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This theoretical study bridges generative modeling and Bayesian inference by demonstrating how diffusion models adapt to the intrinsic dimensionality of high-dimensional clustered data. The authors prove that denoising dynamics effectively concentrate on specific clusters once a critical signal-to-noise ratio is reached, yielding error bounds that scale with intrinsic rather than ambient dimension. These findings advance our understanding of the geometric limits and efficiency of diffusion processes in complex, multimodal distributions.
Diffusion Models for High-Dimensional Clustered Data: Intrinsic-Dimension Adaptivity via Bayesian Classification
arXiv:2608.19067v1 Announce Type: cross Abstract: The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $\Theta (\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.
Abstract:The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $\Theta (\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.
Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
arXiv:2608.19067 [stat.ML]
(or arXiv:2608.19067v1 [stat.ML] for this version)
https://doi.org/10.48550/arXiv.2608.19067 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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