Conditional Diffusion Models

Updated 6 July 2025

Conditional diffusion models are generative methods that condition on auxiliary data to reverse a noising process and produce structured outputs.
They incorporate techniques like concatenation, label embedding, and cross-attention to integrate information such as class labels and textual prompts.
These models are applied in diverse areas including image synthesis, lossy compression, super-resolution, and recommendation for enhanced control and fidelity.

Conditional diffusion models are a class of generative models that produce data samples conditioned on auxiliary information, commonly referred to as “conditions” or “controls.” These models extend the foundational framework of score-based or denoising diffusion probabilistic models, which generate samples by learning to reverse a gradual noising process. By incorporating conditioning variables into training and sampling, conditional diffusion models can generate highly structured outputs—such as images, sequences, or other data—with features directly guided by side information like class labels, continuous attributes, input prompts, or other context.

1. Mathematical Principles and Model Formulation

In conditional diffusion models, the goal is to approximate the conditional distribution $p(x|y)$ , where $x$ denotes the data to generate and $y$ is the conditioning variable. The standard approach involves defining a forward diffusion process that adds noise to $x$ and a reverse process that incrementally denoises, typically parameterized by a neural network and trained with score matching objectives.

The conditional score function at noise level $t$ is defined as:

$s_\theta(x, t, y) \approx \nabla_x \log p_t(x | y)$

where $p_t(x | y)$ is the distribution of $x$ after adding noise up to time $t$ , conditioned on $y$ . Training aims to minimize a denoising score matching loss that ties $s_\theta$ to the true conditional score.

Sampling proceeds by simulating the reverse SDE or denoising Markov chain, where at each step the score network is queried with the current noisy sample and conditioning information. For much of the literature, conditioning is injected into the neural network via concatenation, label embedding, cross-attention, or other architectural mechanisms tailored to the data and the nature of $y$ .

Several methodological variants exist:

Classifier-based and classifier-free guidance for conditional generation, which interpolate between unconditional and conditionally-guided scores or drifts, enabling control over sample-fidelity to the condition.
Conditional forward processes as in ShiftDDPMs (2302.02373), dispersing the effect of the condition across all diffusion timesteps, not merely the reverse process.
Twisted Diffusion Samplers (TDS) (2306.17775), which use sequential Monte Carlo methods to produce asymptotically exact conditional samples via importance-weighted trajectories.

2. Conditioning Strategies and Training Objectives

Approaches to incorporate and enable conditioning in diffusion models vary based on the side information type and task requirements:

Label or Attribute Embedding: For categorical/y continuous scalars (e.g., class labels, pose angles), embedding networks map $y$ to a feature vector that is concatenated/added to network activations or time-step embeddings (2405.03546).
Concatenation and Cross-Attention: For image or sequence conditionings (such as masked, control images, or tokens), models concatenate the condition or use cross-attention mechanisms to enable the score network to access the relevant context (2408.08526, 2410.21967).
Vicinal and Hard-Vicinal Losses for Continuous Conditioning: To deal with data sparsity along continuous conditioning dimensions, loss formulations (e.g., hard vicinal image denoising loss (2405.03546)) consider “vicinity” of labels, aggregating across similar conditions during training for improved sample efficiency and label consistency.
Diffused Taylor Approximations and Polynomial Network Construction: For distributional and statistical theory, conditional score functions can be uniformly approximated via diffused Taylor expansions and implemented using ReLU neural networks, yielding nearly minimax-optimal sample complexity bounds (2403.11968).
Reinforcement Learning for Post-hoc Conditioning: Fine-tuning pre-trained diffusion models to respond to new controls by defining a reward (using a classifier on generated samples) and shaping the drift term by maximizing expected reward subject to a KL-penalty versus the base model (2406.12120).

3. Model Variants and Specialized Architectures

Conditional diffusion models have been systematically extended into multiple directions to meet application-specific challenges:

Continuous Conditioning: CCDM (2405.03546) demonstrates architectures and training objectives specifically suited for generating images conditioned on scalar continuous values, integrating label embeddings at multiple network layers and classifier-free guidance adapted for regression labels.
Dual Conditioning for Sequences: In sequential recommendation (2410.21967), models integrate both implicit (global behavior context) and explicit (stepwise actions) conditions using dual conditional diffusion mechanisms. Cross-attention transformers (DCDT) are used to explicitly inject and integrate history throughout the denoising process.
Autoregressive Diffusion for Capturing Dependencies: AR-diffusion models (2504.21314) generate data patches sequentially, addressing the failure of standard diffusion models to accurately reproduce complex conditional dependence structures. The theoretical framework shows that AR diffusion closes the KL gap for conditional distributions more rapidly than joint-only diffusion.
Cascaded Conditional Diffusion: For multi-resolution design (2408.08526), cascaded models independently train low- and high-resolution conditional diffusion modules, improving design quality by propagating conditional information between stages and supporting flexible upsampling tasks.
Progressive Multi-stage Pipelines: Three-stage structures such as Progressive Conditional Diffusion Models (PCDMs) (2310.06313) first infer global features, then perform coarse inpainting, and finally refine textures---all using conditional diffusion models at their respective stages.

4. Theoretical Properties and Statistical Efficiency

Recent advances establish strong guarantees for conditional diffusion models:

Minimax-Optimality: Under smoothness assumptions, conditional diffusion models achieve minimax-optimal convergence rates in total variation and Wasserstein metrics for conditional distribution estimation (2409.20124).
Manifold Adaptivity: Models adapt to the intrinsic (rather than ambient) dimension of both the data and conditioning variables, thereby maintaining statistical efficiency even when the ambient space is high-dimensional but the true data lie on low-dimensional manifolds (2409.20124).
Sample-Efficient Transfer Learning: When the dependency between $x$ and $y$ factors through a low-dimensional learned representation, the sample complexity of transferring and fine-tuning a CDM to a new target task can be reduced, with explicit error bounds scaling in the intrinsic dimension of the representation, not the raw conditioning dimension (2502.04491).
Conditional Independence Testing: When used to model $P(X|Z)$ in conditional randomization tests (CRT), conditional diffusion models provide accurate, stable approximations for the pseudo-sampling step, outperforming GAN baselines both theoretically (in total variation) and empirically (2412.11744).

5. Practical Applications and Experimental Evidence

Conditional diffusion models have demonstrated utility across a variety of application domains, supported by extensive experimental benchmarks:

Image Synthesis and Editing: Guided by categorical, textual, or continuous attributes, these models enable high-fidelity controlled generation, inpainting, and translation tasks.
Lossy Compression: Compression approaches that use a latent code to guide a conditional diffusion decoder yield superior perceptual metrics (e.g., FID) and support explicit tradeoffs between rate, distortion, and perceptual quality (2209.06950).
Super-Resolution and Restoration: Conditional diffusion enables fast, robust upsampling from noisy or partial observations (e.g., LiDAR upsampling (2405.04889)) and image super-resolution (2307.00781) with accelerated inference via deterministic or higher-order samplers.
Sequential Recommendation: Dual conditional diffusion models significantly improve the accuracy and computational efficiency of session-based recommendation systems, integrating user-item context and explicit histories (2410.21967).
Scientific and Engineering Modeling: In numerical weather prediction, PDE forecasting, and data assimilation, conditional diffusion models—especially when combined with autoregressive sampling and hybrid conditioning—outperform traditional data-driven surrogates (2410.16415).
Protein Design and Motif Scaffolding: In computational biology, the Twisted Diffusion Sampler (TDS) framework (2306.17775) allows for nearly exact conditional sampling under complex spatial constraints, outperforming previous state-of-the-art approaches.
Conditional Independence Testing: CDMs deliver statistically valid and computationally efficient conditional randomization tests for structured and high-dimensional settings (2412.11744).

6. Open-source Ecosystem and Available Software

Several conditional diffusion modeling frameworks and libraries are available for research and application development:

MSDiff library: Open-sourced for implementing and experimenting with diffusion models, including multi-speed and non-uniform diffusion variants (per (2207.09786); no further detail in the data).
CCDM Implementation: An end-to-end pipeline for continuous conditional image generation is released at https://github.com/UBCDingXin/CCDM (2405.03546).
CDM Compression and Super-Resolution Code: Implementations supporting image compression (2209.06950) and accelerated single image super-resolution (2307.00781) are distributed to streamline reproducibility.
CTRL (Conditional Control RL): Code for reinforcement learning based conditional control in diffusion models is made available at https://github.com/zhaoyl18/CTRL (2406.12120).
PCDMs for Pose-Guided Synthesis: Released models and code at https://github.com/tencent-ailab/PCDMs (2310.06313).

7. Limitations, Trade-offs, and Future Directions

Although conditional diffusion models achieve state-of-the-art results in numerous tasks, their deployment and scalability can be challenged by:

Longer inference time compared to GANs and VAEs (alleviated by accelerated samplers, distilled one-step GAN-like models (2405.05967), and AR variants).
Data sparsity for continuous conditions, addressed by specialized training losses (vicinal losses (2405.03546)) and sample-efficient transfer learning (2502.04491).
The need for large, annotated, or multi-modal datasets in downstream or composed conditional tasks, for which reinforcement learning frameworks (CTRL (2406.12120)) and shared low-dimensional representations offer mitigating strategies.

Best practices for practical implementation include:

Carefully selecting the architectural mechanism for conditioning injection (based on the nature of $y$ and the desired controllability).
Employing manifold-adaptive model selection and regularization for data with latent low-dimensional structure.
Using accelerated or autoregressive sampling where capturing conditional dependencies or achieving real-time performance is critical.
Leveraging transfer learning where condition spaces are high-dimensional but share intrinsic structure.

Ongoing research seeks to bridge gaps in inference efficiency, extend theoretical guarantees to broader families of diffusion models, refine approaches for compositional and hierarchical conditioning, and broaden the applicability to data domains beyond vision (e.g., audio, graph data, scientific simulation).