Generative Learning of Densities on Manifolds

• URL: http://arxiv.org/abs/2503.03963v2
• Date: Sat, 19 Apr 2025 04:09:16 GMT
• Title: Generative Learning of Densities on Manifolds
• Authors: Dimitris G. Giovanis, Ellis Crabtree, Roger G. Ghanem, Ioannis G. Kevrekidis,
• Abstract summary: A generative modeling framework is proposed that combines diffusion models and manifold learning.<n>The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space.
• Score: 3.081704060720176
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent) spaces in the high-dimensional data (ambient) space. Two approaches for sampling from the latent data density are described. The first is a score-based diffusion model, which is trained to map a standard normal distribution to the latent data distribution using a neural network. The second one involves solving an It\^o stochastic differential equation in the latent space. Additional realizations of the data are generated by lifting the samples back to the ambient space using Double Diffusion Maps, a recently introduced technique typically employed in studying dynamical system reduction; here the focus lies in sampling densities rather than system dynamics. The proposed approaches enable sampling high dimensional data densities restricted to low-dimensional, a priori unknown manifolds. The efficacy of the proposed framework is demonstrated through a benchmark problem and a material with multiscale structure.

Related papers

• Constrained Diffusion Models via Dual Training [80.03953599062365]
  Diffusion processes are prone to generating samples that reflect biases in a training dataset. We develop constrained diffusion models by imposing diffusion constraints based on desired distributions. We show that our constrained diffusion models generate new data from a mixture data distribution that achieves the optimal trade-off among objective and constraints.
  arXiv  Detail & Related papers  (2024-08-27T14:25:42Z)
• Latent diffusion models for parameterization and data assimilation of facies-based geomodels [0.0]
  Diffusion models are trained to generate new geological realizations from input fields characterized by random noise. Latent diffusion models are shown to provide realizations that are visually consistent with samples from geomodeling software.
  arXiv  Detail & Related papers  (2024-06-21T01:32:03Z)
• 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)
• A Geometric Perspective on Diffusion Models [57.27857591493788]
  We inspect the ODE-based sampling of a popular variance-exploding SDE. We establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm.
  arXiv  Detail & Related papers  (2023-05-31T15:33:16Z)
• Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems [64.29491112653905]
  We propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG with the denoised data ensures the data consistency update to remain in the tangent space. Our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.
  arXiv  Detail & Related papers  (2023-03-10T07:42:49Z)
• Infinite-Dimensional Diffusion Models [4.342241136871849]
  We formulate diffusion-based generative models in infinite dimensions and apply them to the generative modeling of functions. We show that our formulations are well posed in the infinite-dimensional setting and provide dimension-independent distance bounds from the sample to the target measure. We also develop guidelines for the design of infinite-dimensional diffusion models.
  arXiv  Detail & Related papers  (2023-02-20T18:00:38Z)
• 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)
• Your diffusion model secretly knows the dimension of the data manifold [0.0]
  A diffusion model approximates the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that, if the data concentrates around a manifold embedded in the high-dimensional ambient space, then as the level of corruption decreases, the score function points towards the manifold.
  arXiv  Detail & Related papers  (2022-12-23T23:15:14Z)
• Unifying Diffusion Models' Latent Space, with Applications to CycleDiffusion and Guidance [95.12230117950232]
  We show that a common latent space emerges from two diffusion models trained independently on related domains. Applying CycleDiffusion to text-to-image diffusion models, we show that large-scale text-to-image diffusion models can be used as zero-shot image-to-image editors.
  arXiv  Detail & Related papers  (2022-10-11T15:53:52Z)
• The Manifold Scattering Transform for High-Dimensional Point Cloud Data [16.500568323161563]
  We present practical schemes for implementing the manifold scattering transform to datasets arising in naturalistic systems. We show that our methods are effective for signal classification and manifold classification tasks.
  arXiv  Detail & Related papers  (2022-06-21T02:15:00Z)

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.