Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion

• URL: http://arxiv.org/abs/2410.05898v3
• Date: Wed, 16 Oct 2024 09:10:54 GMT
• Title: Manifolds, Random Matrices and Spectral Gaps: The geometric phases of generative diffusion
• Authors: Enrico Ventura, Beatrice Achilli, Gianluigi Silvestri, Carlo Lucibello, Luca Ambrogioni,
• Abstract summary: We analyze the spectrum of eigenvalues of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Our analysis reveals the existence of three distinct qualitative phases during the generative process.
• Score: 8.389423957434818
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we investigate the latent geometry of generative diffusion models under the manifold hypothesis. To this purpose, we analyze the spectrum of eigenvalues (and singular values) of the Jacobian of the score function, whose discontinuities (gaps) reveal the presence and dimensionality of distinct sub-manifolds. Using a statistical physics approach, we derive the spectral distributions and formulas for the spectral gaps under several distributional assumptions and we compare these theoretical predictions with the spectra estimated from trained networks. Our analysis reveals the existence of three distinct qualitative phases during the generative process: a trivial phase; a manifold coverage phase where the diffusion process fits the distribution internal to the manifold; a consolidation phase where the score becomes orthogonal to the manifold and all particles are projected on the support of the data. This `division of labor' between different timescales provides an elegant explanation on why generative diffusion models are not affected by the manifold overfitting phenomenon that plagues likelihood-based models, since the internal distribution and the manifold geometry are produced at different time points during generation.

Related papers

• Convergence of Score-Based Discrete Diffusion Models: A Discrete-Time Analysis [56.442307356162864]
  We study the theoretical aspects of score-based discrete diffusion models under the Continuous Time Markov Chain (CTMC) framework. We introduce a discrete-time sampling algorithm in the general state space $[S]d$ that utilizes score estimators at predefined time points. Our convergence analysis employs a Girsanov-based method and establishes key properties of the discrete score function.
  arXiv  Detail & Related papers  (2024-10-03T09:07:13Z)
• Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models [59.331993845831946]
  Diffusion models benefit from instillation of task-specific information into the score function to steer the sample generation towards desired properties. This paper provides the first theoretical study towards understanding the influence of guidance on diffusion models in the context of Gaussian mixture models.
  arXiv  Detail & Related papers  (2024-03-03T23:15:48Z)
• Dynamical Regimes of Diffusion Models [14.797301819675454]
  We study generative diffusion models in the regime where the dimension of space and the number of data are large. Our analysis reveals three distinct dynamical regimes during the backward generative diffusion process. The dependence of the collapse time on the dimension and number of data provides a thorough characterization of the curse of dimensionality for diffusion models.
  arXiv  Detail & Related papers  (2024-02-28T17:19:26Z)
• Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformization [17.535229185525353]
  We introduce an algorithm leveraging the uniformization of continuous Markov chains, implementing transitions on random time points. Our results align with state-of-the-art achievements for diffusion models in $mathbbRd$ and further underscore the advantages of discrete diffusion models in comparison to the $mathbbRd$ setting.
  arXiv  Detail & Related papers  (2024-02-12T22:26:52Z)
• Time Series Diffusion in the Frequency Domain [54.60573052311487]
  We analyze whether representing time series in the frequency domain is a useful inductive bias for score-based diffusion models. We show that a dual diffusion process occurs in the frequency domain with an important nuance. We show how to adapt the denoising score matching approach to implement diffusion models in the frequency domain.
  arXiv  Detail & Related papers  (2024-02-08T18:59:05Z)
• Diffusion Models are Minimax Optimal Distribution Estimators [49.47503258639454]
  We provide the first rigorous analysis on approximation and generalization abilities of diffusion modeling. We show that when the true density function belongs to the Besov space and the empirical score matching loss is properly minimized, the generated data distribution achieves the nearly minimax optimal estimation rates.
  arXiv  Detail & Related papers  (2023-03-03T11:31:55Z)
• Score Approximation, Estimation and Distribution Recovery of Diffusion Models on Low-Dimensional Data [68.62134204367668]
  This paper studies score approximation, estimation, and distribution recovery of diffusion models, when data are supported on an unknown low-dimensional linear subspace. We show that with a properly chosen neural network architecture, the score function can be both accurately approximated and efficiently estimated. The generated distribution based on the estimated score function captures the data geometric structures and converges to a close vicinity of the data distribution.
  arXiv  Detail & Related papers  (2023-02-14T17:02:35Z)
• Mathematical analysis of singularities in the diffusion model under the submanifold assumption [0.0]
  We show that the analytical mean drift function in DDPM and the score function in SGMally blow up in the final stages of the sampling process for singular data distributions. We derive a new target function and associated loss, which remains bounded even for singular data distributions.
  arXiv  Detail & Related papers  (2023-01-19T05:13:03Z)
• Time-inhomogeneous diffusion geometry and topology [69.55228523791897]
  Diffusion condensation is a time-inhomogeneous process where each step first computes and then applies a diffusion operator to the data. We theoretically analyze the convergence and evolution of this process from geometric, spectral, and topological perspectives. Our work gives theoretical insights into the convergence of diffusion condensation, and shows that it provides a link between topological and geometric data analysis.
  arXiv  Detail & Related papers  (2022-03-28T16:06:17Z)
• AI Giving Back to Statistics? Discovery of the Coordinate System of Univariate Distributions by Beta Variational Autoencoder [0.0]
  The article discusses experiences of training neural networks to classify univariate empirical distributions and to represent them on the two-dimensional latent space forcing disentanglement based on the inputs of cumulative distribution functions (CDF) The representation on the latent two-dimensional coordinate system can be seen as an additional metadata of the real-world data that disentangles important distribution characteristics, such as shape of the CDF, classification probabilities of underlying theoretical distributions and their parameters, information entropy, and skewness.
  arXiv  Detail & Related papers  (2020-04-06T14:11:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.