nIs this method practical?", "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": "5keB59SALN", "reviewer": "Reviewer_4Bsi", "summary": "The authors propose a novel solution to tackling the interesting and challenging problem of diffusion bridges conditioned on singular rewards.\nTo solve this issue, they make use of theory of Malliavin calculus (essentially stochastic calculus of variations from my understanding) to handle the singularities.\nUsing this theory they proposed a generalization of Tweedie's formula which the conditional score function $\\nabla_x \\log p_{T|t}(x_T|x)$ is equal to the conditional expectation of their score process.\nThey then show to configure the variance of the Monte-Carlo estimator of the conditional score function.\nLastly, they use a few toy experiments to illustrate their method.\n\n## Update after rebuttal\nI agree with the other reviewers that the limited demonstration of real world applications is a shortcoming so I adjusted my score.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks a lot for your positive review! We're very happy to hear that you really enjoyed this paper! And we were especially happy that you found the general quality to be on par with some of the seminal papers in this field.\nWe appreciate your detailed comments and questions, and answer them below.\n\n### Section 4Bsi.1 \n_\"Can the claim \"recall the connection of diffusion bridges to Doob’s -transform in Section 2.1, in particular the relevance of conditional score functions\" be a contribution?\"_\n\nIndeed, we agree it is well known. Our intention was to also to use this section to provide an outline to the paper, but we see how this could be confusing. To make it clearer, we will write \"recall the (well-known) connection of diffusion bridges\" and cite the relevant literature. We hope you agree this makes it clearer.\n\n### Section 4Bsi.2: \"Footnote 1 is incomplete.\"\nYou're right -- the footnote currently continues on the next page. We will fix this! Thank you.\n\n### Section 4Bsi.3: Computation of the Score Process (Q2)\n\nThanks for the excellent questions. Indeed, the paper currently lacks some detail on how we compute the score estimator, which may have caused confusion. As you noted, vector-Jacobian products are cheaper via reverse-mode autodiff, and adjoint methods avoid computing the full Jacobian—this is exactly what we do. We simulate paths of the reference SDE (1), then compute the integral (6) in reverse, akin to adjoint ODE methods, but adapted for our setting, picking up $dB_t$ terms along the way instead of only backpropagating a final gradient. We present the full algorithm below for $\\sigma = 1$ and Euler-Maruyama discretization, and will include a detailed explanation in the paper. Thank you for the helpful references—we will cite them in the revised version.\n\n**Given**:\n- $(X_0, X_{\\delta t}, \\ldots, X_T)$\n- $(Z_0, Z_{\\delta t}, \\ldots, Z_T)$, which is the noise which was used to produce $X_{t + \\delta t} = X_t + \\delta t  b(X_t) + \\sqrt{\\delta t}~Z_t$\n\n**Output**:\n- $(S_0, S_{\\delta t}, \\ldots S_T)$, the score process at all times $t$\n\n**Method**:\n_Initialize_: $S_T = 0$\n_Propagate_: $S_t = S_{t+\\delta t}^T (I + \\delta t \\nabla b(X_t))$\n_At the end_: Divide $S_t$ by $T - t$ to normalize it.\n\nWe hope it's now clear that since we left-multiply with the Jacobian, we never need the full Jacobian—just one backprop pass per timestep. This yields  $N = T/\\delta t$ regression targets $S_t$ for gradient updates.\n\nNote that the Jacobian in (6) is transposed before right-multiplying with $dB_t$, which is equivalent to left-multiplication. We agree this wasn’t clear and will revise it.\n\nThanks again for the great question—highlighting this has improved the paper. To show scalability to higher dimensions, we’ve added image experiments (see “Section q3Kt.1” in our reply to Reviewer q3Kt).\n\n\n### Section 4Bsi.4: Choice of $\\alpha$ (Q6 and Q7)\nWe found the choices \"first\", \"optimal\" and \"average\" all to do quite well on most problems (however, average and first are easier to implement than optimal). Which one worked the best was problem dependent. Note that \"optimal\" is only theoretically optimal in a specific setting (a forward process which is a Brownian motion). There are two interesting ways to study optimality:\n- Getting a deeper theoretical understanding of optimal $\\alpha$ choices for different classes of SDEs.\n- Optimizing $\\alpha$ numerically for a given problem.\n\nHowever, these were out of the scope of the current work, but they are certainly very interesting for future research.\n\nNote that in Section 4.2 we do not compare different $\\alpha$ schemes. We just picked the best-performing one for shape spaces (which was \"average\"), and compare it to other algorithms from other publications. We observe that we outperform them. We did not observe this problem to behave very differently with regard to the choice of $\\alpha$ than the other problems.\n\n### Section Other Questions\n-- 1. -- Indeed, it's the derivative with respect to $t$. We will add this to the theorem statement to make it clear! \n\n-- 3. -- Thanks for bringing this for our attention. We have updated this to be \"for $i=1$ to $N$ ...\". \n\n-- 4. --  Yes, thanks - Equation (17) should read $\\alpha'_t$. We've corrected this!\n\n-- 5. -- That's a very interesting comment. In our setting (diffusion bridges), there should be no oscillation between the wells as the conditioning \"pins down\" the endpoint; transitioning back and forth is highly unlikely (even more unlikely than transitioning once!) under the original dynamics. Oscillating back and forth would be expected if the dynamics were set up to equilibrate and target a (spatial) distibution of interest, for example in MCMC type sampling algorithms. Potentially our method could be modified to enhance sampling, but since this is quite different, we believe it is better reserved for future work.\n\nThanks again for the feedback! We hope we managed to answer your questions sufficiently and address your concerns.", "claims_and_evidence": "* Can the claim\n\"recall the connection of diffusion bridges to Doob’s $h$-transform in Section 2.1, in particular the relevance of conditional score functions\"\nbe a contribution?\n\n The connection between Doob's $h$-transform and diffusion bridges is quite well known, see [1-4]\n\n[1] Shi et al., *Diffusion Schrödinger Bridge Matching*, NeurIPS 2023, https://proceedings.neurips.cc/paper_files/paper/2023/file/c428adf74782c2092d254329b6b02482-Paper-Conference.pdf\n\n[2] Somnath et al., *Aligned Diffusion Schrödinger Bridges*, UAI 2023, https://proceedings.mlr.press/v216/somnath23a/somnath23a.pdf\n\n[3] Du et al., *Doob’s Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling*, NeurIPS 2024, https://arxiv.org/pdf/2410.07974\n\n[4] Brekelmans et al., *On Schrödinger Bridge Matching and Expectation Maximization*, Optimal Transport and Machine Learning Workshop at NeurIPS 2023, https://openreview.net/pdf?id=Bd4DTPzOGO", "methods_and_evaluation_criteria": "They make sense.", "theoretical_claims": "I read the main paper thoroughly and performed a quick read of the proofs in the appendix. I could have missed details in the appendix.", "experimental_designs_or_analyses": "From what I read in the main paper the experiments generally make sense, see questions below for any concerns.", "relation_to_broader_scientific_literature": "In section 3.2 the authors provide an excellent description of how their work connects with prior work. I have no issues with it.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "* The paper is well-written, and despite being a mathematically dense paper I found it be quite easy to read.\n* I found the motivations and arguments for the proposed method to be compelling.\n* I found the paper to be rigorous and easy to read, standing next to some of the seminal papers in this field in terms of general quality.", "other_comments_or_suggestions": "* Footnote 1 is incomplete.\n\nNote, I really enjoyed this paper and found it to be quite interesting. I would be happy to improve the score if the questions and concerns I raised are addressed.", "questions_for_authors": "1. In Equation (6) what does $\\alpha\\_t'$ denote? I see $\\alpha\\_t$ defined in the theorem but not $\\alpha\\_t'$. Is it the derivative wrt $t$?\n2, Wouldn't solving equation (5) amount to the continuous-time analog of forward sensitivity equations, *i.e.*, forward-mode autodiff?\n   \n   The work on neural SDEs by Kidger and Li solved the adjoint equations in *reverse-time* corresponding to reverse-mode autodiff. For more details see [1, Section 5.1.4.]\n3. Related to Q2 wouldn't simulating vector-matrix products be more efficient than matrix-vector products?\n4. In Algorithm 1 in line 1 is $B$ and index or Brownian motion? The notation `for i = 1 to B do` seems strange to me when the next line is Sample $\\{X,B\\}_i$ ...\n5. In Equation (17) is it supposed to be $\\alpha_t$ or $\\alpha_t'$?\n6. In the double well experiment, (Figures 3 and 4) shouldn't samples ideally oscillate between the two wells and not just tend from 1 to -1?\n7. In Figures 4 and 5 the impact of different variance schemes (choices for $\\alpha_t'$ or $\\alpha_t$) seems not to matter much.\n8. In Sec 4.2 why does the of variance scheme seem to matter more than in Sec 4.1?\n\n[1] Patrick Kidger, *On Neural Differential Equations*, Ph.D. thesis, https://arxiv.org/pdf/2202.02435", "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": "tsCCdh7Mmp", "reviewer": "Reviewer_q3Kt", "summary": "In this paper, the authors propose a framework based on Malliavin Calculus to address the robustness issue associated with singular rewards. Their work focuses on the problem of diffusion bridges, a type of diffusion process with fixed endpoints. Using Doob's h-transform and Malliavin Calculus, the authors derive the Generalized Tweedie's formula. This theoretical result serves as the foundation for their proposed BEL algorithm. Additionally, the authors analyze the variance of the network's training target and select an optimal parameter to minimize this variance. Finally, they conduct experiments on simple diffusion processes and shape processes to demonstrate the empirical performance of their algorithm.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your review! We're glad you think the method is interesting. We hope we will be able to address your concerns on the scalability and efficiency of our proposed method, which we discuss below!\n\n\n### Section q3Kt.1: Larger Experiments and Flow-Matching\nThank you for the great suggestions. Initially, we mainly targeted the case of conditioning diffusion on a single endpoint, since this can be seen as the most singular, and hence in principle most challenging, reward case (a Dirac delta). However, due to the fact that our method works in these extremal cases, it is actually applicable to a wide range of settings. We followed your proposal and applied our methodology to condition a diffusion model trained on images (could be similarly done for flow matching)!\n\nWe first trained a diffusion model on Fashion-MNIST and then used our methodology to condition on the upper left corner of an image. You can see the results here:\n- https://postimg.cc/bsbNgMY2\n\nThe upper left image of the right grid is the ground truth. We extracted the upper left quarter of that image as conditioning input (seen as the upper left image in the left grid). We then generated samples from the posterior distribution after conditioning on that upper left image. Training the conditioning drift for this particular problem took us about 30-60 minutes on a high-end GPU. Note that after this training time one can condition on *any* upper left corner. \n\nIn general, we can condition on any observation of the form $Y = G(X).$ Here, $Y$ can be discrete (classification) or continuous (upscaling, inpainting, ...), and $G$ can be smooth, but also non-differentiable and even non-continuous, since for diffusion bridges we have the extreme case of $G$ being a Dirac delta. We support both deterministic (noise-free) and stochastic (noisy) versions of G; adding artificial noise is not required but is possible to account for uncertain observations depending on the application. Of course, when $G$ has nice properties, like gradients, it might make sense to include these, but due to our lack of assumptions on the reward, our algorithm is quite generally applicable. \n\nWe understand that our paper wasn't to clear on this, and will put more emphasis on this point in the updated version. However, nearly nothing has to change, one only needs to replace $x_T$ by a general condition $y$ in the neural network input during training and evaluation, since we never make use of the form of the observation $x_T = Id(x_T)$ in any of our steps.\n\n### Section q3Kt.2: Schroedinger Bridges\nIndeed, bridges can be used in the calculation of Schroedinger Bridges. Note that if you can solve the static Schroedinger Bridge problem (Section 3.1 of the paper you linked), and one has access to the Bridges of the reference process, one can interpolate the static coupling of the marginals with the bridges to get the dynamic Schroedinger bridge solution. Therefore, if one has already solved for the bridge process, the problem of solving Schroedinger bridges on path space reduces to just finding the optimal coupling of the marginals!\n\nI hope we were able to dispel your concerns about the applicability of our method to different settings and high dimensional problems. Thank you for motivating us to apply the method to a diffusion model, we are very happy with the results and think this showcases the generality of our method well.\n\n### Section q3Kt.3: Lack of Analysis on Efficiency and Complexity\nWe agree that it maybe was not fully clear how costly the evaluation of the score process is. Therefore we updated our paper to be more detailed on the calculation of the score process. We explain the complexity of the algorithm in the answer to reviewer 4Bsi, you can find it by searching for \"Section 4Bsi.3\" on this page. \n\nWe hope this answers all questions. Furthermore, we hope our experiments on problems with very high energy barries (double well) as well as high dimensions (images), showcase the applicability and efficiency of our method for different kinds of problems. \n\n\n### Summary\nThanks again for the review. We believe we have managed to resolve the issues you raised, but please let us know if this is not the case! We hope you agree that the changes to the paper of including a high-dimensional image experiment alongside more details on computing the loss and its complexity will improve it!", "claims_and_evidence": "I believe the answer is likely no. The authors explicitly state in the paper: 'Unfortunately, the singular nature of such a measure renders most existing approaches unsuitable: gradients are not well-defined, and naive strategies (e.g., approximating the Dirac measure with highly peaked Gaussians) often face numerical instability.' (lines 55–59). The primary goal of the proposed BEL algorithm is to address this issue. Therefore, I think the authors should present experiments demonstrating improvements in stability and performance for tasks such as flow matching [1] and entropic optimal transport [2] (mentioned in lines 49–55) using their method. In my opinion, the experiments conducted on simple toy examples are insufficient to support such a claim.\n\n[1]. Lipman,Y., Chen,R.T., Ben-Hamu,H., Nickel,M., andLe,M. Flow matching for generative modeling. arXiv preprint arXiv:2210.02747, 2022.\n\n[2]. Shi,Y., DeBortoli,V., Campbell,A., and Doucet, A. Diffusion schrodinger bridge matching. Advances in Neural Information Processing Systems, 36, 2024.", "methods_and_evaluation_criteria": "I think the methods is quite interesting, but the authors should conduct experiments on High-Dimensional and Large-Scale datasets (e.g. image datasets) to test the empirical performance of the method.", "theoretical_claims": "I have checked the proofs of the theorems.", "experimental_designs_or_analyses": "As mentioned earlier, I believe the authors should conduct experiments on high-dimensional and large-scale datasets (e.g., image datasets) to evaluate the empirical performance of the proposed method.", "relation_to_broader_scientific_literature": "I believe that the key tool used to derive the algorithm in the paper is Malliavin Calculus. This is a powerful mathematical framework that allows the application of the integration-by-parts formula on path space. The authors leverage this tool to address the problem in this scenario, which is both innovative and novel.", "essential_references_not_discussed": "Nothing.", "other_strengths_and_weaknesses": "1. **Lack of High-Dimensional and Large-Scale Experiments**    \n   While I find the idea presented in the paper both interesting and novel, the current experiments are not sufficient to convince me that the BEL algorithm performs well on high-dimensional data or is scalable. These types of experiments are essential, as downstream tasks of diffusion bridges, such as flow matching, are inherently scalable and often involve high-dimensional generative tasks.  \n   \n2. **Lack of Analysis on Efficiency and Complexity**    \n   An analysis of the computational efficiency and complexity of the BEL algorithm is also necessary. Specifically, I believe that estimating the score process within the algorithm could be time-consuming, and this aspect warrants further discussion and evaluation.", "other_comments_or_suggestions": "Nothing.", "questions_for_authors": "See the sections above.", "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:

Meta-review: This is an interesting paper which explores applications of Malliavin calculus to diffusions. The paper is praised by the reviewers for the novelty but also various suggested are made to improve the final version of the paper. I hope the authors will implement some of these in the final version.
Decision: Accept (poster)