Estimating means on Riemannian manifolds is generally computationally expensive because the Riemannian distance function is not known in closed-form for most manifolds. To overcome this, we show that Riemannian diffusion means can be efficiently estimated using score matching with the gradient of Brownian motion transition densities using the same principle as in Riemannian diffusion models. Empirically, we show that this is more efficient than Monte Carlo simulation while retaining accuracy and is also applicable to learned manifolds. Our method, furthermore, extends to computing the Fr\'echet mean and the logarithmic map for general Riemannian manifolds. We illustrate the applicability of the estimation of diffusion mean by efficiently extending Euclidean algorithms to general Riemannian manifolds with a Riemannian $k$-means algorithm and maximum likelihood Riemannian regression.

翻译：在黎曼流形上估计均值通常计算成本高昂，因为大多数流形的黎曼距离函数没有闭式解。为克服此问题，我们证明了利用与黎曼扩散模型相同的原理，通过布朗运动转移密度梯度的得分匹配，可以高效估计黎曼扩散均值。实验表明，该方法在保持精度的同时比蒙特卡洛模拟更高效，并且适用于学习得到的流形。此外，我们的方法可扩展至计算一般黎曼流形的弗雷歇均值和对数映射。我们通过将欧几里得算法高效扩展至一般黎曼流形——具体以黎曼k均值算法和最大似然黎曼回归为例，阐明了扩散均值估计的适用性。