{
  "id": "2305.19947",
  "version": "v1",
  "published": "2023-05-31T15:33:16.000Z",
  "updated": "2023-05-31T15:33:16.000Z",
  "title": "A Geometric Perspective on Diffusion Models",
  "authors": [
    "Defang Chen",
    "Zhenyu Zhou",
    "Jian-Ping Mei",
    "Chunhua Shen",
    "Chun Chen",
    "Can Wang"
  ],
  "categories": [
    "cs.CV",
    "cs.LG",
    "stat.ML"
  ],
  "abstract": "Recent years have witnessed significant progress in developing efficient training and fast sampling approaches for diffusion models. A recent remarkable advancement is the use of stochastic differential equations (SDEs) to describe data perturbation and generative modeling in a unified mathematical framework. In this paper, we reveal several intriguing geometric structures of diffusion models and contribute a simple yet powerful interpretation to their sampling dynamics. Through carefully inspecting a popular variance-exploding SDE and its marginal-preserving ordinary differential equation (ODE) for sampling, we discover that the data distribution and the noise distribution are smoothly connected with an explicit, quasi-linear sampling trajectory, and another implicit denoising trajectory, which even converges faster in terms of visual quality. We also establish a theoretical relationship between the optimal ODE-based sampling and the classic mean-shift (mode-seeking) algorithm, with which we can characterize the asymptotic behavior of diffusion models and identify the score deviation. These new geometric observations enable us to improve previous sampling algorithms, re-examine latent interpolation, as well as re-explain the working principles of distillation-based fast sampling techniques.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-31T15:33:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "diffusion models",
      "geometric perspective",
      "re-examine latent interpolation",
      "stochastic differential equations",
      "marginal-preserving ordinary differential equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}