Conditioning non-linear and infinite-dimensional diffusion processes
Authors: Elizabeth Louise Baker
Gefan Yang
Michael Lind Severinsen
Christy Anna Hipsley
Stefan Sommer
Venue: NeurIPS 2024 spotlight (2024)
Keywords: Doob's h-transform
bridges
infinite dimensions
conditioning
non-linear stochastic differential equations
Primary area: diffusion_based_models

Abstract: Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, conditioning non-linear processes in infinite dimensions has not been explored. This paper conditions function valued stochastic processes without prior discretisation. To do so, we use an infinite-dimensional version of Girsanov's theorem to condition a function-valued stochastic process, leading to a stochastic differential equation (SDE) for the conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where we discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.

Conditioning non-linear and infnite-diimensional diffusion processes
Elizabeth Louise Baker
Gefan Yang
Department of Computer Science, University of Copenhagen elba@di ku. dk
Department of Computer Science, University Copenhagen gyõdi ku. dk
Michael L. Severinsen
Christy Anna Hipsley Department of Biology, University of Copenhagen christy. hipsley@bio ku. dk
Globe Institute,
University of Copenhagen mi chael. baandesund. ku.
Stefan Sommer
Department of Computer Science,
University of Copenhagen sommer@di. ku.
Abstract
Generative diffusion models and many stochastic models in science and engineering naturally live in infinite dimensions before discretisation. To incorporate observed data for statistical and learning tasks, one needs to condition on observations. While recent work has treated conditioning linear processes in infinite dimensions, con- ditioning non- non-linear processes in infinite dimensions has not been explored. This dnectionvvluuee stochastic processes without prior discretisation.
paper conditions To do so, we use an infinieediimensional version Girsanov's 's theorem to condition function-valued stochastic process, leading to stochastic differential equation (SDE) for conditioned process involving the score. We apply this technique to do time series analysis for shapes of organisms in evolutionary biology, where discretise via the Fourier basis and then learn the coefficients of the score function with score matching methods.
Introduction
When modelling finite-dimensional data, such as temperature or speed, there are well-known methods for incorporating observations into stochastic probabilistic models, for example, those based Gaussian Gaussian-process process regression [Rasmussen and Williams) 2005). For non-linear models, techniques like Doob's 's fi-transform can be used [Rogers and Williams [2000] Chapter 6]. But for data that is function-valued (and thus infiniteddimensional) with non-lineat models, conditioning is still an open problem. This paper introduces way of conditioning infnite-dimmesiinal diffusion processes by introducing an dininse-dimeniionall version of Doob's 's h- h-tramsform. We then discretise the conditioned process and sample from it; that is, we condition and then discretise rather than discretising and then conditioning.
We present methods to condition process hit specific at the end time, also known bridges. This method covers the case of conditioning strong solutions to Hilbert space-valued stochastic differential equations (SDEs). For the conditioning, consider two scenarios. The first is that the
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Trajs. Mcan
--0
t 0.2
t- 0.4
t t= 0.6
0.8
1.0
Figure We condition an SDE between two curves, representing two butterfly species, (starting from red dashed and ending at green dashed). Each time point of the trajectory represents a shape. Row 1: We take the mean over 20 trajectories. Row 2: We plot the individual trajectories, used in the mcan calculation.
transition operators of the SDE solution are smooth, which generally not obvious [Goldys and Maslowskil 2008). In the second scenario, we condition on observations with Gaussian noise. This technique can be applied whenever the solution to the SDE sufficiently differentiable. To condition, we use the define Doob's 's h t-trrnsform analogously finite dimensions. We then use score- more-mntching techniques, allowing us to sample from the conditioned process. We do this by training on the coefficients of the stochastic process, represented in the Fourier basis.
counterparts of Itô's lemma and Girsanov's theorem, enabling us to
One specific use case is modelling changes in morphometry (i.e. shapes) of organisms evolutionary biology. The morphometry of organism be modelled points, curves surfaces embedded in Euclidean space. [elsenstein [985] suggests using Wiener processes to model changes in morphometry, for example, height. Sommer et [2021] propose extending this methodology to whole shapes by using stochastic flows to define diffeomorphisms on Euclidean space [Kunita 1997 Then, changes of the shapes are modelled deterministically
Rd that act on the embeddings (Younes] 2019]. Recent work has generalised this to the stochastic setting by considering diffusion bridges between shapes [Arnaudon et al.) [219 2022]. However, they first discretise the shapes and then find diffusion bridge between the discretised shapes. In this work, we condition, and then we discretise. In doing so, we show that the number of points goes to infinity, the bridge is still well-defined. Moreover, we may use other discretisations for shapes such as Fourier bases.
diffeomorphisms
Related work and contributions
2.1 Related work
In finite dimensions, methods have been developed approximate non-linear bridge processes of the form Equa- tion (6). When conditioning on an end point y at time T. the conditioned process contains an intractable term log x: T. y). called the score term. This term can be replaced with V log p(t. f, x: T. T.y). where is the transi- tion function from another SDE with a known closed form, such as Brownian motion
0.
0.4
other linear processes [De- Iyon and Hu 2006 van der Meulen and Schauer 2022 Then we can sample from the approximation instead, and use Monte Carlo methods using the ratio given by the Radon-Nikodym derivative a likelihood ratio, thereby sampling from the true path distribution [van der Meulen and Schauer 2022
0.2
0.0
0.0
0.5
0.5
1.0
Figure 2: A stochastic process between two butterfly outlines (Papilio polytes in red, Parnassius honrathi in blue).
Recently, Heng al. [2021 adapted the score- matching
methods of Vincent [2011], Song and Ermon] (2019). [Sog et al. [202]] to learn the score term for non-linear bridge processes. To do so, they introduce a new loss function to learn the time reversal of the process. They then learn the time reversal of the time reversal, which gives the forward bridge. Our work uses their method learn the score term after discretising the SDE via truncated sums
of basis elements. Phillips et al. 22022 also consider using truncated sums of basis elements for

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discretising SDEs, however, only for diniiii-adieniinalll Orstein-Uhlenbeeck processes, which are linear.
Recent work on generative modelling has investigated score matching for infrnitedimmnsiional
diffusion processes [Pidstrigach et al. [2023] Franzese al. 2023 [Bond-Taylor and Willcocks [023
Hagemann et al. [203] Lim et al. 2023]. This problem is similar to our task of conditioning an SDE, but not the same: The main difference is that our SDEs are fixed, known a priori, and potentially
nonlinear, whereas in generative modelling the SDE can be chosen freely. Hence, generative modelling often uses linear SDEs because the transition densities are known in closed form. this sense, our problem relates to generative modelling but has different setup.
In shape space, there is interest in defining stochastic bridges between shapes [Armaudon al. 2023]. Shapes are represented in the LDDMM framework (Younes) 2019]. where they are modelled embeddings in Fuclidean space, e-g. curves and surfaces sets of landmarks approximating curve or surface. The deterministic image registration problem of matching two shapes is solved by finding "minimum energy" mapping. More specifically, for two shapes so, S1 IRd Rd we
find a diffcomorphism f: [0, 7" x I + Rd such that f(0,-) so(-) and at f(T,-) :) s1 (-)) and f
optimises given energy functional Bauer et al. 2014
We employ a stochastic version of the LDDMM framework for our experiments. Instead of direct paths from fo to fr that optimise energy functional, we define stochastic paths of diffeomorphisms between two shapes. For infinite-dimen dimensional shapes such as curves, stochastic shape analysis was
the focus of [Trouvé and Vialard, [012 Vialard [013 Arnaudon al., 2019], where stochastic processes are defined in the LDDMM framework. However, these do not explore the problem of
conditioning the defined processes. In [Amaadon al.) 2022). bridges between fnite-dimension
shapes are derived. This is the first work to bridge between mninite-dimensional shapes.
2.2 Contributions
1. We derive Doob's h-trransform for infinit-ddimensiona non-linear processes, allowing conditioning without first discretising the model.
2. We detail two models: one for direct conditioning on data and the second for assuming some observation error.
3. We use score matching to learn the score arising from the h--transform by training on the coefficients of the Fourier basis.
4. We demonstrate method in modelling the changes in the shapes of butterflies over time.
3 Problem Statement
We assume that the data lives in separable Hilbert space (H, ()- and let (9,, F, be probability space with natural filtration {Fr} and take B(H) be the Borel algebra. We take a Wiener process W a separable Hilbert space U (where U can equal H) with covariance given by Q. We then consider Hilbert space-valued SDEs of the form
dx(() (AX()) + (( B (())d(W()
X(0) Eo H,
(I)
where A: ((A) C H H is the infinitesimal generator of a strongly continuous semigroup, and F [0. H H, B [o. T H HS(Q)/2(U), H), where HS(QQ H) denotes the Hilbert-Schmidt operators from Q (U) to H, and Q1/² is the unique, non-negative symmetric, linear operator on U satisfying /2 1/2 = (see Röckner and Claudia 2207 Proposition 2.3.4] for details). Note that F' and B can depend non-linearly on Xtt), so other methods, such as Gaussian process regression, cannot be used to incorporate data. We also need the following assumptions about Equation (T):
Assumption 3.1. Equation I has a unique strong Markov solution denoted X(t, Eo). Assumption 3.2. The solution Xtt, {) is twice Fréchet differentiable with respect to the initial value E. with derivatives continuous on [0, T] x H.
Assumption 3.1 is strong, but satisfied when is bounded and and satisfy certain Lipschitz continuity and boundedness conditions. With some extra Fréchet differentiabiliiy conditions F
3

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and B, Assumption 3.2 will also be satisfied. See Da Prato and Zabczyk] [2014] Part II] for details on solutions. Under the above setup, we consider two problems in conditioning the model on given observational data. First, we tackle the exact matching problem:
Problem 3.1. (Exact matching) Condition X such that X ( Eu) E T where r S H is a set with nonzero measure.
More precisely, we define original expectation conditioned on the event that X (T, Eu) E Т: Bnew = - (T) E r]. Section 5.2.1 discusses this. To solve this, we also require extra assumption:
new probability measure, under which the expectation equals the
Assumption 3.3. The transition operator P(C, 1) == Elór (t, <)) is twice Fréchet differentiable with respect to ç. and once with respect to f with continuous derivatives.
In infinite dimensions Assumption 3.3 is a strong assumption; however, we will study some specific sets I' for which this is satisfied in Sec. [.3 Moreover, with extra conditions on A and Q, there are also SDEs that satisfy this assumption for any nonzero measure Borel set. See Da Prato and Zabczyk [2014] Theorem 9.39 and Theorem 9.43] and Cerrai [200T] Section 6.5 and Section 7.3] for examples. The inexact matching problem does not require Assumption 3.3 and allows us to consider observation
noise:
Problem 3.2. (Inexact matching) Condition X so that X (T', Eo) is "near" an observed function V.
Exact matching could be rephrased as conditioning such that at time T, the distance between X(T, and a set r is equal to O. For the inexact matching, we instead condition such that the distance between X (T,, Eo) and a target set (or target function V) is Gaussian with mean O. More generally, we can also take any differentiable radial basis function on the distance instead of a Gaussian.
For both problems, we show that the process (tt, {u) conditioned exhibit the wanted behaviour at time T, will be process (t, {o) satisfying the SDE
(t) [AX*(1) + F(X (O) + (O) + B(X (1) V (t, (1))
(2)
(0( = Eo.
where Ve (oog(h( {)) is score function. For the exact matching problem, we will see that h(t, {) Elór
(TX E set E) = [X f, - VI mr: 0, o]]. for f Gaussian function, and V target function. The SDE Equation (2) is analogous to the case of conditioning finite dimensions, so the form may not be surprising. However, it is not obvious that this should work for non-linear equations in infinite dimensions.
{)). For the inexact matching problem, we will instead
4 Background
4.1 Strong Markoy solutions
We consider Hilbert space-valued SDEs of the form Equation T for W a Q-Wiener process in a Hilbert space U, where Q can be the identity operator. We only consider strong solutions to Equation [p. An H-valued predictable process X is strong solution Equation ① if e [0, T] it satisfies a well-defined integral
x(1) + J, [AX(s) F(s + S. ((s)
(3)
We refer to Da Prato and Zabczyk 2014] for a discussion on the existence of strong solutions (and more general solutions). However, when F, B satisfy some Lipschitz and linear growth conditions, and when A is bounded, Equation [ has a unique strong solution. Since this is true for any initial value E, we use the notation X (t. <) mean the unique solution of Equation ( with initial value &. This solution is a Markov process and for f H + R bounded function, measurable with respect to the Borel algebra the transition operators P. ( (X (t) satisfy the Markov condition:
()), | F.) (l 8, ()) = ((,((()) ())
(4)
This says that the expected value of the solution at time from a starting value &, given all the information from some previous time 8, is the same as the expected value of the solution started at value X (s, {) at time t

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4.2 Doob's h-transform in finite dimensions
In order to give more intuition for the ceinite-dimensional Doob's s f--transform, we present an informal introduction to the topic. This is all well-known, and the details can be found, for example, in Rogers and Williams (2000 Chapter 6]. Doob's h-transform in finite dimensions is useful theory
for conditioning stochastic differential equations. For example, suppose we have an SDE in IRd
<01) - [(1, + at,, (()) (1), xx0) - Io E R
(5)
and we want to condition this SDE to hit y at time T. This corresponds to finding a measure y/. Doob's h-transform allows us to define such
Q such that [x)x(t)] = ((x((t) x(T)
measure using so called h-functions. Let p(t, y; s,y') be the transition density of Iz, defined as
[0, x Fd + (0, oo) be a function satisfying h(t,,)) = h(t + 8, v)p(t, 1 dy.., such that for 2(t) = h(t, xt)). [[=(T)) = 1. d0
E [x( + s) E A xxt) = yl = P(t, ytt s, s/y Let h Let
Then, =(1) is martingale and there exists a measure Q such that oF Fr Zz. Moreover, under this measure Q, (1) satisfies a new SDE
(x) f(,
oo (l,,="VV log = + =(), x²(0) 20.
(6)
The SDE in Equation (6) can be thought of as a conditioned version of the original SDE in Equation (5) For example, consider what happens when we take 114, x) == p(0,=o:7.x)
p(r.a:T.)) Then for a function ʃ
E [f(((tt)) / f(z) p(t, 2; T,
p(O, ro; T, p(0, zot t, 2)dz = (lf()(t) x<T) y).
This is one example of Doob's h-trransfrmm but other h functions can also be defined, for example, to condition to stay within certain bounds, or not to go above certain value for a certain time period.
p(O. as in conditioning on an end point, there is, in general, no closed form solution for h. Different methods to learn the bridge exist [Delyon and Hu 2006 Schauer al. 2017) More recently, score-based learning methods were proposed to learn the term Va log 2t,
For h(t, := p(t.a::T..))
(Heng al. 2021]. which will adapt the infinite-dimensions setting.
5 Method
We interested in conditioning the stochastic process exhibit particular behaviour at the end time T' We consider two scenarios. The first is exact matching: we condition such that given a set I' C. H, then X (T, {o) E I'. The second is inexact matching: for some Y H we condition such that as & approaches T, X (t, Eo) becomes "close" to Y.
We proceed follows. First, show that can define appropriate random variable. When we have shown that this is possible, we will discuss options for the random variable. We will give specific variables, show that they fit some necessary conditions, and solve Problem and Problem 3.2
probability measure given an
5.1 Doob's h-transform in infinite dimensions
Here, we suppose that we already have an appropriate function f which we show that we can use to rescale our original probability distribution, giving us a conditioned probability.
Theorem 5.1. Let [0, T| H - RR be continuous function twice Fréchet differentiable with respect to & E H and once differentiable with respect to I, with continuous derivatives. Suppose X is the strong solution to the stochastic differential equation in Equation ( Moreover, we assume that Ztt)= == hit, X (t)) is strictly positive martingale, with Z()) 1, and B[Z(T)) = 1.
Then die cTT) defines new probability measure. Moreover, X satisfies the SDE
X(t) =X(0) + S B(X(s)) log h(s, (l))
(8)
1 [AX(s) + FFX()))) S. ((X(s))dW(s))
5

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where W is the Wiener process with respect to the measure
Proof. First, we show that ( h(t, (t)) defines a continuous martingale and apply an infinite- dimensional Itô's theorem followed by Doléans exponential to rewrite Z. We then may apply the coinitediimensiona Girsanov's theorem, and rewrite the original SDE in terms of the resulting Wiener process W. See Theorem C.1 for full details.
5.2 Defining the transforms
Previously, we showed that given a function h satisfying certain conditions, can use this to weight the probability measure, giving us a new conditioned probability measure. Now, we address which h functions to use and the properties of the resulting processes. In infinite dimensions, there is no measure that satisfies all properties of the usual lnite-dimensional Lebesgue measure and so it does not make sense to consider transition densities. However, transition operators of the form P(f(E) = (t,, exist and satisfy the Markov property in Equation {4) so we opt to use these instead. For strictly positive, bounded Borel function e H -> R, take functions of the form
((t, = Cr (X
where Cr = 1/E [o)] is normalising constant. Consider, for example. when the function the Dirac delta function of some set with non-zero measure. Then h(t, E) is the probability that at the end time, the process will be in the set r, given that at the current time, the process is equal E. Functions of the form Equation (9) satisfy some of the necessary assumptions on h for Theorem 5.1 they are positive martingales, with (() 1 and E[Z(T))
Lemma 5.2. Let X be as in Equation ( satisfying Assumption 3.1 Given a function h: [0. T x H + R satisfying Equation (9). Z(t) = h(t, (t, Eo)) is strictly positive martingale such that (0) = and 7(T) Cr (T((T,)))
Proof. Apply the tower property and check the assertions hold. See Lemma C.3 for full details.
The other necessary condition on is differentiability. For this, we consider the specific functions with different vt's separately for the exact and inexact matching cases.
5.2.1
Exact matching
Let H be Borel mcasurable, and suppose that Assumption 3.3 holds. For some instances where
this holds, see Da Prato and Zabezyk [2014] Chapter 9], or Sec. 5.3 Then we define
h(t, = (X - (E((((--(,))))((((XX((((( En)]
(10)
The proof that this h-transform does solve Problem 3.1 is similar to the diiteeddmensoonal version. Note that see:
((( Hence, for some random variable Y
Y /
/
Ydp = E X(T, T).
(11)
(xer)er
5.2.2 Inexact matching
In addition to the exact matching problem, we solve the inexact matching problem by conditioning the process such that, at the end time, it approximates some behaviour. This has two advantages. Firstly, we do not need Assumption 3.3 instead use Assumption (3.2] Assumption 3.22 is satisfied when F and B satisfy Fréchet differentiable conditions. The second advantage that this allows us account for observation noise in models. We condition on the Gaussian distance between the endpoint and some target point or observation by defining function v H + IR that is twice Fréchet differentiable. Then under Assumptions 3.1 and 3.2 (() is strong solution, Markov process and is twice differentiable with respect to the ininal value, the function ht(, {) = E| t, { will satisfy the necessary conditions given the start of Section 5.1
Lemma 5.3. Let vh H + R be continuous function, twice Fréchet differentiable, with continuous derivatives. Then h(t,, 5) = (X - I, E)] is twice Fréchet differentiable & and once differentiable in with continuous derivatives.

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Apply Itô's formula to h and use properties of expectation to differentiate. See Lemma C.4
for the
details.
One such function satisfying Lemma 5.3 is the Gaussian kernel function k: x H lR
20 IX 1 -11:
ko (V, =
exp
(12)
V2Ta
Here, we fix an observation
and parameter o R and vary X X H. The function is twice Fréchet differentiable in each argument, with continuous derivatives; hence the function h(t, g) E| [ (V, (1, satisfies the requirements method of including observation noise in our model. In finite dimensions, to model inexact matching for a stochastic process s(1) E Rdd, one can take the function
Lemma 5.3 Moreover, this gives us
h(t,z)) = Ja(uyy, E)p(t, x; T)
(13)
where v 1Rd is target value, p(t, x;; T, is the transition density of the ---uul stochastic process and fa(:; jr, Σ) is the density of the normal distribution Rd with mean μ E R and covariance Σ E Rdddd See Arnaudon et al. (2022] Section 3.1] for more details. To compare, for fi(:0,0) the density the one-dimensional normal distribution with mean 0 and variance a e R. and P(T - 1") = (T) I' (t) = E). we set
hL4, = Elk,
=
(14)
Defining the function in this way means we condition on distance between (T) and our observation. It also means we can change the distance function to another similarity measure. For example, when the functions represent shapes, we could use another norm that measures the dissimilarity of shapes as in Pennec et al. 20200 Chapter 12].
5.3 Sampling from infiniteddimensional h--transforms
Thus far, we have shown that in infinite dimensions, we can condition stochastic processes either exactly or inexactly, and the conditioned process has form Equation (2). We now turn our attention to sampling from these conditioned processes. For this we discuss how discretise.
For an orthonormal basis {ei of a separable Hilbert space H, let H = span( [ C H.
Let (1,, be strong solution to Equation (1), with X(0) &. Since strong solutions are also weak solutions [Da Prato and Zabezyk 2014, Chapter 6. I.. we can write (11, E) as sum of finite dimensional SDEs, with each finite SDE satisfying
-f (AX(6)
((X(s)), (4X))) / (e,. ((X()))) (15)
(X(1), = (E, (E)
Using Equation 15 we can define an SDE XN as (1 = (X(1), es), where X is the th component of X E R For finite dimensional sets I, C IR we look at the problem conditioning on cylindrical sets of the form
T== P E, 15
(16)
for o; = =ò, es).
Lemma 5.4. Let I' be as in Equation (6 and h [0, x N + R be defined by h(t,
El (X - -t Y) Moreover, define [0.7] x RN + R by g(t,t) = III N ,
Sr (X, (T-
t.y))]. Then (V log htt, Y), e,) = IV g(t, (Y.)
Proof. It follows by noting that E(őr, (Y) = BIIT 8r, (Y:) See Lemma C.5 5 for details.
Since l((1, Y) = ,(, (Yr) the sets N satisfy Assumption 3.3 as long as the sets T2 do in finite dimensions. We have shown that conditioning on sets only depending on the first N eigenvalues equivalent to conditioning the N dimensional projection of the SDE onto the first N basis elements.
7

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Table 1: A comparison of the trained score of the Brownian motion process.
Fourier (num. bases)
Landmarks (num. pts)
8
16
32
16
32
RMSE
5.09
6.66
10.54
7.95
6.08
10.79
Time (s)
105.1
201.8
949.4
95.9
104.8
183.0
Epochs
100
150
300
100
100
100
With this discretisation onto finite dimensions, we adapt a finite-dimensional algorithm sample from the fiittedimennsionll bridges. For this we opt for using the algorithm in Henn et al. (2021), scale up to higher dimensions
since it can be easily applied to Problems 3. and 3.2 and by using a different network architecture. Here, they leverage the diffusion approximation (in this case, Mare-ammaay and score- morthnaa techniques to first learn the time reversal of the diffusion process. Applying the algorithm again on the time reversal, started at the proposed end point, gives the forward- in-timme diffusion bridge.
Experiments
We consider two main setups. Firstly look Brownian motion between shapes and use this to evaluate our method, since for Brownian motion we have a closed form solution for the score function. We then apply this to problems from the shape space literature. There, they are interested in stochastic bridges between shapes which has applications within medical imaging and evolutionary biology [Gerig al. 2001 Amnudon al. 2017 2023]. We expand that body of work, by allowing shapes to be treated as infiiie-dimensional objects when bridging, as in the non-stochastic case [Younes] 2019 Until now, this was impossible for stochastic shape paths, since the theory for this was missing.
The code used for our training and experiments can be found at https:////ii thub com/ Libbyllaaker/nnfseebrdddg and further details on experiments can be found in Appendix B
6.1 Brownian Motion
For Brownian motion between shapes, we look at using both discretisations via landmarks and Fourier bases, for conditioning both exactly and inexactly and compare to the true solution. We train on a target shape of circle with radius 1. For the landmark discretisation, landmarks of the process end at the landmarks of the target shape, and for the Fourier basis we condition such that the chosen basis elements
condition such that the
equal to the Fourier basis of the circle at the end time. In Tab. | we give the mean square error for different numbers of dimensions. For the Fourier basis, we evaluate the score on 100 points, and find the error between this and the true value so that we may compare to the landmark errors. We see that as the dimensions grow, we need a larger training time maintain lower errors. We note that each Fourier basis, contains two parts: the real and imaginary. We train on batches of 50 SDE trajectories, with 40 batches per epoch. For training details see Appendix B.J
6.2 Experiments on shape space
Next we turn to a concrete problem: we model the change in morphometry (shapes) of butterflies over time. Studying changes of morphometry of organisms over time is important to evolutionary biologists. For example. for butterflies, one can ask whether the change in wing shape correlates with change in habitat climate. Rather than extracting nimen-dimensional information from the shapes, such as height or subset of chosen points, we apply the analysis to the entire shape, suggested in Sommer et al.) [202T). Being able condition between shapes is key step in phylogenetic inference, where it will be applied to compute likelihoods of phylogenetic trees from morphological data. Until now, it was only possible for dimensiaimnnional extractions of the shape. Future work will consider extending our methods for parameter estimation of SDEs for phylogenetic inference. This is in order to extend the Brownian motion model of trait evolution to shapes where the SDE models the transitions over the edges in the phylogenetic tree [Felsenstein] 1985].

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6.2.1 SDEs in shape space
For SDEs in shape space we take the SDE defined in [Sommer et al. [2021]
dx(() //- ((X())((() (Qh)
(17)
where for each h E H (2 Q(h) H + H is Hilbert- operator, and & is smooth kernel 1.2 (.2 x R²)
This corresponds to stochastic flow of diffeomorphisms on IR2, with Brownian temporal model. For each æ E 22, X (f, x) models the position of æ IR² at time and the function + X (t, .) is diffeomorphism for all t. To see this we write this in the language of stochastic flows as defined in Kunita [197]]. Define the martingale ((t, x) = QW(t,z), where W the Wiener process on H/ and Q is defined as before. Then, we define the stochastic flow of the martingale F as
p(t,2) x 1-" F(p(r.
(18)
where the integral defined in Kunita 1997, Chapter 3]. Then by Kumita 1997 Theorem 4.6.5] & is smooth, the map p(l, is diffeommrphism for each L. See also Da Prato and Zabezyk (2014]
Chapter 0.1] for general details of lifts of diffusion processes to infinite dimensions.
In Figure g we plot some example trajectories Equation 17 for various parameters, with a circle as the initial value. In Figure 6 we plot one trajectory for butterfly. The trajectories are calculated in terms of a Fourier basis and for Figure we plot the trajectories of subset of points of the shape where we see the temporal Brownian model.
6.2.2 Results
We illustrate our method on butterfly data. To demonstrate, we first use two butterflies with somewhat different shapes [GBIFOOr User 2024). One trajectory between the two butterflies is plotted in Figure In this, we can see the high correlation between neighbouring points, with a Brownian temporal model. In Figure we plot 120 butterfly trajectories, specific time points. For t 0.2 we see that the butterfly outlines are mostly close to the start butterfly in pink, and at time / = 0.8, they closer to the green target butterfly. In Appendix for varying numbers of basis elements Figure 3
B. also plot the score function over time
For the next experiment, take fifty butterfly specimens across five closely related species from the Papilio family (see Figure 5) (Kawahara et al. 2023). The butterflies are aligned via Procrustes, and mean consensus shape obtained using Geomorph [Adams and Otarola-Castille 2013). More details of the butterflies and their processing are in Appendix A We train our model on the mean of the butterflies to learn the time reversal from any given input, to hit the distribution at time T 1.0, which we plot in F
0.5
16
N
24
0.0
0.0
Time
Time
0.4
Time 1
Time
Figure 3: Score fields evaluated on a selection of points at different time steps. general, the score field is expected to "push" the shape towards the target. We show the learned score fields (black arrows) represented by varying numbers of base functions at different time steps, as well as the current shape (blue curves) and the target shape (red curves).

---

Ambrax
Deiphobus
Protenor
Phestus
Polytes
Figure 4: We use dataset of 40 closely related butterflies with five different species. We find mean across the dataset and plot single trajectories between the mean at time = (in blue) and specimen from each species time (in red).
7 Conclusion, limitations and future work
We have proved that Doob's h-traanform can also be used in infiniteddimension for stochastic differential equations with strong solutions. We can condition non-linear function-valued stochastic models on observations, either directly on data or by including observation noise. The conditioned stochastic process satisfies new differential equation involving score function, which we can approximate using score learning. However, due to the reliance on Itô's formula, it would be hard to generalise this proof to non-strong solutions.
To learn the score, we used the architecture detailed in Figure 7 Although this seemed to work well for experiments, more research could go into the network architecture which could further increase the dimensions that we consider. Furthermore, we only do the first step in Heng et al. [202]] and learn the time reversal since eтror compounds in learning the forward bridge. Future work will consider how well the time reversal approximates the forward bridge or how to learn the forward bridge directly. Moreover, as previously mentioned, learning Doob's h- t--rrnnsform only the first step in phylogenetic inference for shapes of species. Future work will consider how to expand the infnite-dimensional bridges inference problems.
Acknowledgments and Disclosure of Funding
The work presented in this article was done at the Center for Computational Evolutionary Morphom- etry and is partly supported by Novo Nordisk Foundation grant NNFI CNNF1000005000, a research grant (VIL40582) from VILLUM FONDEN, and UCPH Data+ Strategy 2023 funds for interdisciplinary research.
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Figure 5: The five closely related species of Papilio. from left to right; Papilio Ambrax, Papilio Deiphobus, Papilio Protenor, Papilio Phestus and Papilio Polytes. subset of the landmarks for each specimen is shown underneath each corresponding image.
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A Butterfly Processing
The Lepidoptera images originate from five closely related species within the genus Papilio from the Papilionidae family (Kawahara et 2023). The images are obtained through gbif.org (gbif.org 2023]. filtering within preserved material from museum collections. The images are segmented with
the Python packages Segment Anything [Kirillvv al. 2023 and Grounding Dino [Liu et al., 2023] The thorax's 's contour is removed from the outline by localising the horizontal local minimum in the four anatomical landmarks where the wing is mounted to the thorax. The separation landmark of the fore and hind wings is set by identifying the vertical valley of the outline on both the left and right sides. The landmarks are used to place 250 evenly spaced semi-landmarks by interpolating the segmentation outline for each wing -mgges with incorrectly placed landmarks, specimens with broken wings, or abnormalities are manually removed during the process.
outline on both the top and bottom sides of the thorax, corresponding
Eight random images were drawn for each of the five species. In total, 40 of 1000 landmarks were aligned using Procrustes alignment. The alignment and the mean consensus shape were obtained by using the package Geomorph v. 4.06 [Adams and Starol--Castill 2013). See Figure 5 for examples of the five species of butterflies and subsample of the aligned landmarks.
B Experiment details and further figures
B.1 Score Learning
Given an SDE discretised over the first N coefficients of a basis function, we learn the score function. do this, we use the algorithm presented in Heng et al.) (202T) for find- ing the score function of a finiteddimensional Markov process s(l)) with transition function p(z(() | x(0)),0 < < T. given by V Vlogp(=(T) a())). Here, they leverage the diffusion approximation (in this case, Euler-Maryyama) and score-matching techniques to
13

---

Table Training configurations for learning scores with different numbers of bases
Input/output dims
Time embedding
Downsampling dims
Upsampling dims
Num. bases
Activation
dims
8
32
32
[64, 32, 16, 8] [128, 64, 32, 16]
[8, 16, 32, 64]
silu silu silu
16
64
64
[16. 32, 64, 128]
32
128
128
[256, 128, 64, 32]
[32, 64, 128, 256]
first learn the time reversal of the diffusion process. Applying the algorithm again on the time reversal, started at the proposed end point, gives the forward-in-time time diffusion bridge.
We found that the neural network architecture outlined in Heng al. 2021] did not scale well to learning higher dimensional SDEs. We therefore used different struc- ture (see Figure 7). We use U-net [Ronneberger al.) 2015) structure with skip connections in the form of fully connected layers scale up the network's capacity. The time step information is encoded by the well-known si- nusoidal embedding proposed in Vaswani et al. [017]] and added element-wise to the outputs of the fully con- nected layers. Finally, the real denoising score matching loss function proposed in Heng et al. [202] is computed and the stochastic gradient descent used update the network parameters.
08
06
0.0
0.2
0.4
11.0
Figure Sample from the SDE of Equa- tion
The exact specification we used for different bases or points given in Table We used the Adam optimiser for the training, with a starting learning rate of 0.0001 and 500 warmup steps. After warming up, the learning rate decreases cosinely until reaches le-6. All the training and evaluation computations were done with one NVIDIA RTX 4090 GPU and one Intel(R) Xeon(R) CPU E5-2650 v4 @ 2.20GHz.
B.2
Shape Spaces
B.2.1
Discretisation
We look at both discretisations in terms of the Fourier basis, and in terms of points. For points we take the discretisation
(t. Do, + Σ ((z(())
(19)
DEG
where G is of grid point in R22 and (1) RR2 Wiener process. For the Fourier basis, writing the SDE in Equation (17) into basis elements, gives
dx =
(en, Q(X(())( (f)ea,
(20)
1,m=1
where en (x) ónnz and gl. (11, x22) m, < M elements. The values in the sum can be approximated follows:
e jizz + immr Then, we truncate the bases n S N and
(en, ((x(((()))))m)))))
R
ine
Σ ((X()))).)))mmmm))))))) DEG2
(21)
2T
where 9i and G2 are grids over |-x, π| and D C IR.. The inner sum can be computed as a fast Fourier transform, and the outer as 2-dimensional, inverse fast Fourier transform. For the function k. we use the Gaussian kernel, with varying values for the covariance. In Appendix we apply this SDE, with various parameters, to circle embedded into IR² in Figure and to butterfly in Figure 6
14

---

B.2.2 Further experiments
we apply the unconditioned SDE in t circle, embedded into R². The parameter is the variance the kernel We see that increasing values of a, the process becomes smoother. This makes the kernels, mean the noise fields are wider and therefore points are more highly correlated leading to smoother shapes.
In Figure 8
since for each point y IR?, we associate noise field. Larger values of a for
We see that increasing the number of basis elements used, initially leads to slightly noisier shapes which is to be expected, since higher basis elements contain the higher frequencies and details. However, the process seems to converge quite quickly, and there appears very little difference between N 16 and N 24 basis elements.
In Figure 6 we show the process started on butterfly, with 0.1 and eight landmarks, where the evolution starts from fixed butterfly shape (in blue) and continues until time = 1 (in red).
Dense SILU
Dosse SIL43
Untoe SAL0
Detee S0L3
Dasser Dasse-SiLL SILAI
Dorse Derse-SiLL SILL
NiLAs
Ueter S050
Detse $IL.3
Enceding
Figure 7: The neural network structure for approximating the discretised score function. A U-net architecture with skip connections (dashed lines) is used. Each layer consists of two dense layers activated by SiLU functions. Batch normalisation is applied to the end of layer (not shown). The time step is encoded using the sinusoidal embedding and added element-wise the outputs of dense layers.
C Proofs
Theorem C.1. (Theorem 5.1 in paper) Let h: 0, H be a continuous function twice Fréchet differentiable with respect & E H and once differentiable with respect with continuous derivatives. Suppose X is the strong solution to the stochastic differential equation in Equation (I). Moreover, we assume that (t) htt, (t)) )) strictly positive martingale, with Z(0) and (T) 1.
Unconditional Forward Trajectories
N=8
=011
N=16
-=1.0 a =055
N=24
10.0
102
10.4
10.66
t=0.8
t=0.0
+0.0
1=02
t-44
1=0.6
t=0.8
t=1.0
Figure 8: We visualise the effect of the SDE on circle, for varying covariance of the Gaussian kernel a, and different numbers of basis elements N.
15

---

Then di- = d(P) defines new probability measure. Moreover, X satisfies the SDE
x(1) =X()) + S. (x (
(22)
+ F(X(s))] + S. B(X(s))dW(s))
where W is the Wiener process with respect to the measure P.
We split the proof into two and start with lemma showing that Z(1) h(,, (()) can be written in terms of an exponential.
Lemma C.2. Let h [0, H R and Z(t) = ((t,X(()) satisfy the assumptions of Theorem 5.1
Then ((t) exp (L(1) LK0
where
L(I) I'
(23)
and [L] is the quadratic variation of L.
Proof. We denote the 7th Fréchet derivative with respect to the 7th argument function by D In case = 1, we will simply write D.J. First, apply the infiniteddimensional Itô's lemma included in Theorem D.2 Brzezniak et al.) 2008). We can do this since we assume that h and its Fréchet derivatives D2 h. Dzh, Dih exist and are continuous. Moreover, X is strong solution, (s) is stochastically integrable, and (x(s) (s) + Fs) is integrable and adapted. Furthermore, since assume that h X (t)) is martingale, the drift terms arising Itô's lemma must disappear. Therefore for (t) == ht, (()).
(()) (0, X(0) + J. (()))(()
(24)
Now, we write Z the form of an exponential. For this, we use Doléans exponential. Set
1 dZ..
L(I) log Z(0)
(25)
7,
By assumption Z is continuous and strictly positive, so via the Doléans-Dade exponential and Williams, 2000 Chapter 3), holds that
Rogers
(t)
Z(t) exp [(t)
(26)
Since defined Z(t) h(t,, (t)), and we assume that Z(0) 1, know that
L(t) = / D2 log log (X)))))(())
(27)
By the Riesz representation theorem, there exists an element in H that denote V log X(ss)(ss)) such that
D2 log (s,, X(s))(X) = (V log Y)H. Hence, we get the claimed value for L(t). Lastly, we find the quadratic variation of L. We can equivalently write L(t) as
(28)
L(t) J' (s)dd(()
(s) = (V ((X(s)
(29)
where (t)) HS(QU/2((), R), i.e. the space of Hilbert-Schmidt operators from Q1)2(U) Then Röckner and Claudia (007 Lemma 2.4.2] states that
RR.
( ((sQ (Xx((Xxx)) (x)) log M(s, (()))lloua)())
(30)
Finally, by Röckner and Claudia 2007 Lemma 2.4.3], it holds that
[ 1B3 (X(()) log h(, X(s))())
[L]
(31)
16

---

The proof of the theorem is now simply an application of Girsanov's theorem, normalising Z by (T)( necessary:
Proof. ((s) (X ho, Then is a Fi predictable process.
By Lemma C.2 it holds that
(l((s),d((s))) dw(()) (f(()))) 1 [ [++() 2
z(t) exp
(32)
Hence, applying Ginsanov's theorem, we can define new measure dF = Z(T)dP, and know the Wiener process with respect has form
((t) = W(1) J B(X(s)) ((s)dss.
(33)
Rewriting as a stochastic equation with respect to P,, we see that X satisfies
x(1) =X(0) I/'
(34)
! [AX(s)+ F(X())) J. B(X(s))
giving the h-transformed process.
Lemma C.3. (Lemma 5.2 in paper) Let be as in Equation
satisffying Assumption 3.7 Given
function h: J0, T] H + R satisfying Equation (9). Z(t) h(t, Xt,, Eo)) is strictly positive
martingale such that (())
and (T) = CT e (T, Eo))
Proof. The functions of form hit, {) are Markov transition operators satisfying Equation (4. Define (t) == htt, (t)). Then Z(T) 1. Further, by the normalisation Cr, it holds Z(0) = 1. Strict positivity of Z holds since vi is strictly positive. To see that Z is martingale, we use the tower property: let Y be a random variable and
(X. XT, {o)] 46(X (T, (a)). Hence, ET)
Hi C H2 C F. Then
EEE H21 B[Y H2...
(35)
Now, for < Fs C Fr, we get
([Z(t) F.] = E[E(r ((X(TE) FFl F.l = Z(s).
(36)
Lemma C.4. (Lemma 5.3 in paper) Let 1b + be a continuous function, twice Fréchet differentiable, with continuous derivatives. Then h(t, == (T is twice Fréchet differentiable in and once differentiable in t, with continuous derivatives.
Proof. As before, we denote the jth Fréchet derivative with respect to the 7th argument of a function by J. If I only has one argument then we will instead write D. First note that by Assumption ((t,E) is twice Fréchet differentiable with respect to & and has continuous derivatives. By assumption that z is twice Fréchet differentiable with continuous derivatives, we know that the composition (t, E) is also twice Fréchet differentiable with second derivative
(),, 9) ( D2X(t,)))
(37)
+ (X ())
This is continuous in [0,7] x H. By Lebesgue's 's dominated convergence theorem and the definition of Fréchet differentiabilityy and noting that this holds for any E [0, T) M(L = (T (, is also twice Fréchet differentiable.
17

---

Next, we show that we can differentiate with respect to t. By Itô's lemma and properties of expectation, it holds that:
(E(((
-15(3 + E J. (D)))(((.E))XX((s.) ((X))))))
(38)
+ 2 2 L. (DBXXXEEN
Note that by assumed continuity properties of X and th Equation 38 is continuous. Therefore, using Lebesgue's dominated convergence theorem and the fundamental theorem of calculus, we see
1 lim (g(t + +, g(t,E))
(39)
r-0 r
1 B I t+r
lim r-0
E I. 1+
(40)
= -IIOMMAACCE))NNNN)
+ (TTxxxitn
and so g(t, {) differentiable with respect to t. Noting that h(t, {) = = -t,)), and T differentiable, we get the result.
t is
Lemma C.5. (Lemma 5.4 in paper) Let I' be in Equation 16 and h 00 H + R
be defined by h(t, Y) Bl (X (T - t, Y)), Moreover, define g [0,7] x EN + R by
g(t, = E ór, (X, (T t, B))] Then (V log h(t, Y), = (V log gt. (1)
Proof. First note that for any Y H,
1(1, Y) Blor -(T-4-Y))
(41)
E
5r, ((X
(42)
i=1
N
br, (x, (T - (x) g(, ((.)
(43)
Now note that for any E E N
+ - Y)t h(t, Y + +E)
g(t. (Y + (c)) gt. (Y))
(44)
|E||HN
[I(:) IR
Hence, by the properties of the Fréchet derivative, it holds (Vlog Y), [V gt, (Y))
D Further background
D.1 Infinite dimensional Itô and Girsanov
In order to condition, we rely heavily on the dimeete-dimensionall analogues of Itô's lemma and Girsanov's theorem. We state the exact version of both that
Girsanov's theorem (Da Prato and Zabezyk 2014, Section 10.2.1] allows us to define a change or reweighting of the measure and also tells us what stochastic processes look like with regard to this
18

---

new measure. Hence we can use this to get change of measure which possesses some wanted behaviour and then write stochastic processes with respect to this measure. we have martingale Z with respect to a probability measure IP, then we define new probability measure dlP = (T)) and we can write the Q- Q-Wiener process of in terms of the original Wiener process.
Theorem D.1. Let W be Q-Wiener process in U and let (b)) be (U))uuue Fr-predictable process. If
z())= = exp (f (o(s),
2 / (s)s)))ss))
(45)
where the inner product and norm are the Hilbert space Q² (U), then B[Z(T)) Then
((t) (W)) L. (s)dss E [0,2]
(46)
a Q-Wiener process with respect to {Fihe2o on (0,F, for df = d(
Another theorem we shall be relying on the mnfnite-dimension analogue of Itô's lemma. This is the analogue of the chain rule for Hilbert space-valued processes. The version use is adapted from a more general version for Banach spaces [Brzezniak et al., 2008 Theorem 2.4). Da Prato et al. [219)] discuss Itô's lemma for Hilbert space-valued SDEs.
Theorem D.2. Let H and E be separable Hilbert spaces. Assume that f x E is of class CLL22 Let 4 : [0, 7 x Ω + HS(Q)/2(U), H) be measurable and stochastically integrable with respect to W, a Q-Wiener process on U. Let 1b [0, T] x Ω H be measurable and adapted with paths in L (0, Tú; H) almost surely, Let Eo S + H be Fo-measurable. Define X [0, T x $2 + H
by
x = Eo + L'' v(s)db+ ( (s)dWW(s)
(47)
Then almost surely for all [0,
( + / Xif(s) / Def(ss
(48)
/ D ( D2f(s, (()
where (s, is defined as
Σ
p3 X())) (()
(49)
J21
for an orthonormal basis {uyby21 U.
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Reviews:
[{"review_id": "DJ0j5EG9Pa", "reviewer": "Reviewer_haqz", "summary": "This paper addresses the challenge of conditioning infinite-dimensional stochastic processes, particularly non-linear ones, without prior discretisation. Traditional methods condition finite-dimensional data but struggle with infinite-dimensional, function-valued data. The authors employ an infinite-dimensional version of Girsanov’s theorem and Doob’s h-transform to condition such processes. This method is applied to time series analysis of shapes in evolutionary biology, specifically modelling changes in the morphometry of organisms. The paper also utilizes score matching techniques to learn the coefficients of the score function in the Fourier basis.", "questions": "What is the computational complexity of the method?", "limitations": "N/A", "rating": 7, "confidence": 1, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper introduces a novel method for conditioning infinite-dimensional non-linear processes without prior discretization, generalizing recent work on linear processes in infinite dimensions.\n- The authors derive Doob’s h-transform for infinite dimensional non-linear processes, allowing conditioning without first discretizing the model. Then, score matching is used to learn the score arising from the h-transform by training on the coefficients of the Fourier basis.\n- The paper demonstrates a practical application to evolutionary biology to model changes in the shapes of organisms.", "weaknesses": "- The empirical experiments focused specifically on modeling the change in the shape of butterflies. Thus, it's unclear how the method performs in more general benchmarks for diffusion processes.\n- Computational complexity could be large for the the proposed method especially for the non-linear setting.\n- The evaluation lacks necessary comparison with related approaches.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for the positive review! \nIn the following, we would like to briefly clarify the lack of related approaches (this work is, to the best of our knowledge, the first to operate in an infinite-dimensional, nonlinear setting)\nand describe the computational complexity:\n\n1. Comparison with related approaches / benchmarks: We are unaware of benchmarks respectively other approaches for nonlinear **and** infinite-dimensional processes -- we are only familiar with nonlinear (but finite-dimensional) (e.g. [1]) as well as infinite-dimensional (but linear) comparisons (e.g. [2]), neither of which directly compare to our work.\n\n2. Complexity: Thanks for asking! The complexity of a single evaluation of the loss of a minibatch of $B$ trajectories with $N$ time steps and in dimension $d$ (which in this case corresponds to the number of basis elements) is $O(B \\cdot N \\cdot d^3)$. The cubic factor comes from covariance-matrix arithmetic, which is common for multi-output stochastic processes (both finite- and infinite-dimensional) (see e.g. [3]). We are planning on bringing that factor down in a follow-up project.\n\nWe thank you again for your positive evaluation! We hope that we were able to clarify a few points and we look forward to the discussion!\n\n[1] Frank van der Meulen and Moritz Schauer. Automatic backward filtering forward guiding for markov processes and graphical models. arXiv preprint arXiv:2010.03509, 2022.\n\n[2] Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich, and Sven Wang. Infinite-dimensional diffusion models for function spaces. arXiv preprint arXiv:2302.10130, 2023.\n\n[3] James Hensman, Nicolò Fusi, and Neil D. Lawrence. Gaussian processes for Big data. Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence. 2013.", "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": "tnucxjeYXe", "reviewer": "Reviewer_bXXx", "summary": "This paper explores the conditioning of non-linear processes in infinite dimensions. To achieve this, the authors introduce an **infinite version of Doob’s $h$-transform** (contribution 1) that relies on the infinite-dimensional counterparts of Itô’s lemma and Girsanov’s theorem. They then discretize the conditioned process and use score-matching techniques to **learn the score** arising from the $h$-transform by training on the coefficients of the Fourier basis, which allows sampling from the conditioned process (contribution 2).\n\nThese mathematical tools are used to condition a process to hit a specific set at the end time, also known as bridges. The authors  **detail two models** based on different scenarios: one for direct conditioning on data (**exact matching**) when the transition operator of the SDE solution is smooth, and the second for assuming some observation error (**inexact matching**) (contribution 3).\n\nThey **illustrate their procedure by modeling changes in the morphometry** (i.e., shapes) of organisms in evolution, specifically the changes in the shapes of butterflies over time (contribution 4).", "questions": "The questions follow the potential identified weaknesses:\n\n**Assumption 3.3** Can you illustrate it by providing conditions on the solution process $(X_t)\\_{t\\in\\mathbb{R}_+}$ itself rather than on the set $\\Gamma$? Perhaps using results on the regularity of the solution process density via the Malliavin derivative (S. Kusuoka and D. Stroock, \"Applications of Malliavin calculus, part II\", Kohatsu-Higa and Tanaka, Annales IHP 2012, D. Nualart, M. Zakai, Séminaire de probabilités 1989), or the parametrix method (Bally and Kohatsu-Higa, AAP 2015)?\n\n**Theoretical analysis** Would it be possible to quantitatively measure the quality of the procedure, i.e.  to provide an upper bound on the error between the conditioning and the true solution in the case of exact matching? In the case of inexact matching, can we measure the impact of the noise on this error?\n \n\n**Proof of Lemma C.4** It seems to me that the proof of Lemma C.4 corresponds to the calculation of the infinitesimal generator associated with the process \n$(X_t)\\_{t\\in\\mathbb{R}_+}$ and not to what is stated. In the statement of the Lemma $h$ is defined as $h(t,\\xi)=\\mathbb E[\\psi(X(T-t,\\xi))]$ whereas in the proof, $h(t,x):=\\mathbb E[\\psi(X(t,\\xi))]$. Adapt the proof maybe by defining $g(t,x)=h(T-t,x)$. \n\n\n### Minor comments \n\nThere are some typographical errors and notational awkwardness. These notation issues recur repeatedly:\n\n- l. 98, 137, 185 etc.: Write $W$ or $\\{W_t\\}$ instead of $W_t$ when dealing with a process. Same remark for $e^{tA}$ (l. 101).\n\\item l. 140, 147, 198 etc.: The initial condition of the SDE is denoted first by $x_0$ (Equation (1)) and later by $\\xi_0$, $x$. Please harmonize the notation.\n- l. 590, Equations (35) and (36): Partial derivatives can be denoted as $D_x$ or $D_2$. Please harmonize the notation.\n\nHere is a non-exhaustive list:\n- l. 80: $f(T,s_0)=s_1,f$. $f$ and ? I don't understand the sentence/ \n- l. 155: Add $x(0)=x_0\\in\\mathbb{R}^d$.\n- l. 156: $X$ should be lowercase.\n- l. 158: What is $p$ ? \n- l. 158: I think we should have $\\mathbb{E}[Z(T)]=1$.\n- l. 159: I think is it $d\\mathbb{Q}/d\\mathbb{P}|\\mathcal F_t$.\n- l. 234: Equation (13), what is $v$?\n- l. 553: Define $[L]$.\n- l. 558 : How do you defined $D_2$ ? I think it's $\\partial_x$ or $h_x$ ?\n- l. 562: Using that $Z(t)=h(t,X(t))$\n- l. 569: What is $H_Q$ ? \n- l. 583: $Z(s)$ instead of $Z_s$\n- l. 583 : $C_T=1/P_T\\psi(\\xi)$ by definition\n- l. 590 equation (35), equation (36) harmonisation of notation  with the reference lemma of Itô formula\n- l. 599 : Lemma C.5, How do you define  $c^i$ and l. 601 $c_i$?\n- l. 618 : $d\\widehat P=Z(T)d\\mathbb{P}$ ($d$ is missing)\n- Define properly the Hilbert space $Q^{1/2}(H)$.", "limitations": "The limitations are briefly discussed in the conclusion, particularly the fact that the procedure would not be applicable to weak solutions of SDEs. Some directions for future research (focus on network architecture to increase the dimension that can be considered, infinite-dimensional bridges to inference problems) are also provided. There is no potential negative societal impact in this work.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The theoretical mathematical contribution, namely the conditioning of non-linear processes in infinite dimensions, is noteworthy and broadens the scope of previous work that focused on approximating non-linear bridge processes in finite dimensions (Delyon and Hu, 2006, van der Maulen and Schauer, 2022). The infinite version of Doob’s $h$-transform, although not surprising in its form (similar to the finite-dimensional case), is of independent interest. The general procedure, based on this transform, allows conditioning without discretizing the model beforehand.\n\nTwo models are developed: exact matching and inexact matching. Inexact matching involves conditioning the process so that at the final time it does not exactly satisfy a final condition but approaches it. This approach is particularly interesting as it incorporates potential observation errors (by introducing noise) and relaxes the restrictive Assumption 3.3, which is unavoidable in the case of exact matching.\n\nThe application to modeling changes in morphometry, using bridge processes between shapes, is highly relevant for illustrating the usefulness of the developed procedure. Unlike previous work on the subject (Arnaudon et al., 2019, 2022), the conditioning precedes discretization, ensuring the proper definition of the bridge even as the number of points tends to infinity.\n\nThe overall presentation of the paper is excellent: the introduction effectively situates the study within the existing literature on related topics (approximation of non-linear bridge processes, learning score functions in generative diffusion models, diffusion bridges in shape spaces), the contributions are clearly outlined, and the tools developed (the infinite version of Doob’s $h$-transform) are introduced in a pedagogical and concise manner without sacrificing rigor.\n\nApart from a few minor confusions in the notation, the proofs seem correct and well-written.", "weaknesses": "### Assumption 3.3\n\nAs mentioned in the article itself, Assumption 3.3—indispensable in the case of exact matching—concerning the regularity of the transition function, is strong. An example of a subset $ \\Gamma$ (finite-dimensional cylinder) matching this condition is provided in Section 5.3. It seems to me that the inherent difficulty of Assumption 3.3 for exact matching in infinite dimension is circumvented by choosing an example where the problem is ultimatly 'reduced to finite dimension'. Maybe having an example with conditions on the solution process itself $(X_t)\\_{t\\in\\mathbb{R}_+}$ or the coefficients of the SDE it satisfies, which illustrate assumption 3.3, would be more interesting.\n\n\n\n### Theoretical achievement \n\nAs it stands, the article develops an interesting method (though perhaps not completely groundbreaking) based on an extension of the Doob $h$-transform and reversing the usual discretization-conditioning steps, allowing for a well-defined bridge despite the difficulty associated with infinite dimensions. Perhaps a theoretical study of the error between  the conditioning and the true solution would strengthen relevance of the approach.\n\n### Notation \n\nIt's not really a weakness, but the notations should be harmonized (for example $ D_2 $ or $D_x$, $x_0$ or $\\xi_0$) to make reading and reviewing the proofs easier. Perhaps a summary table of notations could be included?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the review and the insightful questions. We're glad you liked the paper and find the theoretical contribution noteworthy!\n\nIn the following, we address your questions one by one:\n\n1. Assumption 3.3: Yes, you're right and we agree this would be nice to include!\n    We are aware of a result for SDEs of form\n    $$dX = [AX + F(X)]\\mathrm{d}t + \\sqrt{Q}\\mathrm{d}W(t),$$\n    with $F$ being once differentiable, and $A, Q$ satisfying some extra assumptions (see Theorem 9.39/9.43 of [1] or Section 6.5, 7.3 of [2]; references below). We will include this in the discussion of Assumption 3.3.\n    In general, it would indeed be nice to prove something for other cases, especially in the case of stochastic flows as in [3], which could perhaps be done using the Malliavin derivative as you point out. We plan on looking into this for future work.\n\n2. Theoretical analysis: We agree this kind of result would be interesting! However, deriving such an error estimate would require too much additional analysis for now, which is why we leave it to future work.\n\n3. Proof of lemma C.4: Yes, you're right! We've corrected this now. Concretely: We define $g(t,x)=h(T-t,x)$ (where we before showed that $g$ is differentiable in time). Then, note that the function $t \\to T-t$ is differentiable, and therefore $h(t, x)$ is, too. The spatial Fréchet differentiability still holds, since we showed it holds for all $t \\in [0, T]$.\n\nThanks a lot for the feedback on the notation and for listing the small errors! We've now incorporated these into the paper and made sure all the notation is consistent.\nWe hope that you agree that this improves the presentation.\n\nThank you again for the positive review, and we look forward to further discussion!\n\n[1] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2nd edition, 2014.\n\n[2] Cerrai, S., Second Order PDE’s in Finite and Infinite Dimension: a Probabilistic Approach, Lecture Notes in Mathematics. Springer, 2001.\n\n[3] Hiroshi Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1st edition, 1997.", "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": "Uer2syUSaG", "reviewer": "Reviewer_veYd", "summary": "The paper attempts to derive a means of conditioning a nonlinear diffusion process upon function-valued observations, via the $h$-transforms, adapting the method of _Jeremy Heng, Valentin De Bortoli, Arnaud Doucet, and James Thornton. Simulating diffusion bridges with score matching._ to a function-valued setting.", "questions": "1. What is happening in figure 1? It is not easy to parse the start and end shapes. I can just about work it out from section 6.2.2, but can we add some visual cues in the figure, for example, fading out the starting shape as time does on, and fading in the terminal one?\n\n   > we first use two butterflies with somewhat320 different shapes [GBIF.Org User, 2024]. One trajectory between the two butterflies is   plotted in Figure 2. In this, we can see the high correlation between neighbouring points, with a Brownian temporal model. In Figure 1, we plot 120 butterfly trajectories, at specific time points. For t = 0.2 we see that the butterfly outlines are mostly close to the start butterfly in pink, and at time t = 0.8, they are closer to the green target butterfly\n\n2. 4.2/l159\n\n   > Moreover, under this measure $\\mathbb{Q}, x(t)$ satisfies a new SDE\n   >$$\n   > \\mathrm{d} x^c(t)=f\\left(t, x^c(t)\\right) \\mathrm{d} t+\\sigma \\sigma^T\\left(t, x^c\\right) \\nabla_x \\log h\\left(t, x^c(t)\\right) \\mathrm{d} t+\\sigma\\left(t, x^c(t)\\right) \\mathrm{d} W(t) .\n   >$$\n\n   Can you clarify the relationship between $x$ and $x^c$?\n\n3. l165 confusing phrasing\n\n   > When $h(t, x)=p(t, x ; T, y)$ there is no general closed form solution. Different methods to learn the bridge exist Delyon and Hu [2006], Schauer et al. [2017]. More recently, score-based learning methods were proposed to learn the term $\\nabla_x \\log p(t, x ; T, y)$ Heng et al. [2021].\n\n   Do you mean something like this?\n\n   > For the required  $h(t, x)=p(t, x ; T, y)$ there is no general closed form for $h$. Different methods to learn the bridge exist Delyon and Hu [2006], Schauer et al. [2017]. More recently, score-based learning methods were proposed to learn the term $\\nabla_x \\log p(t, x ; T, y)$ Heng et al. [2021], and it is the infinite-dimensional generalisation of the latter method  that we pursue here\n\n4. eq15:\n\n   >$\\left\\langle X(t, \\xi), e_i\\right\\rangle=\\left\\langle\\xi, e_i\\right\\rangle+\\int_0^t\\left\\langle A X(s)+f(X(s)), e_i\\right\\rangle \\mathrm{d} t+\\int_0^t\\left\\langle e_i, B(X(s)) \\mathrm{d} W(s)\\right\\rangle$.\n\n   Is something wrong with the variables of integration here? $\\int_0^t\\left\\langle A X(s)+f(X(s)), e_i\\right\\rangle \\mathrm{d} t$ is an integral in $t$ and yet the integrand doesn't depend upon t, and it does depend upon $s$ which is a free variable\n\nMinor typos:\n\n* l195\n\n  > However, transition operators of form Sec. 4.1 exist and satisfy the Markov property Equation (4)\n\n  should that be\n\n  > However, transition operators of form Equation (3) exist and satisfy the Markov property Equation (4)", "limitations": "The paper seems to depend upon explicit orthogonal bases (sec 5.3) which is IMO a  restriction in practice, since the diffusion methods of industrial interest frequently have no such explicit basis.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Discrete approximations of notionally continuous objects is a ubiquitous problem in machine learning. By representing conditional nonlinear SDE solutions themselves in function space, this expands the range and type of discretization that can be employed to solve problems which are naturally regarded as functions; in this paper, it enables the use of reasonably general  (separable) Hilbert-space basis functions as the means of discretization, rather than, e.g. a raster grid.\n\nThis problem is interesting and well-posed.", "weaknesses": "There are many small oddities in the style which make this paper a difficult read.\n\nSee below for those.\n\nThe paper presents essentially one result, which is the up-lifting of learned bridge diffusion on a finite dimensional vector space, to ones on a function space with a finitely-truncated basis. This result seems somewhat, if not massively, important.\n\nThe first two pages, before the problem statement, are confusing. If we read the paper in linear order we cannot understand many of the assertions made there without reference to equations which have not been introduced yet, and are not even cross referenced.  e.g. l46/sect 2.1\n\n>Given an SDE, the conditioned SDE contains an intractable score function. This is similar, but46\nslightly different, to the score function that arises in generative diffusion models Vincent [2011],47\nSong and Ermon [2019], Song et al. [2021]. There, the starting distribution is complicated, but the48\nstochastic process is linear. In our case, we are interested in the process itself, particularly non-linear49\nprocesses. In this way, our work generalises the finite-dimensional work on conditioning non-linear50\nSDEs and infinite dimensional score matching, where they consider time reversals of linear SDEs\n\nWhat is doing on? Which score function is intractable? There is a lot of this kind of thing where technical statements are made without reference to the equations that ground them. This becomes (more) clear after reading the whole paper, but in the order that the paper is written, this entire section is hard to parse.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for your positive review and the concrete suggestions. We're glad you appreciate the importance of our result! \n\nIt seems to us that the weaknesses you listed were almost entirely presentational. We have fixed the issues you point out in your review as follows.\n\n**Clarity of introduction and related work:**\n\nThank you for your feedback on the first two pages! We have now edited these sections, some examples of which we give in the following. \n\n> Given an SDE...\n\nThanks for pointing out this paragraph! This section states that for given $T, y$ the score function $\\nabla \\log p(t, x; T, y)$ is intractable, since for nonlinear SDEs there is no known closed solution for $p(t, x; T, y)$. \nWe have now removed this paragraph and instead incorporate it into the related work on infinite dimensional diffusion models. The corresponding subsection in the related work now reads (previously lines 69-74):\n\n> Recent work on generative modelling has investigated score matching for infinite-dimensional diffusion processes [Pidstrigach et al., 2023, Franzese et al., 2023, Bond-Taylor and Willcocks, 2023, Hagemann et al., 2023, Lim et al., 2023]. This problem is similar to our task of conditioning an SDE, but not the same: The main difference is that our SDEs are fixed, known a-priori, and potentially nonlinear, whereas in generative modelling the SDE can be chosen freely. Hence, generative modelling often uses linear SDEs because the transition densities are known in closed form. In this sense, our problem relates to generative modelling, but has a different setup.\n\n\nThe references to generative diffusion models have been moved to the paragraph about score matching for finite-dimensional nonlinear bridges (previously lines 61-68):\n\n> Recently, Heng et al. [2021] adapted the score-matching methods of Vincent (2011), Song and Ermon (2019), and Song et al. (2021) to learn the score term for non-linear bridge processes. To do so, they introduce a new loss function to learn the time reversal of the process. They then learn the time reversal of the time reversal, which gives the forward bridge. Our work uses their method to learn the score term after discretising the SDE via truncated sums of basis elements. Phillips et al. [2022] also consider using truncated sums of basis elements for discretising SDEs, however, only for infinite-dimensional Ornstein-Uhlenbeck processes, which are linear.\n\n\nWe hope that these changes resolve your presentational issues, and that you agree with us that the presentation is improved.\n\n\n**To answer your questions:**\n\n1. Thanks for the suggestion. We've tried to make this figure clearer now (see PDF in the general reply). The caption in the PDF will also be included in the paper, which we hope gives additional clarity.\n\n2. Thanks for the question. The process $x^c$ can be thought of as the conditioned version of $x$. For example if $h(t, x) = \\frac{p(t, x; T, y)}{p(0, x_0; T, y)},$ then for a set $B\\subset\\mathbb{R}^d$, $\\mathbb{P}(x^c(t) \\in B) = \\mathbb{P}(x \\in B \\mid x(T)=y)$ holds. \nWe've refined the corresponding explanation in Section 4.2.\n\n3. Yes, this is precisely what we mean. We've edited it, so it's hopefully more clear now. The new version reads:\n\n    > For $h(t, x) := \\frac{p(t, x; T, y)}{p(0, x_0; T, y)},$ as in conditioning on an end point, there is, in general, no closed form solution for $h$. Different methods to learn the bridge exist (Delyon and Hu [2006], Schauer et al. [2017]). More recently, score-based learning methods were proposed to learn the term $\\nabla_x \\log p(t, x; T, y)$ (Heng et al. [2021]), which we will adapt to the infinite-dimensional setting.\n\n4. That was a typo, thanks for catching it! It is supposed to read\n    $$\\langle X(t, \\xi), e_i \\rangle = \\langle \\xi, e_i\\rangle + \\int_0^t \\langle AX(s) + f(X(s)), e_i\\rangle \\mathrm{d}s + \\int_0^t\\langle e_i, B(X(s))\\mathrm{d}W(s)\\rangle.$$\n    We have corrected this typo in the paper.\n\nWe would also like to clarify the limitation you mentioned. There are some non-standard Hilbert spaces without explicit orthogonal bases (e.g. Sobolev spaces on non-standard manifolds). However, when working in $L^2$ or Sobolev spaces on Euclidean spaces and spheres, one does have access to explicit bases, so this might be less of a limitation than indicated in the review.\n\nWe hope that you agree that the presentational aspects are now improved!\nIn any case, thank you for the positive evaluation and the concrete suggestions for how to improve the presentation -- we've updated the paper accordingly, as outlined above.", "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 thank all reviewers for their reviews and positive evaluations of our work. We are glad you all liked our contribution!

Many weaknesses seem to relate to presentational concerns, which we believe are easy to correct.
Below, we reply to all reviews in separate threads.
Attached is a PDF that contains an update to Figure 1, relating to the review by Reviewer veYd. 

We look forward to the discussion!

Thank you again for the positive reviews, and best wishes,

The authors

Meta-review: The reviewers agree that this is an interesting paper that deserves publication. There were a number of concerns about presentation that the authors should carefully address in the final version. A weakness of this paper is the empirical section, that focuses on a single task (modeling changes in shapes of butterflies)
Decision: Accept (spotlight)