SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Sungbin Lim $^{1,6,7*}$ , Eunbi Yoon $^{1}$ , Taehyun Byun $^{2}$ , Taewon Kang $^{3}$ , Seungwoo Kim $^{4}$ , Kyungjae Lee $^{5}$ , Sungjoon Choi $^{2*}$
$^{1}$ Department of Statistics, Korea University
$^{2}$ Department of Artificial Intelligence, Korea University
$^{3}$ Department of Computer Science and Engineering, Korea University
Artificial Intelligence Graduate School, UNIST
$^{5}$ Department of Artificial Intelligence, Chung-Ang University
$^{6}$ LG AI Research
$^{7}$ SNU-LG AI Research Center
Continuous-time score-based generative models consist of a pair of stochastic differential equations (SDEs)—a forward SDE that smoothly transitions data into a noise space and a reverse SDE that incrementally eliminates noise from a Gaussian prior distribution to generate data distribution samples—are intrinsically connected by the time-reversal theory on diffusion processes. In this paper, we investigate the use of stochastic evolution equations in Hilbert spaces, which expand the applicability of SDEs in two aspects: sample space and evolution operator, so they enable encompassing recent variations of diffusion models, such as generating functional data or replacing drift coefficients with image transformation. To this end, we derive a generalized time-reversal formula to build a bridge between probabilistic diffusion models and stochastic evolution equations and propose a score-based generative model called Hilbert Diffusion Model (HDM). Combining with Fourier neural operator, we verify the superiority of HDM for sampling functions from functional datasets with a power of kernel two-sample test of 4.2 on Quadratic, 0.2 on Melbourne, and 3.6 on Gridwatch, which outperforms existing diffusion models formulated in function spaces. Furthermore, the proposed method shows its strength in motion synthesis tasks by utilizing the Wiener process with values in Hilbert space. Finally, our empirical results on image datasets also validate a connection between HDM and diffusion models using heat dissipation, revealing the potential for exploring evolution operators and sample spaces.
Score-based generative models [52] have shown success in various domains, including the generation of images [25], texts [30], videos [26], motions [31]. [52] proposes a framework for continuous-time score-based generative models that use a pair of stochastic differential equations (SDEs); a forward SDE smoothly transitions data into noise space, and a reverse SDE incrementally eliminates noise from a Gaussian prior distribution to generate samples from the data distribution. Both SDEs have the same marginal distribution when their coefficients satisfy the time-reversal formula [1, 24], which is an ingenious application of the Kolmogorov-Fokker-Planck equation for diffusion processes.
*Corresponding Author. e-mail: sungbin@korea.ac.kr.
This work is done at UNIST.
A Unified Framework with Stochastic Evolution Equations in Hilbert Spaces
(a) SDE in Infinite-dimensions (HDM-SDE):  $\mathbf{B}_t = -b(t)\mathbf{I}$ ,  $\mathbf{G}_t = \sqrt{2b(t)}\mathbf{I}$
(b) Parabolic Stochastic PDE (HDM-SPDE):  $\mathbf{B}_t = b(t)(\Delta - \sigma \mathbf{I}), \mathbf{G}_t = \sqrt{2b(t)} (\sigma \mathbf{I} - \Delta)^{-\frac{k}{2}}$
Figure 1: A unified framework with various stochastic evolution equations to yield score-based generative models for different datasets. (a) SDE in infinite-dimensional spaces can generate functions as samples from the function space. (b) Utilizing a prior distribution generated from heat dissipation-type forward evolution  $\mathbf{B}_t = b(t)(\Delta - \sigma \mathbf{I})$ , the same as IHDM [50], we can generate image samples by adding Gaussian noise ( $\delta$ ) and executing reverse evolution through the parabolic SPDE.
Recently, there has been an active proposal of variations of diffusion models, such as generating functional data [18, 19, 23, 29, 38, 48, 49] or replacing drift coefficients with image transformation [4, 27, 50], which cannot be covered by the original SDE framework [52] formulated in the Euclidean space. Motivated by this problem, we propose a unified framework for continuous-time score-based generative models by using stochastic evolution equations in Hilbert spaces [6, 14, 34, 35],
$$
\mathrm {d} \mathbf {X} _ {t} = \mathbf {B} _ {t} (\mathbf {X} _ {t}) \mathrm {d} t + \mathbf {G} _ {t} \mathrm {d} \mathbf {W} _ {t}, \quad \mathrm {d} \widehat {\mathbf {X}} _ {t} = [ - \mathbf {B} _ {T - t} (\widehat {\mathbf {X}} _ {t}) + \mathbf {S} (T - t, \widehat {\mathbf {X}} _ {t}) ] \mathrm {d} t + \mathbf {G} _ {T - t} \mathrm {d} \mathbf {W} _ {t}, \quad (1)
$$
where  $(\mathbf{X}_t,\widehat{\mathbf{X}}_t)$  is a pair of forward and reverse stochastic processes with evolution operators  $(\mathbf{B}_t,\mathbf{G}_t)$ ,  $\mathbf{W}_t$  is a Wiener process with values in some Hilbert space  $\mathcal{H}$ , and  $\mathbf{S}(t,\cdot):[0,T]\times \mathcal{H}\to \mathcal{H}$  is a score operator, which generalizes the notion of score functions in the Euclidean space. Since we can choose  $\mathcal{H}$  and  $(\mathbf{B}_t,\mathbf{G}_t)$  in an abstract setting, stochastic equations (1) naturally extend the use of original SDEs to the infinite-dimensional setting and include stochastic partial differential equations (SPDEs), such as a parabolic equation driven by white noise (see Figure 1). Consequently, we can expand the SDE framework [52] in two aspects: sample space and evolution operator, enabling it to encompass recent variations of diffusion models in a continuous-time score-based generative model.
Contributions An essential key to identifying the relationship between the forward process  $\mathbf{X}_t$  and the reverse process  $\widehat{\mathbf{X}}_t$  in (1) is the derivation of time reversal formula with time-dependent operators  $\mathbf{B}_t$  and  $\mathbf{G}_t$ , which has been a challenging problem [23, 29, 38, 49] when we utilize coefficient schedulings [46, 52]. Contrary to previous approaches using semigroup theory [14], our approach is grounded in Kolmogorov equations with functional derivatives, well-studied in [2, 3, 7, 15-17], so that we can consider a wide scope of stochastic evolution equations with time-dependent operators.
- We derive a generalized time-reversal formula and unify the SDE framework [52] with stochastic equations in Hilbert spaces. Consequently, we propose a class of continuous-time score-based generative models, Hilbert Diffusion Model (HDM), which enables using SDEs in infinite-dimensional settings (HDM-SDE) and parabolic SPDEs (HDM-SPDE).
- Combining with neural operators [37], HDM-SDE shows the superiority for sampling functions from functional datasets with multi-modalities compared to GP [55], NP [20], DDPM [25], and other diffusion models formulated in function spaces, NDP [18] and SP-SGM [48]. We also validate that HDM-SDE plays a crucial role in successful human motion synthesis compared to DDPM, by utilizing the Wiener process in the Hilbert space.
- We build a bridge between parabolic SPDE and diffusion models using heat dissipation [50]. HDM-SPDE improves the performance in image generation as a continuous-time version of IHDM. Also, the proposed method shows the inductive bias in controllable coarse-to-fine image generation, e.g., inpainting, deblurring, and depixelation, without additional training.
2 A Foundation on Time-Reversal of Stochastic Evolution Equations
We first present a preliminary theory on the stochastic evolution equations in Hilbert space and introduce the logarithmic derivatives of probability measures to propose a Hilbert space valued score operator (see Section 2.2) which generalizes the notion of score functions in the Euclidean space. Then we derive the time-reversal formula (Theorem 2.1), which is a key to obtain the time-reversal of the diffusion processes by using the Kolmogorov equations in Hilbert space.
2.1 Preliminaries: Stochastic Evolution Equations in Hilbert Spaces
In preliminaries, we introduce elements of stochastic evolution equations in Hilbert spaces. See B.1 or we refer [5, 6, 14, 34, 35] for basic setting.
Gelfand Triple and Cameron-Martin Space Let  $\mathcal{H}$  be a separable Hilbert space with the  $\mathcal{H}$ -norm defined by the inner product  $\| \cdot \|_{\mathcal{H}} = \sqrt{\langle\cdot,\cdot\rangle_{\mathcal{H}}}$ . Let  $\mathcal{L}(\mathcal{H})$  denotes the set of all bounded linear operators on  $\mathcal{H}$ . We assume there exists a Radon centered Gaussian measure  $\lambda$  on the Borel  $\sigma$ -field  $\mathcal{B}(\mathcal{H})$  with the nonnegative, symmetric, self-adjoint, and the finite trace covariance operator  $\Lambda \in \mathcal{L}(\mathcal{H})$ , hence there exists eigensystem  $\{(\lambda^{(\ell)},\phi^{(\ell)})\in \mathbb{R}_{+}\times \mathcal{H}:\ell \in \mathbb{N}\}$  such that  $\Lambda (\phi^{(\ell)}) = \lambda^{(\ell)}\phi^{(\ell)}$  holds and  $\mathrm{Tr}(\Lambda) = \sum_{\ell = 1}^{\infty}\lambda^{(\ell)} < \infty$ . We define Cameron-Martin space by  $\mathcal{H}_{\lambda}:= \Lambda^{1 / 2}(\mathcal{H})$  with the inner product  $\langle h_1,h_2\rangle_{\mathcal{H}_\lambda}:= \langle \Lambda^{-1 / 2}(h_1),\Lambda^{-1 / 2}(h_2)\rangle_{\mathcal{H}}$ . Let  $\mathcal{H}^* \subset \mathcal{H}_\lambda^*$  be the dual space of  $\mathcal{H}$  and  $\mathcal{H}_{\lambda}$ , respectively. By identifying  $\mathcal{H}_{\lambda}$  with  $\mathcal{H}_{\lambda}^*$  via the Riesz isomorphism, we can consider a continuous inclusion map  $i_{\mathcal{H}_{\lambda}}:\mathcal{H}^{*}\to \mathcal{H}_{\lambda}$  such that
$$
\varphi (\xi) = \langle i _ {\mathcal {H} _ {\lambda}} (\varphi), \xi \rangle_ {\mathcal {H} _ {\lambda}} = \langle \Lambda^ {- 1 / 2} (i _ {\mathcal {H} _ {\lambda}} (\varphi)), \Lambda^ {- 1 / 2} (\xi) \rangle_ {\mathcal {H}}, \quad \varphi \in \mathcal {H} ^ {*}, \xi \in \mathcal {H} _ {\lambda}. \tag {2}
$$
Thus, we can also identify  $\mathcal{H}^*$  with  $\Lambda^{1/2}(\mathcal{H}_{\lambda})$  (see [8, Example 1.1]). For notational convenience, we write  $i_{\mathcal{H}_{\lambda}}(\varphi)$  as  $\Lambda(\varphi)$  by regarding  $\varphi$  as an element in  $\mathcal{H}$ . In summary,
$$
\mathcal {H} _ {\lambda} = \Lambda^ {1 / 2} (\mathcal {H}), \quad i _ {\mathcal {H} _ {\lambda}} \left(\mathcal {H} ^ {*}\right) = \Lambda (\mathcal {H}), \quad i _ {\mathcal {H} _ {\lambda}} \left(\mathcal {H} ^ {*}\right) \subset \mathcal {H} _ {\lambda} \subset \mathcal {H}. \tag {3}
$$
We refer to  $(\mathcal{H}^*,\mathcal{H}_\lambda ,\mathcal{H})$  as Gelfand triple which provides groundwork for the SDE theory in  $\mathcal{H}$
Example 2.1 (Euclidean space). Let  $\mathcal{H} \coloneqq \mathbb{R}^d$  and  $\lambda = \mathcal{N}(0, \Sigma)$  be a Gaussian measure with a positive-definite covariance matrix  $\Sigma \in \mathbb{R}^{d \times d}$ . Then  $|\Sigma^{-1/2}\mathbf{x}| < \infty$  for  $\mathbf{x} \in \mathbb{R}^d$ , hence  $\mathcal{H}_{\lambda} = \mathcal{H}$ .
Example 2.2 (Reproducing kernel Hilbert space). Let  $\mathcal{H}$  denote the class of square-integrable functions on finite measure space  $(\mathcal{X},\mu_{\mathcal{X}})$  with the usual  $L_{2}$ -norm,  $\| f\|_{\mathcal{H}}\coloneqq \left(\int_{\mathcal{X}}|f(x)|^{2}\mu_{\mathcal{X}}(\mathrm{d}x)\right)^{1 / 2}$ . Due to the assumption on  $\Lambda$ , there exists symmetric and positive definite kernel function  $K_{\lambda}$  such that
$$
\Lambda (f) (\mathbf {x}) = \int_ {\mathcal {X}} K _ {\lambda} (\mathbf {x}, \mathbf {y}) f (\mathbf {y}) \mu_ {\mathcal {X}} (\mathrm {d} \mathbf {y}), \quad \mathbf {x} \in \mathcal {X}, f \in \mathcal {H}. \tag {4}
$$
Hence Cameron-Martin space  $\mathcal{H}_{\lambda}$  is reproducing kernel Hilbert space (RKHS) with  $K_{\lambda}$ .
Wiener Processes and Stochastic Evolution Equations in Hilbert Spaces For  $T > 0$ , let  $(\mathbf{W}_t)_{t\in [0,T]}$  be  $\mathcal{H}$ -valued  $\mathcal{F}_t$ -adapted  $\Lambda$ -Wiener process (see Definition B.1 for details). By the Kosambi-Karhunen-Loève theorem,  $\mathbf{W}_t$  has a series representation:
$$
\mathbf {W} _ {t} := \Lambda^ {1 / 2} \left(\mathcal {W} _ {t}\right) = \sum_ {\ell = 1} ^ {\infty} \sqrt {\lambda^ {(\ell)}} W _ {t} ^ {(\ell)} \phi^ {(\ell)}, \quad t \in [ 0, T ], \tag {5}
$$
where  $\mathcal{W}_t = \{W_t^{(\ell)} : \ell \in \mathbb{N}\}$  is white noise of which the components  $W_t^{(\ell)}$  are independent real-valued Wiener processes. Then  $\mathrm{Cov}(\mathbf{W}_t) = t\Lambda$  holds and  $\mathbb{E}\langle \mathbf{W}_t, \mathbf{W}_s \rangle_{\mathcal{H}} = \min \{t, s\} \cdot \mathrm{Tr}(\Lambda) < \infty$  since  $\Lambda$  has the finite trace. If  $\mathcal{H}_{\lambda}$  is RKHS with  $K_{\lambda}$  (Example 2.2),  $\mathbf{W}_t$  preserves spatial correlation,
$$
\mathbb {E} \left[ \mathbf {W} _ {t} \left(\mathbf {x} _ {1}\right) \mathbf {W} _ {s} \left(\mathbf {x} _ {2}\right) \right] = \min  \{t, s \} K _ {\lambda} \left(\mathbf {x} _ {1}, \mathbf {x} _ {2}\right), \quad \mathbf {x} _ {1}, \mathbf {x} _ {2} \in \mathcal {X}. \tag {6}
$$
Let  $\mathcal{L}_2(\mathcal{H})$  denote the class of Hilbert-Schmidt operator on  $\mathcal{H}$ . Following the usual construction of the stochastic integral with respect to  $\mathbf{W}_t$  (e.g., see [14, Section 4] and [35, Section 2]), we can define the following stochastic evolution equation in  $\mathcal{H}$ ,
$$
\mathbf {X} _ {t} = \mathbf {X} _ {0} + \int_ {0} ^ {t} \mathbf {B} _ {s} \left(\mathbf {X} _ {s}\right) \mathrm {d} s + \int_ {0} ^ {t} \mathbf {G} _ {s} \mathrm {d} \mathbf {W} _ {s}, \quad \mathbf {X} _ {0} \in \mathcal {H}, \quad t \in [ 0, T ], \tag {7}
$$
where  $\mathbf{B}_t:[0,T]\times \mathcal{H}\to \mathcal{H}$  and  $\mathbf{G}_t:[0,T]\rightarrow \mathcal{L}_2(\mathcal{H})$  are progressively measurable satisfying regular conditions [35, Section 3.2] which guarantee solvability in  $\mathcal{H}$ . Under these assumptions, there exists a unique  $\mathcal{H}$ -valued solution  $\mathbf{X}_t$  to (7) for every  $t\in [0,T]$  such that  $\mathbf{X}_0\in \mathcal{H}$  and  $\mathbf{W}_t\in \mathcal{H}$  are independent. We assume  $\mathbf{G}_t$  is constant on  $\mathcal{H}$ , which still recovers the SDE framework of [52].
2.2 Score Operator and Time-Reversal Formula
Now we consider a time-reversal  $\widehat{\mathbf{X}}_t\coloneqq \mathbf{X}_{T - t}$  of (7) and its corresponding diffusion process,
$$
\widehat {\mathbf {X}} _ {t} = \widehat {\mathbf {X}} _ {0} + \int_ {0} ^ {t} \widehat {\mathbf {B}} _ {s} \left(\widehat {\mathbf {X}} _ {s}\right) \mathrm {d} t + \int_ {0} ^ {t} \widehat {\mathbf {G}} _ {s} \mathrm {d} \mathbf {W} _ {s}, \quad \widehat {\mathbf {X}} _ {0} \sim \mathcal {N} (0, \Lambda), \quad t \in [ 0, T ]. \tag {8}
$$
A condition for the existence of the time-reversal process (8) is well-studied for diffusion processes in abstract settings (e.g., see [12]), hence we focus on identifying evolutionary operators  $\widehat{\mathbf{B}}$  and  $\widehat{\mathbf{G}}$ .
Logarithmic Derivative of Probability Measure and Score Operator A score function in the Euclidean space is defined by a gradient of logarithmic probability density  $\nabla_{x}\log p_{t}(x)$ . For Hilbert spaces, we employ logarithmic derivatives of Fomin differentiable probability measures [8, 9]. Let us define the class of smooth cylinder functions [10], which is dense in the Sobolev space on  $\mathcal{H}$ , by
$$
\mathcal {F C} _ {b} ^ {\infty} := \left\{f _ {\varphi_ {1: m}} = f \left(\varphi_ {1}, \dots , \varphi_ {m}\right): \mathcal {H} \rightarrow \mathbb {R}: f \in C _ {b} ^ {\infty} \left(\mathbb {R} ^ {m}\right), \varphi_ {i} \in \mathcal {H} ^ {*}, m \in \mathbb {N} \right\}. \tag {9}
$$
For  $f_{\varphi_{1:m}} \in \mathcal{FC}_b^\infty$ , we compute the  $\mathcal{H}_{\lambda}$ -gradient at  $u \in \mathcal{H}$  in the direction of  $\xi \in \mathcal{H}_{\lambda}$  by
$$
\langle D f _ {\varphi_ {1: m}} (u), \xi \rangle_ {\mathcal {H} _ {\lambda}} = \sum_ {i = 1} ^ {m} \partial_ {i} f (\varphi_ {1} (u), \dots , \varphi_ {m} (u)) \langle \xi , \Lambda (\varphi_ {i}) \rangle_ {\mathcal {H} _ {\lambda}}, \quad u \in \mathcal {H}, \tag {10}
$$
where  $Df_{\varphi_{1:m}}$  denotes a Gâteaux derivative. A differentiable measure  $\mu$  on  $\mathcal{H}$  along a vector  $\xi \in \mathcal{H}_{\lambda}$  has the partial logarithmic derivative  $\rho_{\xi}^{\mu} \in L_1(\mathcal{H}, \mu)$  if the following integration by parts holds,
$$
\int_ {\mathcal {H}} \left\langle D f _ {\varphi_ {1: m}} (u), \xi \right\rangle_ {\mathcal {H} _ {\lambda}} \mu (\mathrm {d} u) = - \int_ {\mathcal {H}} f _ {\varphi_ {1: m}} (u) \rho_ {\xi} ^ {\mu} (u) \mu (\mathrm {d} u), \quad f _ {\varphi_ {1: m}} \in \mathcal {F C} _ {b} ^ {\infty}. \tag {11}
$$
If there exists a Borel map  $\rho_{\mathcal{H}_{\lambda}}^{\mu}(\cdot):\mathcal{H}\to \mathcal{H}$  such that  $\langle \rho_{\mathcal{H}_\lambda}^\mu (u),\xi \rangle_{\mathcal{H}_\lambda} = \rho_\xi^\mu (u)$  for  $\xi \in \mathcal{H}_{\lambda}$ , then  $\rho_{\mathcal{H}_{\lambda}}^{\mu}$  is called the vector logarithmic derivative of  $\mu$  associated with  $\mathcal{H}_{\lambda}$ . For instance, if  $\mu$  is a Gaussian measure with mean  $m_{\mu}\in \mathcal{H}$  and covariance operator  $\Lambda_{\mu}\in \mathcal{L}(\mathcal{H})$  satisfying  $\Lambda (\mathcal{H})\subset \Lambda_{\mu}(\mathcal{H})$ , then  $\rho_{\mathcal{H}_{\lambda}}^{\mu}$  is computed by (see [8, Example 2.5] and [14, Proposition 2.26]),
$$
\rho_ {\mathcal {H} _ {\lambda}} ^ {\mu} (u) = \Lambda \circ \Lambda_ {\mu} ^ {- 1} \left(m _ {\mu} - u\right), \quad u \in \mathcal {H}, \tag {12}
$$
which generalizes the score function  $\nabla_{\mathbf{x}}\log p(\mathbf{x})$  of a Gaussian density  $p(\mathbf{x})\sim \mathcal{N}(m_{\mu},\Sigma_{\mu})$ . For a sequence of probability measures  $(\mu_t)_{t\in [0,T]}$  of the solution to (7) with  $\mu_0 = \mathbb{P}_{\mathrm{data}}$ , we define a score operator  $\mathbf{S}_{\lambda}(t,\cdot):[0,T]\times \mathcal{H}\to \mathcal{H}$  by multiplying  $\mathbf{G}_t\mathbf{G}_t^*$  to the vector logarithmic derivative  $\rho_{\mathcal{H}_{\lambda}}^{\mu_t}$ ,
$$
\mathbf {S} _ {\lambda} (t, u) := \mathbf {G} _ {t} \mathbf {G} _ {t} ^ {*} \rho_ {\mathcal {H} _ {\lambda}} ^ {\mu_ {t}} (u). \tag {13}
$$
Hence the score operator  $\mathbf{S}_{\lambda}(t,\cdot)$  is analogous to  $g^{2}(t)\nabla_{\mathbf{x}}\log p_{t}(\mathbf{x})$  in the reverse SDE of [52].
Kolmogorov Equations and Time-Reversal Formula Score-based generative model through SDE is grounded on the Fokker-Planck-Kolmogorov equations [1, 24]. Following [5, 6, 8-10], we define Kolmogorov operator  $\mathcal{L}_t$  on the class of smooth cylinder functions  $\mathcal{FC}_b^\infty$  by
$$
\mathcal {L} _ {t} f _ {\varphi_ {1: m}} (u) := \frac {1}{2} \operatorname {T r} _ {\mathcal {H} _ {\lambda}} \left(\mathbf {A} _ {t} (u) \circ \Lambda \circ D ^ {2} f _ {\varphi_ {1: m}} (u)\right) + \langle D f _ {\varphi_ {1: m}} (u), \mathbf {B} _ {t} (u) \rangle_ {\mathcal {H} _ {\lambda}}, \tag {14}
$$
where  $\mathbf{A}_t\coloneqq \mathbf{G}_t\mathbf{G}_t^*$  and  $D^{2}f_{\varphi_{1:m}}$  denotes the second-order Gâteaux derivative. The Kolmogorov operators  $(\mathcal{L}_t)_{t\in [0,T]}$  describes evolution of  $(\mu_t)_{t\in [0,T]}$  and becomes a generator of Markov diffusion process [5, 6, 24]. Now we present our main result on the time-reversal formula.
Theorem 2.1 (Time-Reversal Formula). Time reversal  $\widehat{\mathbf{X}}_t$  satisfying (8) has generator
$$
\widehat {\mathcal {L} _ {t}} f _ {\varphi_ {1: m}} (u) := \frac {1}{2} \operatorname {T r} _ {\mathcal {H} _ {\lambda}} \left(\widehat {\mathbf {A}} _ {t} \circ \Lambda \circ D ^ {2} f _ {\varphi_ {1: m}} (u)\right) + \langle D f _ {\varphi_ {1: m}} (u), \widehat {\mathbf {B}} _ {t} (u) \rangle_ {\mathcal {H} _ {\lambda}}, \tag {15}
$$
where  $\widehat{\mathbf{A}}_t$  and  $\widehat{\mathbf{B}}_t$  satisfy the following time-reversal formula
$$
\widehat {\mathbf {A}} _ {t} = \mathbf {A} _ {T - t}, \quad \widehat {\mathbf {B}} _ {t} (u) = - \mathbf {B} _ {T - t} (u) + \mathbf {S} _ {\lambda} (T - t, u). \tag {16}
$$
See Appendix B.1 for the proof of Theorem 2.1. The proof is based on the Kolmogorov forward equation and the Kolmogorov backward equation with functional derivatives to show that (15) is the generator of the time-reversal diffusion process (8), which is analogous to the proof in the Euclidean space [24]. Due to (16), we can rewrite (8) by
$$
\widehat {\mathbf {X}} _ {t} = \widehat {\mathbf {X}} _ {0} + \int_ {0} ^ {t} \left[ - \mathbf {B} _ {T - s} \left(\widehat {\mathbf {X}} _ {s}\right) + \mathbf {S} _ {\lambda} (T - s, \widehat {\mathbf {X}} _ {s}) \right] d s + \int_ {0} ^ {t} \mathbf {G} _ {T - s} d \mathbf {W} _ {s}. \tag {17}
$$
Hence we can generate a sample from  $\hat{\mathbf{X}}_T\sim \mathbb{P}_{\mathrm{data}}$  by running (17) if we obtain  $\mathbf{S}_{\lambda}(t,\cdot)$  for  $t\in [0,T]$ .
2.3 Applications of Theorem 2.1
This section presents two important applications of Theorem 2.1; SDEs in infinite-dimensional space and parabolic SPDEs in finite-dimensional space. We utilize these stochastic evolution equations to propose a unified framework of continuous-time score-based generative models in Section 3.
2.3.1 SDEs in Infinite-Dimensions
We first generalize the SDE framework [52] in the infinite-dimensions based on Theorem 2.1. Let  $\{\lambda^{(\ell)},\phi^{(\ell)}\}_{\ell \in \mathbb{N}}$  be an eigensystem of  $\mathcal{H}$  and consider the following SDE system with values in  $\mathcal{H}$ ,
$$
\mathrm {d} X _ {t} ^ {(\ell)} = b _ {t} ^ {(\ell)} X _ {t} ^ {(\ell)} \mathrm {d} t + \sqrt {\lambda^ {(\ell)}} \sigma_ {t} ^ {(\ell)} \mathrm {d} W _ {t} ^ {(\ell)}, \quad t \in [ 0, T ], \ell \in \mathbb {N}, \tag {18}
$$
where  $b_{t}^{(\ell)}:[0,T]\to \mathbb{R}$  and  $\sigma_t^{(\ell)}:[0,T]\to \mathbb{R}_+$  are real-valued functions for each  $\ell \in \mathbb{N}$ . Set  $\beta_t^{(\ell)}\coloneqq \exp \left(\int_0^t b_s^{(\ell)}\mathrm{d}s\right)$  and  $\Sigma_{\beta}^{(\ell)}(t)\coloneqq \left(\sigma_t^{(\ell)} / \beta_t^{(\ell)}\right)^2$ . Conditioning at  $X_0^{(\ell)}$ , the law of  $X_{t}^{(\ell)}$  follows Gaussian distribution, hence we can apply [44, Theorem 5.3] to (18), which induces  $\widehat{\mathbf{B}}_t(\mathbf{x})$  by
$$
\widehat {\mathbf {B}} _ {t} (\mathbf {x}) = \sum_ {\ell \in \mathbb {N}} \left(- b _ {T - t} ^ {(\ell)} x ^ {(\ell)} + \Sigma_ {\beta} ^ {(\ell)} (T - t) \frac {\beta_ {T - t} ^ {(\ell)} X _ {0} ^ {(\ell)} - x ^ {(\ell)}}{\int_ {0} ^ {T - t} \Sigma_ {\beta} ^ {(\ell)} (s) d s}\right). \tag {19}
$$
Indeed, (19) is a special case of (16) (see Appendix C.1).
Example 2.3 (VP-SDE and sub-VP-SDE in Infinite-Dimensions). Set  $b_{t}^{(\ell)} = -\frac{1}{2} b(t)$  and  $\sigma_t^{(\ell)} = \sqrt{b(t)}$  for every  $\ell \in \mathbb{N}$ . Then we have  $\beta_{t} = e^{-\frac{1}{2}\int_{0}^{t}b(s)\mathrm{d}s}$  and  $\Sigma_{\beta}(t) = b(t)e^{\int_0^t b(s)\mathrm{d}s}$ . Thus,
$$
m _ {\mu_ {t} | \mathbf {X} _ {0}} = e ^ {- \frac {1}{2} \int_ {0} ^ {t} b (s) \mathrm {d} s} \mathbf {X} _ {0}, \quad \Lambda_ {\mu_ {t} | \mathbf {X} _ {0}} = \left(1 - e ^ {- \int_ {0} ^ {t} b (s) \mathrm {d} s}\right) \Lambda . \tag {20}
$$
Hence we can compute the conditional score function  $\mathbf{S}_{\lambda}(t,\mathbf{X}_t|\mathbf{X}_0)$  by using (19). As a result, (20) generalizes VP-SDE [52] in infinite-dimensions. Similarly, we can recover the sub-VP-SDE in infinite-dimensions, hence (19) recovers SDEs proposed in [52] when  $\Lambda = \mathrm{Id}$  and  $\mathcal{H} = \mathbb{R}^d$
2.3.2 Parabolic SPDEs
We introduce parabolic SPDEs [34, 35] based on Theorem 2.1. For  $k > -1$ , we define
$$
\mathrm {d} u _ {t} = b (t) \left[ \Delta u _ {t} - \sigma u _ {t} \right] \mathrm {d} t + \sqrt {2 b (t)} \left(\sigma \mathbf {I} - \Delta\right) ^ {- \frac {k}{2}} \mathrm {d} \mathcal {W} _ {t}, \quad t \in [ 0, T ], \tag {21}
$$
on a bounded domain  $\mathcal{O}$  with a zero-Neumann boundary condition. By following [50], we use the discrete cosine transformation (DCT) from a coordinate vector  $u_{t}$  expressed in the standard basis to a coordinate vector  $\hat{u}_t$ , with the eigendecomposition  $\Delta \triangleq \mathbf{VDV}^{\top}$ . Hence we can rewrite (21) as