DeRaDiff: Triple Diffusion Methodologies

• DeRaDiff for diffusion models enables dynamic sample-time control through geometric mixture interpolation, maintaining photorealism with error metrics under 0.5%.
• The Deep Random Difference Method replaces Hessians with random finite differences, achieving scalable high-dimensional PDE solutions with relative errors of 10⁻³–10⁻².
• Dereflection Any Image employs one-step diffusion and a large DRR dataset to robustly remove image reflections with PSNR up to 27.25 dB and SSIM of 0.930.

DeRaDiff refers to three distinct methodologies in contemporary machine learning and computational mathematics: (1) Denoising Time Realignment of Diffusion Models for sample-time alignment control in diffusion-based generative models (Manujith et al., 28 Jan 2026); (2) the Deep Random Difference Method for scalable solution of high-dimensional quasilinear parabolic partial differential equations (PDEs) and Hamilton-Jacobi-Bellman (HJB) equations (Cai et al., 25 Jun 2025); and (3) Dereflection Any Image with Diffusion Priors and Diversified Data for robust single-image reflection removal (Hu et al., 21 Mar 2025). Each “DeRaDiff” framework addresses a unique problem domain—conditional generative modeling, scientific computing, and image restoration—by leveraging advanced stochastic or diffusion-based neural techniques with theoretical and practical guarantees. The following sections present an in-depth treatment of each interpretation.

1. Denoising Time Realignment of Diffusion Models for Sample-Time Alignment Control

The DeRaDiff framework for diffusion models addresses the challenge of selecting the appropriate regularization strength $\beta$ when aligning pretrained text-to-image diffusion models to human preference rewards—e.g., PickScore—via KL-regularized reward maximization,

$$\max_{p} \;\; \mathbb{E}_{c\sim\mathcal{D},\,x_0\sim p(\cdot\mid c)}\bigl[r(c,x_0)\bigr] \;-\; \beta\,\mathrm{KL}\!\bigl[p(x_0\mid c)\,\|\,p_{\mathrm{ref}}(x_0\mid c)\bigr].$$

Large $\beta$ yields weak alignment, while small $\beta$ leads to reward hacking. Traditional approaches require grid searches over $\beta$, retraining at each value—a prohibitive computational cost.

DeRaDiff sidesteps repeated retraining by enabling dynamic, sample-time control of the effective alignment strength via a “geometric mixture” of the reverse-step posteriors of the anchor-aligned and pretrained reference models. At each denoising iteration $t$, the backward transition is computed as

$$q_\lambda(x_{t-1}|x_t)\propto q_0(x_{t-1}|x_t)^{1-\lambda}q_{[\beta]}(x_{t-1}|x_t)^{\lambda},$$

with $q_0$ and $q_{[\beta]}$ Gaussian, where $\lambda\in[0,1]$ interpolates continuously between maximal ($\lambda=0$) and nominal ($\lambda=1$) preference alignment. Under standard schedulers (DDPM/DDIM), this yields a closed-form Gaussian with mean and covariance given by

$$\Sigma_{\mathrm{new}}^{-1} = (1-\lambda)\Sigma_0^{-1}+\lambda\Sigma_\beta^{-1},\qquad \mu_{\mathrm{new}} = \Sigma_{\mathrm{new}}\left[(1-\lambda)\Sigma_0^{-1}\mu_0+\lambda\Sigma_\beta^{-1}\mu_\beta\right].$$

A single forward pass requires only a modest increase in computation over the baseline guided sampler; all reference and aligned posterior terms are already obtained in vanilla classifier-free guidance. At each denoising step: (1) obtain predictions from both models; (2) compute posterior statistics; (3) apply the precision-weighted interpolation; and (4) sample from the new Gaussian or use its mean for deterministic decoding.

The result is sample-time, on-the-fly modulation of effective regularization, with error on human-preference (PickScore, HPS v2), semantic alignment (CLIP), and FID competitive with fully retrained models (mean absolute errors below $10^{-3}$ and relative errors below $0.5\%$) (Manujith et al., 28 Jan 2026). DeRaDiff also ameliorates reward hacking by interpolating away from misaligned anchors without retraining, restoring photorealism to hacked models. The framework achieves linear compute savings in $\beta$-sweeps.

2. Deep Random Difference Method for High-Dimensional PDEs

The DeRaDiff/DRDM (“Deep Random Difference Method”) framework is designed for the scalable solution of parabolic PDEs and HJB equations in extremely high dimensions ($d\sim10^5$) (Cai et al., 25 Jun 2025). Standard deep PDE approaches rely on automatic differentiation to estimate second derivatives (Hessians), which becomes computationally infeasible for very high $d$.

The DRDM replaces explicit Hessians by a random-difference operator: for any suitably regular solution $v(t,x)$,

$$\mathcal{D}_h v(t,x):=E\left[\frac{v(t+h,x+\mu h+\sqrt{h}\sigma\xi)-v(t,x)}{h}\right]=\mathcal{D}v(t,x)+O(h).$$

Here, the expectation is over a random vector $\xi$ with zero mean, identity covariance, and finite fourth moments. The second derivatives are subsumed within finite differences, evaluated by Monte Carlo path-sampling (Euler–Maruyama scheme). All second-order terms are replaced by first-order differences inside expectations, allowing high-dimensional parallelism.

Galerkin weak enforcement and adversarial minimax optimization eliminate the need for pointwise evaluation of expectations, further reducing computational overhead. Neural networks $v_\theta$ (for solution) and $\rho_\eta$ (for test function) are trained via the objective

$$\min_\theta\max_\eta \left(\int_0^{T-h} E\left[\rho(t,X_t)R(t,X_t,\xi;v_\theta)\right]\,dt\right)^2,$$

where $R$ encodes the residual of the random-difference Euler–Maruyama step against the PDE nonlinearity.

Extensions to HJB equations use policy iteration: controls are alternately updated by minimization of target Hamiltonians, with each PDE residual evaluated by random differences, requiring only first derivatives throughout.

Theoretical analysis establishes $O(\Delta t)$-order global error bounds in the mean-square pathwise sense. Empirical results (PDEs to $d=10^5$) show relative $L^1$-errors from $10^{-3}$ to $10^{-2}$, runtimes below $1.5$ hours, and memory/computation scaling linear in $d$ and in sample size $M$, not $d^2$ (Cai et al., 25 Jun 2025).

3. Dereflection Any Image with Diffusion Priors and Diversified Data

The DeRaDiff (“Dereflection Any Image with Diffusion Priors and Diversified Data”) methodology targets single-image reflection removal, which requires both data diversity and robust modeling (Hu et al., 21 Mar 2025). Its contributions include a large-scale Diverse Reflection Removal (DRR) dataset and a single-step diffusion-based restoration model coupled with a progressive three-stage training regime.

The DRR dataset comprises 23,303 real image pairs (plus over 20,000 synthetic pairs), acquired by videoing scenes through rotated glass plates (3 mm, 8 mm) with controlled variation in thickness, angle, reflection intensity, lighting, and content. Frames from videos with and without glass are aligned with SIFT+RANSAC to construct ground-truth and mixed images. The synthetic component adopts a blending model with random coefficients: $$M = \gamma_1 T + \gamma_2 R - \gamma_1\gamma_2 (T\circ R),\quad \gamma_1\in[0.8,1.0],\ \gamma_2\in[0.4,1.0].$$

The dereflection model operates in a single forward diffusion step: let $z_0$ be the clean latent, apply noise up to $z_T$, then use a U-Net+ControlNet $\mu_{\theta,\phi}$ to recover $\hat z_0$ in one shot. The foundational loss is

$$\mathcal{L}_{\mathrm{diff}} = \mathbb{E}_{z_T, M} \left\| z_T - \mu_{\theta, \phi}(z_T, c, t, f_\phi(\mathcal{E}(M))) \right\|_2^2,$$

with $c$ as text prompt embedding and $f_\phi$ as a mixed-image encoder.

Training consists of: (i) foundational diffusion pretraining (all data/loss above), (ii) reflection-invariant finetuning (enforce denoising consistency across different reflections for the same scene via an additional loss), and (iii) cross-latent decoder fine-tuning for detail refinement, using a combination of $L_1$-, SSIM-, and LPIPS-based reconstruction losses.

Architecturally, the model is built on the Stable Diffusion v2.1 U-Net with synchronized ControlNet downsampling and cross-latent decoder shortcut paths. While the model is stochastic during training, inference is deterministic and runs at approximately 1 second for a 768$\times$768 image on an NVIDIA 3090.

Empirical evaluation on Nature, Real, SIR², and both DRR-S/-C benchmarks demonstrates PSNR up to 27.25 dB and SSIM up to 0.930. Ablations reveal additive performance gains from the cross-latent decoder, DRR pretraining, and reflection-invariant finetuning. Qualitative results indicate strong reflection removal and detail preservation even in highly complex scenes (Hu et al., 21 Mar 2025).

4. Theoretical Properties and Algorithmic Significance

The three frameworks share an emphasis on denoising, stochastic latent evolution, and scalable control (either at inference or optimization time) via closed-form or expectation-based updates.

• The Denoising Time Realignment approach exposes a geometric structure in the space of diffusion posteriors, yielding efficient and theoretically transparent interpolation among alignment regimes with no retraining overhead (Manujith et al., 28 Jan 2026).
• The Deep Random Difference Method guarantees first-order consistency in the temporal discretization for high-dimensional PDEs, rigorously quantifying global error and scaling, due to the elimination of Hessian dependence (Cai et al., 25 Jun 2025).
• The Dereflection Any Image framework exploits single-step denoising and staged curriculum learning to secure both restoration accuracy and operational efficiency in open-set, real-world image tasks (Hu et al., 21 Mar 2025).

5. Applications, Limitations, and Future Directions

These distinct DeRaDiff methods have immediate applications in generative model alignment (e.g., text-to-image synthesis), high-dimensional scientific modeling (e.g., financial mathematics, stochastic control), and image restoration (e.g., removing reflections from photography, digital forensics).

A limitation in Denoising Time Realignment is the requirement for a pretrained anchor-aligned model; shifts unsupported by the initial anchor cannot be extrapolated without further training. The Random Difference method, while eliminating Hessian computation, incurs stochastic variance and is still subject to sampling error for low $M$. The reflection removal pipeline’s deterministic inference does not capture uncertainty, which could affect results in highly ambiguous cases.

A plausible implication is that these techniques will generalize to broader families of diffusion- or SDE-driven models, encourage hybrid strategies (e.g., dynamic inference-time regularization, end-to-end differentiable PDE solvers), and foster further research in curriculum-based restoration strategies or scalable control in generative systems.

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Summary Table: Distinct DeRaDiff Frameworks

Methodology/Domain	Core Technique	Key Reference
Diffusion Model Alignment	Denoising time geometric mixture interpolation	(Manujith et al., 28 Jan 2026)
High-Dimensional Scientific PDEs	Deep random difference (finite-difference MC)	(Cai et al., 25 Jun 2025)
Single-Image Reflection Removal	One-step diffusion, staged curriculum, DRR data	(Hu et al., 21 Mar 2025)