SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We begin our analysis of the forward-reverse conditional SDE framework by examining the case where the prior of  $X_0$  is a Gaussian measure. This case provides illuminating insight, not only because it is possible to get an analytic formula of the score, but also because it offers a full characterization of SDMs in the infinite-dimensional setting, showing under which conditions we are sampling from the correct target conditional distribution and how fast the reverse SDE converges to it. We also show that the conditional score can have a singular behavior at small times when the observations are noiseless, in contrast with the unconditional score under similar hypotheses.
We assume that  $\Phi = 0$  in (9). All distributions in play are Gaussian:
$$
X _ {0} \sim \mathcal {N} (0, C _ {\mu}), \tag {10}
$$
$$
X _ {t} \mid X _ {0} \sim \mathcal {N} \left(e ^ {- t / 2} X _ {0}, (1 - e ^ {- t}) C\right), \tag {11}
$$
$$
\left. X _ {0} \right| Y \sim \mathcal {N} \left(M _ {o} Y, C _ {o}\right), \tag {12}
$$
$$
\left. X _ {t} \right| Y \sim \mathcal {N} \left(e ^ {- t / 2} M _ {o} Y, e ^ {- t} C _ {o} + (1 - e ^ {- t}) C\right), \tag {13}
$$
where  $M_{o} = C_{\mu}A^{*}(AC_{\mu}A^{*} + C_{B})^{-1}$  and  $C_{o} = C_{\mu} - C_{\mu}A^{*}(AC_{\mu}A^{*} + C_{B})^{-1}AC_{\mu}$ . By Mercer theorem [52], there exist  $(\mu_j)$  in  $[0, + \infty)$  and an orthonormal basis  $(v_{j})$  in  $H$  such that  $C_\mu v_j = \mu_j v_j \forall j$ . We consider the infinite-dimensional case with  $\mu_j > 0 \forall j$ . We assume that  $C_\mu$  is trace class so that  $\sum_{j}\mu_{j} < + \infty$ . We assume that the functions  $v_{j}$  are eigenfunctions of  $C$  and we denote by  $\lambda_{j}$  the corresponding eigenvalues.
We assume an observational model corresponding to observing a finite-dimensional subspace of  $H$  spanned by  $v_{\eta(1)}, \ldots, v_{\eta(n)}$  corresponding to  $g_k = v_{\eta(k)}$ ,  $k = 1, \ldots, n$ , where  $g_j \in H$  is such that  $(Af)_j = \langle g_j, f \rangle$ . We denote  $\mathcal{I}^{(n)} = \{\eta(1), \ldots, \eta(n)\}$ . We assume moreover  $C_B = \sigma_B^2 I_n$ . Let  $Z_t$  be the solution of reverse-time SDE:
$$
d Z _ {t} = \frac {1}{2} Z _ {t} d t + S (T - t, Z _ {t}, y) \mathrm {d} t + \sqrt {C} d W _ {t}, \quad Z _ {0} \sim X _ {T} | Y = y. \tag {14}
$$
We want to show that the reverse SDE we have just formulated in (14) indeed constitutes a reversal of the stochastic dynamics from the forward SDE in (1) conditioned on  $y$ . To this aim, we will need the following lemma:
Lemma 2. We define  $Z^{(j)} = \langle v_j,Z\rangle$ ,  $p^{(j)} = \lambda_j / \mu_j$  for all  $j$ . We also define  $y^{(j)} = y_{\eta (j)}$  for  $j\in \mathcal{I}^{(n)}$  and  $y^{(j)} = 0$  otherwise, and  $q^{(j)} = \mu_j / \sigma_B^2$  for  $j\in \mathcal{I}^{(n)}$  and  $q^{(j)} = 0$  otherwise. Then we can write for all  $j$
$$
d Z _ {t} ^ {(j)} = \mu^ {(x, j)} (T - t) Z _ {t} ^ {(j)} d t + \mu^ {(y, j)} (T - t) y ^ {(j)} d t + \sqrt {\lambda_ {j}} d W ^ {(j)}, \tag {15}
$$
with  $W^{(j)}$  independent and identically distributed standard Brownian motions,
$$
\mu^ {(x, j)} (t) = \frac {1}{2} - \frac {e ^ {t} p ^ {(j)} \left(1 + q ^ {(j)}\right)}{1 + \left(e ^ {t} - 1\right) p ^ {(j)} \left(1 + q ^ {(j)}\right)}, \quad \mu^ {(y, j)} (t) = \frac {e ^ {t / 2} p ^ {(j)} q ^ {(j)}}{1 + \left(e ^ {t} - 1\right) p ^ {(j)} \left(1 + q ^ {(j)}\right)}. \tag {16}
$$
Proof. The proof is a Gaussian calculation. It relies on computing  $\langle S, v_j \rangle$ , which yields an analytic formula. See Appendix B.
Lemma 2 enables us to discuss when we are sampling from the correct target conditional distribution  $X_0|Y \sim \mathcal{N}(M_oY, C_o)$ . We can make a few remarks:
- In the limit  $T \to \infty$ , we get  $\mu^{(x,j)}(T - t) \to -1/2$  and  $\mu^{(y,j)}(T - t) \to 0$ .
- If  $j \notin \mathcal{I}^{(n)}$  then we have the same mode dynamics as in the unconditional case. Thus we sample from the correct target distribution if  $T$  is large or if we start from  $Z_0^{(j)} \sim \mathcal{N}(0, \Sigma_0^{(j)})$  for  $\Sigma_0^{(j)} = \mu_j e^{-T} + \lambda_j (1 - e^{-T})$ , which is the distribution of  $X_T^{(j)} = \langle X_T, v_j \rangle$  given  $Y = y$ .
- If  $j \in \mathcal{I}^{(n)}$  and we start from  $Z_0^{(j)} \sim \mathcal{N}(\bar{z}_0^{(j)}, \Sigma_0^{(j)})$ , then we find  $Z_T^{(j)} \sim \mathcal{N}\big(\bar{z}_T^{(j)}, \Sigma_T^{(j)}\big)$  with
$$
\begin{array}{l} \Sigma_ {T} ^ {(j)} = \Sigma_ {0} ^ {(j)} \left(\frac {e ^ {T}}{(1 + (e ^ {T} - 1) p ^ {(j)} (1 + q ^ {(j)})) ^ {2}}\right) + \frac {\mu_ {j}}{1 + q ^ {(j)}} \left(1 - \frac {1}{1 + (e ^ {T} - 1) p ^ {(j)} (1 + q ^ {(j)})}\right), (17) \\ \bar {z} _ {T} ^ {(j)} = \frac {\bar {z} _ {0} ^ {(j)} e ^ {T / 2}}{1 + (e ^ {T} - 1) p ^ {(j)} (1 + q ^ {(j)})} + \frac {y ^ {(j)} q ^ {(j)}}{1 + q ^ {(j)}} \left(1 - \frac {1}{1 + (e ^ {T} - 1) p ^ {(j)} (1 + q ^ {(j)})}\right). (18) \\ \end{array}
$$
The distribution of  $X_0^{(j)} = \langle X_0, v_j \rangle$  given  $Y = y$  is  $\mathcal{N}(y^{(j)}q^{(j)} / (1 + q^{(j)}), \mu_j / (1 + q^{(j)})$ . As  $\bar{z}_T^{(j)} \to y^{(j)}q^{(j)} / (1 + q^{(j)})$  and  $\Sigma_T^{(j)} \to \mu_j / (1 + q^{(j)})$  as  $T \to +\infty$ , this shows that we sample from the exact target distribution (the one of  $X_0$  given  $Y = y$ ) for  $T$  large.
- If we start the reverse-time SDE from the correct model
$$
\bar {z} _ {0} ^ {(j)} = \frac {e ^ {- T / 2} y ^ {(j)} q ^ {(j)}}{1 + q ^ {(j)}}, \quad \Sigma_ {0} ^ {(j)} = \frac {\mu_ {j} e ^ {- T}}{1 + q ^ {(j)}} + \lambda_ {j} (1 - e ^ {- T}), \tag {19}
$$
then indeed  $Z_T^{(j)} \sim \mathcal{N}(y^{(j)}q^{(j)} / (1 + q^{(j)}),\mu_j / (1 + q^{(j)}))$ . This shows that, for any  $T$ ,  $Z_T$  has the same distribution as  $X_0$  given  $Y = y$ , which is the exact target distribution. We can show similarly that  $Z_{T - t}$  has the same distribution as  $X_{t}$  given  $Y = y$  for any  $t \in [0,T]$ .
- In the case that  $\sigma_B = 0$  so that we observe the mode values perfectly for  $j \in \mathcal{I}^{(n)}$ , then
$$
\mu^ {(x, j)} (t) = \frac {1}{2} - \frac {e ^ {t}}{e ^ {t} - 1}, \quad \mu^ {(y, j)} (t) = \frac {e ^ {t / 2}}{e ^ {t} - 1}, \tag {20}
$$
and indeed  $\lim_{t\uparrow T}Z_t^{(j)} = y^{(j)}$  a.s. Indeed the  $t^{-1}$  singularity at the origin drives the process to the origin like in the Brownian bridge.
Our analysis shows that, as long as we start from the invariant distribution of the diffusion process, we are able to sample from the correct target conditional distribution and that happens exponentially fast. This proves that the score of Definition 2 is the reverse drift of the SDE in (14). Additionally, the analysis shows that the score is uniformly bounded, except when there is no noise in the observations, blowing up near  $t = 0$ .
Remark 2. Note that, for  $q^{(j)} = 0$ , we obtain the unconditional model:
$$
d Z _ {t} ^ {(j)} = \mu^ {(j)} (T - t) Z _ {t} ^ {(j)} d t + \sqrt {\lambda_ {j}} d W ^ {(j)}, \text {w i t h} \mu^ {(j)} (t) = \frac {1}{2} - \frac {e ^ {t} p ^ {(j)}}{1 + (e ^ {t} - 1) p ^ {(j)}}. \tag {21}
$$
If  $C = C_{\mu}$ , the square expectation of the norm and the Lipschitz constant of the score are uniformly bounded in time:  $\sup_{j,t\in [0,T]}\left|\mu^{(j)}(t)\right| = 1 / 2$
Proposition 1. The score is  $S(t,x,y) = \sum_{j}S_{G}^{(j)}(t,\langle x,v_{j}\rangle ,y^{(j)})v_{j}, S_{G}^{(j)}(t,x^{(j)},y^{(j)}) = \left(\mu^{(x,j)}(T - t) - 1 / 2\right)x^{(j)} + \mu^{(y,j)}(T - t)y^{(j)}$  and it satisfies
$$
\mathbb {E} [ \| S (t, X _ {t}, y) \| _ {H} ^ {2} | Y = y ] = \sum_ {j} \frac {e ^ {t} \left(1 + q ^ {(j)}\right)}{1 + \left(e ^ {t} - 1\right) p ^ {(j)} \left(1 + q ^ {(j)}\right)} \frac {\lambda_ {j} ^ {2}}{\mu_ {j}}. \tag {22}
$$
Proof. The proof is a Gaussian calculation given in Appendix B.
In the unconditional setting, we have  $\mathbb{E}[\| S(t,X_t)\| _H^2 ] = \sum_j\frac{e^t}{1 + (e^t - 1)p^{(j)}}\frac{\lambda_j^2}{\mu_j}$  which is equal to  $\sum_{j}\lambda_{j}$  when  $C = C_{\mu}$ . It is indeed uniformly bounded in time.
In the conditional and noiseless setting  $(\sigma_B = 0)$ , we have  $\mathbb{E}[\| S(t,X_t,y)\| _H^2 |Y = y] = \sum_{j\notin \mathcal{I}^{(n)}}\frac{e^t}{1 + (e^t - 1)p^{(j)}}\frac{\lambda_j^2}{\mu_j} +\sum_{j\in \mathcal{I}^{(n)}}\frac{\lambda_j}{1 - e^{-t}}$ , which blows up as  $1 / t$  as  $t\to 0$ . This result shows that the extension of the score-based diffusion models to the conditional setting is not trivial.
5 Well-posedness for the reverse SDE for a general class of prior measures
We are now ready to consider the case of a general class of prior measures given as a density with respect to a Gaussian measure. The analysis of this case resembles the one of Pidstrigach et al. [24] for the unconditional setting. The main challenge is the singularity of the score for small times, an event that in the Gaussian case was observed in the noiseless setting. In this section we will provide a set of conditions to be satisfied by  $\Phi$  in (9), so that the conditional score is bounded uniformly in time. The existence of this bound is needed to make sense of the forward-reverse conditional SDE, and to prove the accuracy and stability of the conditional sampling.
We start the analysis by recalling that, in the infinite-dimensional case, the conditional score is (8). It is easy to get a first estimate:
$$
\mathbb {E} [ \| S (t, X _ {t}, y) \| _ {H} ^ {2} | Y = y ] \leq (1 - e ^ {- t}) ^ {- 1} \operatorname {T r} (C). \tag {23}
$$
The proof follows from Jensen inequality and the law of total expectation, see Appendix C. Note that (23) is indeed an upper bound of (22) since  $\operatorname{Tr}(C) = \sum_{j} \lambda_{j}$ .
Note that the bound (23) is also valid for the unconditional score  $S(t,x) = -(1 - e^{-t})^{-1}\big(x - e^{-t / 2}\mathbb{E}[X_0|X_t = x]\big)$ . We can observe that the upper bound (23) blows up in the limit of small times. We can make a few comments:
- The bound (23) is convenient for positive times, but the use of Jensen's inequality results in a very crude bound for small times. As shown in the previous section, we know that there exists a bound (21) for the unconditional score in the Gaussian case that is uniform in time.
- The singular behavior as  $1 / t$  at small time  $t$  is, however, not artificial. Such a behavior is needed in order to drive the state to the deterministic initial condition when there are exact observations. This behavior has been exhibited by (20) and (22) in the Gaussian case when  $\sigma_B = 0$ . This indicates that the following assumption (24) is not trivial in the conditional setting.
For Definition 2 to make sense in the more general case where the prior of  $X_0$  is given as a density with respect to a Gaussian measure, we will need to make the following assumption.
Assumption 1. For any  $y \in \mathbb{R}^n$ , we have
$$
\sup  _ {t \in [ 0, T ]} \mathbb {E} \left[ \| S (t, X _ {t}, y) \| _ {H} ^ {2} | Y = y \right] <   \infty . \tag {24}
$$
We are now ready to state the analogous result to Pidstrigach et al. [24, Theorem 1].
Proposition 2. Under Assumption 1, the solution of the reverse-time SDE
$$
d Z _ {t} = \frac {1}{2} Z _ {t} d t + S (T - t, Z _ {t}, y) d t + \sqrt {C} d W _ {t}, \quad Z _ {0} \sim X _ {T} | Y = y, \tag {25}
$$
satisfies  $Z_{T}\sim X_{0}|Y = y$
Proof. Given Assumption 1, the proof follows the same steps as the one given in [24] for the unconditional score. See Appendix C for the full proof.
Assumption 1 is satisfied under some appropriate conditions. In the following proposition, we provide a set of conditions that ensure the satisfaction of this assumption. It shows that it is possible to get an upper bound in (23) that is uniform in time provided some additional conditions are fulfilled.
Proposition 3. We assume that  $C_{\mu}$  in (9) and  $C$  in (1) have the same basis of eigenfunctions  $(v_j)$  and we define  $X_t^{(j)} = \langle X_t, v_j \rangle$  and  $S^{(j)}(t,x,y) = \langle S(t,x,y), v_j \rangle$  so that in (1)  $S(t,x,y) = \sum_j S^{(j)}(t,x,y)v_j$ . We assume an observational model as described in Section 4 and that the  $p^{(j)}(1 + q^{(j)})$  are uniformly bounded with respect to  $j$  and that  $C$  is of trace class. We make a modified version of assumption in (9) as follows. We assume that 1) the conditional distribution of  $X_0$  given  $Y = y$  is absolutely continuous with respect to the Gaussian measure  $\mu$  with a Radon-Nikodym derivative proportional to  $\exp(-\Psi(x_0,y))$ ; 2) we have  $\Psi(x_0,y) = \sum_j \Psi^{(j)}(x_0^{(j)},y)$ ,  $x_0^{(j)} = \langle x_0, v_j \rangle$ ; 3) for  $\psi^{(j)}(x^{(j)}, y) = \exp(-\Psi^{(j)}(x^{(j)}, y))$  we have
$$
\frac {1}{K} \leq | \psi^ {(j)} \left(x ^ {(j)}, y\right) | \leq K, \quad | \psi^ {(j)} \left(x ^ {(j)}, y\right) - \psi^ {(j)} \left(x ^ {(j) ^ {\prime}}, y\right) | \leq L \left| x ^ {(j) ^ {\prime}} - x ^ {(j)} \right|, \tag {26}
$$
where  $K$  and  $L$  do not depend on  $j$ . Then Assumption 1 holds true.
Proof. The proof is given in Appendix C.
To use the new score function of Definition 2 for sampling from the posterior, we need to define a way to estimate it. In other words, we need to define a loss function over which the difference between the true score and a neural network  $s_{\theta}(t,x_t,y)$  is minimized in  $\theta$ . A natural choice for the loss function is
$$
\mathbb {E} _ {t \sim U (0, T), x _ {t}, y \sim \mathcal {L} \left(X _ {t}, Y\right)} \left[ \| S (t, x _ {t}, y) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right], \tag {27}
$$
however it cannot be minimized directly since we do not have access to the ground truth conditional score  $S(t, x_t, y)$ . Therefore, in practice, a different objective has to be used. Batzolis et al. [39] proved that, in finite dimensions, a denoising score matching loss can be used:
$$
\mathbb {E} _ {t \sim U (0, T), x _ {0}, y \sim \mathcal {L} \left(X _ {0}, Y\right), x _ {t} \sim \mathcal {L} \left(X _ {t} \mid X _ {0} = x _ {0}\right)} \left[ \| C \nabla_ {x _ {t}} \ln p \left(x _ {t} \mid x _ {0}\right) - s _ {\theta} (t, x _ {t}, y) \| ^ {2} \right]. \tag {28}
$$
This expression involves only  $\nabla_{x_t}\log p(x_t|x_0)$  which can be computed analytically from the transition kernel of the forward diffusion process, also in infinite dimensions. In the following proposition, we build on the arguments of Batzolis et al. [39] and provide a proof that the conditional denoising estimator is a consistent estimator of the conditional score in infinite dimensions.
Proposition 4. Under Assumption 1, the minimizer in  $\theta$  of
$$
\mathbb {E} _ {x _ {0}, y \sim \mathcal {L} (X _ {0}, Y), x _ {t} \sim \mathcal {L} (X _ {t} | X _ {0} = x _ {0})} \left[ \| - (1 - e ^ {- t}) ^ {- 1} \left(x _ {t} - e ^ {- t / 2} x _ {0}\right) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right] \tag {29}
$$
is the same as the minimizer of
$$
\mathbb {E} _ {x _ {t}, y \sim \mathcal {L} \left(X _ {t}, Y\right)} \left[ \| S (t, x _ {t}, y) - s _ {\theta} (t, x _ {t}, y) \| _ {H} ^ {2} \right]. \tag {30}
$$
The same result holds if we add  $t \sim \mathcal{U}(0, T)$  in the expectations.
Proof. The proof combines some of the arguments of Batzolis et al. [39] and steps of the proof of Lemma 2 in [24], see Appendix C.  $\square$
Remark 3. A statement of robustness can be written as in [24, Theorem 2].
To put the presented theoretical results into practice, we provide two examples. The first stylized example aims at showcasing (i) the ability of our method in capturing nontrivial conditional distributions; and (ii) the discretization-invariance property of the learned conditional SDM. In the second example, we sample from the posterior distribution of a linearized seismic imaging problem in order to demonstrate the applicability of our method to large-scale problems. In both examples, in order to enable learning in function spaces, we parameterize the conditional score using Fourier neural operators [53]. Details regarding our experiment and implementation $^2$  are presented at Appendix D.
Stylized example Inspired by Phillips et al. [31], we define the target density via the relation  $x_0 = ay^2 + \varepsilon$  with  $\varepsilon \sim \Gamma(1, 2)$ ,  $a \sim \mathcal{U}\{-1, 1\}$ , and  $y \in [-3, 3]$ . Figure 1a illustrates the samples  $x_0$  evaluated on a fine  $y$  grid. After training (details in Appendix D), we sample the conditional distribution on uniformly sampled grids between  $[-3, 3]$ , each having 20 to 40 grid points. Figures 1b and 1c show the predicted samples for grid sizes of 25 and 35, respectively. The marginal conditionals associated with  $y = -1.0, 0.0, 0.5$  are shown in Figures 1d-1f, respectively. The gray shaded density in the bottom row of Figure 1 indicates the ground truth density, and colored estimated densities correspond to different discretizations of the horizontal axis. The visual inspection of samples and estimated densities indicates that our approach is indeed discretization-invariant.
Linearized seismic imaging example In this experiment, we address the problem of estimating the short-wavelength component of the Earth's subsurface squared-slowness model (i.e., seismic image; cf. Figure 2a) given surface measurements and a long-wavelength, smooth squared-slowness model (cf. Figure 2b). Following Orozco et al. [54], in order to reduce the high dimensionality of surface measurements, we apply the adjoint of the forward operator, the Born scattering operator, to the measurements and use the outcome (cf. Figure 2c) instead of measured data to condition the SDM. After training, given previously unseen observed data, we use the SDM to sample  $10^{3}$  posterior
$^{2}$ Code to reproduce results can be found at https://github.com/alisiahkoohi/csgm.
Figure 1: A depiction of the method's discretization invariance. Top row displays ground truth (a) and predicted samples (functions) on a uniformly sampled grid with (b) 25 and (c) 35 grid points. Bottom row shows conditional distribution marginals for (d)  $y = -1.0$ , (e)  $y = 0.0$ , and (f)  $y = 0.5$ .
(d)
Figure 2: Seismic imaging and uncertainty quantification. (a) Ground-truth seismic image. (b) Background squared-slowness. (c) Data after applying the adjoint Born operator. (d) Conditional (posterior) mean. (e) Pointwise standard deviation. (f) Absolute error between Figures 2a and 2d.
samples to estimate the conditional mean (cf. Figure 2d), which corresponds to the minimum-variance estimate [55], and the pointwise standard deviation (cf. Figure 2e), which we use to quantify the uncertainty. As expected, the pointwise standard deviation highlights areas of high uncertainty, particularly in regions with complex geological structures—such as near intricate reflectors and areas with limited illumination (deep and close to boundaries). We also observe a strong correlation between the pointwise standard deviation and the error in the conditional mean estimate (Figure 2f), confirming the accuracy of our Bayesian inference method.
We introduced a theoretically-grounded method that is able to perform conditional sampling in infinite-dimensional Hilbert (function) spaces using score-based diffusion models. This is a foundational step in using diffusion models to perform Bayesian inference. To achieve this, we learned the infinite-dimensional score function, as defined by Pidstrigach et al. [24], conditioned on the observed data. Under mild assumptions on the prior, this newly defined score—used as the reverse drift of the diffusion process—yields a generative model that samples from the posterior of a linear inverse problem. In particular, the well-known singularity in the conditional score for small times can be avoided. Building on these results, we presented stylized and large-scale examples that showcase the validity of our method and its discretization-invariance, a property that is a consequence of our theoretical and computational framework being built on infinite-dimensional spaces.
JG was supported by Agence de l'Innovation de Défense - AID - via Centre Interdisciplinaire d'Etudes pour la Défense et la Sécurité - CIEDS - (project 2021 - PRODIPO). LB, AS, and MVdH acknowledge support from the Simons Foundation under the MATH + X program, the Department of Energy under grant DE-SC0020345, and the corporate members of the Geo-Mathematical Imaging Group at Rice University. KS was supported by Air Force Office of Scientific Research under grant FA9550-22-1-0176 and the National Science Foundation under grant DMS-2308389.
[1] J. Hadamard. Lectures on the Cauchy's Problem in Linear Partial Differential Equations. Yale University Press, 1923.
[2] R. C. Aster, B. Borchers, and C. H. Thurber. Parameter estimation and inverse problems. Elsevier, 2018.
[3] K. Ito and B. Jin. Inverse Problems. World Scientific, 2014.
[4] M. S. Lehtinen, L. Paivarinta, and E. Somersalo. Linear inverse problems for generalised random variables. Inverse Problems, 5(4):599, 1989.
[5] A. M. Stuart. Inverse problems: A Bayesian perspective. Acta Numerica, 19:451-559, 2010.
[6] A. Tarantola. Inverse problem theory and methods for model parameter estimation. SIAM, 2005.
[7] B. T. Knapik, A. W. van der Vaart, and J. H. van Zanten. Bayesian inverse problems with Gaussian priors. The Annals of Statistics, 39(5):2626 - 2657, 2011.
[8] A. F. Bennett. Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press, 2002.
[9] A. Stuart. Uncertainty quantification in Bayesian inversion. In 2014 SIAM Conference on Uncertainty Quantification, 2014.
[10] Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
[11] B. Kawar, G. Vaksman, and M. Elad. SNIPS: Solving noisy inverse problems stochastically. In Advances in Neural Information Processing Systems, volume 34, pages 21757-21769, 2021.
[12] Y. Song, L. Shen, L. Xing, and S. Ermon. Solving inverse problems in medical imaging with score-based generative models. In International Conference on Learning Representations, 2022.
[13] B. D. Anderson. Reverse-time diffusion equation models. Stochastic Processes and their Applications, 12(3):313-326, 1982.
[14] P. Dhariwal and A. Nichol. Diffusion models beat GANs on image synthesis. In Advances in Neural Information Processing Systems, volume 34, pages 8780-8794, 2021.
[15] D. Gnaneshwar, B. Ramsundar, D. Gandhi, R. Kurchin, and V. Viswanathan. Score-based generative models for molecule generation, 2022. URL https://arxiv.org/abs/2203.04698. Preprint.
[16] H. Lee, J. Lu, and Y. Tan. Convergence of score-based generative modeling for general data distributions. In Proceedings of The 34th International Conference on Algorithmic Learning Theory, volume 201, pages 946-985, 2023.
[17] Y. Song and S. Ermon. Generative modeling by estimating gradients of the data distribution. In Advances in Neural Information Processing Systems, volume 32, 2019.