Safe and Stable Diffusion (S²Diff)

• Safe and Stable Diffusion (S²Diff) is a family of methods that integrate rigorous safety, traceability, and stability constraints directly into diffusion models.
• Key approaches include invisible watermarking in image synthesis, training-free modifications for safe denoising, and Lyapunov-guided sampling for control tasks.
• Empirical evaluations demonstrate improved watermark recoverability, reduced unsafe content, and enhanced control stability across various benchmarks.

Safe and Stable Diffusion (S²Diff) encompasses a family of advanced frameworks that rigorously enforce safety, stability, or traceability constraints within diffusion models. These approaches share the objective of tightly integrating safety-related mechanisms—such as provable region avoidance, strong watermark traceability, or Lyapunov-based guarantees—directly into the generative or control pipeline, moving beyond conventional prompt engineering or black-box postprocessing.

1. Core Methodological Variants and Problem Domains

S²Diff refers to several methodological advances across separate domains:

• Safe-SD: A framework for copyright and provenance assurance via invisible generative watermarking within image synthesis, simultaneously ensuring fidelity and high traceability (Ma et al., 2024).
• Training-Free Safe Denoisers: A theoretically grounded, training-free modification of the denoising process to strictly avoid “unsafe” regions in the data manifold, enabling robust content filtering at inference time without retraining (Kim et al., 11 Feb 2025).
• Lyapunov-Guided S²Diff for Control: A framework embedding Lyapunov-theoretic certificate functions inside trajectory-level diffusion sampling, providing almost-sure safety and stability for planning and dynamical systems control (Cheng et al., 29 Sep 2025).

Each variant targets a distinct notion of “safety.” In generative imaging, safety means high-fidelity watermark permanence or avoidance of unsafe content; in control, safety and stability are defined via formal state-space criteria.

2. Architectural and Theoretical Frameworks

A. Safe-SD: Invisible Watermarking for Diffusion Models

Safe-SD introduces a two-stage design:

• First-Stage VAE (Injector/Detector):
  • A shared encoder $E$ maps images $x$ and graphical watermarks $w$ (e.g., QR codes) into latents $z_i, z_w \in \mathbb{R}^{h\times w\times d}$.
  • A 1×1 convolution $f_c$ fuses $z_i$ and $z_w$ into a mixed latent $z_m$.
  • Two decoders: $D_i$ (parameters $\theta_f$ frozen) reconstructs the watermarked image; $D_w$ (trainable parameters $\theta_t$) extracts the watermark from $z_m$.
• Second-Stage Conditional Latent Diffuser (U-Net $\epsilon_\theta$ wrapped by the VAE):
  • Text-conditional generation guided by CLIP embeddings.
  • Watermark injection is temporally randomized via a binary $\lambda$-sampling schedule and secured by $\lambda$-encryption, such that injection steps are cryptographically masked.

Mathematically, the injection and detection tasks are optimized jointly using a composite reconstruction and adversarial loss: $$\mathcal L_{s^1} = \|\,x - \hat x\,\|_2^2 + \gamma\,\|\,w - \hat w\,\|_2^2 + \mathcal L_{\rm adv}$$ A forward process stochastically injects watermark information at $\lambda$ selected diffusion steps, with the key $m \in \{0,1\}^T$ encoding the schedule (Ma et al., 2024).

B. Training-Free S²Diff: Safe Denoisers via Negation Sets

This approach formulates safety as exclusion of unsafe regions in data space. The optimal safe denoiser is proven to be

$$E_{\text{safe}}[x|x_t] = E_{\text{data}}[x|x_t] + \beta^*(x_t)\left(E_{\text{data}}[x|x_t] - E_{\text{unsafe}}[x|x_t]\right)$$

where $E_{\text{unsafe}}$ describes the conditional mean over the unsafe (negation set) manifold and the weight $\beta^*(x_t)$ grows as $x_t$ approaches unsafe regions. This yields provably safe samples at the cost of a controlled repulsion from the negated set (Kim et al., 11 Feb 2025).

The entire method is training-free, involves no model retraining, and operates only at sampling time.

C. Lyapunov-Guided S²Diff for Safe Planning and Control

Safe and stable diffusion for control tasks is built on trajectory-level diffusion, guided by a Control–Lyapunov–Barrier Function (CLBF) $V$ parameterized by a neural network. The certificate $V$ enforces:

1. $V(x_*) = 0$ at equilibrium,
2. $V(x) > 0$ for $x \neq x_*$,
3. $V(x) \leq c$ for all $x \in \mathcal{X}_s$ (safety),
4. Uniform dissipation: $$\inf_{u \in \mathcal{U}} [\mathcal{L}_f V(x,u) + \lambda V(x)] \leq 0$$ for $x \neq x_*$ (stability).

The diffusion process is steered by a Gibbs-type target density over control-trajectory space: $$p(U) \propto p_{\text{safe}}(U) \cdot p_{\text{stable}}(U) \cdot p_{\text{cost}}(U)$$ where $p_{\text{safe}}$ encodes region constraints, $p_{\text{stable}}$ penalizes Lyapunov violations, and $p_{\text{cost}}$ encodes task objectives. The denoising process thus enforces safety and stability via direct modification of the trajectory sampling distribution (Cheng et al., 29 Sep 2025).

3. Algorithmic and Optimization Details

A. Injection, Triggering, and Detection in Safe-SD

• Pretraining: The VAE (encoder $E$, decoders $D_i, D_w$, fusion $f_c$) is trained on $(x, w)$ pairs to simultaneously optimize for high-fidelity reconstruction and watermark recoverability, subjected to $\ell_2$ and adversarial losses.
• Diffusion Fine-tuning: The U-Net component is fine-tuned with $\lambda$-masking and $\lambda$-encryption to ensure that watermark information is present at only selected noise-injection steps.
• Inference: At test-time, a user prompt and optional input image trigger injection of the selected graphical watermark according to a prompt-conditioned policy.

B. Training-Free Safe Denoising

• Safe Denoiser Algorithm:

1. At each sampling step $t$, compute a kernel-weighted repulsion from the negation set in data space.
2. Apply the safe denoiser correction only during “critical” timesteps, typically in structural formation phases, to avoid detail degradation.
3. Hyperparameters (negation set size $N$, RBF kernel width $\sigma$, repulsion scale $\eta$, threshold $\beta_{\text{thresh}}$) control the fidelity–safety tradeoff.

C. Lyapunov-Guided Sampling for Safe Control

• Alternating optimization:

1. Guided diffusion samples state–action trajectories with scores shaped by the learned CLBF.
2. The CLBF neural network is updated via backpropagation on a composite loss, incorporating equilibrium, positivity, safe-set, Lyapunov, and discrete-difference penalties.

• Proof framework: Almost-sure safety and exponential stability are established under mild, measure-based dissipation assumptions.

4. Quantitative and Qualitative Evaluation

A. Safe-SD: Watermarking

Dataset	PSNR ↑	FID ↓	LPIPS ↓	CLIP ↑
LSUN-Churches	33.17	18.89	0.232	88.15
FFHQ	32.73	19.36	0.215	93.99

Detection of injected QR-code watermarks is $>99\%$ robust to severe manipulations (rotation, resize, brightness, crop, combinations). Multi-watermark scenarios confirm that two logos can be extracted simultaneously with negligible quality loss. Pixel-difference visualizations show imperceptibility and localization in high-texture regions (Ma et al., 2024).

B. Training-Free Safe Denoiser

• NSFW Avoidance: Attack Success Rate (ASR) on “Ring‐A‐Bell” nudity set reduced from $0.797$ (vanilla SD) to $0.127$ (SAFREE + SafeDenoiser), with CLIP/FID scores largely preserved.
• Bias Mitigation: On FFHQ$\rightarrow$CelebA, female bias reduced with improved fidelity.
• Category Negation: On ImageNet256, class removal (e.g., “Chihuahua”) performed without drastic loss of diversity or precision for other classes, outperforming sparse repellency.

C. Safe Control

Controller	Safety Rate (%)	Terminal Error ↓	Inference Time (ms)
S²Diff	98.8	0.226	45
MPC	66.3	>0.38	249
rCLBF-QP	78.8	>0.38	<10
MBD	73.8	>0.38	–

Notably, on F-16 (non-affine), S²Diff achieved $100\%$ safety and cut error versus plain model-based diffusion, with faster inference (Cheng et al., 29 Sep 2025).

5. Security, Cryptographic, and Theoretical Guarantees

• Safe-SD: $\lambda$-encryption randomizes which diffusion steps are watermarked; knowledge of the binary mask $m$ is required for perfect removal, making adversarial erasure combinatorially complex.
• Training-Free S²Diff: Provable removal of unsafe regions is achieved as the denoising trajectory is actively repelled from a representative negation set; if this set is incomplete, residual risk remains.
• Control S²Diff: Under structural assumptions (smooth $f, V$, small “almost” violation set), almost-sure global safety and exponential convergence in the CLBF are rigourously guaranteed.

6. Limitations and Potential Extensions

• Safe-SD: The current $\lambda$-sampling uses uniform random selection; learnable schedules or context-dependent injection could enhance robustness. Attacks via inversion, model fine-tuning, or GAN-based restoration remain challenges; integration with differential privacy or robust optimization is a possible extension. The method is directly portable to other latent-diffusion architectures (e.g., DALL·E 2, Imagen) with retraining (Ma et al., 2024).
• Training-Free S²Diff: Strictly limited by the coverage of the user-supplied negation set and sensitive to hyperparameter tuning. Computational cost scales with the negation set cardinality; efficient approximations are an open direction (Kim et al., 11 Feb 2025).
• Control S²Diff: The framework’s practical performance is dictated by the expressivity and correctness of the learned CLBF; improving neural certificate learning or extending to high-dimensional or partially observable domains are ongoing research topics (Cheng et al., 29 Sep 2025).

7. Conclusion

Safe and Stable Diffusion (S²Diff) establishes state-of-the-art methods for embedding safety, traceability, or stability properties into the core of diffusion models—achieving high-fidelity outputs alongside mathematically or cryptographically rigorous assurance. Applications span invisible generative watermarking, provable exclusion of unsafe or biased generations, and safe control synthesis, each with domain-specific optimization and theoretical foundations. S²Diff methods surpass or complement existing baselines in avoidance, recoverability, safety, and stability, marking a critical advance in the deployment of diffusion models within mission-critical, legally constrained, or ethically sensitive domains (Ma et al., 2024, Kim et al., 11 Feb 2025, Cheng et al., 29 Sep 2025).