Conditioning non-linear and infinite-dimensional diffusion processes

Thank you for the review and the insightful questions. We're glad you liked the paper and find the theoretical contribution noteworthy!

In the following, we address your questions one by one:

1. Assumption 3.3: Yes, you're right and we agree this would be nice to include! We are aware of a result for SDEs of form $dX = [AX + F(X)]\mathrm{d}t + \sqrt{Q}\mathrm{d}W(t),$ 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. 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.

2. 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.

3. 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 $t \in [0, T]$.

Thanks 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. We hope that you agree that this improves the presentation.

Thank you again for the positive review, and we look forward to further discussion!

[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.

[2] Cerrai, S., Second Order PDE’s in Finite and Infinite Dimension: a Probabilistic Approach, Lecture Notes in Mathematics. Springer, 2001.

[3] Hiroshi Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1st edition, 1997.

Thanks for your positive review and the concrete suggestions. We're glad you appreciate the importance of our result!

It 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.

Clarity of introduction and related work:

Thank you for your feedback on the first two pages! We have now edited these sections, some examples of which we give in the following.

Given an SDE...

Thanks 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 $p(t, x; T, y)$. We 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):

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.

The references to generative diffusion models have been moved to the paragraph about score matching for finite-dimensional nonlinear bridges (previously lines 61-68):

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.

We hope that these changes resolve your presentational issues, and that you agree with us that the presentation is improved.

To answer your questions:

1. 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.

2. 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. We've refined the corresponding explanation in Section 4.2.

3. Yes, this is precisely what we mean. We've edited it, so it's hopefully more clear now. The new version reads:

   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.

4. That was a typo, thanks for catching it! It is supposed to read $\langle X(t, \xi), e_i \rangle = \langle \xi, e_i\rangle + \int_0^t \langle AX(s) + f(X(s)), e_i\rangle \mathrm{d}s + \int_0^t\langle e_i, B(X(s))\mathrm{d}W(s)\rangle.$ We have corrected this typo in the paper.

We 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.

We hope that you agree that the presentational aspects are now improved! In 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.

Thanks for the positive review! In 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) and describe the computational complexity:

1. 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.

2. 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.

We 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!

[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.

[2] Jakiw Pidstrigach, Youssef Marzouk, Sebastian Reich, and Sven Wang. Infinite-dimensional diffusion models for function spaces. arXiv preprint arXiv:2302.10130, 2023.

[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.