Diffusion Models Learn Low-Dimensional Distributions via Subspace Clustering

• URL: http://arxiv.org/abs/2409.02426v1
• Date: Wed, 4 Sep 2024 04:14:02 GMT
• Title: Diffusion Models Learn Low-Dimensional Distributions via Subspace Clustering
• Authors: Peng Wang, Huijie Zhang, Zekai Zhang, Siyi Chen, Yi Ma, Qing Qu,
• Abstract summary: diffusion models can effectively learn the image distribution and generate new samples. We provide theoretical insights into this phenomenon by leveraging key empirical observations. We show that the minimal number of samples required to learn the underlying distribution scales linearly with the intrinsic dimensions.
• Score: 15.326641037243006
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Recent empirical studies have demonstrated that diffusion models can effectively learn the image distribution and generate new samples. Remarkably, these models can achieve this even with a small number of training samples despite a large image dimension, circumventing the curse of dimensionality. In this work, we provide theoretical insights into this phenomenon by leveraging key empirical observations: (i) the low intrinsic dimensionality of image data, (ii) a union of manifold structure of image data, and (iii) the low-rank property of the denoising autoencoder in trained diffusion models. These observations motivate us to assume the underlying data distribution of image data as a mixture of low-rank Gaussians and to parameterize the denoising autoencoder as a low-rank model according to the score function of the assumed distribution. With these setups, we rigorously show that optimizing the training loss of diffusion models is equivalent to solving the canonical subspace clustering problem over the training samples. Based on this equivalence, we further show that the minimal number of samples required to learn the underlying distribution scales linearly with the intrinsic dimensions under the above data and model assumptions. This insight sheds light on why diffusion models can break the curse of dimensionality and exhibit the phase transition in learning distributions. Moreover, we empirically establish a correspondence between the subspaces and the semantic representations of image data, facilitating image editing. We validate these results with corroborated experimental results on both simulated distributions and image datasets.

Related papers

• Constrained Diffusion Models via Dual Training [80.03953599062365]
  We develop constrained diffusion models based on desired distributions informed by requirements. 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)
• Integrating Amortized Inference with Diffusion Models for Learning Clean Distribution from Corrupted Images [19.957503854446735]
  Diffusion models (DMs) have emerged as powerful generative models for solving inverse problems. FlowDiff is a joint training paradigm that leverages a conditional normalizing flow model to facilitate the training of diffusion models on corrupted data sources. Our experiment shows that FlowDiff can effectively learn clean distributions across a wide range of corrupted data sources.
  arXiv  Detail & Related papers  (2024-07-15T18:33:20Z)
• Training Class-Imbalanced Diffusion Model Via Overlap Optimization [55.96820607533968]
  Diffusion models trained on real-world datasets often yield inferior fidelity for tail classes. Deep generative models, including diffusion models, are biased towards classes with abundant training images. We propose a method based on contrastive learning to minimize the overlap between distributions of synthetic images for different classes.
  arXiv  Detail & Related papers  (2024-02-16T16:47:21Z)
• The Journey, Not the Destination: How Data Guides Diffusion Models [75.19694584942623]
  Diffusion models trained on large datasets can synthesize photo-realistic images of remarkable quality and diversity. We propose a framework that: (i) provides a formal notion of data attribution in the context of diffusion models, and (ii) allows us to counterfactually validate such attributions.
  arXiv  Detail & Related papers  (2023-12-11T08:39:43Z)
• Steerable Conditional Diffusion for Out-of-Distribution Adaptation in Medical Image Reconstruction [75.91471250967703]
  We introduce a novel sampling framework called Steerable Conditional Diffusion. This framework adapts the diffusion model, concurrently with image reconstruction, based solely on the information provided by the available measurement. We achieve substantial enhancements in out-of-distribution performance across diverse imaging modalities.
  arXiv  Detail & Related papers  (2023-08-28T08:47:06Z)
• Reflected Diffusion Models [93.26107023470979]
  We present Reflected Diffusion Models, which reverse a reflected differential equation evolving on the support of the data. Our approach learns the score function through a generalized score matching loss and extends key components of standard diffusion models.
  arXiv  Detail & Related papers  (2023-04-10T17:54:38Z)
• 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)
• Fast Inference in Denoising Diffusion Models via MMD Finetuning [23.779985842891705]
  We present MMD-DDM, a novel method for fast sampling of diffusion models. Our approach is based on the idea of using the Maximum Mean Discrepancy (MMD) to finetune the learned distribution with a given budget of timesteps. Our findings show that the proposed method is able to produce high-quality samples in a fraction of the time required by widely-used diffusion models.
  arXiv  Detail & Related papers  (2023-01-19T09:48:07Z)
• From Denoising Diffusions to Denoising Markov Models [38.33676858989955]
  Denoising diffusions are state-of-the-art generative models exhibiting remarkable empirical performance. We propose a unifying framework generalising this approach to a wide class of spaces and leading to an original extension of score matching.
  arXiv  Detail & Related papers  (2022-11-07T14:34:27Z)

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.