Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance

• URL: http://arxiv.org/abs/2212.06359v1
• Date: Tue, 13 Dec 2022 03:48:01 GMT
• Title: Score-based Generative Modeling Secretly Minimizes the Wasserstein Distance
• Authors: Dohyun Kwon, Ying Fan, Kangwook Lee
• Abstract summary: We show that score-based models also minimize the Wasserstein distance between them under suitable assumptions on the model. Our proof is based on a novel application of the theory of optimal transport, which can be of independent interest to the society.
• Score: 14.846377138993642
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Score-based generative models are shown to achieve remarkable empirical performances in various applications such as image generation and audio synthesis. However, a theoretical understanding of score-based diffusion models is still incomplete. Recently, Song et al. showed that the training objective of score-based generative models is equivalent to minimizing the Kullback-Leibler divergence of the generated distribution from the data distribution. In this work, we show that score-based models also minimize the Wasserstein distance between them under suitable assumptions on the model. Specifically, we prove that the Wasserstein distance is upper bounded by the square root of the objective function up to multiplicative constants and a fixed constant offset. Our proof is based on a novel application of the theory of optimal transport, which can be of independent interest to the society. Our numerical experiments support our findings. By analyzing our upper bounds, we provide a few techniques to obtain tighter upper bounds.

Related papers

• Theory on Score-Mismatched Diffusion Models and Zero-Shot Conditional Samplers [49.97755400231656]
  We present the first performance guarantee with explicit dimensional general score-mismatched diffusion samplers. We show that score mismatches result in an distributional bias between the target and sampling distributions, proportional to the accumulated mismatch between the target and training distributions. This result can be directly applied to zero-shot conditional samplers for any conditional model, irrespective of measurement noise.
  arXiv  Detail & Related papers  (2024-10-17T16:42:12Z)
• Unveil Conditional Diffusion Models with Classifier-free Guidance: A Sharp Statistical Theory [87.00653989457834]
  Conditional diffusion models serve as the foundation of modern image synthesis and find extensive application in fields like computational biology and reinforcement learning. Despite the empirical success, theory of conditional diffusion models is largely missing. This paper bridges the gap by presenting a sharp statistical theory of distribution estimation using conditional diffusion models.
  arXiv  Detail & Related papers  (2024-03-18T17:08:24Z)
• Statistically Optimal Generative Modeling with Maximum Deviation from the Empirical Distribution [2.1146241717926664]
  We show that the Wasserstein GAN, constrained to left-invertible push-forward maps, generates distributions that avoid replication and significantly deviate from the empirical distribution. Our most important contribution provides a finite-sample lower bound on the Wasserstein-1 distance between the generative distribution and the empirical one. We also establish a finite-sample upper bound on the distance between the generative distribution and the true data-generating one.
  arXiv  Detail & Related papers  (2023-07-31T06:11:57Z)
• 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)
• 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)
• Convergence of denoising diffusion models under the manifold hypothesis [3.096615629099617]
  Denoising diffusion models are a recent class of generative models exhibiting state-of-the-art performance in image and audio synthesis. This paper provides the first convergence results for diffusion models in a more general setting.
  arXiv  Detail & Related papers  (2022-08-10T12:50:47Z)
• Deep Generative Learning via Schr\"{o}dinger Bridge [14.138796631423954]
  We learn a generative model via entropy with a Schr"odinger Bridge. We show that the generative model via Schr"odinger Bridge is comparable with state-of-the-art GANs.
  arXiv  Detail & Related papers  (2021-06-19T03:35:42Z)
• Goal-directed Generation of Discrete Structures with Conditional Generative Models [85.51463588099556]
  We introduce a novel approach to directly optimize a reinforcement learning objective, maximizing an expected reward. We test our methodology on two tasks: generating molecules with user-defined properties and identifying short python expressions which evaluate to a given target value.
  arXiv  Detail & Related papers  (2020-10-05T20:03:13Z)

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.