提出高效扩散模型，让对称流形生成更快速准确。
Efficient Diffusion Models for Symmetric Manifolds
- 用投影欧氏布朗运动替代热核计算，降低复杂度。
- 每步仅需常数次梯度计算，算术操作近线性增长。
- 适用于环面、旋转群等对称流形，适合高维生成任务。
我们提出一种针对d维对称流形（包括环面、球面、特殊正交群和酉群）的高效扩散模型框架。现有方法依赖热核，缺乏闭式表达，每步需d次梯度评估或指数级算术操作。新模型引入空间可变协方差，通过投影欧氏布朗运动绕过热核计算。训练算法基于伊藤引理推导的新目标函数，每步仅需O(1)梯度评估和近似O(d^{1.19})算术操作，显著缩小对称流形与欧氏空间扩散模型的效率差距。流形对称性保证扩散满足‘平均情况’李普希茨条件，实现高精度高效采样。实验表明，该模型在环面、特殊正交群和酉群的合成数据集上，训练速度更快，样本质量更优。
原文摘要 · Abstract (English)
We introduce a framework for designing efficient diffusion models for $d$-dimensional symmetric-space Riemannian manifolds, including the torus, sphere, special orthogonal group and unitary group. Existing manifold diffusion models often depend on heat kernels, which lack closed-form expressions and require either $d$ gradient evaluations or exponential-in-$d$ arithmetic operations per training step. We introduce a new diffusion model for symmetric manifolds with a spatially-varying covariance, allowing us to leverage a projection of Euclidean Brownian motion to bypass heat kernel computations. Our training algorithm minimizes a novel efficient objective derived via Ito's Lemma, allowing each step to run in $O(1)$ gradient evaluations and nearly-linear-in-$d$ ($O(d^{1.19})$) arithmetic operations, reducing the gap between diffusions on symmetric manifolds and Euclidean space. Manifold symmetries ensure the diffusion satisfies an "average-case" Lipschitz condition, enabling accurate and efficient sample generation. Empirically, our model outperforms prior methods in training speed and improves sample quality on synthetic datasets on the torus, special orthogonal group, and unitary group.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。