Fixed-Point-Corrected Numerical Schemes for Reverse-Time Diffusion Sampling: Stability and Error Decomposition

Abstract

We study fixed-point-corrected numerical schemes for reverse-time diffusion sampling. Instead of treating the reverse sampler only as an explicit Euler–Maruyama discretization, we formulate each reverse-time step as a local implicit equation and approximate its solution by a finite number of fixed point corrections. After fixing the backward-time drift convention, we prove well-posedness of the local implicit step, contraction of the inner solver for sufficiently small step sizes, and a conditional global error decomposition separating terminal initialization, score approximation, assumed implicit time-discretization error, and fixed point truncation error. The estimate clarifies how additional inner corrections reduce numerical residuals at the cost of extra score evaluations. Multi-seed exact-score experiments, implicit-gap diagnostics, controlled score perturbations, a fitted Gaussian-mixture score example, and a neural-score Fashion-MNIST experiment with inpainting demonstrate the distinction between algebraic solver consistency and distributional sample quality. The paper is intended as a solver-level numerical analysis of reverse diffusion dynamics, not as a new large-scale generative architecture or a complete convergence theorem for diffusion models.

1. Introduction

Diffusion-based generative models have become a central class of modern generative methods. Their common principle is to transform a data distribution into a tractable noise distribution through a forward noising process and then to generate new samples by approximately reversing these dynamics [1,2]. In the continuous-time formulation, this reverse process is described by a stochastic differential equation whose drift depends on the score function of the time-dependent marginal distribution [3,4]. Denoising diffusion probabilistic models and their continuous-time score-based extensions have motivated a wide literature on sampling procedures, probability-flow ordinary differential equations, predictor–corrector methods, and accelerated numerical solvers [5,6,7,8,9,10,11].

Although these models are usually implemented through neural networks, their sampling stage is fundamentally a numerical approximation problem for stochastic or deterministic differential equations. Thus, the quality of generated samples depends not only on the learned score, but also on the discretization of the reverse-time dynamics, the terminal initialization error, and the stability of the numerical solver. Recent theoretical works have begun to quantify convergence properties of SDE and probability-flow ODE samplers and to separate score approximation effects from time-discretization effects [12,13,14,15,16,17]. These results provide end-to-end sampling guarantees under different assumptions on the data distribution, score error, and sampler dynamics. The present paper is complementary: it focuses on the local nonlinear algebraic problem created by an implicit reverse-time step and quantifies the additional error introduced when that algebraic equation is solved only approximately.

The purpose of this paper is to investigate a fixed-point-corrected discretization of reverse-time diffusion sampling. Given a time step of the reverse dynamics, we replace the fully explicit update with a local implicit equation. Since this equation is generally nonlinear, we solve it approximately by a finite number of fixed-point corrections. This construction produces a family of samplers interpolating between explicit Euler-type schemes and fully implicit schemes. The main question is how the finite number of inner fixed-point iterations contributes to the total sampling error.

The method should be distinguished from high-order or exponential-integrator samplers such as PNDM, DEIS, and DPM-Solver, whose primary goal is to reduce the number of reverse ODE evaluations by exploiting the semilinear structure of diffusion dynamics [6,7,11]. The fixed point correction considered here is a solver-level mechanism for a local implicit equation. It can therefore be interpreted as an inner numerical stabilization step, orthogonal to score training and potentially combinable with more sophisticated outer discretizations.

The main contributions of the paper are as follows:

• We formulate a local implicit reverse-time diffusion step and define a fixed-point-corrected sampler with a finite number of inner iterations.
• We prove local well-posedness of the implicit step under a contraction condition on the time step and the Lipschitz constant of the score-induced drift.
• We derive a solver error estimate showing geometric decay with respect to the number of fixed-point corrections.
• We give a conditional global error decomposition separating terminal initialization error, score approximation error, an assumed implicit time-discretization error, and fixed-point solver error.
• We report reproducible multi-seed numerical experiments with exact scores, implicit-gap diagnostics, controlled score perturbations, a fitted parametric score example, and a data-driven neural-score experiment on Fashion-MNIST with an inpainting task.

The analysis is intentionally modest in scope. We do not attempt to prove a complete convergence theorem for large-scale image-generation pipelines, nor a new strong convergence theorem for the fully implicit reverse SDE discretization. Instead, we isolate a tractable solver-level numerical-analysis mechanism for reverse diffusion dynamics under standard smoothness and stability assumptions.

The fixed point analysis used below is deliberately classical. We do not claim a new abstract fixed point theorem; the Banach contraction principle is used as a transparent tool for the local nonlinear equation generated by one implicit reverse step. Recent fixed point literature has developed Banach-type results in perturbed metric spaces and probabilistic cone b-metric spaces [18,19]. Such generalized spaces are mathematically interesting, but the sampler analyzed here is formulated pathwise in ${\mathbb{R}}^d$ and the endpoint error is measured through Euclidean $L^2$ couplings and Wasserstein distance. Extending the residual-based solver analysis to generalized probabilistic metric structures would require replacing both the local residual metric and the distributional endpoint distance, and is left as a separate problem.

2. Reverse-Time Diffusion Dynamics

Let ${\left(X_t\right)}_{t\in [0,T]}$ be the solution of the forward stochastic differential equation

$$d{X}_t=f(X_t,t)dt+g\left(t\right)d{W}_t,$$

(1)

where $f:{\mathbb{R}}^d\times [0,T]\to {\mathbb{R}}^d$ is a drift field, $g:[0,T]\to (0,\infty )$ is a scalar diffusion coefficient, and $\left(W_t\right)$ is a standard Brownian motion in ${\mathbb{R}}^d$. The forward process transports the data law $p_0$ toward a tractable terminal law $p_T$, which is often close to a Gaussian distribution.

Under suitable regularity assumptions, the reverse-time process satisfies the Anderson reverse SDE

$$d{Y}_t=\left[f(Y_t,t)-g{\left(t\right)}^2{\nabla }_{x}log{p}_t\left(Y_t\right)\right]dt+g\left(t\right)d{\overline{W}}_t,$$

(2)

where the time variable in (2) decreases from T to 0, $p_t$ denotes the marginal density of $X_t$, and $\left({\overline{W}}_t\right)$ is a Brownian motion in reverse time [4,20]. For numerical integration with a positive step size, it is useful to isolate the drift that appears when moving from $t_{n+1}$ down to $t_n$. We therefore define the backward-time score drift

$$b^{*}(x,t):=g{\left(t\right)}^2{\nabla }_{x}log{p}_t\left(x\right)-f(x,t).$$

(3)

Equivalently, after the change of variables $\tau =T-t$, the reverse process is an ordinary forward-time SDE in $\tau$ with drift $b^{*}(x,T-\tau )$ and diffusion coefficient $g(T-\tau )$.

For clarity, the positive reverse-time parametrization can be written explicitly as

$${\overline{Y}}_{\tau }:=Y_{T-\tau },\tau \in [0,T],$$

(4)

so that, after relabeling the reverse Brownian motion,

$$d{\overline{Y}}_{\tau }=b^{*}({\overline{Y}}_{\tau },T-\tau )d\tau +g(T-\tau )d{B}_{\tau }.$$

(5)

The theoretical estimates below are written in this positive reverse-time convention. To keep notation compact, we continue to denote the original grid points by $t_n$ and write $b^{*}(·,t_n)$ for the drift used in the update from $t_{n+1}$ to $t_n$.

In practical score-based models, the exact score ${\nabla }_{x}log{p}_t\left(x\right)$ is replaced by an approximation $s_{\theta }(x,t)$. We define the score-induced reverse drift by

$$b_{\theta }(x,t):=g{\left(t\right)}^2{s}_{\theta }(x,t)-f(x,t).$$

(6)

All numerical schemes below use this positive-step backward drift. This convention avoids a sign ambiguity: the coefficient in (2) is $f-g^2\nabla log{p}_t$ because the equation is written in decreasing original time, whereas the update from $t_{n+1}$ to $t_n$ uses $b^{*}=g^2\nabla log{p}_t-f$.

The corresponding probability-flow ODE, introduced in the score-based diffusion framework and subsequently analyzed as a deterministic sampler, is formally given by [4,16]

$${\displaystyle \frac{d{Z}_t}{dt}}=f(Z_t,t)-{\displaystyle \frac{1}{2}}g{\left(t\right)}^2{s}_{\theta }(Z_t,t),$$

(7)

again when t is interpreted as decreasing original time. The corresponding positive-step backward ODE drift is ${\displaystyle \frac{1}{2}}g{\left(t\right)}^2{s}_{\theta }(x,t)-f(x,t)$. The fixed point correction idea developed below applies to both the reverse SDE and the probability-flow ODE; however, we focus on the SDE notation for concreteness.

3. Fixed-Point-Corrected Schemes

Let $0=t_0<t_1<\cdots <t_N=T$ be a uniform partition with $h=T/N$. Since sampling proceeds backward from T to 0, the numerical update maps $X_{n+1}$ to $X_n$.

A direct Euler–Maruyama discretization of the positive-step reverse SDE [21,22] gives the explicit reverse step

$$X_n^E=X_{n+1}+h{b}_{\theta }(X_{n+1},t_{n+1})+g\left(t_{n+1}\right)\sqrt{h}{\xi }_n,$$

(8)

where ${\left({\xi }_n\right)}_{n=0}^{N-1}$ are independent standard Gaussian random vectors in ${\mathbb{R}}^d$.

Motivated by implicit Euler–Maruyama discretizations for SDEs [21,22], we define the implicit reverse step by

$$X_n^I=X_{n+1}+h{b}_{\theta }(X_n^I,t_n)+g\left(t_n\right)\sqrt{h}{\xi }_n.$$

(9)

For fixed $X_{n+1}$ and ${\xi }_n$, this is a nonlinear algebraic equation for $X_n^I$. Introduce the local fixed point map

$${\Phi }_{n,\xi }\left(z\right):=X_{n+1}+h{b}_{\theta }(z,t_n)+g\left(t_n\right)\sqrt{h}{\xi }_n.$$

(10)

Then the implicit step is equivalent to

$$X_n^I={\Phi }_{n,\xi }\left(X_n^I\right).$$

(11)

The fixed-point-corrected scheme uses only $M\ge 0$ inner iterations. We initialize with the explicit step

$$X_n^{\left(0\right)}=X_{n+1}+h{b}_{\theta }(X_{n+1},t_{n+1})+g\left(t_{n+1}\right)\sqrt{h}{\xi }_n,$$

(12)

then compute

$$X_n^{(m+1)}={\Phi }_{n,\xi }\left(X_n^{\left(m\right)}\right),m=0,1,\dots ,M-1.$$

(13)

The numerical update is defined by

$$X_n^{h,M}:=X_n^{\left(M\right)}.$$

(14)

Algorithm 1 summarizes the resulting fixed-point-corrected reverse diffusion sampler.   

Algorithm 1: Fixed-Point-Corrected Reverse Diffusion Sampler.

Given a time grid $0=t_0<t_1<\cdots <t_N=T$, step size $h=T/N$, terminal reference law ${\pi }_T$, and number of inner corrections M, proceed as follows. Step 1. Initialize $X_N\sim {\pi }_T$. Step 2. For $n=N-1,N-2,\dots ,0$, draw ${\xi }_n\sim \mathcal{N}(0,I_d)$ and set $$X_n^{\left(0\right)}=X_{n+1}+h{b}_{\theta }(X_{n+1},t_{n+1})+g\left(t_{n+1}\right)\sqrt{h}{\xi }_n.$$ Step 3. For $m=0,1,\dots ,M-1$, apply the fixed point correction $$X_n^{(m+1)}=X_{n+1}+h{b}_{\theta }(X_n^{\left(m\right)},t_n)+g\left(t_n\right)\sqrt{h}{\xi }_n.$$ Step 4. Set $X_n=X_n^{\left(M\right)}$ and continue to the next reverse-time step. Step 5. Return $X_0$.

Remark 1. 

For $M=0$, the method reduces to an explicit reverse Euler–Maruyama-type step. As $M\to \infty$, the method approaches the fully implicit step whenever the local fixed point iteration converges. Hence M provides a computational knob between per-step cost and numerical stability.

4. Assumptions and Local Fixed Point Analysis

The following Lipschitz, linear-growth, one-sided stability, boundedness, and moment hypotheses are standard types of assumptions in SDE discretization and stability analyses [21,22]. In the present work, they are used as controlled sufficient conditions for isolating the algebraic error of the inner fixed point solver, rather than as universal assumptions on trained neural score fields.

Assumption 1 

(Lipschitz drift). There exists $L>0$ such that

$$∥b_{\theta }(x,t)-b_{\theta }(y,t)∥\le L∥x-y∥$$

(15)

for all $x,y\in {\mathbb{R}}^d$ and $t\in [0,T]$.

Assumption 2 

(Linear growth). There exists $C_b>0$ such that

$$∥b_{\theta }(x,t)∥\le C_b(1+∥x∥)$$

(16)

for all $x\in {\mathbb{R}}^d$ and $t\in [0,T]$.

Assumption 3 

(One-sided stability). There exists $\mu \in \mathbb{R}$ such that

$$〈b_{\theta }(x,t)-b_{\theta }(y,t),x-y〉\le \mu {∥x-y∥}^2$$

(17)

for all $x,y\in {\mathbb{R}}^d$ and $t\in [0,T]$. The same bound is assumed for the exact drift $b^{*}$ whenever exact and approximate reverse processes are compared.

Assumption 4 

(Bounded diffusion coefficient). The function g is bounded on $[0,T]$, i.e.,

$$0<g_{min}\le g\left(t\right)\le g_{max}<\infty .$$

(18)

Assumption 5 

(Time regularity and local moments). There exist constants $L_t,L_g,K_2>0$ such that

$$∥b_{\theta }(x,t)-b_{\theta }(x,s)∥\le L_t(1+∥x∥)|t-s|,|g\left(t\right)-g\left(s\right)|\le L_g|t-s|$$

(19)

for all $x\in {\mathbb{R}}^d$ and $s,t\in [0,T]$. In the local estimates below, the exact implicit chain and the corrected chain are assumed to satisfy the uniform second-moment bound

$$\underset{n}{sup}\mathbb{E}{∥X_n^I∥}^2+\underset{n}{sup}\mathbb{E}{∥X_n^{h,M}∥}^2\le K_2.$$

(20)

The moment condition in Assumption 5 is used as an assumed moment-stability hypothesis for the local solver estimates. Under stronger dissipativity or Lyapunov conditions it can be derived by standard SDE-discretization arguments, but proving such a result for general learned score fields is outside the scope of this paper.

Theorem 1 

(Local well-posedness of the implicit reverse step). Assume Assumption 1. If $hL<1$, then for every fixed $X_{n+1}\in {\mathbb{R}}^d$ and every fixed ${\xi }_n\in {\mathbb{R}}^d$, the map ${\Phi }_{n,\xi }$ defined by (10) is a contraction on ${\mathbb{R}}^d$. Consequently, the implicit Equation (9) admits a unique solution $X_n^I$.

Proof. 

For any $z_1,z_2\in {\mathbb{R}}^d$, Assumption 1 gives

$$∥{\Phi }_{n,\xi }\left(z_1\right)-{\Phi }_{n,\xi }\left(z_2\right)∥=h∥b_{\theta }(z_1,t_n)-b_{\theta }(z_2,t_n)∥\le hL∥z_1-z_2∥.$$

Since $hL<1$, ${\Phi }_{n,\xi }$ is a contraction. The conclusion follows from the Banach contraction theorem [23].    □

Theorem 2 

(Inner fixed point solver error). Under the assumptions of Theorem 1, let $X_n^I$ be the unique implicit solution and let ${\left(X_n^{\left(m\right)}\right)}_{m\ge 0}$ be generated by (13). Then

$$∥X_n^{\left(M\right)}-X_n^I∥\le {\left(hL\right)}^M∥X_n^{\left(0\right)}-X_n^I∥.$$

(21)

Proof. 

Since $X_n^I={\Phi }_{n,\xi }\left(X_n^I\right)$ and ${\Phi }_{n,\xi }$ is $hL$-contractive,

$$∥X_n^{(m+1)}-X_n^I∥=∥{\Phi }_{n,\xi }\left(X_n^{\left(m\right)}\right)-{\Phi }_{n,\xi }\left(X_n^I\right)∥\le hL∥X_n^{\left(m\right)}-X_n^I∥.$$

Iterating this inequality yields (21).    □

Lemma 1 

(Residual control and explicit-start consistency). Assume the hypotheses of Theorem 1 and set $q=hL<1$. For any approximate inner iterate Z, define its fixed point residual by

$$R\left(Z\right):=Z-{\Phi }_{n,\xi }\left(Z\right).$$

Then the distance from Z to the implicit solution satisfies the a posteriori estimate

$$∥Z-X_n^I∥\le {\displaystyle \frac{∥R\left(Z\right)∥}{1-q}}.$$

(22)

Moreover, if Assumption 5 holds and $0<h\le 1$, then the explicit initializer (12) satisfies

$$\mathbb{E}{∥X_n^{\left(0\right)}-X_n^I∥}^2\le C_{\mathrm{init}}{h}^3.$$

(23)

Equivalently, the explicit initializer is $O\left(h^{3/2}\right)$ accurate in root mean square as a local approximation of the implicit step. In particular, since $h\le 1$, (23) implies the weaker $O\left(h^2\right)$ mean-square bound used later for a uniform global solver estimate.

Proof. 

Since $X_n^I={\Phi }_{n,\xi }\left(X_n^I\right)$,

$$\begin{array}{cc}\hfill ∥Z-X_n^I∥& \le ∥Z-{\Phi }_{n,\xi }\left(Z\right)∥+∥{\Phi }_{n,\xi }\left(Z\right)-{\Phi }_{n,\xi }\left(X_n^I\right)∥\hfill \\ & \le ∥R\left(Z\right)∥+q∥Z-X_n^I∥.\hfill \end{array}$$

Moving the last term to the left gives (22). This estimate is useful computationally because it turns the residual reported in the experiments into a direct upper bound on the local algebraic solve error up to the factor ${(1-q)}^{-1}$.

For the initializer, write $A=X_{n+1}$ and $Z_I=X_n^I$. Subtracting (9) from (12) gives

$$X_n^{\left(0\right)}-Z_I=h\{b_{\theta }(A,t_{n+1})-b_{\theta }(Z_I,t_n)\}+\{g\left(t_{n+1}\right)-g\left(t_n\right)\}\sqrt{h}{\xi }_n.$$

Using Assumptions 1 and 5,

$$∥X_n^{\left(0\right)}-Z_I∥\le hL∥A-Z_I∥+h^2{L}_t(1+∥A∥)+L_g{h}^{3/2}∥{\xi }_n∥.$$

The implicit equation also gives

$$∥A-Z_I∥\le h∥b_{\theta }(Z_I,t_n)∥+g_{max}\sqrt{h}∥{\xi }_n∥.$$

Assumptions 2, 4, and 5, together with $\mathbb{E}{∥{\xi }_n∥}^2=d$, imply

$$\mathbb{E}{∥A-Z_I∥}^2\le Ch$$

for a constant C depending on $C_b,g_{max},d$, and $K_2$. Squaring the previous bound for $X_n^{\left(0\right)}-Z_I$ and using ${(a+b+c)}^2\le 3(a^2+b^2+c^2)$ yields

$$\mathbb{E}{∥X_n^{\left(0\right)}-Z_I∥}^2\le C^{\prime }(h^3+h^4+h^3)\le C_{\mathrm{init}}{h}^3$$

for $0<h\le 1$. This proves (23).    □

Remark 2. 

The condition $hL<1$ is a sufficient local condition for the plain Picard correction used here. In applications, L may be large for learned scores, especially near small noise levels. This is precisely where damped corrections or semi-implicit splitting can be combined with the same local-error bookkeeping. Assumption 3 is used for propagation estimates; it is weaker than global contraction and is standard in stability analyses of SDE discretizations.

In practical neural-score samplers, the global constant L is usually unknown and may not be finite on the whole state space. Therefore, the condition $hL<1$ should be read as a sufficient mathematical certificate rather than as an operational rule that can always be verified. In implementations, one can reduce the outer step size, monitor the fixed point residual in (45), and stop the inner iteration once the residual no longer decreases meaningfully relative to the other error sources. If the residual stagnates or grows, the contraction theorem should no longer be interpreted as a certificate for that local step; the calculation is then a stabilized heuristic implicit update rather than a guaranteed Picard solve.

5. Global Error Decomposition

Let $Y_0^{*}$ denote the ideal reverse process at data time 0 driven by the exact score and initialized from the exact terminal law $p_T$. Then $\mathcal{L}\left(Y_0^{*}\right)=p_0$. Let ${\tilde{Y}}_0^{*}$ denote the exact-score reverse process initialized instead from the reference terminal law ${\pi }_T$, and let $Y_0^{\theta }$ denote the continuous-time reverse process driven by the approximate score $s_{\theta }$ and initialized from ${\pi }_T$. Finally, let $X_0^{h,I}$ denote the fully implicit time-discrete approximation of $Y_0^{\theta }$, and let $X_0^{h,M}$ denote the fixed-point-corrected approximation with M inner corrections per time step.

By the triangle inequality in $W_2$ [24], we have

$$\begin{array}{c}\hfill W_2(\mathcal{L}\left(X_0^{h,M}\right),p_0)\le W_2(\mathcal{L}\left(X_0^{h,M}\right),\mathcal{L}\left(X_0^{h,I}\right))\end{array}$$

(24)

$$\begin{array}{cc}\hfill & +W_2(\mathcal{L}\left(X_0^{h,I}\right),\mathcal{L}\left(Y_0^{\theta }\right))\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & +W_2(\mathcal{L}\left(Y_0^{\theta }\right),\mathcal{L}\left({\tilde{Y}}_0^{*}\right))\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & +W_2(\mathcal{L}\left({\tilde{Y}}_0^{*}\right),\mathcal{L}\left(Y_0^{*}\right)).\hfill \end{array}$$

(27)

The four terms represent, respectively, the fixed-point solver error, the time-discretization error, the score approximation error, and the terminal/modeling error.

Assumption 6 

(Score approximation error). There exists ${\epsilon }_{\mathrm{score}}\ge 0$ such that

$${\left({\int }_0^T\mathbb{E}{∥s_{\theta }({\tilde{Y}}_t^{*},t)-\nabla log{p}_t\left({\tilde{Y}}_t^{*}\right)∥}^{2}dt\right)}^{1/2}\le {\epsilon }_{\mathrm{score}}.$$

(28)

Integrated $L^2$-type score error assumptions of this form are consistent with recent convergence analyses that separate score approximation from sampler and discretization errors [12,14,17].

In Propositions 1 and 3, the time variable is understood in the positive reverse-time parametrization (4) and (5). Thus, the coupled processes start from the terminal law and evolve over an interval of length T toward data time. The endpoint at positive reverse time T corresponds to original data time 0.

Proposition 1 

(Stability with respect to score error). Assume Assumptions 3 and 4. Couple $Y^{\theta }$ and ${\tilde{Y}}^{*}$ synchronously, using the same Brownian path and the same initial draw from ${\pi }_T$. Then there exists a constant $C_{\mathrm{score}}>0$, depending only on T, μ, and $g_{max}$, such that

$$W_2(\mathcal{L}\left(Y_0^{\theta }\right),\mathcal{L}\left({\tilde{Y}}_0^{*}\right))\le C_{\mathrm{score}}{\epsilon }_{\mathrm{score}}.$$

(29)

Proof. 

We write the estimate in the positive reverse-time parametrization; the notation t below therefore runs over an interval of length T from the terminal law toward the data law. Let $D_t=Y_t^{\theta }-{\tilde{Y}}_t^{*}$ under the synchronous coupling. Since the same Brownian path is used for both processes, the diffusion terms cancel in the difference equation. Splitting the drift difference gives

$$b_{\theta }(Y_t^{\theta },t)-b^{*}({\tilde{Y}}_t^{*},t)=b_{\theta }(Y_t^{\theta },t)-b_{\theta }({\tilde{Y}}_t^{*},t)+g{\left(t\right)}^2\{s_{\theta }({\tilde{Y}}_t^{*},t)-\nabla log{p}_t\left({\tilde{Y}}_t^{*}\right)\}.$$

Applying the chain rule to ${∥D_t∥}^2$, using Assumption 3 for the first term, and using Young’s inequality $2\langle u,v\rangle \le {∥u∥}^2+{∥v∥}^2$ for the score-error term gives

$${\displaystyle \frac{d}{dt}}\mathbb{E}{∥D_t∥}^2\le (2\mu +1)\mathbb{E}{∥D_t∥}^2+g_{max}^4\mathbb{E}{∥s_{\theta }({\tilde{Y}}_t^{*},t)-\nabla log{p}_t\left({\tilde{Y}}_t^{*}\right)∥}^2.$$

The two processes have the same terminal initialization under this coupling, so the initial difference in the positive reverse-time variable is zero. Gronwall’s inequality yields

$$\mathbb{E}{∥D_T∥}^2\le g_{max}^4{e}^{(2\mu +1)T}{\int }_0^T\mathbb{E}{∥s_{\theta }({\tilde{Y}}_t^{*},t)-\nabla log{p}_t\left({\tilde{Y}}_t^{*}\right)∥}^{2}dt.$$

Taking square roots and using Assumption 6 gives (29), for example with $C_{\mathrm{score}}=g_{max}^2{e}^{(2\mu +1)T/2}$. The Wasserstein bound follows because the synchronous coupling is an admissible coupling of the two endpoint laws.    □

Assumption 7 

(Time discretization error). For the fully implicit scheme driven by $b_{\theta }$, there exist constants $C_{\mathrm{disc}}>0$ and $r>0$ such that

$$W_2(\mathcal{L}\left(X_0^{h,I}\right),\mathcal{L}\left(Y_0^{\theta }\right))\le C_{\mathrm{disc}}{h}^r.$$

(30)

For reverse SDEs with globally Lipschitz coefficients, one typically has $r=1/2$ in strong error, while deterministic probability-flow ODE variants may give $r=1$ under standard smoothness assumptions [21,22].

Assumption 7 is deliberately separated from the fixed point solver analysis. The present paper does not aim to replace the standard convergence theory of implicit Euler–Maruyama schemes; instead, it isolates the additional algebraic solver error created by terminating the fixed point iteration after finitely many corrections. The following proposition records a classical setting in which Assumption 7 is satisfied, so that the decomposition in Theorem 3 becomes unconditional with respect to the time-discretization term in the standard globally Lipschitz additive-noise regime.

Proposition 2 

(A sufficient condition for the implicit time-discretization estimate). Consider the positive reverse-time SDE

$$d{Y}_{\tau }=a(Y_{\tau },\tau )d\tau +\sigma \left(\tau \right)d{B}_{\tau },a(x,\tau )=b_{\theta }(x,T-\tau ),\sigma \left(\tau \right)=g(T-\tau ).$$

(31)

Assume that a is globally Lipschitz in x with constant L, has linear growth, is Lipschitz in time in the sense of Assumption 5, and that the diffusion is state-independent, with σ bounded and Lipschitz. Assume also the following second-moment and temporal regularity bounds for the exact solution,

$$\underset{\tau \in [0,T]}{sup}\mathbb{E}{∥Y_{\tau }∥}^2<\infty ,\mathbb{E}{∥Y_{\tau }-Y_s∥}^2\le C_Y|\tau -s|.$$

Let ${\left({\widehat{Y}}_k\right)}_{k=0}^N$ be the fully implicit Euler–Maruyama approximation

$${\widehat{Y}}_{k+1}={\widehat{Y}}_k+ha({\widehat{Y}}_{k+1},{\tau }_{k+1})+\sigma \left({\tau }_{k+1}\right)\Delta B_k,{\tau }_k=kh,$$

(32)

with the initial value coupled to the exact solution by ${\widehat{Y}}_0=Y_0$ almost surely and with $hL\le q_0<1$. Then there exists a constant $C_{\mathrm{disc}}$, independent of h, such that

$${\left(\mathbb{E}{∥{\widehat{Y}}_N-Y_T∥}^2\right)}^{1/2}\le C_{\mathrm{disc}}{h}^{1/2}.$$

(33)

Consequently, after relabeling the positive reverse-time grid back to the original notation, Assumption 7 holds with $r=1/2$.

Proof. 

The proof is the standard strong-error argument for backward Euler–Maruyama with state-independent diffusion, included here to clarify the role of Assumption 7. Write one exact step as

$$Y_{{\tau }_{k+1}}=Y_{{\tau }_k}+ha(Y_{{\tau }_{k+1}},{\tau }_{k+1})+\sigma \left({\tau }_{k+1}\right)\Delta B_k+{\rho }_{k+1},$$

where the local defect is

$${\rho }_{k+1}={\int }_{{\tau }_k}^{{\tau }_{k+1}}\{a(Y_s,s)-a(Y_{{\tau }_{k+1}},{\tau }_{k+1})\}ds+{\int }_{{\tau }_k}^{{\tau }_{k+1}}\{\sigma \left(s\right)-\sigma \left({\tau }_{k+1}\right)\}d{B}_s.$$

The Lipschitz and time-regularity assumptions, the $L^2$ temporal regularity of Y, and Itô isometry give

$${\left(\mathbb{E}{∥{\rho }_{k+1}∥}^2\right)}^{1/2}\le C{h}^{3/2}.$$

Couple the exact and numerical paths with the same initial value and the same Brownian increments, and set

$${\delta }_k:={\widehat{Y}}_k-Y_{{\tau }_k},e_k:={\left(\mathbb{E}{∥{\delta }_k∥}^2\right)}^{1/2}.$$

Subtracting the exact perturbed implicit step from the numerical step gives

$${\delta }_{k+1}={\delta }_k+h\{a({\widehat{Y}}_{k+1},{\tau }_{k+1})-a(Y_{{\tau }_{k+1}},{\tau }_{k+1})\}-{\rho }_{k+1}.$$

Hence, by the spatial Lipschitz bound,

$$(1-hL)∥{\delta }_{k+1}∥\le ∥{\delta }_k∥+∥{\rho }_{k+1}∥.$$

Taking $L^2$ norms and using the local-defect estimate gives

$$e_{k+1}\le {(1-hL)}^{-1}{e}_k+{(1-hL)}^{-1}C{h}^{3/2},e_0=0.$$

Since $hL\le q_0<1$, the second factor is bounded by ${(1-q_0)}^{-1}$. Iterating this recurrence over $N=T/h$ steps and using ${(1-hL)}^{-N}\le exp(LT/(1-q_0))$ gives $e_N\le C_{\mathrm{disc}}{h}^{1/2}$. Since the synchronous coupling is admissible for $W_2$, the Wasserstein bound in Assumption 7 follows.    □

For deterministic probability-flow ODE variants, the same argument without the stochastic integral gives the usual first-order implicit Euler estimate under corresponding smoothness assumptions. Thus the conditional form of Theorem 3 is mainly a modular bookkeeping device: it allows the solver-error term to be combined with whichever standard time-discretization estimate is appropriate for the chosen reverse dynamics.

Assumption 8 

(Terminal initialization error). If the sampler is initialized from ${\pi }_T$ instead of the exact terminal law $p_T$, then

$$W_2({\pi }_T,p_T)\le {\epsilon }_{\mathrm{init}}.$$

(34)

Proposition 3 

(Propagation of terminal initialization error). If the exact backward drift satisfies Assumption 3 with constant ${\mu }_{*}$, then

$$W_2(\mathcal{L}\left({\tilde{Y}}_0^{*}\right),\mathcal{L}\left(Y_0^{*}\right))\le e^{{\mu }_{*}T}{\epsilon }_{\mathrm{init}}.$$

(35)

Proof. 

Couple the two exact-score reverse processes by an optimal coupling of their terminal initializations and by the same Brownian path. If $D_t$ denotes the difference of the two coupled trajectories in the positive reverse-time parametrization, then the diffusion terms cancel. Assumption 3 for the exact drift gives

$${\displaystyle \frac{d}{dt}}\mathbb{E}{∥D_t∥}^2=2\mathbb{E}\langle D_t,b^{*}(Y_t,t)-b^{*}({\tilde{Y}}_t,t)\rangle \le 2{\mu }_{*}\mathbb{E}{∥D_t∥}^2.$$

By Gronwall’s inequality,

$$\mathbb{E}{∥D_T∥}^2\le e^{2{\mu }_{*}T}\mathbb{E}{∥D_0∥}^2.$$

The initial coupling was chosen optimally, so $\mathbb{E}{∥D_0∥}^2=W_2{({\pi }_T,p_T)}^2\le {\epsilon }_{\mathrm{init}}^2$. Taking square roots yields (35).    □

Lemma 2 

(Accumulated fixed point solver error). Suppose Assumptions 1–4 hold and assume the uniform step-size condition

$$q:=hL\le q_0<1.$$

(36)

Assume that the initialization of the inner solver satisfies the conditional local consistency bound

$$\mathbb{E}\left[{∥{\Gamma }_{h,0}(a,{\xi }_n,t_n)-{\Psi }_h(a,{\xi }_n,t_n)∥}^2|a\right]\le C_{\mathrm{loc}}{h}^2$$

(37)

uniformly in n and in the admissible input values a reached by the coupled chains. Here ${\Psi }_h$ denotes the exact implicit transition and ${\Gamma }_{h,0}$ denotes the explicit initializer used before the fixed point corrections. Then there exists a constant $C_T>0$, depending on T, L, $q_0$, $C_{\mathrm{loc}}$, and moment bounds but not on h or M, such that

$$W_2(\mathcal{L}\left(X_0^{h,M}\right),\mathcal{L}\left(X_0^{h,I}\right))\le C_T{q}^M.$$

(38)

Proof. 

Let ${\Psi }_h(a,\xi ,t_n)$ denote the exact implicit transition, i.e., the unique solution z of

$$z=a+h{b}_{\theta }(z,t_n)+g\left(t_n\right)\sqrt{h}\xi .$$

If $z={\Psi }_h(a,\xi ,t_n)$ and $\overline{z}={\Psi }_h(\overline{a},\xi ,t_n)$ are driven by the same Gaussian increment, then

$$z-\overline{z}=(a-\overline{a})+h\{b_{\theta }(z,t_n)-b_{\theta }(\overline{z},t_n)\}.$$

Therefore

$$∥z-\overline{z}∥\le ∥a-\overline{a}∥+q∥z-\overline{z}∥,∥z-\overline{z}∥\le {\displaystyle \frac{1}{1-q}}∥a-\overline{a}∥.$$

Thus the exact implicit transition is Lipschitz in its input with factor ${(1-q)}^{-1}$ under synchronous noise.

By Theorem 2, the local truncation of the inner fixed point solve, with the same input a and the same $\xi$, satisfies

$${\left(\mathbb{E}{∥{\Gamma }_{h,M}(a,\xi ,t_n)-{\Psi }_h(a,\xi ,t_n)∥}^2\right)}^{1/2}\le q^M{C}_{\mathrm{loc}}^{1/2}h,$$

where ${\Gamma }_{h,M}$ denotes the M-step fixed-point-corrected transition and (37) is used in the last factor.

Now couple the corrected chain $X_n^{h,M}$ and the fully implicit chain $X_n^{h,I}$ with the same terminal draw and the same Gaussian increments. Set

$$e_n:={\left(\mathbb{E}{∥X_n^{h,M}-X_n^{h,I}∥}^2\right)}^{1/2}.$$

Using the preceding two estimates and Minkowski’s inequality,

$$e_n\le q^M{C}_{\mathrm{loc}}^{1/2}h+{\displaystyle \frac{1}{1-q}}{e}_{n+1},e_N=0.$$

Iterating the recurrence over $N=T/h$ steps gives

$$e_0\le q^M{C}_{\mathrm{loc}}^{1/2}h{\displaystyle \sum _{j=0}^{N-1}}{(1-q)}^{-j}\le q^M{C}_{\mathrm{loc}}^{1/2}T{(1-q)}^{-N}.$$

The uniform condition (36) gives

$${(1-q)}^{-N}={(1-hL)}^{-T/h}\le exp\left({\displaystyle \frac{LT}{1-q_0}}\right),$$

where we used $-log(1-x)\le x/(1-q_0)$ for $0\le x\le q_0<1$. Hence $e_0\le C_T{q}^M$. Since the synchronous coupling is admissible for $W_2$, (38) follows.    □

Theorem 3 

(Conditional total error decomposition under an assumed implicit discretization estimate). Under Assumptions 1–8, and under the local solver estimate of Lemma 2, the fixed-point-corrected reverse sampler satisfies

$$W_2(\mathcal{L}\left(X_0^{h,M}\right),p_0)\le C_1{\epsilon }_{\mathrm{init}}+C_2{\epsilon }_{\mathrm{score}}+C_3{h}^r+C_4{\left(hL\right)}^M,$$

(39)

where the constants $C_1,C_2,C_3,C_4$ depend on stability and moment bounds of the reverse dynamics.

Proof. 

Apply the decomposition (24). The first term is bounded by Lemma 2. The second term is bounded by Assumption 7. The third term is controlled by Proposition 1. The fourth term is controlled by Proposition 3. Combining these estimates gives (39).    □

Remark 3. 

Theorem 3 separates four distinct error sources. This separation is useful experimentally: exact-score Gaussian-mixture tests isolate discretization and solver errors, while perturbed-score tests reveal the interaction between score error and numerical stability.

6. Numerical Experiments

This section first reports a compact set of low-dimensional experiments intended to validate the error decomposition, and then a data-driven neural-score experiment on Fashion-MNIST. The low-dimensional tables report means and standard deviations over 12 independent seeds, with 2200 generated samples and 2200 reference samples per run unless otherwise stated. These experiments are diagnostic rather than competitive benchmarks for large-scale image generation.

The low-dimensional calculations were run on a local CPU machine with an Intel Core i5-8265U processor. The neural-score Fashion-MNIST training and evaluation were run on a RunPod cloud GPU instance with one NVIDIA L4 GPU with 24 GB VRAM, 4 vCPUs, and 19 GB system RAM. The neural implementation used PyTorch 2.4.1+cu124 with CUDA acceleration. The reported neural run used 512 generated samples for the generation diagnostics and 128 test images for the inpainting diagnostics.

6.1. Exact-Score Gaussian-Mixture Target

We use a variance-exploding Brownian forward process

$$d{X}_t=\sqrt{{\sigma }_T^2}d{W}_t,t\in [0,1],$$

(40)

so that the accumulated forward variance is ${\sigma }_T^{2}t$. The target is an eight-component Gaussian mixture in ${\mathbb{R}}^2$,

$$p_0\left(x\right)={\displaystyle \frac{1}{K}}{\displaystyle \sum _{j=1}^K}\phi (x;m_j,{\sigma }_0^{2}I),K=8,$$

(41)

where the means $m_j$ are equally spaced on a circle of radius 2, ${\sigma }_0^2=0.08$, and ${\sigma }_T^2=3$. The noised marginal is

$$p_t\left(x\right)={\displaystyle \frac{1}{K}}{\displaystyle \sum _{j=1}^K}\phi (x;m_j,({\sigma }_0^2+{\sigma }_T^{2}t)I).$$

(42)

Hence, the exact score is available in closed form

$$\nabla log{p}_t\left(x\right)=-{\displaystyle \frac{x-{\sum }_{j=1}^K{\rho }_j(x,t)m_j}{{\sigma }_0^2+{\sigma }_T^{2}t}},{\rho }_j(x,t)={\displaystyle \frac{\phi (x;m_j,({\sigma }_0^2+{\sigma }_T^{2}t)I)}{{\sum }_{\ell =1}^K\phi (x;m_{\ell },({\sigma }_0^2+{\sigma }_T^{2}t)I)}}.$$

(43)

This experiment isolates time-discretization and fixed point solver effects because no learned-score approximation is present.

We compare the explicit method ($M=0$), fixed-point-corrected methods ($M=1,3,8$), and a reference implicit solve approximated by 60 inner iterations. We report sliced $W_2$ and negative log likelihood (NLL) as distributional diagnostics, together with the pathwise implicit gap

$${\left(\mathbb{E}{∥X_0^{h,M}-X_0^{h,I}∥}^2\right)}^{1/2}$$

(44)

computed under the same terminal samples and Brownian increments. This directly measures the first term in (24). We also report the average fixed point residual

$${\displaystyle \frac{1}{N}}{\displaystyle \sum _{n=0}^{N-1}}\mathbb{E}∥X_n^{\left(M\right)}-{\Phi }_{n,\xi }\left(X_n^{\left(M\right)}\right)∥.$$

(45)

Table 1. Exact-score Gaussian-mixture experiment over 12 seeds. The implicit gap estimates (44); the residual column measures (45).

Method	N	M	SW2	NLL	Implicit Gap	Residual
FPC	40	1	$0.1244\pm 0.0210$	$1.7108\pm 0.0066$	$2.06\times {10}^{-1}\pm 6.7\times {10}^{-3}$	$1.09\times {10}^{-2}\pm 4.5\times {10}^{-5}$
FPC	40	3	$0.1179\pm 0.0227$	$1.7403\pm 0.0063$	$1.63\times {10}^{-1}\pm 5.7\times {10}^{-3}$	$6.78\times {10}^{-3}\pm 4.5\times {10}^{-5}$
FPC	40	8	$0.0821\pm 0.0278$	$2.2668\pm 0.0117$	$1.04\times {10}^{-1}\pm 1.7\times {10}^{-3}$	$4.51\times {10}^{-3}\pm 3.4\times {10}^{-5}$
explicit	40	0	$0.0899\pm 0.0195$	$3.0060\pm 0.0179$	$2.45\times {10}^{-1}\pm 3.4\times {10}^{-3}$	$2.73\times {10}^{-2}\pm 8.0\times {10}^{-5}$
implicit	40	imp.	$0.0947\pm 0.0248$	$1.9650\pm 0.0080$	$0.00\pm 0.0$	$1.50\times {10}^{-4}\pm 1.4\times {10}^{-6}$
FPC	80	1	$0.0810\pm 0.0308$	$2.0251\pm 0.0147$	$7.65\times {10}^{-2}\pm 4.1\times {10}^{-3}$	$1.53\times {10}^{-3}\pm 6.1\times {10}^{-6}$
FPC	80	3	$0.0767\pm 0.0347$	$2.0971\pm 0.0154$	$3.94\times {10}^{-2}\pm 5.2\times {10}^{-3}$	$1.96\times {10}^{-4}\pm 1.5\times {10}^{-6}$
FPC	80	8	$0.0764\pm 0.0340$	$2.1159\pm 0.0155$	$1.84\times {10}^{-2}\pm 4.9\times {10}^{-3}$	$4.35\times {10}^{-6}\pm 3.7\times {10}^{-7}$
explicit	80	0	$0.0741\pm 0.0311$	$2.6734\pm 0.0227$	$1.37\times {10}^{-1}\pm 2.6\times {10}^{-3}$	$8.94\times {10}^{-3}\pm 3.1\times {10}^{-5}$
implicit	80	imp.	$0.0780\pm 0.0357$	$2.1149\pm 0.0156$	$0.00\pm 0.0$	$4.87\times {10}^{-9}\pm 1.5\times {10}^{-8}$
FPC	160	1	$0.0844\pm 0.0223$	$2.2161\pm 0.0174$	$2.57\times {10}^{-2}\pm 2.6\times {10}^{-3}$	$2.43\times {10}^{-4}\pm 1.2\times {10}^{-6}$
FPC	160	3	$0.0836\pm 0.0216$	$2.2376\pm 0.0179$	$1.31\times {10}^{-2}\pm 3.0\times {10}^{-3}$	$8.52\times {10}^{-6}\pm 2.7\times {10}^{-7}$
FPC	160	8	$0.0839\pm 0.0231$	$2.2370\pm 0.0178$	$6.14\times {10}^{-3}\pm 2.2\times {10}^{-3}$	$2.49\times {10}^{-7}\pm 7.9\times {10}^{-8}$
explicit	160	0	$0.0816\pm 0.0233$	$2.5280\pm 0.0235$	$7.13\times {10}^{-2}\pm 2.1\times {10}^{-3}$	$3.04\times {10}^{-3}\pm 7.0\times {10}^{-6}$
implicit	160	imp.	$0.0840\pm 0.0238$	$2.2366\pm 0.0179$	$0.00\pm 0.0$	$1.11\times {10}^{-10}\pm 9.1\times {10}^{-11}$

shows the expected solver behavior. For example, with $N=80$, the implicit gap decreases from approximately $1.37·{10}^{-1}$ for the explicit method to $1.84·{10}^{-2}$ for $M=8$, while the residual decreases from $8.94·{10}^{-3}$ to $4.35·{10}^{-6}$. The distributional metrics remain of comparable scale once the solver residual is small, which is consistent with Theorem 3: after solver error is reduced, discretization and sampling variability dominate.

Representative target and sampler outputs are shown in Figure 1.

6.2. Controlled Perturbed-Score Test

To test the score-error term in Theorem 3, we perturb the exact score by a bounded field

$$s_{\theta }(x,t)=s^{*}(x,t)+\delta r(x,t),$$

(46)

where $s^{*}=\nabla log{p}_t$, r has a componentwise hyperbolic-tangent form, and $\delta \ge 0$ controls the score error level.

Table 2. Perturbed-score experiment with $N=80$ and $M=3$. NLL denotes the negative log likelihood under the exact target mixture.

$\mathit{\delta }$	SW2	MMD2	NLL	Residual
0.00	$0.0776\pm 0.0218$	$0.0012\pm 0.0011$	$2.1070\pm 0.0118$	$1.96\times {10}^{-4}\pm 1.8\times {10}^{-6}$
0.05	$0.0739\pm 0.0205$	$0.0008\pm 0.0004$	$2.1159\pm 0.0131$	$1.95\times {10}^{-4}\pm 1.8\times {10}^{-6}$
0.10	$0.0848\pm 0.0216$	$0.0010\pm 0.0006$	$2.1265\pm 0.0151$	$1.94\times {10}^{-4}\pm 1.8\times {10}^{-6}$
0.20	$0.1303\pm 0.0270$	$0.0024\pm 0.0015$	$2.1506\pm 0.0136$	$1.93\times {10}^{-4}\pm 1.2\times {10}^{-6}$
0.40	$0.2356\pm 0.0247$	$0.0050\pm 0.0033$	$2.2268\pm 0.0150$	$1.89\times {10}^{-4}\pm 1.7\times {10}^{-6}$

uses $N=80$ and $M=3$. The solver residual remains essentially unchanged, while sliced $W_2$, MMD, and NLL deteriorate for larger perturbations. For $\delta =0.40$, sliced $W_2$ increases to about $0.236$, whereas the residual remains near $1.9·{10}^{-4}$. This is consistent with the decomposition: once the inner solver error is controlled, the score term becomes visible in distributional quality.

6.3. Parametric Learned-Score Example

We next replace the exact score by a fitted parametric score. For each seed, we draw 1200 training samples from the target distribution, fit an eight-component isotropic Gaussian mixture by k-means initialization followed by expectation–maximization, and use the analytically convolved fitted mixture to define a learned score at every time t. The score RMSE in

Table 3. Parametric learned-score experiment with $N=80$. The fitted Gaussian-mixture score is trained separately for each seed.

Score Model	N	M	Score RMSE	SW2	NLL	Residual
Exact	80	3	$0.0000\pm 0.0000$	$0.0857\pm 0.0234$	$2.0969\pm 0.0176$	$1.96\times {10}^{-4}\pm 3.0\times {10}^{-6}$
Learned GMM	80	0	$0.4753\pm 0.1852$	$0.1267\pm 0.0236$	$3.3730\pm 0.3455$	$7.24\times {10}^{-3}\pm 8.0\times {10}^{-4}$
Learned GMM	80	3	$0.4753\pm 0.1852$	$0.1281\pm 0.0225$	$2.9623\pm 0.4317$	$4.89\times {10}^{-5}\pm 5.7\times {10}^{-5}$
Learned GMM	80	8	$0.4753\pm 0.1852$	$0.1255\pm 0.0224$	$2.9649\pm 0.4261$	$1.06\times {10}^{-6}\pm 1.1\times {10}^{-6}$

is estimated on independently noised validation samples. This experiment is intentionally modest, but it introduces a genuine score-estimation component rather than a hand-designed perturbation.

The learned score has a nonzero error and worse distributional metrics than the exact-score sampler. Nevertheless, fixed point corrections still reduce the numerical residual: for the learned score, the residual decreases from approximately $7.24·{10}^{-3}$ at $M=0$ to $1.06·{10}^{-6}$ at $M=8$. The NLL improves substantially from $M=0$ to $M=3$, after which score approximation error dominates. This supports the interpretation that the method controls the numerical solve independently of, but not instead of, score quality.

6.4. Neural-Score Fashion-MNIST Experiment and Inpainting

The previous experiments used either an exact score or a parametric score in a two-dimensional controlled model. To test the method in a genuinely data-driven setting, we also train a time-conditioned convolutional score network on Fashion-MNIST [25]. Images are scaled to $[-1,1]$ and perturbed by a variance-exploding Gaussian noise schedule with ${\sigma }_{min}=0.01$ and ${\sigma }_{max}=10$. The score network is trained for 120 epochs by denoising score matching, using the target $-\eta /\sigma$ for perturbed samples $x+\sigma \eta$. Sampling uses 60 reverse noise levels, and the near-implicit reference is approximated by $M_{\mathrm{ref}}=12$ fixed-point iterations with the same Gaussian increments.

Table 4. Neural-score Fashion-MNIST generation experiment. The score network is trained by denoising score matching for 120 epochs. KID is computed in the feature space of a separately trained classifier and scaled by ${10}^3$.

Method	M	Evals/Step	KID $\times {10}^3$	Entropy	Conf.	Implicit Gap	Residual
explicit	0	1	2.223	1.588	0.840	$3.482\times {10}^{-1}$	$1.798\times {10}^{-1}$
FPC	1	2	1.913	1.684	0.900	$1.155\times {10}^{-1}$	$2.803\times {10}^{-1}$
FPC	3	4	2.187	1.597	0.904	$5.777\times {10}^{-3}$	$2.712\times {10}^{-1}$
FPC	5	6	2.349	1.535	0.897	$9.096\times {10}^{-4}$	$2.705\times {10}^{-1}$
FPC	8	9	2.224	1.586	0.904	$5.530\times {10}^{-4}$	$2.707\times {10}^{-1}$

reports a feature-space kernel inception distance (KID) computed from a separately trained Fashion-MNIST classifier, together with class entropy, classifier confidence, implicit gap, and fixed point residual. The classifier contributes a 256-dimensional penultimate feature vector. We standardize the real and generated feature clouds using the real-feature mean and variance, normalize feature vectors to unit length, and compute the degree-three polynomial KID on the resulting features. The KID value is scaled by ${10}^3$ and is used only as a relative diagnostic, following the use of kernel two-sample metrics in generative modeling [26,27]. The key solver-level quantity is the implicit gap: it decreases from $3.48·{10}^{-1}$ for the explicit update to $5.53·{10}^{-4}$ for $M=8$. The feature KID and entropy do not improve monotonically, which is expected because the neural score error and the finite reverse-time grid remain present. Unlike the exact-score contraction experiment, the raw residual is not monotone for this learned score; this is consistent with the fact that the neural drift is not guaranteed to satisfy the global contraction condition used in the local theorem.

Figure 2 provides a visual comparison of real and generated Fashion-MNIST samples.

We also test a simple inverse imaging task. A central $12\times 12$ region is removed from Fashion-MNIST test images, and the sampler is run with hard data consistency on the observed pixels at each noise level. This is not intended as a new inverse-problem algorithm; it is a stress test for the same solver mechanism under a learned diffusion prior, in the spirit of diffusion-based posterior sampling for inverse problems [28].

Table 5. Fashion-MNIST center-inpainting experiment using the learned neural score. Missing PSNR is measured only on the removed $12\times 12$ central region.

Method	M	PSNR	Missing PSNR	Implicit Gap	Residual
Explicit	0	19.01	11.65	$1.127\times {10}^{-1}$	$1.773\times {10}^{-1}$
FPC	1	21.04	13.68	$2.999\times {10}^{-2}$	$6.196\times {10}^{-1}$
FPC	3	21.00	13.64	$1.514\times {10}^{-3}$	$6.129\times {10}^{-1}$
FPC	5	20.65	13.29	$2.350\times {10}^{-4}$	$6.111\times {10}^{-1}$
FPC	8	21.03	13.67	$1.757\times {10}^{-4}$	$6.118\times {10}^{-1}$

shows that one or more fixed point corrections improve the reconstruction PSNR from $19.01$ dB for the explicit update to about 21 dB, while the implicit gap again decreases by several orders of magnitude. Because hard data consistency is imposed after each step, the reported residual should be interpreted as a diagnostic of the unconstrained neural reverse update rather than as a monotone contraction certificate for the constrained inverse problem.

The corresponding masked observations and reconstructions are shown in Figure 3.

6.5. Interpretation Relative to the Theory

The experiments should be read against the four-term decomposition in (24). In the exact-score Gaussian-mixture setting, the score-error term is removed and the terminal law is sampled directly from the known forward marginal. The implicit gap in (44) therefore targets the first term in the decomposition, namely the difference between the finite-correction chain and the fully implicit chain. In this controlled contraction regime, both the residual and the implicit gap decrease as M increases, matching Theorem 2, Lemmas 1 and 2. In the neural-score Fashion-MNIST experiment, the implicit gap still decreases strongly, but the raw residual is not monotone; this highlights the conditional nature of the contraction theory and the practical effect of using an unconstrained learned score.

The distributional metrics are intentionally interpreted more cautiously. Sliced $W_2$, MMD, feature KID, NLL, and reconstruction PSNR measure final sample or image quality, so they contain time-discretization, Monte Carlo, and score-estimation effects. Hence, these metrics need not improve monotonically with M once the algebraic solver gap is already small. The perturbed-score, fitted-score, and Fashion-MNIST experiments confirm this separation: fixed point corrections can reduce the pathwise gap to the implicit reference while imperfect scores and finite discretization remain visible in distributional quality. This behavior supports the central claim of the paper, which is solver stabilization and error separation rather than a guarantee that every quality metric improves after every additional inner correction.

6.6. Computational Cost and Practical Choice of Inner Corrections

Each fixed-point-corrected step uses one score evaluation for the explicit initializer and M additional score evaluations for the inner corrections. Thus, the raw score-evaluation cost is $M+1$ evaluations per reverse step, compared with one evaluation for the explicit Euler–Maruyama update. This is the main computational overhead of the method.

Table 6. Cost–benefit summary for the neural Fashion-MNIST generation experiment. Relative time is normalized by the explicit sampler. The table highlights that the implicit gap can decrease substantially even when feature-space KID does not improve monotonically.

M	Evals/Step	Time (s)	Rel. Time	Implicit Gap	KID $\times {10}^3$
0	1	40.1	1.00	$3.482\times {10}^{-1}$	2.223
1	2	42.9	1.07	$1.155\times {10}^{-1}$	1.913
3	4	48.0	1.20	$5.777\times {10}^{-3}$	2.187
5	6	53.2	1.33	$9.096\times {10}^{-4}$	2.349
8	9	61.0	1.52	$5.530\times {10}^{-4}$	2.224

reports the measured generation time for the neural Fashion-MNIST experiment. The wall-clock increase is smaller than the raw score-evaluation factor in this batched implementation, but the asymptotic score-evaluation cost remains linear in $M+1$.

The experiments suggest a practical stopping rule rather than a universal optimal value of M. One should increase M only until the implicit gap or fixed point residual is small compared with the expected score and time-discretization errors. In the exact-score Gaussian-mixture experiments, larger M visibly reduces both the residual and the pathwise implicit gap, as predicted by the contraction estimate. In the learned-score experiments, most of the solver-level improvement occurs for small values such as $M=1$ or $M=3$; in Table 6, moving from $M=0$ to $M=3$ reduces the implicit gap by roughly two orders of magnitude with a measured time increase of about $20\%$, whereas increasing from $M=3$ to $M=8$ yields a smaller solver-level gain and no monotone KID improvement. This supports using M as an adaptive numerical tolerance parameter rather than as a sample-quality knob.

7. Discussion and Scope

The proposed method should be interpreted as a numerical correction mechanism for reverse diffusion dynamics. It is not a new training objective, not a new neural architecture, and not a Wasserstein proximal sampler. Its mathematical role is to replace an explicit reverse update by a locally implicit equation solved approximately through contraction iterations.

This distinction is important. A Wasserstein gradient-flow approach studies evolution in the space of probability measures, often through variational proximal steps. In contrast, the present paper studies sample-path-level numerical discretization of reverse-time SDEs or ODEs. The fixed point argument is therefore local in time and finite-dimensional, appearing at each implicit step of the sampler.

The main assumptions should also be read as controlled mathematical hypotheses. Global Lipschitz continuity of the score-induced drift and the uniform contraction condition $hL\le q_0<1$ are stronger than what can generally be verified for high-dimensional neural score models, particularly near small noise levels or manifold-supported data [3,15]. The Fashion-MNIST experiment illustrates this limitation: the pathwise implicit gap can still be reduced by fixed point corrections, while the raw fixed point residual and feature-based quality metrics are affected by the learned score and by the absence of a verified contraction constant. The value of the analysis is therefore not a fully general guarantee for practical image-generation pipelines, but a transparent separation of algebraic solver error from score, terminal, and time-discretization errors in settings where these quantities can be cleanly defined.

Possible extensions include adaptive choices of M, damped fixed point corrections, semi-implicit splitting of stiff score terms, and probability-flow ODE variants. Another direction is to replace global Lipschitz assumptions by monotonicity, dissipativity, or one-sided Lipschitz conditions, which are more realistic for score-induced drifts.

The generalized fixed-point setting mentioned above gives another possible mathematical extension. Perturbed metric spaces and probabilistic cone b-metric spaces provide abstract environments in which Banach-type arguments can be reformulated [18,19]. Applying such frameworks to reverse diffusion sampling is not immediate, because the present proof uses pathwise Euclidean residuals, synchronous Brownian couplings, and Wasserstein endpoint distances. A meaningful extension would have to specify how stochastic trajectories, residuals, and distributional errors are represented in the chosen generalized metric structure.

The fast-sampler literature also suggests a complementary research direction. DDIM, PNDM, DEIS, DPM-Solver, EDM, and DPM-Solver++ improve the outer reverse-time discretization by exploiting deterministic ODE structure, pseudo numerical methods, exponential integrators, higher-order formulas, or guidance-aware parameterizations [5,6,7,8,10,11]. A natural next problem is to combine such outer discretizations with residual-controlled inner implicit corrections, choosing both the outer step size and the number of inner corrections adaptively from an a posteriori solver residual.

8. Conclusions

We introduced a fixed-point-corrected discretization framework for reverse-time diffusion sampling. By formulating each reverse step as a local implicit equation and approximating it by a finite number of fixed-point iterations, we obtained a conditional error decomposition into initialization, score, assumed implicit discretization, and algebraic solver components. The resulting estimates suggest a clear numerical trade-off: increasing the number of inner corrections reduces the pathwise gap to the implicit reference at the cost of additional score evaluations. The exact-score experiments show the predicted residual decay in a controlled contraction regime, while the neural-score Fashion-MNIST experiment shows that fixed point corrections may reduce the implicit gap without producing monotone improvement in every diagnostic. This solver consistency improvement does not automatically eliminate score approximation, terminal mismatch, time discretization, or Monte Carlo variability. The framework provides a mathematically transparent route for studying the stability of reverse diffusion samplers under controlled assumptions.

9. Reproducibility

The numerical tables and figures were generated by the accompanying reproducibility code supplied in the Supplementary Materials. The low-dimensional implementation fixes all random seeds, constructs exact and fitted Gaussian-mixture scores, runs the explicit, fixed-point-corrected, and implicit samplers, and writes the outputs from which the compact tables and figures in Section 6 are selected. The neural implementation trains the Fashion-MNIST score network and classifier, evaluates the fixed-point-corrected samplers, and writes the generation and inpainting outputs. The reported neural run used settings corresponding to 120 score-training epochs, 60 reverse sampling levels, 512 generation samples, 128 inpainting samples, and $M_{\mathrm{ref}}=12$ for the near-implicit reference.