用黎曼流形上的群形态卷积提升扩散模型的几何特征提取能力
DEGMC: Denoising Diffusion Models Based on Riemannian Equivariant Group Morphological Convolutions
- 基于黎曼流形设计具有旋转对称性的形态卷积
- 在多个数据集上生成质量优于基线扩散模型
- 适合需要几何对称性建模的任务如图像生成
本文针对去噪扩散概率模型（DDPM）中的两大问题：几何关键特征提取与网络等变性。由于DDPM预测网络依赖于仅具备平移等变性的U-Net结构，本文引入结合欧氏群（包含旋转、反射和置换）等变性的几何方法。提出黎曼流形上的群形态卷积，其源自一阶哈密顿-雅可比型偏微分方程的粘性解，实现多尺度膨胀与腐蚀操作。通过加入对流项并用特征线法求解，更有效捕捉非线性、表征细长几何结构，并将对称性融入学习过程。在MNIST、RotoMNIST和CIFAR-10数据集上的实验表明，相比基线DDPM模型有显著性能提升。
原文摘要 · Abstract (English)
In this work, we address two major issues in recent Denoising Diffusion Probabilistic Models (DDPM): {\bf 1)} geometric key feature extraction and {\bf 2)} network equivariance. Since the DDPM prediction network relies on the U-net architecture, which is theoretically only translation equivariant, we introduce a geometric approach combined with an equivariance property of the more general Euclidean group, which includes rotations, reflections, and permutations. We introduce the notion of group morphological convolutions in Riemannian manifolds, which are derived from the viscosity solutions of first-order Hamilton-Jacobi-type partial differential equations (PDEs) that act as morphological multiscale dilations and erosions. We add a convection term to the model and solve it using the method of characteristics. This helps us better capture nonlinearities, represent thin geometric structures, and incorporate symmetries into the learning process. Experimental results on the MNIST, RotoMNIST, and CIFAR-10 datasets show noticeable improvements compared to the baseline DDPM model.
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