Bayesian diffusion models memorize training data when mutual information between restricted observations and training data exceeds log dataset size, and generalize otherwise.
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Convergence of denoising diffusion models under the manifold hypoth- esis.arXiv preprint arXiv:2208.05314
14 Pith papers cite this work. Polarity classification is still indexing.
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abstract
Denoising diffusion models are a recent class of generative models exhibiting state-of-the-art performance in image and audio synthesis. Such models approximate the time-reversal of a forward noising process from a target distribution to a reference density, which is usually Gaussian. Despite their strong empirical results, the theoretical analysis of such models remains limited. In particular, all current approaches crucially assume that the target density admits a density w.r.t. the Lebesgue measure. This does not cover settings where the target distribution is supported on a lower-dimensional manifold or is given by some empirical distribution. In this paper, we bridge this gap by providing the first convergence results for diffusion models in this more general setting. In particular, we provide quantitative bounds on the Wasserstein distance of order one between the target data distribution and the generative distribution of the diffusion model.
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JET is a conditional flow matching framework that generates EEG as continuous raw sequences with added constraints for spectral and temporal properties, achieving over 40% lower TS-FID than prior discrete denoising methods on three benchmarks.
MM-SOLD is a training-free particle sampler whose large-particle limit converges to a moment-matched Gibbs distribution obtained by exponentially tilting a score-smoothed target.
PGM framework links diffusion to proximal regularization for closed-form Moreau-score sampling in Bayesian inverse problems, learned only from prior samples.
The choice of closeness measure in diffusion reward alignment determines the computational primitives and tractable reward classes, with linear exponential tilts sufficing for KL with convex rewards and proximal oracles for Wasserstein with concave or low-dimensional Lipschitz rewards.
First-order asymptotic expansions of weak and Fréchet discretization errors in diffusion sampling are derived, explicit under Gaussian data through covariance geometry and robust to other data geometries.
Defines diffusion processes on implicit data manifolds via proximity-graph approximations to the infinitesimal generator and carré-du-champ operator, proves convergence in law to the continuous manifold process, and provides an Euler-Maruyama integrator validated on synthetic and MNIST manifolds.
A variance-aware anisotropic OU drift keeps multimodal and correlated structure longer in diffusion, and bounds reverse initialization error by local cluster variance rather than global variance.
Recasting diffusion noise schedule design as optimal control on Fisher information yields sufficient conditions for O(d/n) sampling error and parametric closed-form schedules that generalize exponential/sigmoid ones and improve empirical performance.
Diffusion models on manifold-supported data admit score decompositions whose statistical rates are controlled by intrinsic dimension and curvature.
The paper proves Hölder continuity of optimal transport maps for PDE-induced measures via doubling conditions and derives excess-risk bounds for one-step generative models like DeepParticle.
Reusing source latent spaces in diffusion models under distribution shift produces target score error set by principal-angle misalignment and diffusion-time-amplified ambient noise.
Review of neural scaling laws and their relation to constraints and inductive biases when applying machine learning to physics problems.
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Geometry-Aware Discretization Error of Diffusion Models
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Recasting diffusion noise schedule design as optimal control on Fisher information yields sufficient conditions for O(d/n) sampling error and parametric closed-form schedules that generalize exponential/sigmoid ones and improve empirical performance.
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Diffusion Model for Manifold Data: Score Decomposition, Curvature, and Statistical Complexity
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