拓展非高斯扩散模型，提升图像与金融数据生成效果
Tweedie's Formulae and Diffusion Generative Models Beyond Gaussian
- 将Tweedie公式推广至几何布朗运动等非高斯过程
- 基于新公式构建的模型在图像与金融时间序列生成上表现更优
- 适用于需要非高斯建模的场景，如金融数据分析
扩散模型在从未知数据分布中生成样本方面取得了显著成功。主流基于随机微分方程的扩散模型通过添加高斯噪声将目标分布转化为简单先验，并利用去噪得分匹配（得分函数学习）生成干净样本，该方法源于Tweedie公式。然而，具有状态依赖扩散系数的非高斯扩散模型及其对应的Tweedie公式尚未被充分探索。本文将Tweedie公式扩展到重要的非高斯过程，包括几何布朗运动（GBM）、平方贝塞尔（BESQ）过程和柯克斯-因格索尔-罗斯（CIR）过程，从而推导出相应的去噪得分匹配目标。我们进一步将所得公式应用于基于GBM和CIR的图像与金融时间序列生成，并在BESQ设定下进行经验贝叶斯估计。实验结果表明，非高斯模型具备良好潜力。
原文摘要 · Abstract (English)
Diffusion models have achieved remarkable success in generating samples from unknown data distributions. Most popular stochastic differential equation-based diffusion models perturb the target distribution by adding Gaussian noise, transforming it into a simple prior, and then use denoising score matching, a consequence of Tweedie's formula, to learn the score function and generate clean samples from noise. However, non-Gaussian diffusion models with state-dependent diffusion coefficient have been largely underexplored, as have the corresponding Tweedie's formulae. In this work, we extend Tweedie's formula to important non-Gaussian processes, including geometric Brownian motion (GBM), squared Bessel (BESQ) processes, and Cox-Ingersoll-Ross (CIR) processes, thereby yielding the corresponding denoising score-matching objectives. We then apply the derived formulae to image and financial time series generation using GBM- and CIR-based diffusion models, and to empirical Bayes estimation under the BESQ setting. The reported experimental results demonstrate the potential of non-Gaussian models.
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