Stochastic Optimal Control for Diffusion Bridges in Function Spaces
Authors: Byoungwoo Park
Jungwon Choi
Sungbin Lim
Juho Lee
Venue: NeurIPS 2024 poster (2024)
Keywords: stochastic optimal control
bridge matching
diffusion model
Primary area: diffusion_based_models

Abstract: Recent advancements in diffusion models and diffusion bridges primarily focus on finite-dimensional spaces, yet many real-world problems necessitate operations in infinite-dimensional function spaces for more natural and interpretable formulations. 
In this paper, we present a theory of stochastic optimal control (SOC) tailored to infinite-dimensional spaces, aiming to extend diffusion-based algorithms to function spaces. 
Specifically, we demonstrate how Doob’s $h$-transform, the fundamental tool for constructing diffusion bridges, can be derived from the SOC perspective and expanded to infinite dimensions. 
This expansion presents a challenge, as infinite-dimensional spaces typically lack closed-form densities. 
Leveraging our theory, we establish that solving the optimal control problem with a specific objective function choice is equivalent to learning diffusion-based generative models. 
We propose two applications: 1) learning bridges between two infinite-dimensional distributions and 2) generative models for sampling from an infinite-dimensional distribution. 
Our approach proves effective for diverse problems involving continuous function space representations, such as resolution-free images, time-series data, and probability density functions.

Stochastic Optimal Control for Diffusion Bridges in Function Spaces
Byoungwoo Park', Jungwon Choi', Sungbin Lim 2im Juho Leel KAIST Korea University", LG Al Research
(bw.park, jungwon. choi, juholee (kalleeekkist.... ac. ac.kr, sungbin@korea.ee ac
Abstract
Recent advancements in diffusion models and diffusion bridges primarily focus on finite-dimensional spaces, yet many real-world problems necessitate operations in infinite-dimensional function spaces for more natural and interpretable formu- lations. In this paper, we present a theory of stochastic optimal control (SOC) tailored infinieediimensional spaces, aiming to extend diffusion-basee algo- rithms to function spaces. Specifically, we demonstrate how Doob's ' h-traansfrmm the fundamental tool for constructing diffusion bridges, can be derived from the SOC perspective and expanded to infinite dimensions. This expansion presents challenge, as infnite-dimensional spaces typically lack closed-form densities. Leveraging theory, we establish that solving the optimal control problem with a specific objective function choice equivalent to learning diffusion-based gen- erative models. We propose two applications: 1) learning bridges between two nnfnite-dimensional distributions and 2) generative models for sampling from an infinite-dimensonna distribution. Our approach proves effective for diverse prob- lems involving continuous function space representations, such as resolution free images, time-series data, and probability density functions. Code is available at https:/ //gi /i thub. com/bw-mee//ccccc http//bwaarr///ee park/DBFS.
Introduction
Stochastic Optimal Control (SOC) is designed to steer a noisy system toward a desired state by minimizing specific cost function. This methodology finds extensive applications across various fields in science and engineering, including rate event simulation [33, 35], stochastic filtering and data assimilation [47, 55, 72), non-convex optimization [9], modeling population dynamics [8, 43). SOC is also related to diffusion-based sampling methods that are predominant in machine learning literature. Specifically, if we choose the terminal cost of control problem as the log density ratio between target distribution and simple prior distribution, solving the optimal control reduces to learning diffusion-basee generative models [56, 73, 77] built upon the Schrödinger bridge problem [12, 54].
While SOC associated diffusion based generative models have been well-established for finite- dimensional spaces [5, 10. 11, 44], their theoretical foundations and practical algorithms for infinite- dimensional spaces remain underexplored. There a growing interest in developing generative models in function spaces. Examples include learning neural operators for partial differential equations (PDEs) [37, 39, 40], interpreting images as discretized functions through implicit neural representations (INRs) [21, 63], and operating in function spaces for Bayesian neural networks (BNNs) [65, 74]. Models that operate in function spaces are more parameter-efficient as they avoid resolution-specifc parameterizations. In response to this demand, several extensions of
Equual advising.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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function spaces have been proposed [3, 25, 32, 41, 42, 53]. However, SOC theory for building diffusion bridges in function spaces is still demanding in the community of generative modeling.
diffusion-basee generative models for
To address these challenges, this work introduces extension of SOC for diffusion bridges in limnitt-dimmnniooaal Hilbert spaces, particularly focusing on its applications in sampling problems. Specifically, we demonstrate the idea of Doob's h-traasffrmm [13, 59] can be derived from SOC theory and extend its formulation into Hilbert spaces. Due to the absence of a density with respect to the Lebesgue measure in is nontrivial task separated from the spaces, building diffusion bridge between function spaces cases [51, 61]. To this end, we propose a Radon-Nike Madon-Nikodym derivative relative a specified Gaussian reference measure in Hilbert space. Leveraging the
Doob's ' fi-transform and SOC, we then formulate diffusion bridge- based sampling algorithms function spaces. While the infinitediimensional Doob's 's h-transform has already been derived in [2, 26, 31], our main goal is not merely to derive it. Instead, we aim to generalize various finite-dimensional sampling problems [51, 61, 73, 77] into the infmite-dimensional space by exploiting the theory of infinite-dimensionall SOC.
To demonstrate the applicability of our theory, we present learning algorithms for two representative problems. First, we introduce an cimnnte-dimensionall bridge-matching algorithm as an extension of previous methods [51, 61] into Hilbert spaces, which learns a generative model to bridge two distributions defined in function spaces. As an example, we show that our framework can learn smooth transitions between two image distributions in a resolution-free manner. Second, we propose simulation-basee Bayesian inference algorithm [73, 77] that operates in function space. Instead of directly approximating the target Bayesian posterior, our algorithm learns a stochastic transition from the prior to the posterior within the function space. We demonstrate the utility of this approach by inferring Bayesian posteriors of stochastic processes, such as Gaussian processes. We summarize our contributions follows:
Based the SOC theory, we derive the Doob's h-transform in Hilbert spaces. We propose a h function as a Randon-Nikodym derivative with respect to a Gaussian measure in infinite- dimensional space.
Based on the infinitediimensional extension of the Doob's h-transform, we present the diffusion bridge and simulation-based Bayesian inference algorithm in function spaces.
We demonstrate our method for various real-world problems involving function spaces, including resolution-free image translation and posterior sampling for stochastic processes.
Notation. Consider real and separable Hilbert space H with the norm and inner product denoted by Il-llz and (-- Throughout the paper, we consider a path measure denoted by PP(-) on the space of all continuous mappings Ω C((O, T), H). The stochastic processes associated with this path measure pp()) are denoted by x(), and their time-narginal distribution at time t E [0, 7] as
fush-forward measure 1h : (x IP(-), Furthermore, for function V [0. 7] x H - RR we define D.V, Dxx V as the first and second order Fréchet derivatives with respect to the variable x E H. respectively, and 0, V as the derivative with respect to the time variable E [0,T].
2 A Foundation on Stochastic Optimal Control for Diffusion Bridges
In this section, we first present a brief introduction to the theory of stochastic optimal control (SOC) in infinite dimensions and the Verification Theorem (Lemma 2),,, which is the key to understanding the theoretical connection between SOC and the diffusson bridges. Then we propose Doob's h-transform in Hilbert spaces based the SOC theory (Theorems 2.2 and 2.3).
2.1 Preliminaries
and (H. B(H)) be two mea- surable spaces and consider H-valued random variable X : Ω + H. such that the forh-forward measure ja := X y Q induced by X Gaussian, i.e., a real-valued random variable (u, X)x H fol- lows Gaussian distribution on IR for any u E H. Then, there exist unique mean mx H given by (mx, u) E,, [(X, u) and nonnegative, symmetric, and self-adjoint covariance opera- tor Q H + 7. defined by (u, Qu)H E,, [(X mx, u) H. (X -mx v) for any u, E H.
Gaussian Measure and Cameron-Martin Space. Let

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We write = -m Q) and say X is centered if mx = 0. For a covariance operator Q, we assume H-valued centered X follows the distribution (0, Q) supported on so that Q is guar- anteed to be compact. Hence, there exists an aigen-system {()(k) ((h)) E x H: E N such that (()(()
(A) (A) holds and Tr(Q)= = Ik= A(M) < cc. We define the Cameron- space by Ho == (H),. Ho _ H is a separable Hilbert space endowed with the inner product (u, u) == (Q- u, Q 1)2
Stochastic Differential Equations in Hilbert Spaces. The standard -aluue Wiener process has independent increments W - We ~ NN(, A,Ta). In the case of ciinitedimmessinal Hilbert space Ha, however, such identity covariance may not be trace class. We consider larger space Ho H1 such that Ho is embedded into M1 with a Hilbert-Schmic embedding J Ho - H1. Let Q = JJ and we define Hi-valued Q- Q-Wiener process [17, Proposition 4.7] W = Σ oc We focus on the Cameron-Matte
k=1 ( (k) space Ho where the Wiener process has increments (0, A,Ix).
and larger space H- H. where the Wiener process has increments (O, A) Then we define path measure IP and associated stochastic differential equation (SDE) in H as follows
dX, AX, + odW?, Xo E H, [0,
(1)
where A: H H. is linear operator, a constant a > 0 and a Q-Wiener process Wº defined on a probability space (S2, F, (Fi)e2o. P). For a more comprehensive understanding, see [17, 29].
2.2 Stochastic Optimal Control in Hilbert Spaces
From the uncontrolled SDE introduced in equation (1), various sampling problems in Rd including density sampling [5, 73, 77] and generative modeling [10, 11] can be solved by adjusting the SDE with proper drift (control) function
Rd Motivated by these approaches, introducing stochastic optimal control (SOC) to solve real-world sampling problems, we aim to introduce SOC to the infinite-dimensional Hilbert space H. We consider that controlled path measures _Ptt is induced by following
SDE defined as follows:
dx;
[AX: + aQ dt + odW? X-= = Xo.
E [0.
where ar.) [0 H + H. is infinite-dimensional Markov control (see Section A.1.2 for more details). We refer to the SDE in equation (2) as controlled SDE, The controlled SDE can be exploited to the various problems. In general, it can be done by finding the optimal control function that minimizes the objective functional with suitably chosen cost functionals depending on the problem:
I Jt, = Epa [."
[[I((a))) [R(a.)] ds G(X)) = X
where R: : H. R are running cost and G H. R is terminal cost. The measurable function J(t, x, a), representing the total cost incurred by the control a over the interval [t, given that the control strategy from the interval [0, I) has resulted in X x. The objective is minimize the objective functional in (3) over all admissible control policies a E U4 where 14 is the Hilbert space of all squane-integrable H-valued processes adapted to Wº defined on [0, T). Then we define the value function Vft, = inf J (t, x, a). the optimal costs conditioned on (t, x) E [0, x H. By using the dynamic programming [23], we can solve the Caciiitoo--acoi---aaie
(HJB) equation, (4)
V,v. + -VV + inf (a,
+ R| 00, V x) = G(x),
aeu
where CV, = (X,,, AD&V.) + + [o2Q v. We demonstrate that with specific choices of cost functionals R and G in (3), the optimal control a* of the minimization problem aligns with the proper drift function for sampling problems we will discuss later. To do so, we start with how the HJB equation can characterize the optimal controls.
Lemma 2.1 (Verification Theorem). Let V be solution of HJB equation (4) with R(a) } ||a||
satisfying the assumptions in A.I. Then, we have V(t, x) & Tt,, for every a E l and
(1,x) x) E [0, T" x H. Let xa"",""" be an admissible pair such that
a arg inf (a =212222)) + las las = /2V aeu
(5)
for almost every s [t, I and P-almost surely. Then (a" " satisfying V(t,x) [t,xx,a").
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Lemma 2.1 demonstrate that with specific choices of running costs, the solution to the HJB equation in (4) the optimal cost of the minimization problem in (3) with the closed-form optimal control of -0 DV, In the subsequent subsection, we reveal the connection between the optimal controlled process, characterized by the controlled SDE with optimal control a², and the conditioned SDE in H. Additionally, we show various problems depending on the choice of the terminal cost G.
2.3 Doob's h--raansfrrm for Diffusion Bridges in Hilbert Spaces
The conditioned SDE is stochastic process that is guaranteed to satisfy a set constraint defined over the interval [0, T.. For instance, Diffusion Bridge is stochastic process that satisfies both terminal constraints X,- = Xg- XT = XT for any Xo, XT E H. We will show that with a proper choice of the terminal cost G. the conditioned SDE is special case of controlled SDE. For this, let define the
function h [0, x H IR:
J,x ((z) CAx.Qr, (dz) = TE G(XT))X, =
lut,, x)
where Noltx.Q, is Gaussian measure with mean function exxxx and covariance operator Q+ So el(=))AQelts)AAi and G == e- for the function in (3). The function h evaluates the future states X using & which propagated by (1) for given initial x at time E [0, It can be shown that the function satisfies the Kolmogorov-backe backkord equation [17] with terminal condition,
0,h, + Lh, = 0, x(T,x) G(x).
Now, we employ the Hopf-Cole transformation [24] to establish an inherent connection between two classes of PDEs, the linear PDE in (7) and the HJB equation in (4), which provide us a key insight to deriving the Doob's h-transform in function spaces utilizing the SOC theory.
Theorem 2.2 (Hopf- Cole Transform). Let V/ -log hr. Then V. satisfies the HJB equation:
1 |0
D&V. /2V 2 H
V.V. + CV:
=0, V(T,x) = -Gxx).
(8)
According to Theorem 2.2, the solution of linear PDE in equation (7) is negative exponential to the solution of the HJB equation in (8). Given that it already verified that the optimal control a* results the value function Ve, which has explicit form as described in (8), we find that the relationship of two PDES through V; = - log he leads to a distinet form of optimal control a* = -0Q1/2 D = oQ 1/2 Dx log h, where Dx log h == Dxh/h. Consequently, it yields another class of SDE as follows:
dx t = [AX] dt + o² DDQDx log h(t, x()] dt + odw? X = =x=-
(9)
where T is Q-Wiener process P This representation is consistent with infinite-dimensine dimensional conditional SDE [2, 26] which induces an expansion of Doob's /--rranformm [59] in Hilbert space H.
The Doob's hi-transform in mnite-dimensionll spaces is well-established to construct the diffusion bridge process under the assumption that 11(1, x) = P(X, E dxr| X = x) has an explicit Radon- Nikodym density function [46, 60]. enable simulate the bridge SDEs. In contrast, although the choice of G(x) 1dxr (x7) in (6) yields same representation of htt,, x) = PPX E dx x), we cannot easily define the Radon-Nikodym density in H. due to the absence of an equivalent form of Lesbesgue measure which hinder the computation of Dx log h(t, X(') in (9) explicitly. Hence, to define the diffusion bridge process in H. we need to identify an explicit density form of h(t,x) P(Xq E dxy X, x). The following theorem reveals the explicit form of h function and becomes a key ingredient in deriving infinite dimensional diffusion bridge processes.
Theorem 2.3 (Explicit Representation of h). For any t t> O and any E H. the measure NatAx.Q: and No. are equivalent, where No, is an invariant meusure of P in (1) as + oo where
Qx = -AA- Moreover, for any x, y E H, the Radon-Nikodyy density dN, dN.1A..0. (-) = (x(,..) is given by
dNo.
(xx =
(,) - exp 2 ((1 -10 Q- 1-112
(10)
+ ((1- -11-0))-1 1/2 Q- 1/2 yy'n
1
(0 -,)- 2= 0-1/3)) (11)
2
where 0 = Q2, 20 ",, "",","""000""",,,,,"""""""""""""

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Theorem 2.3 states that the marginal distribution of certain classes of SDEs described in (1) has an explicit
density with respect their invariant measure. Therefore, by the time-homogeneity of the process in (1), it allows us to define the h function explicitly:
((t, x) = / Gla)N..T-AAA.e--... =
=
(dz).
(12)
This framework enables the construction of in (12) properly. Below, we demonstrate the infiniteddimension diffusion bridge.
bridge process, by selecting Ō
Example 2.4 (Diffusion Bridge in HD. Let ((()) E R x H k E N) be eigen-system of H. Then for each N, the SDE system equation (1) can be represented as:
ix (b) = = (K) de aov Xwdd!!
( x
(13)
where Astk) (h) Qolk) = (k) ((k), X (Xt, ( H and w! (We, (00)) H. Then,
for any XT H. the conditional law of XT given X is Gaussian N(m x M 2n1 with
(k)
(k)
F = 0 (T--))
E o² Au
(1 (1-e e-an- -2aa(T-t)).
(14)
2ak
Now, by setting the terminal condition in (6) G(x) 1xp(x) (i.e., Dirac delta of xT) then h(t, x) = -r--e(x, xx). Thus, for each coordinate h. we get following representation:
dx!
-axx
(k) 2age 2age-oa(T-3 (T-))
2Tx (XT (-) x dt +ooo////W (15)
with two end points conditions X (k) =x("
(k) and X (M) (K)
XiT
2.4 Approximating path measures
Since the function h is intractable for general terminal cost G in (12), simulating the conditioned SDEs in (9) requires some approximation techniques. As observed in Theorem 2.2, finding the function is equal to learning the control function a such that fa is equal to F := _pa" = 1p Therefore, with parametrization
a(-,0) O) the approximation can be done by neural network parameterization i.e., ax Re a with local minimum 0* = arg ming D(P*) whereD(Pes [[P* is a divergence between IPa and P" For example, the cost functional described in equation (3) can be represented as relative-cntropy loss (PP ||P*) = Ex- [log 1 Therefore, the training loss for 0 can be estimated by first simulating the parameterized control path and then calculating (3) for specified cost functional R. G. Moreover, we can access to the P*, one can define the variational
optimization [69] where the loss is defined as cross-entropy loss Deross (IPa [IP*) Ep-
log dP
aro
See [20, 48] for more details about the approximation technique and other loss functions.
3 Simulating Diffusion Bridges in Infinite Dimensional Spaces
Leveraging the SOC theory within H. we show how our approach generalizes existing diffusion- based sampling methods. Specifically, incorporating the relation between controlled SDEs (2) and conditioned SDEs (9), we introduce two learning algorithms that allow us to simulate various diffusion bridge-based sampling algorithms.
3.1 Infinite Dimensional Bridge Matching
In this section, our objective is to learn a control a that yields pa such that (X:) [e((, satisfies 44 zs Eri for all pre-specified μą over the interval t E [o, 77. Specifically, we assume that the end-point marginals 1ú and μ. follow the laws of two data distributions To and Nr, respectively, and the intermediate marginals {μC FrE(o.T) are defined as a mixture diffusion bridge paths. This
See See A.5 for details
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Algorithm/ Bridge Matching of DBHS
Algorithm 2 Bayesian Learning in H
Input: Coupling 1, Bridge Pjo.r for v = +, .N do
Input: Initial condition xo. energy functional i4
for = 1, 1,,r. ,N do
Sample (xo,Xr) Xr) ~ Io.
Simulate (o,T ~ pap with Xi Xo Compute Laaye (fx) with (24) Update Px+1 with Vo,, Laves (0m) end for
Simulate X 0.71 Fjo,7 with (Xu,xr)
Compute (aM( wtih (20)
Update On+1 with Ve,, (O.) end for
Output: Approximated optimal control a
Output: Approximated optimal control 0*
learning problem is referred to as the Bridge Matching (BM) algorithm [45, 50, 61] and can be expressed as a solution of the SOC problem structured as
inf ((F*)P"), such that dX; = [AX; +a0 dt + odW? X To.
(16)
In (16), various divergences can be chosen for the same learning problem, as discussed in Section 2.4. Here, we will choose the cross-entropy because the relative entropy requires the appropriate selection of the terminal cost G in (3), which is intractable since we do not have access to the distributional form of To, Tr [44]. Furthermore, keeping the entire computational graph of F with parameterized a can become resource- resource-intensiig intensive, especially for digher-dimensional datasets like images [11].
Now, we specify the optimal path measure 1P* for problem in (16). Let P P be path measure induced by (15) and M+j0.T be a marginal distribution Pjo.T- Moreover, let IP* Io.. for an independent coupling No,r = & Then the optimal path measure IP* is defined as Mixture of bridge. Under regular assumptions, the optimal control a' that induces the optimal path measure P can be constructed
mixture of functions h in (9) by choosing G(x) 1x,(x))
Theorem 3.1 (Mixture of Bridges in H). Let us consider a marginal distribution IP at E T1,
μi (dx,) f f (*jo. (dx, Ho, (dxo, dxr) ) has density pt with respect to some Gaussian reference measure Moe i.e., P8 (dx:)/ / Mne = pt (Xe). Then the optimal path measure Pf associated with:
dx; [AX; + Bx (dxr) [o²QDx NogN(xx:: x Erye) dt + adw?,
(17)
where mrj: = (T'A and Ery = 0² _-- eT (0 and X pe for t E [0,
Objective Functional for Bridge Matching. With the structure of FP specified in (17) we can estimate the divergence between the optimal target path measure 1p* and path measure IP,. To accomplish this, we first define
(t, x: 0) = Q2/2 E Exprr (1der/ (dx)))x)
=2 log N my Enps] at. x; 0]
(18)
Then, by applying the Girsanov theorem' which provides us the Radon-Nikodye derivative between Fs* and po. we can derive the cross-entropy loss in equation (16):
dF+
[
17 X;; 0) ds [
D Deross (Ipa"
IP*)
Ex.
log
Egg.
(19)
d'pa"
Then, under the neural network parameterization of control function a we can reformulate the SOC problem in (16) as a learning problem with the training loss function represented by:
(o)(O) Z Ep (xprodder xx)
2
//2 Dx log (xT;; my Xi Erje) all,, X;:
H
(20)
The (0) in (20) yields the infinite-dimensionaa BM summarized in Alg 1.
3.2 Bayesian Learning in Function Space
In the previous section, we observed that by appropriately defining a terminal cost functional G in (3), the SOC problem aligns with the sampling problem, where optimal control effectively steers
See See A.5 for details.

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the distribution from To to the target distribution TT, where we can access samples from To and Ty. However, accessing samples from unknown target distribution Fy generally not feasible. For instance, for a иT, posterior distribution over function. In this case, although direct samples from Tr are unattainable, its distributional representation is given as [3, 66]:
dur
(XT) & exp (-U(XT)), Pprior
N (mprior: Qprior),
(21)
diprrior
where is a energy function. Here, our primary objective is to sample from a distribution over function тт == pr by simulating the controlled diffusion process {X; [e over finite horizon [0, 2] with T < co. It can be represented as solution of the following SOC problem:
inf (PP")P") such that dx; AX; + aQ dt + odW? X - Xu-
(22)
The following theorem implies that with suitable terminal cost functional G in (3), it is possible to achieve X Wr expansion of [54, 71] for infinite dimensional space H.
Theorem 3.2 (Exact sampling in H). Consider that the initial distribution μi is given as the Dirac
measure 0x, for some Xg H and the following objective functional
[.'3 lalllll ds log
duT
I(a) EEg
(X
(23)
dur
where μr = -e Xx, QT) as a marginal distribution of XT in (1) with a well-defined terminal cost dayr by Theorem 2.3. Then, X. Ty.
dpr
Objective Functional for Bayesian Learning. Unlike problem in (16) where we can access to the drr dur
(XT) X, = x does not have explicit solution. Therefore, we will use the relative-entropy loss our training loss function:
target path measure IP* directly, is not feasible here because (t,, x) Ep
duT Bayes (0) = Dra((o' PT) Epp. [ a(ss x² :00 ds log
(24)
dur
The key difference from previous algorithms [73, 77] is that the Radon-Nikodym derivative duz (X.) dur
may not be well-defined on H due to the absence of the Lesbesgue measure. However, the Theo- rem 2.3 suggests that by choosing certain classes of Gaussian measure /.ee,//priurr = NN Qc), the Radon-Nikodym density of d'tyeor equivalent. Thus, using the chain rule, the terminal cost for any x E H can be computed as follows:
dæy and dur has explicit form since /'prior and (/ Qxx) are
dppece
dut (x) = -x(x) - logg d0 Qxx) (x) + log aN(o,
dfspriior
dut (xx)
log dur
(25)
With Bayys (0) in (24), the deiiite-dimensione bayesian learning algorithm is summarized in Alg 2.
Related Work
Most diffusion models operate within the framework of time-reversal [1], where the generation process is learned from its corresponding time-reversed SDEs [64]. In contrast, diffusion models based conditioned SDEs, such diffusion bridges, built upon the theory of Doob's 's l--transform, offer a conceptually simpler approach as they solely rely on a forward process. [50] proposes generative models with this concept, showing that the mixture of forward diffusion bridge processes effectively transports between couplings of two distributions. [76] introduces a family of first hitting
diffusion models that generate data with forward diffusion process at a random first hitting time based on Doob's 's h--rransform. Combining time-reversal with the h-transform, [46] proposes diffusion bridge process on constrained domains. Moreover, [51, 61] presented that the Schrödinger bridge problem can be solved by an iterative algorithm, which is improved by [18] to enhance
efficiency. Furthermore, [44] generalizes the Schrödinger bridge matching algorithm by introducing an approximation scheme with a non-trivial running cost. Compared to prior works, which primarily focus on fiiie-dimmenional spaces, our work extends the formulation of Doob's 's h-transform into Hilbert space, enabling the development of various sampling algorithms in function spaces.
7

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Experiments
This section details the experimental setup and the application of the proposed Diffusion Bridges in Function Spaces (DBFS) for generating functional data. We interpret the data from functional perspective, known as field representation [75, 79], where data are seen as a finite collection of function evaluations [pp p. Here, a function Y maps points p: from a coordinate space X' to a signal space y, i.e., Y: X - V. Additional experimental details are provided in Appendix A.8.
5.1 Bridge Matching
First, we present empirical results for the infiniteddimensional BM algorithm discussed in Sec 3. applied to ID and 2D data. For ID data, we consider X' and y R. For 2D data, we assume X' 2 and y = I. for probability density or grayscale images, and y = R for RGB images.
Bridging Field. We begin by validating our bridge matching Algorithm in Alg on bridging probability density function within H. Specif- ically, we set To := 6yo with ring-shaped density function Po and Tr == ópr character- ized by a Gaussian mixture density function pr. The functions map each grid points p; to the probability in y R. Therefore, both density functions can be represented their field rep- resentations {po [p:], }. (pr pi], p. re- spectively. Figure 1 illustrates the progressive propagation of the target optimal bridge process F' from po to pr. Despite the a* is trained the functions generated from IP+ which are evaluated on a coarse grid [p:),²² pa" is capable of producing accurate functional evaluations on a finer grid (p5), This resolution-invariance prop- erty indicates that our method is adept learning continuous functional representations, rather than merely memorizing the discrete evaluations.
Xo po
X. Pojo.7
=
Figure 1: (Top) Diffusion Bridge P evaluated on 32 (Bottom) Learned process p evaluated on 256²
Betca
Data
54
(a) Quadratic
(b) Melbourne
(c) Gridwatch
Figure 2: Results on ID function generation. (Left) Real data and (Right) generated samples from our model.
ID function generation. We conducted an experiment 1D function generation task,
Table 1: A Power(%) of kernel two-sample test.
NDP [22]
SP-SGM [52]
DBES (Ours)
comparing our bascline methods [22, 52] on
Quadrati
99.0
0.7
51
three datasets: Quadratic, Melbourne, and Grid- watch, following the setup from [52]. For gen-
Melboure
12.8 1 0.4
5,3 1 0.7
9.67 1 0.45
Gridwatch
16.3
4.7 0.5
crative modeling, we set the initial distribution To as (O Q) with RBF kernel for the covariance operator Q and the terminal distribution as data distribution Tr, respectively. We employing the bridge matching algorithm in Alg 1. For quantitative evaluation, we used the power of kernel two-sample hypothesis test to distinguish between generated and ground-truth samples. Table I shows that our method performs comparably to baseline infinite-dimensional methods. Additionaly, The generated samples compared to the ground-truth for each dataset are provided in Figure 2.
Unpaired Image Transfer. We compare our proposed model with finite (fixed) dimeessai baseline through an experiment on unpaired image transfer between the MNIST and EMNIST datasets at 32 resolution, as well as wild and cat images from the AFHQ dataset [14], down- sampled to 642 resolution (AFHQ-64). Specifically, we evaluate the performance of [51, 61] and our DBFS model. For a fair comparison, we follow the iterative training scheme of [51] based on
8

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(a) 32 (observed resolution)
(b) 64 (unseen resolution)
(c) 128 (unseen resolution)
(d) 612 (observed resolution)
(c) 128 (unseen resolution)
Figure 3: Results on Unpaired image transfer task. (Up) EMNIST + MNIST (Down) AFHQ-64 64 Wild + Cat. (I.eft) Real data and (Right) generated samples from our model. For generation at unseen resolutions, the images within the red and blue boxed initial conditions were upsampled (using bi-linear transformation) from the observed resolution (322) for EMNIST and (64") for AFHQ-64 Wild, respectively.
the public repository", where two forward and backward control networks are trained alternately. For quantitative evaluation, we estimate the FID score between the generated samples and real datasets. We set a = 1 for both [51] and our method, while FID scores for [61] are taken from [18]. Table 2 shows that our method performs comparably dimensional method. Additionally, we provide generated samples at vari- ous unseen resolutions in Figure 3 demonstrate the pooluutioniivariann property of our have slightly lower FID scores compared to finite-dimensional baselines,
the finite-
Table 2: Test FID on un- paired image transfer task. (A) EMNIST -> MNIST, (B) AFHQ-64 Wild -> Cat.
models. We note that our method may
which may align with the observation in [79] that resolution-agnostie methods tend to have lower FID scores compared to resolution-specific methods can incorpo-
Method
(A) (B)
TDBM [51]
8.2
ones. This could be because resolution-specie rate nomain-speciffe design features their score networks. Samples generated from the reverse direction can be found in Figure A.2.
DSBM [61]
6.0
25.4
DBFS (Ours)
9.1
44.4
result from [18).
5.2 Bayesian Learning
We validate our Bayesian learning algorithm for modeling functional data. Specifically, we will consider the temporal data as a function. We denote Y(O] = (Y(p))) as a collection of function evaluation on a set of 1-dimensional observation grid = { !o) where S Po <... < < S We assume that each observed time series approximates a corresponding underlying continuous function X : IR Rd as the number of observations increases i.e., { (Y [p:]]]
0
X. For given observations Y YOO], our goal is to infer the posterior distribution on some set of unobserved grid T [0, /] O i.e., IP( [[[[T] Y (O) and therefore modeling distribution over X on [0, /]. Please refer to Section A.8.2 for further details.
Functional Regression To verify the effectiveness of the proposed DBFS in generating functions in the ID domain, conducted regression experiments using synthetic data generated from the Gaussian Process (GP) by following the experimental settings in [38]. Figure 4 shows the sampled trajectories of a controlled dynamics X for t [0, Z 2'] trained on data generated from GP with RBF covariance kernel. The stochastic process begins from the deterministic function XT- Xo at t and propagates towards the conditional posterior distribution X P t T.
under MIT License.

---

Table 4: Regression results, "context" and "target" refer to the log-likelhoods O and T, respectively.
RBF
Matérn 5/2
Periodic
Method
context
target
context
target
context
target
CNP
0.97 + 0.01
0.45 + 0.01
0.85+ 0.01
0.21 + 0.02
-0.16+ 0.01
-1.75 0.02
NP
0.90 + 0.01
0.42 + 0.01
0.77 + 0.01
0.20 + 0.03
-0.18 + 0.01
-1.34 + 0.03
DBFS
1.02 + 0.01
0.47 + 0.01
0.93 + 0.01
0.25 + 0.01
-0.15 + 0.01
-1.88 + 0.02
Misdin
Figure Sampled functions from a learned stochastic process X evaluated on [0, [] for E [0, 7.7]. The grey line represents the mean function E[X;] and the blue-shaded region represents the confidence interval. (Left) GP with RBF kemel. (Right) Physionet.
Imputation. We evaluate our method against recent diffusion-based imputation method where the goal is to infer the conditional distribu- tion p( [T) Y [O]) of unobserved grid Y [ give observations Y[O]. CSDI [68] utilizes DDPM [34] to learn the reverse process by treat- ing the temporal data Y(O] as a O) xd dimen-
Table 3: Test imputation RMSE on Physionet.
Model
10%
50%
90%
CSDI [68]
0.60 + 0.27
0,66
0.06
0.84 + 0.04
DSDP-GPT
0.52 :: 0.04
0.64 +: 0.05
0.81 +: 0.03
DBFS (Ouns)
0.50 0.04
0.04
::
results from [6].
sional feature. Extending this, DSDP-GP [6]
enhances CSDI by incorporating noise derived from a stochastic process, instead of simple Gaussian noise. We maintained the same training setup as these models, including random seeds and the model architecture for control a" in (2). Consistent with their methodology. we employed the Physionet dataset [30], which comprises medical time-series data collected on an hourly rate. Since the dataset inherently contains missing values, we selected certain degrees of observed values to create an imputation test set for evaluation. We then reported the results on this test set, varying the degrees of missingness. Table 3 shows that we outperform the previous methods even though it solely relies on forward propagation of controlled SDEs in (9) without denoising procedure.
Conclusion and Limitation
In this work, we shed light on the application of the crinite-dimensionns Doob's h--transform, exploiting SOC theory in fiinite-dimensione spaces. By developing an explicit Nikoo-Ni density, we address the challenge posed by the absence of an equivalent to the Lebesgue measure. With specified cost functions for control objectives, it enables us to extend previous algorithm based on the dinteedimeeniooal Doob's 's li-transform into dimenseadimeniionall function spaces, such as resolution-fiee free unpaired image transfer and functional Bayesian posterior sampling.
Compared to the recent infinite- -iiensi-dimensionall score-based diffusion model [42], our work restricts the coefficients for the stochastic dynamics be tine-independent. This limitation prevents us from defining noise schedule for the diffusion model [78], which may hinder performance improvements. Additionally, in Bayesian learning, computing the gradient of the proposed training loss function (24) can be computationally demanding. Thus, developing more scalable algorithm would be an interesting direction for future work. Furthermore, as our model can be applied to any functional domain, we have limited our experiments to regular ID and 2D domains, leaving the extension to more general domains for future work.
Broader Societal Impact
Similar to other works in the literature, our proposed method holds the potential for both beneficial outcomes, such as automated data synthesis, and adverse implications, such as the deep fakes, depending on how it is used. We adhere to ethical standards for using our model in generative AL.
10

---

Acknowledgements
This work was partly supported by Institute of Information communications Technology Planning & Evaluation(IITP) grant funded by the Korea (MV)))) (No.RS- 20.RS-2001-11190073 Artificial No.2022-0-001 Development and Study of Al
Intelligence Graduate School Program(KAIST]) Technologies to Inexpensively Conform to Evolving Policy on Ethics, No. 2022-0-00612. Geometric and Physical Commonsense Reasoning based Behavior Intelligence for Embodied Al) and the National Research Foundation of Korea(NRF) grant funded by the Korea governnen((MSIT) (NRF- 2021M355D0020030, SRRF-2022RIASA708830
RS 2024-0041008))
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15

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A Appendix
A.1 Verification Theorem and Markov control
For the derivation, we will use the following assumption
Assumption A.1. The function V [0, uniformly continuous on bounded subsets of [0. I] x H. and (0, x H. respectively. Moreover, for all (t, x) (0,T) H. there exists C1, C2 > 0 such that
H. R and its derivatives D&V, Dxx V..v o,v are
(tt, x) + [tx x)1 (t, x)] Dxx x)I IA' C1 (1111+x))))) (x)) (A.1)
where A" is adjoint operator of A.
The HJB equation (4) can be derived by the following theorem.
Theorem A.2. Let assumptions A. hold and let the function V with V(T,x) G(xx satisfying the dynamic programming principle for every 0 < t < t' < TT
(tt, x) = EE [T" [(s, + v(as)) V(L X.
[(a)]
(A.2)
Then is a solution of the following equation:
(o0% V, a) + 1 |la|? =0, V(T.x) G(x),
0, + LV + inf
(A.3)
aell
Proof.
The proof can be found in [23, Theorem 2.34].
A.1.1 Proof of Lemma 2.1
Proof. To begin the proof, we can formally compute the minimum of F(DxV). Since [15]
F(x) = inf aeld (x,
+ || ||a = 2 |x||
(A.4)
Therefore, ((oQ1/2D2V) takes the infimum at a² -0Q1/2 Next, applying Itô's for- mula [23, Proposition 1.165] V and taking expectation both sides, we get
(tt, + E/.00 5. (2.V(s,x))+
(V(s + + (oQ) X;), an) ds
(A.5)
where we denote B [+]
Eps [-]X, = x] By incorporating the fact that V satisfies the equation
(A.3), we can derive the following by adding Efa
[T" ||a,,1? ds to both terms. The LHS
equation (A.5) becomes:
[ || ds al)
E
V(T, X
+ Byxx
(A.6)
=G(X})
And for the RHS of the equation (A.5):
T
J.
(tt, x) + E
||a.11 ds
+ +E
( Dx,,,
(0Q)
x3) ab) ds
=V(,,)) + Ex,x L." (X)V LV,,
+ 1 |la.112 ds
((1(11//222166 X3) ( 11as 2 F(oQQ/22 x.i) ds
V(1,x) x) + x
(A.7)
16

---

where the last equation can be derived by adding and subtracting E/ S (s, X;)ss and incorporating the fact that V satisfies the equation in (A.3) again. Hence, we get the following equation:
a(t, = v(t,x))+
1aaal - F(oQ (22s, ds
(A.8)
Since, by definition
(oQU DxV X;), a.) + ||,112 - F(Dx x:)) 20
(A.9)
Therefore, by taking the infimum over 1 in the RHS of (A.8),
Jtt, x, a.) 2 V(t,x)
(A.10)
Moreover, since we already verified that the F(oQu/2D2) has infimum at a* = = Therefore, by choosing u = ax, we have
(0Q1/2 D.VDD&Vsss xx), ue) + ||ul| 2
=0
(A.11)
Thus, get
[t,, u) = V(t, x).
(A.12)
Therefore, together with (A.8), this implies that (ax", xx") ) is optimal at (1,x) E [0, 7) x H This concludes the proof.
A.1.2 Markov Control Formulation
Now, we introduce the following corollary that states the Markov control formulation.
Corollary A.3 (Markov Control [23]). us consider the measurable function de : (t, x -
which admit a mild solution X c: of the following closed-loop equation:
dX
AA th + 1Q21222200 (..x..x))) ds dw? X. =
(A.13)
Then the pair (a x)),, where the control a is defined by the Markov feedback law at 6(8 x is admissible and it is optimal al (1, x) for all 8 E [.
Therefore, in the context of the initial value problem, such as in case, consider the form of the Markov control ag == ag = 6(s, X for E [0,T]. The proof and details can be found in [23, Chap 2.5.1].
A.2 Proof of Theorem 2.2
Proof. Let us consider the function Vtt, r) = log
0,h -hav, Dh - D²h hDxV & D&V hDxxV,
(A.14)
Recall that h satisfy the KBE in equation (7):
0,h + Ch = 0, hr = G.
(A.15)
Since h -V hence 0,h - - - Then,
0,Vh = Ch
(A.16)
(Dxh, Tr o02 DxxhQ1
(A.17)
=-(hDRV,AN + Tr 002 -
(A.18)
= =- + 2
Tr [o2 (hD2V @ Q
-r O hQ)
(A.19)
17

---

We can simplify the last equation as
0,v - (.xV.AX, + Tr (D&V Q]
xxx
(A.20)
2
= =-VV+ + Tr [o (D&V & D2V)
(A.21)
2
Following [69], the second term of RHS can be derived as follows
TT (D&V Q
Σ
Oo2 D2V) .l0))
(A.22)
2
KEN
1
(o Dx D2V.DAV.Q0
(A.23)
2
KEN
00 DxV, Q6( 6()))
(A.24)
2
KEN
(o2QDVV ((x)) 6(0))
(A.25)
2
KEN
2 (0
QDV))DVV
(A.26)
(oQ1/2 DxV 2
(A.27)
Therefore, combining the above results, we have
0. + LV 2 1000/222))1 V4 =.
1
(A.28)
H
Since (A.28) coincides with (4) with (f+):= == & 11-1112- this concludes the proof.
A.3 Proof of Theorem 2.3
Proof. A proof of Theorem 2.3 is based [16, Chap. 10.3]. Since Qoo = -1AA-- is trace class, we define a trace class operator O, as follows:
c = Q (Q, 1/2 M
12 """
(A.29)
for all 2 Since Q: = Qx - elA Qooe '1'
we can rewrite Q in terms O,,
Q: -xx e²AQ 2A0"0
(A.30)
Q1//2 [ (2x Qud
(A.31)
Q1/2(1 222
(A.32)
1/2 x for all x E Ho- It implies that ((1 ()))xxx)Hx Thus, we have (1- 0,)x Q 12 2=00==
2
the non-negativity of (1 - O,). Moreover it also implies that (1 - 0,) invertible:
(1 - -1= -
(A.33)
Consequently, yields the following formula [16, Proposition. 1.3.11]
qe (0, = - d) exp | - (0, (
(1)1-)))) 0,) -10 00,0011,2110 yHH
(A.34)
Now, for the general case, by using the chain rule, have:
yx dN dNo (y) dN (0,))
(A.35)
dNo.e. dNo
dNo.e.
18

---

and utilizing Cameron-Martin theorem [16, Theorem. 1.3.6], we get:
dNanx00
(y) exp (Qi 1/22" AA
x,
1/11//100 y)H
1/2
1
Qi 1/2 el
12
(A.36)
dNo.2.
2
71
= exp
(4 1/2 e Qi
ynn (1113))
Q2 1/2 exMxx,
x, Qi 1/2 Q:'12 x)
(A.37)
= exp
(Q! -00 e x, 200 1/2 y//
10 Qi 20= eeaxx, x, Q=0 1/2 1001/8/ xx
(A.38)
1
((1 - θ₂) (1)-0)
- 1/2 x, /
((1 0.) 10= c e
exp
2
(A.39)
where (1) follows from (A.33), (1 - 0,) -,)-- 10 ) 1/2 Q
Q, Thus, by
substituting (A.34) and (A.39) into (A.35), obtain the following result:
yx -) exp
(0, ()) -00 1=)
(A.40)
(1--,))- 1 xx 1011))) 2 7(1 0,) 1-000/1//200 // (2
1
-
(A.41)
concludes the proof.
A.4 Derivation of Example 2.4
For diffusion bridge process, let us define the h function
= xi] - f G(2, G (1z. (z)
x xx) htt, Xx,, = Ep (x)
(A.42)
we choose y( = ldy (x). Then, for any y E H and E [O, Theorem 2.3 implies that
/ ((N,
h(0, t, xo.
(A.43)
/ G(Z,y) d. dN (2) (z)dNo.
(A.44)
dNo
/ ((z,))a(x 2))
(A.45)
yx.
(A.46)
Moreover, with an eigen-system of H. {(1(k), (((k)) as [62]:
E ER H k NJ, qi can be represented
(x
dr
(k)
(k) (1()),
(A.47)
KEN
where for each coordinated k, qe (8) (x), (k) ) has following representation:
(1())
-1/2
(y(()) azt
- x2)2
(x)))
4 (k) (A), (1))
- - 222))))
exp
(A.48)
2ax
2x(1) - 2022))
21(k)
AΦ (K) = (
(x) (1) (K)
(2)
(R)
= (x, (k)
Yk yk = (y, (()))
(A.49)
Therefore, since Dx log het, T, xT) Dx log (xx,, xT), by projecting Dx log Qr (xx,
each coordinate ((k), we obtain the following results:
dx (k) log -
d
(x)
xx
(k)
A(+ e (0x(T (x) (k)
2ate (T+))
-os (e)))
(A.50)
19

---

A.5 Deriving Divergence Between Path Measures
Here we present an infiite-dimensionall generalization of Girsanov's theorem [17, Theorem 10. 10.14], which plays crucial role in estimating the divergence between two path measures discussed in See 2.4. The theorem is formulated as follows:
Theorem A.4 (Girsanov's Theorem H). Let y be a Ho-valued Fe-predictable process such that
P ((%
1 ( (1 - 2 2 1l li dt ) )
Inallizo ds < 00 1, E exp
(A.51)
Then the process W Wo - So rsds is a Q-Wiener process with respect to {FiJe20 on the probability space (Ωt, Q) where
dQ exp (J (T No 2 2. Inooliio dt
) d...
(A.52)
Or we can derive alternative formulation, by substituting w we + S0 Ysds to (A.52),
dQ exp (1 (T dW) + f'' Invalizoo at ) dp.
(A.53)
Now, will apply Girsanov's theorem path measures related (1) and (9). Let Q Ipa in (A.53). Then Ys Q1/2a,, and we get
di exp (f + 1 l"" 1 12 7o0
d
(A.54)
dP
Proof. Proof can be founded in [17, Theorem 10.14]
Since our goal is find an optimal control a* such that X satisfying terminal constraints which is represented by function G in (6), may define target path measure 1 dF- where 2 Ep ((x))) Then, we compute the logarithm of Radon Nikodym derivative
G [57],
dis d..
dF
dIp dP
log
= log
+
log
Z
(A.55)
IP
L (a.aa?
2 2.10 Q//2a,
1 ("
Mo +
dt + G(-).
(A.56)
Ho
Since W is Q- Q-Wiener process on pa we can compute the relative entropy loss in See 2.4:
dp
T |as 22 dt G(X+)
(Ps((P((Pt) = Eza
log dIp=
= Ezo
2 2
(A.57)
where we denote 11(-) 1|Q-1/2()) 1 This representation matches with (3) when R(-) I1-1122 Similarly, the cross entropy loss in (19) can be derived by set Q p and P := |P in (A.53). Then "Ye can be defined as (18) and w = w + So" (f)dss is Q- D-Wenner process on 1 where po satisfies the Badon-Nikodyye derivative:
dl
exp (J. (TT0))) dw? + // (1>,(0))1e dt )
(A.58)
dPo
Hence, the cross-entropy loss can be computed as in (19).
20

---

A.6 Proof of Theorem 3.1
Proof. Let consider that the marginal distributions {uz hear satisfying the following relation in a weak sense [4]:
a, / [(x))) - - [(x)) - / (x)(()) (dxx)
(A.59)
where D is adjoint operator of L.. Now, let us denote (x)) = H4 (x, xx) and consider factorizable marginal distribution 14 (dxx) Su (x, XT xT) [d dxr), where 14 (x,|Xo,XX xT)
satisfying the following relation:
ae / f(x)) (dx; XT) S LejorT f ) (xx) (dx: xxo xT)
= =fnr (LAXC, (xi)) + (xx)),((x)), Dxi()))) + 1 Tn se (dxx (x, XT),
where we denote Jojo (xx, == dDQDx log (t,xxx). Now, obtain the Kolmogorov operator associated with the diffusion process associated with the marginal distributions my (dxx) as follows
[(x))) = 0. Jn f(x))) (dx, xx [[(xx, dxr)
a: H
(A.60)
(x (Ases (atss,Dffix)) Dxf(xx)) + (xx),,, (xf()))) + 2 (o (xx)] u (dxx))
(A.61)
-fa (x (dxx)) - / 1 (2)
(A.62)
where we have defined JN h:7 (x)))( (dx²) = Sn by 1u (dx,
(ixx)/Xu Xx; XT II)xx dx It implies
that the diffusion dynamics dX; associated with the Kolmogorov operator = (Ax,, Dx (xx)) (xx(),,, (xx)) + Tr Dxx fxx)] also associated with Fokker-Planck equation [7] Ci (dxx) = O, meaning X; ~ pe for all t [0, Now, for some reference measure foet where the Radon-Nikodym derivatives under Met hü can be defined as follows:
dpter (xx) (x(x)) dujor (x+) = (x), exist. Then, djtee
duf
Sn (
dxT)
h (xx)
(A.63)
Px(x:)
Therefore, the diffusion process associated with marginal distributions 14, (x,) has following repre- sentation:
dx = AX; +
(or) xT)] dt + odw? X; ~ n (A.64)
This concludes the proof.
A.7 Proof of Theorem 3.2
Proof. Let us assume the initial condition fixed to deterministic point X E H and define reference
measure Peef (0, ). Since our goal is X satisfying the terminal condition G, define P*
dr = (.), where 2 Ep (((x))) Therefore, we have the following relation for marginal
distributions duc = 1/2 G x ddur. Moreover, since we have defined G(XT) = - log
dxz (XT)
dur
in (23), then it result htt, x) Ep
dxz dur
(XT) x by following Theorem 2.2, we get G(XT)
(T, Xr). Hence, we have for any Borel set B E B(H).
f
{ hT, Xx)
(u)
dpr
(A.65)
B h0, xo)
8
drT (XT)
[ JN
der
dur / drg
(A.66)
der
(X
dyr
Now, given that we have confirmed that the conditioned SDE in (9) correspond to the controlled
process (9) with optimal control, we can establish the result P P(X. E B) = TT (B). This concludes the proof.
21

---

Encoder x
K
Cross Attention
Self Attention
Decoderx x
Latent array
K
Attention
Self Attention
Projection
Cross
GFFT
(p)
Query encoding
Figure A.I: Transformer based network architecture.
A.8
Experimental Details
A.8.1 Experiment on 2D-Domain
Synthetic Experiment. In synthetic experiment, we computed the log-probability of po and pr across a uniformly sampled grid of 642 points, with each point p, ranging within [-7,7]a For po- the jog-probability generated using 8-Gaussian mixture model as specified in [70]. For pr. we employed the following log-density function:
log logPr(p) m p 11--111-3111 3112 13122,1p. ||p 10--11 5
(A.67)
0.05
Training was conducted using the Adam optimizer with a learning rate of le - 3. The network was trained with a batch size of 24 for total of 1000 iterations. We set 0.2 in (1) for this experiment and set 100 discretization steps. We use single A6000 GPU for this experiment.
The control function was parameterized using 4-layer FNO-2D [40], with the cutoff number of Fourier modes set at 8 and each convolution layer having width 32.
Simulation of DBFS. We follow the simulation scheme introduced in [42, 58]. For x (æ1, x2) R²2 we use the discrete cosine transformation (DCT) for projection. Specifically, the eigenvector (x()(x) (x) and eigenvalue (k) of the negative Laplacian operator - A, which is positive definite and Hermitian, and satisfies the zero Neumann boundary condition, given by:
A (x) (1)) (x)
(A.68)
(x)))(x)
(o(() (x) 0,
(A.69)
Oz
Oz2
It implies that the orthonormality of (.) with respect to the associated inner product, thereby enables computations in (A.47-A.50). Now, considering rectangular domain with Cartesian coordi-
nates, where pixels in the image are sampled from an underlying regular domain, the eigenbasis is given separable cosine basis:
(l,m) (I1, 12)
cos
T
cos
T
(A.70)
H
(m.m) = 2
n
m
(A.71)
W
2
Then, the negative Laplacian - A can then by represented by eigen decomposition -A EDE". where E is the projection matrix for the DCT i.e., X; E X, DCT(X,), and D is digonal
22

---

Algorithm 3 DBFS sampling for Bridge Matching
Input:
Linear operator A, trace-class operator Q, learned control a, initial distribution no. discrete and
inverse discrete cosine transforms DCT, iDCT, target resolution grid T trained resolution grid O
Sample the initial condition Xi TT o O² then
Xg mT
Upsample X [X6 P. h= to [X0
end
for t = =0 , do
Estimate the control oj = =²)
Sample Gaussian noise { (0,
Discrete cosine transform the initial condition, estimated control, Gaussian noise
(x(²)
1k=1 )
DC;),
{ā(
jkmI = DCT(a)), [E()) T
1= DCT(E)
for 1. 1,... T do parallel
X (1)
[+ -a ( + + (1) A. A
end for
Inverse discrete cosine transform, X iDCT((X (4) X Xi-a,
(4A, 1E
Output: X ~ FT
matrix containing the eigenvalues X (n.m), Hence the controlled SDEs (2) with Q I can be rewritten
as
dx -DX; σād dt odW., t E [0,
(A.72)
where X, = E. E,,X, E ar, and W, W, for all i E [0,
Sampling Algorithm. The sampling algorithm for DBFS in bridge matching, discussed in Section 5.1, is provided in detail in Algorithm 3.
Unpaired dataset Transfer Experiment. we followed the setup described in [51].
For the experiment involving transfer between dataset,
(A) For the EMNIST and MNIST datasets, the initial distribution, Tp. was set as the MNIST dataset, while for the terminal distribution, Ty, used the EMNIST dataset with the first five lowercase and uppercase characters, as outlined by [19]. The iterative training scheme proposed by [51]. was adopted, which involved two neural networks, (t, 0) and ag (t, v)), each with around 20.7 million parameters. These networks approximate mixtures of bridges for the forward (To - TT) and reverse (лT my) directions, respectively. The SDE a noise schedule. The DBFS model was trained for 60 iterations, with each iteration comprising 5,000 gradient updates. Additionally, 2,560 cached images were used for training each network, updated every 250 steps. We used the Adam optimizer with a learning rate of le-4 and a batch size of 0.999. The complete DBFS training for the MNIST experiment took approximately 15 hours on a single A6000 GPU.
discretized into 30 steps without
128. with the EMA rate
(B) For the AFHQ dataset [14] evaluated DBFS between the wild and cat classes on a 642 grid, with each class containing approximately 5,000 samples. The control networks each contained about 120.3 million parameters. We discretized the SDE into 100 steps without using noise schedule. The Adam optimizer was used with a learning rate of le-4, and the EMA rate was set to 0.999. We used a batch size of 64 and trained for 20 iterations, with a total of 400,000 gradient steps. Additionally, 2,560 cached images were used for training each network, updated every 1,000 steps. The training took approximately 8 days, using 8 A6000 GPUs for the experiment.
For each control network a, we used a transformer network architecture inspired by PerceiverIO [36] from public repository" to model functional representation. This transformer architecture was chosen for its efficiency in evaluating field representations over a large number of grid points and its
under CC BY-NC 4.0 License. pytoochh, under MIT License.
23

---

Table A.1: Network Hlper-paramettes
Dataset
Latent dim
Position dim
#heads
#ene blocks
#dec blocks
#self attm. per block # parameters
MNIST
256
256
4
6
2
20.7M
AFHQ
512
512
120.3M
ability to map arbitrary input arrays to arbitrary output arrays a domain-agnostic way, Additionally, we adopted the attention mechanism proposed in [49]. The decoding cross-attention mechanism was also modified, inspired by [39], with the output query set the target grid points, which were transformed as Gaussian Fourier features [67].
In practice, we start by passing the evaluations X through a single MLP layer and combine with the Fourier feature embeddings of the grid points This combined representation is then input into the encoder blocks as keys and values, where QKV attention is first applied with the latent array the query, followed by self-attention for each block. We implement the attention block proposed by [49], embedding the time into latent space. Next, the target grid points, represented as Fourier feature embeddings of the grid points p, are fed into the decoder blocks as queries. Here, QKV attention applied with the encoded latent array serving as keys and values, followed by self-attention in each decoder block. Finally, the decoded array is mapped to grey scale or RBF channels using a single linear layer. Conceptual illustrations of the proposed network are presented in Figure 1. and detailed network hyperparameters listed in Table A.I.
5
6
8
e
C 6
O
3
9
C
C
A
D
9
2
A
C
9
0
6
7
9
b
C
e
O
G
8
O
b
e
8
C
e A C
3
8
2
B
e
A
C C
6
C
7
&
2
T
C
B
b
/
S
0
e
6
e
(a) 32 (observed resolution)
(b) 64
(unseen resolution)
(c) 128 (unseen resolution)
61t (observed resolution)
(c) 128 (unseen resolution)
Figure A.2: Results on Unpaired image transfer task. (Up) MNIST + EMNIST (Down) AFHQ-66 64 Cat + Wild. (Left) Real data and (Right) generated samples from our model. For generation unseen resolutions, the images within the red and blue boxed initial conditions upsampled (using bi-linear transformation) from the observed resolution (322) for EMNIST and (64²) AFHQ-64 Cat, respectively.
(a) 64 (observed resolution)
(b) 1282 (unseen resolution)
Figure A.3: Results on Unpaired image transfer task. AFHQ-64 Cat + Dog.
24

---

A.8.2 Experiment on 1D-Domain
For the experiments on ID-domain, we consistently set A := - and Q == cxp(-
cp p /n)
and set 7 0.2 and 7 0.02 for GP regression and imputaion, respectively. The choice of hyper-parameter, we search over the set [0, 01, 0.2, 0.5, 1.0) and find optimal value for GP regression. For imputation, we set 7 = 0.02 by following [6].
Terminal Cost Computation For all experiments conducted the ID domain, we implemented a parameterized initial condition which takes as input the observed sequences X = x (Y()). We employed the energy functional 2/ as the Gaussian negative log-likelihood (NLL). For each evaluation point on T, U can be computed as follows:
T
X PP|
Yppill 20
u(x, = - log N( 00)
(A.73)
where of is set as an output from the neural network in accordance with [38] for GP regression, and fixed as og = 0.5 for imputation, to establish a loss function analogous to [6]. Additionally, we specified a learnable prior distribution Aprror (N Ix" ,Qr). Consequently, the terminal cost retains only the NLL term, simplifying the computation.
GP regression For the GP regression, we borrow the experiment setting from [38]. The model trained with curves generated from GP with RBF kernel and tested in various settings such as data generated from GP with other type of kernel (Matér 5/2, Periodic). We generated p uniformly on the
interval [-2, and generated Y[p] from using RBF kernel k(P:, Pу) 13 exp( I|p, P; I1 /13) with 1 ~ On.1 1.0) and 12 ~ Unif(0.1, 1, 0.6) and the white noise { ~ (0, le 2) is added. We set | randomly from Uni and TT from Unif(3, 50 - |O)) For the other test data, we define <(P,: p,) = (1 + /5d /22 + 5d2 /(31))))) -V5d/l2) with d = (llP, Pyll). 1 Unif(0.1, 1.0) and l2 ~ 10.1, 0.6) for Matérn kernel and < <(P.. P2) = & exp(p= 2 sinn ||P: p,ll /p)/t2 with 1 1ni 1.0), 12 ~ Uni 0.6) and p Unif(0.1, 0.5) for periodic kerne.
We set batch size of 100 and trained for 100, 000 iterations. The Adam optimizer is used, the initial learning rate 5e-4 decayed with cosine annealing scheme. For testing, evaluated the trained models using 3,000 batches, each consisting of 16 samples. We report the mean and standard deviation for five runs. We single A6000 GPU for this experiment.
The architectures for NP [28] and CNP [27], we use the same setting as described in [38]'. In our approach, we adapted the CNP architecture incorporate parameterized initial condition x" (add one linear layer to output xº). The total number of parameters similar across all three models.
Physionet Imputation The Physionet [30] contains 4000 clinical time series with 35 variables for 48 hours from intensive care unit. Following [68], we preprocess this datasets to hourly time- series which have 48 time steps. Since the dataset already contains around 80% of missingness, randomly choose 10/50/900 /90% of observed values as test data.
In the imputation experiments, we employed the same experimental setup as [6] which is slight modification of original CSDI model. In this setup, the Gaussian noise of the DDPM model was replaced with GP noise employing approach to record the actual elapsed time rather than rounding to the nearest hour, better capturing the inherent timing characteristics of the Physionet dataset. We use single A6000 GPU for this experiment.
RBF kernel. Additionally, we adjusted the measurement
under MIT License.
'nttps://github. Diffusion, under Apache 2.0 License.
Process__
25

---

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Justification: Yes, the main claims made in the abstract and introduction are consistent with the scope of the paper.
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Question: Does the paper discuss the limitations of the work performed by the authors?
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26

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Justification: Yes, we provided the required resourees in the experimental details section.
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27


Reviews:
[{"review_id": "DfLtMYUDK9", "reviewer": "Reviewer_WKVm", "summary": "The paper presents stochastic control in function spaces with applications in diffusion bridges and Bayesian learning. Since the Lebesgue measure does not exist in infinite dimensional space, the authors derive Doob-h function with the Radon-Nikodym density with respect to a   suitable Gaussian measure and conduct bridge matching experiments under this setup.", "questions": "Comments and major questions: \n\n1. As the paper introduced in section 2.1, one can instead consider the cylindrical Wiener processes on the Cameron-Martin space. What will break down in the current results? Will the cylindrical Wiener processes set-up bring convenience to the experiments as it is implemented in finite dimensions?\n2. How do you arrive at equation(21)? What assumptions are required, and what are the regularity requirements for the energy function? It would also be good to remind the readers what mu_T is here.\n3. In equation(25), how are energy function U and covariance operator Q determined in general? What are the particular choices used in the presented experiments? \n4. What are the challenges of using time-dependent diffusion processes in the current method? \n\nMinor points: \n\n1. The paper repeatedly refers to Lemma 2.2, but the authors seem to be referring to Theorem 2.2.\n2. The paper should include a proof of Theorem 2.3 for its completeness and rigor.\n3. Why is it H_0 instead of H in Theorem 3.2?", "limitations": "The authors have adequately addressed the limitations.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Overall, the paper is well-motivated and well-written. The paper reviews the stochastic control in function space and the connection of Doob's h-transform with stochastic control and bridge matching in Section 2. It transits smoothly to Section 3, where it proposes an algorithm for diffusion bridges in function space, and an extension for Bayesian learning.", "weaknesses": "As the theory exists for stochastic control in function space, and there is a recent work on h-transform [1] and generative model in infinite dimensional space, the novelty mainly lies in the application to bridge matching and Bayesian learning. These applications are interesting and important, however, the weaknesses are in the discussion on Bayesian learning, and the experiments on bridge matching. In particular, there should be a comparison with finite-dimensional bridge matching; several arguments in section 3.2 about Bayesian learning need more clarification (see questions for the details of this point).\n\n[1] Baker, Elizabeth Louise, et al. \"Conditioning non-linear and infinite-dimensional diffusion processes.\" arXiv preprint arXiv:2402.01434 (2024).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely appreciate your interest in our research and acknowledgment of its significant contributions. We are also grateful for the insightful questions raised by the reviewer, to which we have provided detailed responses in the subsequent text.\n\n-----\n\n**1.Comparison with finite-dimensional bridge matching**\n\n- Thank you for your valuable suggestion to improve our work. In general response, we have included a PDF file containing additional experiments and a comparison with finite-dimensional method [1]. We will incorporate these results into the paper.\n\n-----\n\n**2.Cylindrical Wiener process**\n\n- In theory, selecting a cylindrical Wiener process for infinite-dimensional SDEs can also be a viable option. Indeed, when prior knowledge of the data domain isn't available, opting for a cylindrical Wiener process, as noted by the reviewer, can facilitate the construction of a bridge model. However, the choice of Q determines the geometric structure of the Hilbert space where functional data resides. Therefore, If we model Q so that our Hilbert space contains characteristics of the target data, such as smoothness or curvature information, it can be beneficial. \n- In practice, where our objective is to model complex data as a function, the choice of the covariance operator Q can significantly impact the ability of learned neural network operators (control in our case). For instance, in time-series imputation, our baselines [2, 3] differ only in their selection of Q (Technically “kernel” since they are finite-dimensional models) —[2] using a standard Wiener process and [3] employing an RBF kernel to construct the Wiener process— leading to performance improvements. Furthermore, in generative modeling, studies [4] empirically demonstrate that selecting an appropriate operator Q enhances the correctness of generation. Specifically,  [4] highlights that opting for a cylindrical Wiener process may lead to issues such as mode collapse.\n\n-----\n\n**3.Regularity condition for U and choice of U and Q**\n\n-  Equation (21) is coming from a Bayes formula [5, Section 2] $\\frac{d\\pi_T}{d\\mu_{prior}} \\propto \\exp(-\\mathcal{U}(\\mathbf{X}))$, where $\\mu_{prior} = \\mathcal{N}(m_{prior}, Q_{prior})$ and the potential $\\mathcal{U}$ is a measurable mapping $\\mathcal{H} \\to \\mathbb{R}$ for a given $\\mathbf{X} \\sim \\mu_{prior}$. Hence $\\mu_{prior}$ in equation (21), equation (25) and lines (245-246, 751) should be changed into. We apologize for the typo and for any confusion it may have caused in understanding the paper. The potential $\\mathcal{U}$ typically chosen as a negative log-likelihood function (also referred to as the potential energy); in our case, we set it as a negative Gaussian log likelihood function. For Bayesian learning, we choose $\\mathcal{A}=-\\frac{1}{2}$ and covariance operator $Q$  as an RBF kernel. We have detailed the setting in Appendix A.9.2.\n-----\n\n**4.Time-dependent SDEs**\n\n- The main challenges of using a time-dependent diffusion process in our method are proving the existence and uniqueness of the invariant measure. For an explicit form of the h-function, as stated in Theorem 2.3, we need to define a certain class of Gaussian measures, where the collection of time-dependent Gaussian measures is equivalent to its invariant measure over a long-term period. This class of Gaussian measures should be defined by a linear SDE.\n\n-----\n\n**5.Minor comments**\n\n- We appreciate the reviewer pointing out the typo related to Lemma 2.2 and $\\mathcal{H}_0$​ in Theorem 3.2. It should be expressed with respect to the norm in $\\mathcal{H}$. We will make the necessary corrections in the revised manuscript. Moreover, as reviewer suggested, we will include the proof of Theorem 2.3.\n\n-----\n\n\t[1] Peluchetti, “Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling.”\n\t[2] Tashiro et al., “CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation”\n\t[3] Bilos et al., “Modeling Temporal Data as Continuous Functions with Stochastic Process Diffusion”\n\t[4] Hagemann et al., “Multilevel Diffusion: Infinite Dimensional Score-Based Diffusion Models for Image Generation“\n\t[5] Hairer et al., \"Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods\"", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FKRtPaMs4R", "reviewer": "Reviewer_m5FK", "summary": "This article proposes a perspective on diffusion-based generative models based on stochastic optimal control, with objective functions based on the log density ratio between objectives.", "questions": "Thanks to the authors for addressing my comments during the rebuttal.", "limitations": "Presentation and motivation - adressed by authors during rebuttal.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 1, "contribution": 2, "strengths": "As far as I could evaluate, the mathematics are correct, and this particular mathematical perspective is new (to the best of my knowledge).", "weaknesses": "I found this submission to have a weak presentation. It reads more like a stochastic calculus journal article than a machine learning conference submission. \n\nThis perspective is not clearly motivated: what is gained by considering an infinite dimensional perspective compared to the wide literature already approaching diffusion-based approaches through the length of stochastic optimal control? Since there are several elements that are infinite-dimensional in nature in this problem (distribution of random variables, score matching functions, etc), some early-on explanation and clarification of the approach considered here would be helpful.\n\nFurther, while a lot of the writing is centered around an infinite-dimensional perspective, this is then converted to a parametric model, with finitely many parameters. How much of the infinite-dimensional perspective is then lost? Is this important?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We gratefully thank the reviewers for their valuable feedback and suggestions. Here, we address the concerns raised by the reviewer.\n\n-----\n\n**1.Early-on explanation and clarification**\n\n- Following the reviewers’ suggestions, we have further clarified our motivation in the general response. Please kindly refer to that section for more details. We hope this provides the clarity you need.\n\n-----\n\n**2.Finite-dimensional Approximation**\n\n- We would like to point out that the model having finite number of parameters does not mean that it should be modeling finite-dimensional models; for instance, Gaussian process regression, one of the most popular infinite-dimensional stochastic process models, effectively requires finite number of parameters to be estimated from the observed data. Usually, the function is evaluated only at a countable set of sampling points that are assumed to be generated from an infinite-dimensional stochastic process. In this case, we can approximate the infinite-dimensional function by fitting the models to the finite sampling points with finite-dimensional parameters.\n\n- The “finite dimensional approximation” happening in our model would be in the part where we are approximating the covariance operator Q. Specifically, we approximate Q via the truncation, that is, choosing the finite number of eigenfunctions among infinitely many ones. This approximation may incur approximation error, but it does not alter the nature of our model as an infinite-dimensional model; To see this, note that our model can model image data in a resolution-agnostic way. This is possible because ours deals with the infinite-dimensional stochastic processes, so it can model a varying number of finite sampling points.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "PSwYz0DDmB", "reviewer": "Reviewer_jSMz", "summary": "The authors investigate the notion of h-transform in infinite dimensional state spaces and provide a novel representation (Theorem 2.3) based on connections to stochastic optimal control.\n\nThe authors introduce two approaches to using this h transform derivation - firstly in something resembling bridge matching whereby both marginals are known and secondly by simulating the process with network parameterized h transform and taking gradients through the simulation.\n\nThe authors then apply this to image super resolution and Bayesian inference tasks in function space.", "questions": "See weaknesses.", "limitations": "See weaknesses.", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- Derivations appear correct\n- Although the h-transform has been described for infinite dimension through Hilbert spaces in the context of diffusion models in Baker et al 2024 (https://arxiv.org/pdf/2402.01434); as far as I am aware this connection to optimal control is novel.\n- Experiments are reasonable compared to other infinite dimensional methods (see below) but still not on the same level of fixed dimension methods for e.g. superresolution.\n  - Spectral diffusion processes, Phillips et al 2023: https://arxiv.org/abs/2209.14125\n  -  Neural Diffusion Processes, Dutordoir et al 2022, https://arxiv.org/abs/2206.03992\n  - Baker et al 2024 (https://arxiv.org/pdf/2402.01434)", "weaknesses": "- Motivation for infinitedimensional diffusion bridge is not very strong and experiments are not very convincing. \nI am not so familiar with the Bayesian inference experiments and what is SOTA. There are a few baselines missing as noted below. For the superresolution task there are stronger and simpler methods which have not been discussed. I think some stronger use-case in scientific applications would be needed for a higher score.\n\n- More discussion with Baker et al 2024 (https://arxiv.org/pdf/2402.01434) would be appreciated\n- As the authors note, the second training method for Bayesian learning problems (Alg 2) requires taking gradients through the simulated diffusion which can be slow, unstable and memory intensive. This goes against much of the diffusion model philosophy of splitting the generative problem into smaller problems through time and solving each jointly. I fear this will not be very scalable beyond 2D.\n\n**Experiments**\n- FID score or other quantitative metrics are note provided for superresolution tasks.\n- There are no baseline or comparisions to other methods. There are many superresolution, and infinite dimensional diffusion methods.\n- The authors compare to neural processes but there are more recent and comparable baselines for similar infinite dimensional / functional/ Bayesian experiments  which do not rely on gradients through the simulated process, such as:\n  - Spectral diffusion processes, Phillips et al 2023: https://arxiv.org/abs/2209.14125\n  -  Neural Diffusion Processes, Dutordoir et al 2022, https://arxiv.org/abs/2206.03992", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the recognition of our paper's strengths and extend our thanks to the reviewer for their comprehensive review and insightful comments. Below, we provide detailed responses to address each valuable comment.\n\n-----\n\n**1.Motivation, Comparison with baselines**\n\n- We agree with the reviewer’s concerns regarding motivation and experiments. To address these, we have included a PDF file in the general response with additional experiments comparing unpaired image transfer methods with finite-dimensional baselines [1] and 1D functional generation with infinite-dimensional baselines [2, 3]. Additionally, we have further clarified our motivation. Please kindly refer to that section for more details.\n\n-----\n**2.Comparison with [4]**\n\n- [4] primarily focused on developing the conditional diffusion process in function space. To achieve this, they defined a Doob’s h transform in infinite-dimensional space using Itô’s lemma and Girsanov’s theorem. This approach has its merits, such as enabling the conditioning of non-linear SDEs (while our work has considered the linear SDEs). However, simulating conditioned non-linear SDE often faces challenges because the conditional distribution of such SDE  is generally intractable. Therefore, they require the approximation algorithm presented in [5].\n\n- While our approach shares the idea of developing an infinite-dimensional Doob’s h transform, as the reviewer already mentioned, our primary goal is not merely to derive Doob’s h transform but to generalize various finite-dimensional sampling problems [6, 7] into the infinite-dimensional space by exploiting the theory of infinite-dimensional stochastic optimal control. \n\n- In practice, the choice of linear SDEs to develop the relevant theory might appear to be a strict limitation for modeling complex distributions. However, similar to most recent diffusion-based models, the linear form may be sufficient for modeling. Moreover, this choice can be beneficial as it allows for more scalable algorithms due to the closed-form solution of the conditional distribution. In this light, by leveraging the theory of stochastic optimal control along with the choice of linear dynamical systems, our contribution also includes proposing tractable learning algorithms for real-world sampling problems.\n\n-----\n\n**3.Computational concerns**\n\n-  As the reviewer pointed out and as we stated in our paper, Algorithm 2 may induce computational difficulties. While it is possible to consider a more computationally favorable approach, such as implementing the adjoint solver [8] for memory efficiency or using the variance reduction technique proposed in [9], in the current work we focus on the theoretical property that the optimal control still yields Bayesian posterior sampling despite being defined on function space. Proposing a more scalable algorithm will be an interesting direction for future work.\n\n-----\n\n\t[1] Peluchetti, “Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling.”\n\t[2] Phillips et al., “Spectral Diffusion Processes”\n\t[3] Durordoir et al., “Neural Diffusion Processes”\n\t[4] Baker et al., “Conditioning non-linear and infinite-dimensional diffusion processes”\n\t[5] Heng et al., “Simulating Diffusion Bridges with Score Matching”\n\t[6] Zhang et al., “Path Integral Sampler: A Stochastic Control Approach For Sampling”\n\t[7] Shi et al., “Diffusion schrodinger bridge matching”\n\t[8] Li et al., “Scalable Gradients for Stochastic Differential Equations”\n\t[9] Xu et al., “Infinitely Deep Bayesian Neural Networks with Stochastic Differential Equations”", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ms9xcCVSAB", "reviewer": "Reviewer_uBiL", "summary": "The paper uses a stochastic optimal control to derive Doob's h-transform in infinite dimensions, and it shows the relation between solving the optimal control problem and learning diffusion generative models. The approach applies both to bridge sampling and for generative modelling. The approach is demonstrated on infinite dimensional problems, including bridges between images and bridges between probability distributions.", "questions": "no questions", "limitations": "yes", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Strengths:\n- well-written and interesting paper\n- the stochastic optimal control approach to deriving Doob's h-transform is well-founded and interesting\n- the authors derive a Bayesian inference algorithm using a reference measure\n- the method is tested on simple examples", "weaknesses": "Weaknesses:\n- Doob's h-transform in the infinite dimensional setting has been derived using other methods in previous papers (both in the linear and non-linear cases, e.g. ref [2,47], https://arxiv.org/abs/math/0610386). I believe the list of contributions and the introduction does not clearly show that the current paper is not the first to do this, e.g. [2] is first mentioned much later in the paper. I am not sure the introduction and list of contributions adequately reflects this, something that should be addressed before acceptance", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We would like to express our gratitude to the reviewer for their thorough evaluation of our work. We appreciate your recognition of the merits of our research.\n\n-----\n\n**1.Clarification on prior work**\n\n- We agree that clarifying the relationship between previous papers on Doob's h-transform and our work is essential to avoid any unintended confusion. Our contribution involves leveraging stochastic optimal control to derive Doob's h-transform and extend finite-dimensional SOC problems based on conditional diffusion into infinite-dimensional spaces. In the revised manuscript, we will explicitly outline how our contribution compares to previous works, particularly [1].\n\n-----\n\t[1] Baker et al., \"Conditioning non-linear and infinite-dimensional diffusion processes.\"", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We sincerely appreciate the time and effort the reviewers have dedicated to evaluating our paper. In response to their valuable and insightful feedback, we have provided some general responses that address comments common to all reviewers. **The attached PDF file includes relevant figures and tables for additional experiments.**

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**1.Comparison with baselines** 

- In line with the reviewers' suggestions, we conduct additional experiments to demonstrate the applicability of our method to various real-world problems.

- First, for a comparison with recent infinite-dimensional baselines, we conduct an experiment on 1D function generation task. We evaluated our method against baselines on three datasets: Quadratic, Melbourne, and Gridwatch, following the setting provided in [1]. For generative modeling, we set the initial distribution as a centered Gaussian distribution with covariance operator $Q$ and the terminal distribution as the target data distribution and utilize the bridge matching algorithm in Alg 1. We used an RBF kernel for $Q$. For quantitative evaluation, we employed the power of a kernel two-sample hypothesis test which attempts to distinguish between the dataset from generated samples. Table 2 in the attached PDF file shows that our method is comparable to the baselines. Moreover, we provide a generated sample compared to the ground-truth for each dataset in Figure 1.

- Second, we compare our proposed model with a finite(fixed)-dimensional baseline. We conduct an experiment on unpaired image transfer between MNIST dataset and EMNIST dataset. We compare the performance of the [2] and our DBFS. For a fair comparison, we adhere to the iterative training scheme proposed by [2]  where two forward control and two backward control models are learned alternately. We set $\sigma=1$ for all methods. For quantitative evaluation, we estimate the FID score between the generated data samples and real datasets. Table 1 in the attached PDF file shows that our method is comparable to the finite-dimensional method. 
 Furthermore, we provide additional generated samples at various unseen resolutions in Figure 2 to demonstrate the resolution-invariant property inherent in proposed infinite-dimensional models. We want to stress that our method may have slightly lower FID scores compared to finite-dimensional baselines. This may reflect observation in [3], where resolution-agnostic methods often have lower FID scores compared to resolution-specific methods. They argue that this is because resolution-specific methods can incorporate domain-specific design choices into their score networks (e.g., translation equivariance in CNNs for images). An interesting direction for future work would be to develop well-designed score operators for infinite-dimensional diffusion models.

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**2.Motivation**

- Traditionally, there has been interest in sampling from a probability measure on infinite-dimensional Hilbert spaces, particularly in Bayesian inverse problems [4]. Recently, the modeling of data as continuous functions has become increasingly popular within the machine learning community. This functional representation avoids the need for discretization, enabling the handling of data at arbitrary resolutions. Consequently parameterizing these functions with neural networks provides memory efficiency and the flexibility to represent various data forms [5]. For example, we can regard an image as a continuous function, where the function takes a 2-dimensional grid pixel location as input and outputs grayscale or RBF channels. Therefore, it is an infinite-dimensional object, as a continuous function can produce outputs for any 2-dimensional input defined on some domain.

- Since diffusion-based models are powerful inference tools for various tasks, researchers have been working to extend these models to handle functional data representation. To achieve this, they have generalized the framework of previous diffusion models by extending their formulation into infinite-dimensional Hilbert spaces, also known as function spaces [6, 7]. However, previous diffusion-based generative models typically focus on sampling from a target data distribution. This framework cannot easily address various sampling problems, such as distribution transfer or exact sampling from a posterior distribution (in functional form as in equation (21)). 

- In the finite-dimensional case, these problems can be solved by exploiting the theory of stochastic optimal control (SOC) [8]. This motivates us to extend and generalize finite-dimensional SOC into the infinite-dimensional case to meet the demands of sampling problems from a functional perspective. In practice, by generalizing previous SOC-related problems into infinite-dimensional space, our model can naturally achieve resolution-free data transfer between any two image distributions, perform posterior sampling from a distribution over functions such as GP-posterior, and modeling irregular time series.

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	[1] Phillips et al., “Spectral Diffusion Processes”
	[2] Peluchetti, “Diffusion bridge mixture transports, Schrödinger bridge problems and generative modeling.”
	[3] Zhuang et al., “Diffusion probabilistic fields”
	[4] Hairer et al., “Signal Processing Problems on Function Space: Bayesian Formulation, Stochastic PDEs and Effective MCMC Methods”
	[5] Dupont et al., “From data to functa: Your data point is a function and you can treat it like one”
	[6] Franzese et al., “Continuous-Time Functional Diffusion Processes”
	[7] Lim et al., “Score-based Generative Modeling through Stochastic Evolution Equations in Hilbert Space”
	[8] Zhang et al., “Path Integral Sampler: A Stochastic Control Approach For Sampling”

Meta-review: This paper tackles the problem of extending diffusion models and diffusion bridges to infinite-dimensional function spaces. The authors propose a novel approach that leverages stochastic optimal control (SOC) to derive Doob's h-transform, a key tool for constructing diffusion bridges, and extend it to infinite dimensions. This provides an interesting framework for learning bridges between infinite-dimensional distributions and generating samples from them.

The reviewers acknowledge the paper's significant contributions in bridging the gap between diffusion models, infinite-dimensional function spaces and SOC, and the potential of the method for handling complex problems involving continuous function space representations. However, when preparing the final version of the paper, the authors should carefully address the remaining concerns of the reviewers: better position their work in relation to existing work on Doob's h-transform and include more extensive experiments. This will improve the paper's impact.
Decision: Accept (poster)