Score SDEs and reverse-time generation

Let $d\ge 1$ be an integer, let $I\subseteq \mathbb{R}$ be a time interval, and let $P_t$ be a probability law on ${\mathbb{R}}^d$ for each $t\in I$. Suppose $P_t$ has a density $p_t$ with respect to Lebesgue measure, meaning $P_t(A)={\int }_A{p}_t(x)dx$ for every Borel set $A\subseteq {\mathbb{R}}^d$. On the set of points $x$ where $0<p_t(x)<\mathrm{\infty }$ and the map $x\mapsto p_t(x)$ is differentiable, the marginal score of $P_t$ is

$s_t(x):={\mathrm{\nabla }}_x\log p_t(x)\mathrm{.}$

The score is a spatial vector field; this definition asserts no value where the density is zero, infinite, or not differentiable.

Song et al., Section 3.3

Raginsky, Section 7.2

Let $d\ge 1$, $T>0$, and let $P_t$ be Borel probability laws on ${\mathbb{R}}^d$ for $0\le t\le T$. Suppose each $P_t$ has a positive finite Lebesgue density $p_t$ with ${\int }_{{\mathbb{R}}^d}{p}_t(x)dx=1$. Let $b_t(x)\in {\mathbb{R}}^d$ and ${\sigma }_t(x)\in {\mathbb{R}}^{d\times m}$, where $m\ge 1$, and put $a_t(x)={\sigma }_t(x){\sigma }_t(x{)}^{\mathsf{T}}$. Assume $a_t$ is symmetric, all displayed derivatives exist continuously, and time derivatives at the endpoints are one-sided. Suppose $p_t$ satisfies the forward Fokker–Planck equation

${\mathrm{\partial }}_t{p}_t=-{\sum }_i{\mathrm{\partial }}_i(b_{t,i}{p}_t)+\frac{1}{2}{\sum }_{i,j}{\mathrm{\partial }}_i{\mathrm{\partial }}_j(a_{t,ij}{p}_t)\mathrm{.}$

Define $s_t(x)={\mathrm{\nabla }}_x\log p_t(x)$ and $(\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot a_t{)}_i(x)={\sum }_j{\mathrm{\partial }}_j{a}_{t,ij}(x)$, and set $v_t(x)=b_t(x)-\frac{1}{2}(\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot a_t)(x)-\frac{1}{2}{a}_t(x)s_t(x)$. Assume $v$ has a global $C^1$ flow of diffeomorphisms ${\mathrm{\Phi }}_{t,s}:{\mathbb{R}}^d\to {\mathbb{R}}^d$ for $0\le s,t\le T$, with ${\mathrm{\Phi }}_{s,s}$ the identity, ${\mathrm{\Phi }}_{t,r}\circ {\mathrm{\Phi }}_{r,s}={\mathrm{\Phi }}_{t,s}$, ${\mathrm{\partial }}_t{\mathrm{\Phi }}_{t,s}(x)=v_t({\mathrm{\Phi }}_{t,s}(x))$, and ${\mathrm{\partial }}_s{\mathrm{\Phi }}_{t,s}(x)=-D_x{\mathrm{\Phi }}_{t,s}(x)v_s(x)$, where $D_x$ denotes the Jacobian matrix in the state variable. For $Z_0$ with law $P_0$, let $Z_t={\mathrm{\Phi }}_{t,0}(Z_0)$. Then $Z_t$ has law $P_t$ for every $t\in [0,T]$. Thus the deterministic probability-flow ordinary differential equation (ODE) has the same one-time marginals as the probability-law solution of the displayed forward equation; no equality of path laws is asserted.

Write $C_c^1({\mathbb{R}}^d)$ for continuously differentiable real-valued functions with compact support. Here a time-dependent family of probability laws ${\mu }_t$ weakly solves ${\mathrm{\partial }}_t{\mu }_t+\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot (v_t{\mu }_t)=0$ if, for every continuously differentiable $\phi (t,x)$ with a common compact spatial support,

$\int \phi (t,x){\mu }_t(dx)-\int \phi (0,x){\mu }_0(dx)={\int }_0^t\int ({\mathrm{\partial }}_r\phi (r,x)+v_r(x)\cdot {\mathrm{\nabla }}_x\phi (r,x)){\mu }_r(dx)dr\mathrm{.}$

Song et al., Section 4.3 and Appendix D.1, equations (35)–(38)

Särkkä and Solin, Section 5.2, Theorem 5.4 and proof

Raginsky, Section 2.3

Assume the Axiom of Choice. Fix $T>0$, $0<ϵ<T$, an integer $d\ge 1$, and an integer $N\ge 1$. Let $X_t\in {\mathbb{R}}^d$ be a continuous nonexplosive Markov diffusion on $[ϵ,T]$, where nonexplosive means its paths remain in ${\mathbb{R}}^d$ on the whole interval, and Markov means its conditional future law given the past depends only on its current state. Write $P_t(A)=P(X_t\in A)$ for its law on each Borel set $A$. On a complete filtered probability space with the usual conditions, let $W$ be an adapted standard $d$-dimensional Brownian motion whose increments are independent of the earlier filtration and have laws $N_d(0,(t-s)I_d)$. Let $b_t(x)\in {\mathbb{R}}^d$ and ${\sigma }_t(x)\in {\mathbb{R}}^{d\times d}$ be Borel maps. The process is a continuous Brownian Itô process in the sense that, up to indistinguishability,

$X_t=X_{ϵ}+{\int }_{ϵ}^t{b}_r(X_r)dr+{({\sum }_{j=1}^d{\int }_{ϵ}^t{\sigma }_{ij}(r,X_r)d{W}_r^j)}_{i=1}^d,$

where the drift integrals are pathwise Lebesgue integrals and the coordinate stochastic integrals are localized Itô integrals; assume ${\int }_{ϵ}^T\mathrm{\mid }{b}_r(X_r)\mathrm{\mid }dr<\mathrm{\infty }$ and ${\int }_{ϵ}^T\mathrm{\parallel }{\sigma }_r(X_r){\mathrm{\parallel }}_F^{2}dr<\mathrm{\infty }$ almost surely, with $\mathrm{\parallel }M{\mathrm{\parallel }}_F^2={\sum }_{i,j}\mathrm{\mid }{M}_{ij}{\mathrm{\mid }}^2$. Assume also that ${\sigma }_t(x)$ is invertible with ${\sup }_{t,x}\mathrm{\parallel }{\sigma }_t(x{)}^{-1}\mathrm{\parallel }<\mathrm{\infty }$. Put $a_t(x)={\sigma }_t(x){\sigma }_t(x{)}^{\mathsf{T}}$. A transition density $q(s,x;t,y)$ means that $P(X_t\in dy\mid X_s=x)=q(s,x;t,y)dy$.

Assume the marginal densities $p_t$ and transition densities $q(s,x;t,y)$ are positive, finite, and normalized with respect to Lebesgue measure; $p_t$ has one continuous time derivative and two continuous state derivatives. For $ϵ\le s<t\le T$, assume

$p_t(y)={\int }_{{\mathbb{R}}^d}{p}_s(x)q(s,x;t,y)dx,q(s,x;t,y)={\int }_{{\mathbb{R}}^d}q(s,x;r,z)q(r,z;t,y)dz(s<r<t)\mathrm{.}$

For $L_t\varphi =b_t\cdot \mathrm{\nabla }\varphi +\frac{1}{2}{\sum }_{i,j}{a}_{t,ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_j\varphi$, define $L_t^{\ast }r=-{\sum }_i{\mathrm{\partial }}_i(b_{t,i}r)+\frac{1}{2}{\sum }_{i,j}{\mathrm{\partial }}_i{\mathrm{\partial }}_j(a_{t,ij}r)$. Assume ${\mathrm{\partial }}_t{p}_t=L_t^{\ast }{p}_t$ and ${\mathrm{\partial }}_{t}q(s,x;t,y)=L_t^{\ast ,y}q(s,x;t,y)$ classically. For every compactly supported smooth test, assume transition integrals and their time derivatives are absolutely integrable, differentiation under those integrals is valid, and integration by parts in space has no boundary term.

For $0\le u<T-ϵ$, set $s=T-u$ and define the reverse score $s_s(x)={\mathrm{\nabla }}_x\log p_s(x)$. With $a=\sigma {\sigma }^{\mathsf{T}}$ and $(\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot a{)}_i={\sum }_j{\mathrm{\partial }}_j{a}_{ij}$, define the reverse drift supplied by the positive-density reversal interface:

${\tilde{b}}_u(x)=-b_{T-u}(x)+(\mathrm{\nabla }\text{\hspace{0.17em}⁣}\cdot a_{T-u})(x)+a_{T-u}(x)s_{T-u}(x),{\tilde{\sigma }}_u(x)={\sigma }_{T-u}(x)\mathrm{.}$

Write $C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ for infinitely differentiable real functions with compact support. For $\varphi \in C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$, define ${\tilde{L}}_u\varphi ={\tilde{b}}_u\cdot \mathrm{\nabla }\varphi +\frac{1}{2}{\sum }_{i,j}{a}_{T-u,ij}{\mathrm{\partial }}_i{\mathrm{\partial }}_j\varphi$. Assume that for every reverse initial law the martingale problem for ${\tilde{L}}_u$ has a unique nonexplosive continuous solution in law; this means that each compact-test compensated process $\varphi (Y_u)-\varphi (Y_0)-{\int }_0^u{\tilde{L}}_r\varphi (Y_r)dr$ is a local martingale and its law is unique. The current reversal supplier's proof uses AC through its Brownian-reconstruction lemma, so AC is an explicit algorithm assumption here.

Set $S=T-ϵ$, $h=S\mathrm{/}N$, $u_k=kh$, and $t_k=T-u_k$ for $0\le k\le N$. Given step sizes ${\eta }_k>0$, the predictor-corrector score-sampling algorithm takes an initial draw $Y_0\sim P_T$ and mutually independent vectors $G_k,{\xi }_k\sim N_d(0,I_d)$, independent of $Y_0$. For $k=0,\dots ,N-1$, perform the following two operations:

1. Predictor. Compute one Euler–Maruyama step for the reverse equation: ${\widehat{Y}}_{k+1}=Y_k+h{\tilde{b}}_{u_k}(Y_k)+\sqrt{h}{\tilde{\sigma }}_{u_k}(Y_k)G_k\mathrm{.}$

2. Corrector. Let $U_{k+1}(x)=-\log p_{t_{k+1}}(x)$, so $e^{-U_{k+1}}$ is the normalized target density $p_{t_{k+1}}$. Apply one unadjusted Langevin update with step size ${\eta }_k$: $Y_{k+1}={\widehat{Y}}_{k+1}-{\eta }_k\mathrm{\nabla }{U}_{k+1}({\widehat{Y}}_{k+1})+\sqrt{2{\eta }_k}{\xi }_k={\widehat{Y}}_{k+1}+{\eta }_k{s}_{t_{k+1}}({\widehat{Y}}_{k+1})+\sqrt{2{\eta }_k}{\xi }_k\mathrm{.}$

Return $Y_N$, the finite-step state at forward time $ϵ$. The procedure requires an exact marginal score on $[ϵ,T]$ and starts from the actual terminal law $P_T$; it does not replace $P_T$ by $N_d(0,I_d)$ unless they are equal.

The positive-density Bayes-kernel reversal theorem. Apply it with horizon $S=T-ϵ$ to ${\bar{X}}_r=X_{ϵ+r}$, $0\le r\le S$, on the shifted filtration ${\bar{\mathcal{F}}}_r={\mathcal{F}}_{ϵ+r}$ and with Brownian motion ${\bar{W}}_r=W_{ϵ+r}-W_{ϵ}$. Brownian increments remain independent of ${\bar{\mathcal{F}}}_r$ by the original increment hypothesis, and the shifted filtration retains the usual conditions. The shifted coefficients are ${\bar{b}}_r(x)=b_{ϵ+r}(x)$ and ${\bar{\sigma }}_r(x)={\sigma }_{ϵ+r}(x)$; changing variables in the Lebesgue and elementary stochastic integrals, then passing to their localized limits, gives $d{\bar{X}}_r={\bar{b}}_r({\bar{X}}_r)dr+{\bar{\sigma }}_r({\bar{X}}_r)d{\bar{W}}_r$. Its marginal densities are ${\bar{p}}_r=p_{ϵ+r}$ and its transition densities are $\bar{q}(r,x;s,y)=q(ϵ+r,x;ϵ+s,y)$. Positivity, normalization, Markov identities, forward equations, uniform nonsingularity, integrability, and the assumed reverse martingale problem transfer by this deterministic time translation. Its reverse process is ${\bar{X}}_{S-u}=X_{T-u}$ with natural reversed filtration ${\mathcal{G}}_u=\sigma (X_{T-v}:0\le v\le u)$, and its Bayes kernels become ${\bar{K}}_{u,v}(y,dx)=\frac{p_{T-v}(x)q(T-v,x;T-u,y)}{p_{T-u}(y)}dx,0\le u<v\le S\mathrm{.}$ The supplier therefore gives precisely the displayed reverse coefficients and a Brownian driver in this reversed filtration, retaining Borel time dependence of $\sigma$.

The marginal score definition. It sets $s_t(x)={\mathrm{\nabla }}_x\log p_t(x)$ where the density is positive and differentiable.

The finite-dimensional Brownian convention.

The Brownian Itô-process convention. These conventions fix the increment, filtration, and localized integral assumptions used in the forward diffusion input.

The unadjusted Langevin algorithm. It supplies the one-step corrector used at each time level.

Song et al., Section 4.2 and Appendix G, Algorithm 1

Särkkä and Solin, Algorithm 8.1

Durmus and Moulines, Section 1, equation (2)

Raginsky, Section 7.2

This definition supplies no sampler convergence rate, no solver error bound, and no exactness claim for the finite predictor or unadjusted Langevin corrector. The corrector need not preserve its target density at a fixed positive step size. Errors from estimated scores, choosing a positive cutoff, substituting a terminal law, discretizing the reverse equation, and numerical integration require separate bounds.