Preconditioned Langevin Dynamics with Score-based Generative Models for Infinite-Dimensional Linear Bayesian Inverse Problems

We would like to thank the Reviewer for their constructive feedback.

We are happy to clarify our manuscript in response to the Reviewer's questions. We hope that this could lead to an improvement in their assessment of the paper.

1. Figure 1. We thank the reviewer for spotting this typo. What we intended to say is that the posterior covariance is not trace class; that is, the eigenvalues of its second moment do not decay at infinity---not that the modes exhibit growing variance. As a result, posterior samples do not belong to $H$, which is theoretically inconsistent with the formulation of the problem. Indeed, a Gaussian measure is supported in $H$ only if its second moment is finite on $H$, which is not the case in Figure 1, since the posterior variance is not decaying at fine scales. We will fix this typo in the camera-ready version of the paper.

2. Is $A_N$ really non-singular? We thank the reviewer for pointing out this assumption, which is a leftover from our first analysis attempts, where we wanted to keep the setting as simple as possible. In fact, as the reviewer may have noticed, the final analysis presented in the present manuscript does not rely on the non-singularity of $A_N$; nowhere in the main results or proof do we require that $(A_N^\top A_N)\_{jj}\neq0$. The only part that may require a minor adjustment is lines 234-235, but this does not pose any issue: if $(A_N^\top A_N)\_{jj}=0$, then the convergence rate for that mode is simply $\lambda\_j/\mu\_j$. We will remove this assumption in the camera-ready version of the paper.

3. Assuming $A^\top A$ is diagonalizable. This is a common assumption in many linear Bayesian inverse problem settings and their theoretical analysis; see, for example, the foundational works of Stuart [3] and Knapik et al. [4]. Moreover, in many classical linear inverse problems where $A$ is compact---such as the heat equation, tomography, or inverse scattering for Schr"odinger-type operators under the Born approximation---one can always find a basis in which $A_N^\top A_N$ is diagonal. That said, the diagonalization assumption is not essential to our analysis; it is adopted primarily to keep the presentation as clean, interpretable, and accessible as possible. For example, the asymptotic expansion in Eq. (10) of Theorem 3.1 can be extended to non-diagonal $A_N^\top A_N$, after some leg-work, by replacing scalar expansions with the corresponding matrix-series expansions; the part regarding the higher-order modes remains the same. Likewise, in Section 4 one can verify that, under a perfect score function, the optimal preconditioner still takes the form $C=[C_\mu^{-1} + \sigma^{-2} A^\top A]^{-1}$. We will add a remark including these additional considerations.

4. Missing definition of the mean reversion rate. Thank you for noticing this; we will include the definition in the camera-ready version.

5. Conditions on score approximation error (answer to questions 3,6,7). The affine-error decomposition in Assumption 1 is natural and follows from the co-diagonalization assumption for $C$ and $C_\mu$, which is standard in the literature on infinite-dimensional diffusion models [20,48]. In lines 194--199, we provide a qualitative analysis of the impact of $\epsilon^a_j$ and $\epsilon^b_j$ in Proposition 3.1. From the remarks, it is clear that assuming $p_j \epsilon_j^a$ to be small and $\lambda_j^{-1} \epsilon_j^b$ to be bounded is essential for achieving an accurate sampling of the posterior. Theorem 3.1 then makes these observations more quantitative, with assumptions that directly reflect the insights of Proposition 3.1. Since formula (10) is an equality, any significant deviation of $\epsilon^b_j$ and $\epsilon^a_j$ from the stated assumptions would necessarily result in a large sampling error at leading order, preventing an accurate recovery of the posterior mode. In this sense, Theorem 3.1 highlights the interplay between the data spectral properties and the design of the preconditioned Langevin sampler. It provides a sharp characterization of the link between the score approximation error and the posterior sampling error, showing that if the score approximation error is too large, it can significantly impact the overall posterior sampling error, potentially leading to catastrophic consequences for the error bound. To our knowledge, we are the first ones to provide sufficient conditions that ensure the boundedness of this error in the setting of infinite dimensional Bayesian linear inverse problems. Whether these conditions on the score approximation error are easy to satisfy in practice is an important question for future research, especially given that there is currently no literature on how to control the error arising from training infinite-dimensional score-based generative models to approximate the true score function, see [20, 42, 43, 45, 48]. With this paper, we hope to have laid down not only the theoretical framework but also the practical motivation for why such an analysis is essential to fully understand the infinite-dimensional setting. We thank the reviewer for bringing up this point; in the camera-ready version, we will include these considerations in the Discussion and Future Work section, clarify the connection between the discussion in lines 194--199 and the assumptions in Theorem 3.1, and elaborate on the implications of Remark 3.1.

We would like to thank the Reviewer for their positive feedback.

We are happy to clarify our manuscript in response to the Reviewer's thoughtful remarks and questions. We hope this could lead to an increase in their rating (or confidence score).

1. Explaining Figure 4. Figure 4 shows the empirical autocorrelation function (ACF) of the first ten modes, comparing preconditioned versus vanilla Langevin sampling. The $x$-axis is the lag, and the $y$-axis is the autocorrelation, which measures how strongly each mode at step $k$ remains correlated with its initial value. Under preconditioning with the optimal $C$, all the modes ACFs show uniform and rapid decay, meaning that they converge uniformly and quickly to the posterior, as the theoretical analysis of Section 4 anticipated. In contrast, vanilla Langevin displays slow, non-uniform decay. We agree this figure would benefit from a clearer explanation, and we will use the extra page in the camera-ready version to add a more detailed description. We thank the reviewer for the suggestion.

2. Synthetic examples. The synthetic examples are proposed to illustrate the theoretical results, which are the main contributions of the paper. This led us to choose examples that satisfy the key assumptions---namely, the conditions on the score error from Assumption 1, Theorem 3.1 and Theorem 4.1. For this reason, rather than using a learned score model, we simulate an imperfect score by perturbing the analytical score of the prior. This is a common approach in the literature; see, for example, the extended numerical experiments of [24], which we cite in the Introduction of our paper.

3. Assumptions on co-diagonalization of $C$, $C_\mu$ and $A^\top A$. The co-diagonalizability assumption for $C$ and $C_\mu$ is standard in the literature on infinite-dimensional diffusion models [20, 21, 48]. As for the assumption that $A^\top A$ can be diagonalized, it is common in many linear Bayesian inverse problem settings and their theoretical analysis; see, for example, the foundational works of Stuart [3] and Knapik et al. [4]. Moreover, in many classical linear inverse problem where $A$ is compact---such as the heat equation, tomography, or inverse scattering for Schr"odinger-type operators under the Born approximation---one can always find a basis in which $A_N^\top A_N$ is diagonal. However, we emphasize that the main conclusions of our analysis remain valid even without the diagonalization assumption. For example, the asymptotic expansion in Eq. (10) of Theorem 3.1 can be extended to non-diagonal $A_N^\top A_N$, after some leg-work, by replacing scalar expansions with the corresponding matrix-series expansions; the part regarding the higher-order modes remains the same. Likewise, in Section 4 one can verify that, under a perfect score function, the optimal preconditioner still takes the form $C=[C_\mu^{-1} + \sigma^{-2} A^\top A]^{-1}$. We will add a remark including these additional considerations and thank the reviewer for pointing out the need and relevance of these comments.

4. Non-Gaussian setting and illustrations. The paper presents analysis for both Gaussian and non-Gaussian settings. But since the key insights are closely aligned in both cases (see, for example, lines 194-199; 230-236; 272-281), we preferred focusing on the Gaussian case in the illustrations. This allows for explicit expressions for the mean reversion rate, avoiding the added complexity of working with upper and lower bounds of $\partial^2_j \phi_j$, and enabling a cleaner illustration of the main results of the paper. That said, our framework does allow for a non-Gaussian examples; we will explore this if such an example adds further insight into the core theoretical contributions of the paper.

5. Choosing a preconditioner in practical settings. A simple and practical choice is to select a preconditioner that is as close as possible to the prior covariance; as shown in the analysis of Theorem 4.1, for the higher-order modes, the leading order-term of the preconditioner is simply the inverse of the prior covariance. This choice is especially reasonable when the prior decays quickly, in which case $\mu_j^{-1}\gg \sigma^{-2} (A^T_N A_N)_{jj}$for the lower-order modes. Ideally, any available information about the posterior covariance and the score approximation error can be used to refine this first approximation. We can add a remark on this point, although we emphasize that a detailed investigation of the computational cost of approximating the optimal preconditioner lies beyond the scope of this work.

Dear Reviewer pjRJ,

We hope this message finds you well. We are very grateful for the time and effort you've dedicated to reviewing our work. We did our best to address your comments thoroughly in our rebuttal, and we hope our responses were clear and helpful.

Please don’t hesitate to let us know if anything remains unclear or if further clarification is needed. We would be happy to elaborate.

We thank the Reviewer for their positive feedback and thoughtful questions. We are happy to clarify their doubts and hope this could lead to an increase in their score.

1. Similarities and differences with Theorem 14 of [48]. We thank the reviewer for drawing our attention to this specific result in the foundational work by Pidstrigach et al. We are happy to elaborate on the differences and similarities between the two results, and we plan to include this discussion in the camera-ready version of the paper. As the reviewer correctly notes, the settings are different: Pidstrigach et al. study the problem of sampling an unknown data distribution using a score learned from samples of that distribution, whereas we use a score-based generative model trained to learn the prior in order to sample the posterior. Setting these differences aside, and assuming in Pidstrigach et al. that

with $C_\mu= \text{Diag }(\mu\_j)$, and, for simplicity, $A = \text{Diag}({A}\_{jj})$, one can show that the squared Wasserstein distance between $\mu_{\text{data}}$ and $\mathcal{N}(0,C)$, with $C=\text{Diag}(\lambda_j)$, satisfies

We would like to thank the Reviewer for their positive feedback.

We are happy to clarify our manuscript in response to the Reviewer's remarks and questions.

1. Assuming $A^\top A$ is diagonalizable. This assumption is common in many linear inverse problem settings and their theoretical analysis [3, 4]. We adopted it because we believe that focusing on the diagonal case provides the clearest exposition, keeping our analysis as clean, interpretable, and insightful as possible. Moreover, for any linear inverse problem where $A$ is compact, one can always find a basis in which $A_N^\top A_N$ becomes diagonal. However, what we wrote in the Discussion and Future Work section remains true---the main takeaway of our analysis holds even without the diagonalization assumption. The asymptotic expansion in Eq. (10) of Theorem 3.1 can be extended to non-diagonal $A_N^\top A_N$, after some leg-work and under appropriate assumptions on the structure of the score approximation error, by replacing scalar expansions with the corresponding matrix-series expansions; the part concerning the higher-order modes remains unchanged. Similarly, in Section 4 one can verify that, under a perfect score function, the optimal preconditioner still takes the form $C=[C_\mu^{-1} + \sigma^{-2} A^\top A]^{-1}$. We can add a remark including these additional considerations.

2. Computing the preconditioner. Indeed, exactly computing the optimal preconditioner would require the diagonalization of $A_N^\top A_N$, which can be computationally expensive. However, a simple and practical alternative would be to select a preconditioner that is as close as possible to the prior covariance; as shown in the analysis of Theorem 4.1, for the higher-order modes, the leading order-term of the preconditioner coincides with the inverse of the prior covariance. This choice is particularly justified when the prior decays quickly, in which case $\mu_j^{-1}\gg \sigma^{-2} (A^T_N A_N)_{jj}$ for the lower-order modes. Any additional information about the posterior covariance or the score approximation error can then be used to refine this initial approximation. We can add a remark on this point, although we emphasize that a detailed investigation of the computational cost of constructing the optimal preconditioner lies beyond the scope of this paper and is left for future work.

3. Bias and implications. In the numerical examples, we set $\epsilon^b\_j =0$. Since formula (10) is an equality, a large value of $\epsilon^b\_j$ would necessarily lead to a large sampling error, preventing an accurate recovery of the posterior mode. In fact, learning an incorrect score can significantly impact the posterior sampling---a challenge that already exists in finite dimensions but can become catastrophic when attempting to control the global error in the infinite-dimensional setting. In this sense, our result highlights the interplay between the data spectral properties and the design of the preconditioned Langevin sampler; understanding how these spectral properties, together with the structure of the inverse problem, influence the learnability of the score is an important question for future research, especially given that there is currently little literature on how to control the error arising from training infinite-dimensional score-based generative models to approximate the true score function (see [20, 42, 43, 45, 48]). Our contribution here is a first step toward a theoretical understanding of this issue, in a setting that allows for rigorous analysis of when a learned prior can provide useful insight.