train diffusion models

Existing surveys on diffusion models predominantly focus on high-level architectures, neglecting the design principles underlying fundamental components—namely, the forward process, reverse process, and sampling procedure. To address this gap, we present the first fine-grained, systematic analysis of the key designable elements across these three core components, including noise scheduling, network architectures, loss functions, and sampling strategies. We propose a unified taxonomy that enables abstract consolidation and cross-work comparative analysis. Furthermore, we establish the first comprehensive survey framework explicitly centered on “design fundamentals,” thereby bridging the critical void in low-level principle coverage. This work provides structured theoretical knowledge and practical guidance for component-level analysis, task-driven customization, and efficient implementation of diffusion models. (128 words)

This work unifies diffusion sampling and stochastic localization under a single theoretical framework, addressing the lack of rigorous theoretical connections between them and the limited applicability of existing algorithms. Methodologically, we establish the first formal equivalence between diffusion processes and stochastic localization via stochastic process analysis and statistical mechanical modeling; we introduce a generalized stochastic localization framework wherein standard denoising diffusion is shown to be a specific instance, and extend it to broader distribution families by parameterizing the drift term with neural networks. Key contributions include: (1) a theoretical proof that multiple classes of diffusion samplers—including DDPM, DDIM, and score-based SDEs—are instantiations of stochastic localization; (2) derivation of novel, computationally efficient sampling algorithms grounded in this equivalence; and (3) a new analytical perspective on mixing properties and convergence rates via Poincaré inequality characterization, substantially deepening the understanding of the dynamical mechanisms underlying generative models.

The field of diffusion models for visual generation lacks systematic, pedagogically structured educational resources. Method: This project develops a teaching-oriented unified framework targeting undergraduate and graduate students, systematically integrating foundational probabilistic modeling—包括 forward diffusion, reverse denoising, stochastic differential equation (SDE) solvers, and score matching—with state-of-the-art conditional image and video generation. The framework emphasizes structured exposition of modeling principles, training paradigms, and sampling mechanisms to establish a clear, reproducible conceptual foundation. Contribution/Results: It significantly lowers the entry barrier for learners and fills a critical gap in introductory, comprehensive tutorials on diffusion models. The framework has become a widely adopted pedagogical benchmark and cross-disciplinary reference for both diffusion model instruction and applied research.

This work addresses the theoretical complexity and inconsistent interpretations of diffusion models by proposing a concise, self-contained unifying framework grounded in signal processing. Methodologically, it abandons conventional Markov chain and reverse stochastic differential equation (SDE) formulations, instead modeling generation as a stochastic walk coupled with the Tweedie formula—thereby decoupling score estimation, noise scheduling, and sampling, and enabling likelihood-free conditional generation. Key contributions include: (1) the first self-contained theoretical interpretation independent of reverse SDEs or probability flows; (2) full decoupling of noise scheduling between training and sampling, enhancing flexibility in conditional synthesis; and (3) faithful reproduction and unification of major models—including DDPM, DDIM, and Score SDE—while maintaining state-of-the-art performance on image generation and inverse problems, significantly improving both theoretical parsimony and practical interpretability.

This work addresses the problem of efficiently and accurately bridging arbitrary probability density functions within a bounded time horizon. Methodologically, it introduces a unified generative modeling paradigm based on stochastic interpolation processes, seamlessly integrating flow-based and diffusion-based models—supporting both deterministic ordinary differential equation (ODE) paths and stochastic differential equation (SDE) paths with tunable noise. A novel score-matching objective is derived for the first time; theoretical analysis proves that optimizing only a quadratic loss suffices for likelihood control, overcoming the traditional limitation of deterministic models requiring additional Fisher divergence regularization. By unifying Schrödinger bridge theory, the Fokker–Planck equation, and variational inference, the framework rigorously recovers the Schrödinger bridge solution under optimal interpolation and provides a unified estimator for both likelihood and cross-entropy.

Conditional image synthesis with diffusion models suffers from a lack of systematic understanding due to architectural complexity, task heterogeneity, and diverse conditional mechanisms. Method: This paper introduces the first unified taxonomy for conditional diffusion modeling, categorizing approaches by *where* conditioning is injected—either into the denoising network architecture or the sampling process—and formalizes three paradigmatic stages: training, reuse, and specialization. It further classifies six mainstream sampling-time conditioning strategies. Contributions: Based on a structured analysis of over 100 works, the paper establishes a comprehensive knowledge framework and open-sources an authoritative resource repository (GitHub Awesome-Conditional-Diffusion-Models). It identifies persistent bottlenecks—including limited generalization, inefficient inference, and coarse-grained control—and proposes principled directions toward scalable, modular, and fine-grained conditional modeling.

This work addresses the limitation of conventional Monte Carlo methods for simulating stochastic differential equations (SDEs), which rely on explicit parametric modeling of drift and diffusion coefficients. We propose a model-free, data-driven framework for SDE path generation. To our knowledge, this is the first application of conditional diffusion models to SDE simulation—enabling learning of latent dynamics directly from limited observed trajectories without prior parameter specification. Our approach integrates sequential conditional modeling with generative adversarial training, substantially improving path fidelity and temporal coherence. Experiments demonstrate superior performance over baselines—including neural SDEs—in preserving pathwise statistical properties, recovering dynamic structure, and generalizing to unseen regimes. Furthermore, when deployed in reinforcement learning–driven continuous-time portfolio optimization, our method yields significant improvements in decision-making performance. This work establishes a novel paradigm for black-box stochastic process modeling in finance and related domains.

Existing theoretical analyses of diffusion generative models are fragmented and suffer from inconsistent notation, hindering unified understanding and principled development. Method: This work establishes a rigorous, unified mathematical framework grounded in fundamental properties of Gaussian distributions. It systematically derives the closed-form marginal distribution of the forward noising process, the analytical form of the reverse posterior, and the variational lower bound, ultimately yielding an optimization objective equivalent to noise prediction. Contribution/Results: The framework reveals the intrinsic equivalence between DDIM and rectified flow; provides a unified probabilistic interpretation of classifier-guided and classifier-free guidance; and integrates SDE/ODE formulations, the Fokker–Planck equation, flow matching, and multi-scale modeling—ensuring both theoretical coherence and practical implementability. Validated on mainstream models including Stable Diffusion, the framework enables efficient sampling and precise modeling while unifying disparate theoretical perspectives.

Diffusion models aim to construct invertible generative paths from a noise prior to the data distribution. This paper proposes a unified tripartite framework—integrating variational inference, score-based modeling, and flow matching—to reveal their shared underlying principle: continuous generative trajectories governed by time-dependent velocity fields. By formulating both the forward noising and reverse denoising processes as ordinary differential equations (ODEs), we establish a rigorous, computationally tractable continuous-time generative theory. The framework enables direct pointwise mapping at arbitrary times, flexible conditional generation with explicit control, and seamless integration with energy-based models and time-dependent neural architectures. Experimental and theoretical analyses demonstrate that this unification substantially improves sampling efficiency, controllability, and model interpretability. Our work provides a foundational theoretical framework for deepening the understanding of diffusion models and guiding the design of novel architectures.

This work addresses the limitation of conventional diffusion models, which rely on independent noise injection and struggle to capture complex distributions with spatial correlations. The authors propose a novel diffusion mechanism that replaces independent sampling in both forward and reverse processes with Markov chain Monte Carlo (MCMC) dynamics informed by known interaction structures. Notably, they explicitly embed Ising couplings into the diffusion process for the first time. This approach naturally aligns with probabilistic-bit (p-bit)-based p-computer hardware architectures, thereby expanding the design space for stochastic kernels in diffusion models. Experiments on the 2D ferromagnetic Ising model and the 3D Edwards–Anderson spin glass demonstrate that the proposed method generates samples closer to the ground-truth MCMC reference distribution while achieving higher sampling throughput and energy efficiency.

This paper investigates the theoretical behavior of generative models under finite training samples. For both deterministic and stochastic generation processes, it derives closed-form solutions for the velocity field and score function within a stochastic interpolation framework—revealing that the former exactly recovers training samples, while the latter corresponds to adding Gaussian noise to them. It introduces the first formal definitions of underfitting and overfitting for generative models, proving that, in the presence of model estimation error, stochastic generation amounts to convex combinations of training samples corrupted by a mixture of noise sources. These theoretical findings are empirically validated on downstream classification tasks, confirming that the characterized noise structure aligns with observed generalization performance. The core contribution is an analytical theory of generative processes under finite-sample regimes, unifying the explanatory frameworks for sample recovery and perturbation, and providing verifiable criteria for diagnosing underfitting and overfitting.