Conditional Denoising Diffusion Probabilistic Models

• Conditional DDPMs are generative models that integrate conditioning signals into a progressive denoising process to synthesize data with high fidelity and diversity.
• They employ various conditioning strategies—such as reverse-only and forward-and-reverse approaches—to control the generation process for tasks like inpainting and image restoration.
• Empirical results show that conditional DDPMs outperform traditional generative methods in applications including medical imaging, super-resolution, and LiDAR scan completion.

A conditional denoising diffusion probabilistic model (DDPM) is a generative model framework that leverages a progressive, stepwise denoising process to synthesize new data samples that are consistent with one or more conditioning signals. The conditional DDPM paradigm has become a foundation for a wide range of applications requiring controlled generation, inpainting, image restoration, and data synthesis with complex multi-modal structure. Conditional DDPMs have demonstrated superior ability to model high-dimensional, structured distributions in areas where competing approaches such as GANs and autoregressive models often face mode collapse, limited diversity, or poor generalization.

1. Core Principles of Conditional DDPMs

Conditional DDPMs extend the original DDPM framework by incorporating external information (conditions) into the generative process, enabling the sampling of data consistent with prescribed specifications. The unconditional DDPM consists of a forward diffusion process, which progressively corrupts a sample $x_0$ with Gaussian noise over $T$ timesteps, and a reverse process, parameterized by a neural network, which learns to denoise and invert this trajectory. The reverse kernel is typically parameterized as

$$p_\theta(x_{t-1}|x_t, c) = \mathcal{N}(x_{t-1}; \mu_\theta(x_t, t, c), \Sigma_\theta(x_t, t, c)),$$

where $c$ denotes the conditioning signal and the mean $\mu_\theta$ is a nonlinear function (often via a U-Net) of the noisy input, timestep, and condition.

Conditioning can be incorporated at different stages:

• Reverse-only conditioning: Only the denoising network is provided with the conditional input (e.g., class label, segmentation map) at each step, while the forward process remains unchanged (Lugmayr et al., 2022, Xu et al., 2023, Krishna et al., 7 Sep 2024).
• Forward-and-reverse conditioning: The conditioning affects both the forward and reverse processes, for example by shifting the mean of the forward diffusion at each step to allocate a distinct trajectory for every condition (Zhang et al., 2023).
• Guided sampling: External guidance, such as classifier gradients or low-pass projections, is injected during the reverse process to steer the generation toward fulfilling more complex or multiple constraints (Krishna et al., 7 Sep 2024).

The learning objective is typically the expected squared error between the true noise injected and the model’s noise prediction, optionally augmented with Kullback–Leibler or other structural regularizations:

$$\mathbb{E}_{x_0, t, \epsilon} \left[ \|\epsilon - \epsilon_\theta(x_t, t, c)\|^2 \right].$$

2. Algorithms and Conditioning Mechanisms

Conditional DDPMs have been instantiated using multiple methodologies:

Reverse-only Conditioning

• Methods such as RePaint (Lugmayr et al., 2022) use a pretrained unconditional DDPM for inpainting, by altering only the reverse process at every denoising step: the known pixel regions are resampled from the forward process using their prescribed (unmasked) values, while missing regions are resampled given the model's predictions. This is formalized as

$$x_{t-1} = m \odot x_{t-1}^{\text{known}} + (1-m) \odot x_{t-1}^{\text{unknown}},$$

where $m$ is the binary mask, $x_{t-1}^{\text{known}}$ is sampled from the forward process, and $x_{t-1}^{\text{unknown}}$ from the model posterior.

Forward-and-Reverse Conditioning

• ShiftDDPMs (Zhang et al., 2023) inject condition-dependent shifts $s_t = k_t \cdot E(c)$ at each step in the forward process, resulting in

$$q(x_t | x_0, c) = \mathcal{N}(\sqrt{\bar{\alpha}_t}x_0 + s_t, (1-\bar{\alpha}_t)\Sigma),$$

with $k_t$ controlling the influence schedule and $E(c)$ mapping the condition to latent shift directions.

Multi-Conditional and Guided Sampling

• mDDPM (Krishna et al., 7 Sep 2024) performs multi-conditional guided sampling, modifying each denoising step by matching the low-pass filtered version of the sample with that of multiple guidance images. Each step is augmented as

$$x_{t-1} \gets x_{t-1} + \sum_s \phi_{n_s}(y_{s, t-1}) - \phi_{n_s}(x_{t-1}),$$

with $\phi_{n_s}$ a low-pass operator and $y_s$ the $s$-th conditional reference.

• Classifier-free guidance, introduced in text-to-image synthesis, is extended to allow classifier-free and multi-source conditioning in medical and anatomical image generation (Krishna et al., 7 Sep 2024).

Joint Latent Prior Modeling

• RestoreGrad (Lee et al., 19 Feb 2025) improves efficiency and sample quality for restoration tasks by learning a data-driven prior $p(e|y)$ for the latent variable $e$, using VAE-style KL-regularization between $q(e|x_0, y)$ and $p(e|y)$. This provides a latent code distribution that is closer to the true posterior, resulting in faster convergence and fewer required sampling steps.

3. Applications and Empirical Performance

Conditional DDPMs have demonstrated state-of-the-art performance in a range of application domains:

Application	Conditioning	Notable Results
Free-form Inpainting	Partial image, arbitrary mask	Outperforms autoregressive and GAN-based methods on LPIPS, human studies, and generalizes to unseen masks (Lugmayr et al., 2022)
Blind Super-Resolution	LR image, degradation kernel	Dual-DDPM approach improves PSNR, LPIPS, and FID vs. SOTA; enables distributional, not deterministic, super-resolved outputs (Xu et al., 2023)
Medical Image Synthesis	Multiple guidance images (anatomy, labels)	mDDPM produces synthetic lung CTs indistinguishable from real images in radiologist Visual Turing Tests and surpasses GANs on FID/SSIM (Krishna et al., 7 Sep 2024)
Image Restoration (Microscopy)	Noisy/low-resolution image	Consistently outperforms baseline UNet-RCAN, pix2pix, CARE, and Noise2Void across diverse datasets (Osuna-Vargas et al., 18 Sep 2024)
LiDAR Scan Completion	Partial semantic point cloud	Improves scene reconstruction and semantic completion IoU/mIoU, outperforms LMSCNet and JS3C-Net (Cao et al., 26 Sep 2024)
Speech Enhancement/Image Denoising	Noisy signal	RestoreGrad achieves faster convergence and improved PESQ, LPIPS, with robustness to fewer sampling steps (Lee et al., 19 Feb 2025)

In every domain, conditional DDPMs excel both in fidelity and diversity, with enhanced mode coverage and structural consistency across challenging mask types, clean-guided restoration, and complex multi-conditional requirements.

4. Architectural and Theoretical Advances

Several research directions have emerged to address the computational cost and theoretical properties of conditional DDPMs:

• Sampling Efficiency: Semi-implicit DDPMs (SIDDMs) decouple the denoising process into implicit (adversarial) and explicit (forward-conditional) matching, enabling very large denoising jumps and drastically faster sampling without losing quality (xu et al., 2023).
• Learned Latent Priors: RestoreGrad (Lee et al., 19 Feb 2025) integrates a prior learned jointly with the diffusion process, replacing the standard Gaussian. This results in improved convergence, higher robustness under constrained sampling steps, and stronger alignment with data-driven structures.
• Theoretical Guarantees: Explicit convergence bounds have been established under general noise schedules (Nakano, 3 Jun 2024) and in Wasserstein-2 distance under constant-variance score noise (Arsenyan et al., 11 Jun 2025). These studies show that errors in the estimated score function and time discretization decay as the number of steps increases, and DDPMs are statistically optimal in certain log-concave settings.
• Multi-modal, Non-Euclidean, and Structure-aware Extensions: SPD-DDPM (Li et al., 2023) generalizes the framework to SPD matrix-valued data and preserves the underlying Riemannian structure, relevant for complex geometric datasets. Heat Diffusion Models (HDM) (Zhang et al., 28 Apr 2025) integrate the discrete heat equation to explicitly model local pixel dependencies in spatial domains, improving detail preservation. Iso-Diffusion (Fernando et al., 25 Mar 2024) introduces additional isotropy regularization in the loss to enforce structural noise constraints, enhancing sample fidelity.

5. Limitations, Challenges, and Comparative Analysis

Despite strong performance, conditional DDPMs present several trade-offs:

• Sampling Complexity: High-quality generation typically requires dozens to hundreds of iterative denoising steps. Strategies such as SIDDMs, continuous U-Net parameterizations (Calvo-Ordonez et al., 2023), and learned priors (Lee et al., 19 Feb 2025) partially mitigate this but Gaussian assumptions can break down for large-step or highly multimodal transitions.
• Condition Integration: Most conventional models inject conditioning only in the reverse process, which can restrict influence on latent representations to a narrow interval. Approaches such as ShiftDDPMs (Zhang et al., 2023) that shift forward trajectories offer more entangled control, but require re-specification of the entire diffusion process.
• Guided Sampling Trade-offs: Aggressive guidance (multi-conditional, classifier-based, isotropy regularization) can enhance fidelity at the cost of diversity and sometimes induces artifacts or reduces coverage. The choice of guidance parameters (e.g., isotropy regularization $\lambda$ in Iso-Diffusion) requires empirical trade-offs based on the task (Fernando et al., 25 Mar 2024).
• Comparisons with GANs/VAEs: Conditional DDPMs are empirically more robust to mode collapse, can interpolate between data manifolds, and stabilize clustering and latent structure learning better than GAN- or VAE-based models (Yan et al., 2023, Deshpande et al., 2023). Their likelihood-based training promotes better coverage and generalization across mask types, conditions, and target domains.

6. Future Directions and Broader Implications

Research on conditional DDPMs is converging toward several high-impact trajectories:

• Unified Conditioning Frameworks: The integration of forward- and reverse-conditioning schemes, combined with advanced forms of guidance (multi-source, attention-based, classifier-free, and geometric-aware) is likely to enable even tighter control of generative outputs and improved sample quality across arbitrary conditional configurations.
• Accelerated Sampling: Reduced-step and non-Gaussian reverse process constructions, plug-and-play denoisers, and ODE/SDE-based solvers are expected to yield further improvements in computational efficiency without sacrificing diversity or quality.
• Latent Structured Priors: Incorporating learnable priors (as in RestoreGrad) or explicit geometry-/manifold-aware operations (as in SPD-DDPM, HDM) represents a promising direction for domains where structural consistency and domain adaptation are critical.
• Generalization and Reliability: The capacity of conditional DDPMs to generalize beyond fixed training mask distributions, synthesize in between statistically disparate samples, and interpolate smoothly between multiple conditions is likely to drive further adoption in medical imaging, remote sensing, communications, and autonomous systems.
• Theoretical Understanding: Advances in explicit bounds for sample quality under noisy score estimates and time discretization support the development of provably reliable generative algorithms suitable for mission-critical or high-stakes domains.

Conditional DDPMs have established themselves as a central generative paradigm for controlled, high-fidelity, and structurally consistent data synthesis, with ongoing innovations addressing their computational, statistical, and practical challenges. The framework is adaptable, theoretically grounded, and empirically validated across tasks demanding generative, restoration, and conditional transformation capabilities.