Deviation-Space Diffusion Models

• Deviation-space diffusion models are frameworks that factorize uncertainty into a deviation subspace while preserving deterministic substrate components for targeted synthesis.
• They utilize a DDPM for image pathology synthesis and fractional SPDEs with space-time noise to achieve robust, controlled deviation modeling.
• Optimization incorporates joint objectives with edge-aware regularization and large deviation principles, leading to improved diagnostic metrics and convergence.

A Deviation-Space Diffusion Model refers to any generative or stochastic modeling paradigm where the uncertainty or stochasticity is factorized into a deviation field or subspace, with deterministic substrate variables preserved by construction. Two fundamentally distinct threads instantiate this concept: (i) a conditional diffusion generative model for image-pathology synthesis via additive deviation fields restricted by spatial masks (Wang et al., 29 Dec 2025) and (ii) a theoretical framework for stochastic space-time fractional diffusion equations with associated large deviation principles and super-convergent time integrators (&&&1&&&).

1. Mathematical Structure and Factorization Principles

Both approaches center on an explicit decomposition separating deterministic anatomical or substrate components from stochastic deviation fields. In the generative imaging context, the observed pathological image $x \in \mathbb{R}^{H \times W}$ is decomposed as:

• Subject-specific substrate $x_{\text{sub}}: \Omega \to \mathbb{R}$, optimized via masked inpainting:

$$L_{\text{sub}}(x_{\text{sub}};x,m) = \lambda_{\text{out}} \| (x_{\text{sub}} - x) \odot \bar{m} \|_2^2 + \lambda_{\text{in}} \| (x_{\text{sub}} - \text{Inp}(x,m)) \odot m \|_2^2$$

where $m$ is the binary lesion mask, $\bar{m} = 1 - m$, and $\text{Inp}(x,m)$ is healthy inpainting inside $m$.

• Deviations $r_0 = (x - x_{\text{sub}}) \odot m$, projected by $\tanh$ to a dynamic range.

In stochastic PDE settings, substrate fields $X(t)$ are governed by:

$$\partial_t^\alpha X(t) + A^\beta X(t) = F(X(t)) + {_t^\gamma \dot{W}}(t)$$

where $A^\beta$ is the fractional Laplacian operator and additive stochastic deviation is modeled as fractional integrals of Wiener processes.

This additive factorization enables the generative search space or stochastic action to be projected entirely to the deviation field, typically of lower dimension than the global substrate space.

2. Diffusion Processes and Noise Modeling in Deviation Subspaces

In image synthesis, a $T$-step Denoising Diffusion Probabilistic Model (DDPM) is employed on the deviation field $r_0$, with all injected noise spatially masked:

• Forward Markov chain: $$q(r_t|r_{t-1}) = \mathcal{N}(\sqrt{\alpha_t} r_{t-1}, \beta_t I)$$, with $\alpha_t = 1 - \beta_t$ and a linear $\beta_1 = 10^{-4} \to \beta_T = 0.02$ schedule for $T = 1000$.
• Noising: $$r_t = \sqrt{\bar{\alpha}_t} r_0 + \sqrt{1 - \bar{\alpha}_t} \epsilon, \quad \epsilon \sim \mathcal{N}(0, I)$$.
• Masking: $r_t \leftarrow r_t \odot m, \ \forall t$.

Reverse steps denoise using a learned network $\epsilon_\theta(r_t, x_{\text{sub}}, m, t)$, with means and variances from standard DDPM parameterization. Sampling is performed by ancestral updates, followed by mask enforcement at every step.

In fractional diffusion SPDEs, noise is introduced as space-time fractional integrals of cylindrical Wiener processes. The covariance operator $Q$ and the regularity parameters $(\alpha, \beta, \gamma)$ control the diffusion characteristics. Large deviation theory quantifies the likelihood of rare deviation paths via Freidlin–Wentzell rate functions.

3. Optimization Objectives and Regularization

Image-generation models employ a joint objective:

$$L = L_{\text{sub}} + \lambda_{\text{diff}} L_{\text{diff}} + \lambda_{\text{dev}} L_{\text{dev}} + \lambda_{\text{syn}} L_{\text{syn}}$$

where

• $L_{\text{diff}}$ (DDPM L_simple): $$E_{r_0, \epsilon, t}[ \| \epsilon - \epsilon_\theta(r_t, x_{\text{sub}}, m, t) \|_2^2 ]$$
• $L_{\text{dev}}$: pathology regularization enforcing consistency inside $m$, at the lesion ring, and zero outside; employs a Gaussian-blurred soft map $S$ and ring weight $w_{\text{ring}}$.
• $L_{\text{syn}}$: seam-aware synthesis loss measuring boundary blending integrity.

SPDE models impose rate functions for pathwise control on deviations; for a path $\phi$,

$$I(\phi) = \frac12 \int_0^T \| Q^{-1/2} (\partial_t^\alpha \phi + A^\beta \phi - F(\phi))_\gamma \|^2 dt$$

interpreted as the minimal “energy” needed for controlled deviations.

4. Algorithmic and Architectural Specification

PathoSyn uses a two-stage U-Net pipeline:

• Anatomical estimator $f_{\text{sub}}$: Four-stage U-Net with skip connections.
• Noise predictor $\epsilon_\theta$: Wide-ResNet blocks, 16x16 self-attention bottleneck, sinusoidal timestep embeddings via FiLM, conditioned by channel-wise concatenation of $r_t$, $x_{\text{sub}}$, $m$.
• AdamW optimizer, learning rate $10^{-4}$, weight decay $10^{-5}$, epochs $300$; loss weights detailed for all regularizers.

Space-time fractional SPDEs utilize spectral approximation on $L^2(D)$ and the Mittag–Leffler Euler (MLE) integrator. Key is the convolution decomposition for temporal regularity:

$$\Upsilon_G(t) = \int_0^t S_0(t-s) G(0) ds + \int_0^t S_0(t-s) [G(s) - G(0)] ds$$

MLE error bound: $$\| X(t_m) - Y_m \|_{L^2} \leq C t_m^{2\alpha-1} h^{\rho}$$, with exact $\rho$ from regularity parameters.

5. Inference, Fusion, and Stabilization Mechanisms

During generative inference:

• At each reverse step, enforce $r_{t-1} \leftarrow r_{t-1} \odot m$ to suppress extraneous stochasticity outside the lesion domain.
• After DDPM reversal, seam-aware fusion blends recovered deviations onto the anatomical substrate: $\hat{x} = x_{\text{sub}} + S \odot r_0$, where $S$ is a Gaussian-smoothed mask, eliminating visible seams.

No further post-hoc correction is required, as $L_{\text{dev}}$ and $L_{\text{syn}}$ ensure anatomical fidelity and edge artifact suppression by construction.

6. Theoretical Guarantees, Quantitative Evaluation, and Practical Significance

Deviation-space factorization entails a reduction of search space dimensionality, e.g., only lesion-space degrees of freedom are sampled while global structure is fixed. This translates probabilistically as factorization $p(x) = p(x_{\text{sub}}) p(r|x_{\text{sub}})$.

Orthogonality metrics empirically demonstrate disentanglement:

• Cosine similarity $C \approx 0.072$ and Mutual Information $\text{MI} \approx 0.21$ for PathoSyn, versus $C \approx 0.425$, $\text{MI} \approx 1.24$ for holistic baselines.

Clinical utility is substantiated:

• Diagnostic realism (AUC): $0.55$ for PathoSyn vs $0.72$ for Brain-LDM.
• Segmentation DSC (nnU-Net): $+6.5\%$ absolute improvement ($0.845$ vs $0.780$).
• Classification AUROC (ResNet-50): $0.915$ vs $0.860$ baseline; Expected Calibration Error (ECE) $\downarrow$ from $0.070$ to $0.045$.

In SPDE models, Freidlin–Wentzell LDPs and $\Gamma$-convergence of discrete rate functions demonstrate mathematical equivalence of continuum and numerical deviation-space action functionals.

7. Extensions, Limitations, and Applicability

Deviation-space approaches provide mathematically principled and interpretable generative paradigms for localized stochastic modeling, with boundary-aware regularization and convergence guarantees (Wang et al., 29 Dec 2025, Dai et al., 2022). Applicability covers patient-specific imaging synthesis, benchmark data augmentation for deep learning, precision intervention planning, and controlled rare-event analysis in fractional PDEs.

A plausible implication is extensibility to multimodal and spatio-temporal domains, provided deviation masks and substrates are correctly constructed and regularity conditions (noise, nonlinearity) are satisfied. The principal limitation is reliance on accurate substrate estimation and mask definition, as degeneracy in either compromises disentanglement and fidelity.

Deviation-space diffusion frameworks thus offer an operational and theoretical foundation for both practical generative modeling in biomedical imaging and rigorous stochastic analysis in fractional SPDEs.