Computing sample means on Riemannian manifolds is typically computationally costly. The Fr\'echet mean offers a generalization of the Euclidean mean to general metric spaces, particularly to Riemannian manifolds. Evaluating the Fr\'echet mean numerically on Riemannian manifolds requires the computation of geodesics for each sample point. When closed-form expressions do not exist for geodesics, an optimization-based approach is employed. In geometric deep-learning, particularly Riemannian convolutional neural networks, a weighted Fr\'echet mean enters each layer of the network, potentially requiring an optimization in each layer. The weighted diffusion-mean offers an alternative weighted mean sample estimator on Riemannian manifolds that do not require the computation of geodesics. Instead, we present a simulation scheme to sample guided diffusion bridges on a product manifold conditioned to intersect at a predetermined time. Such a conditioning is non-trivial since, in general, manifolds cannot be covered by a single chart. Exploiting the exponential chart, the conditioning can be made similar to that in the Euclidean setting.

翻译：计算器样本在里曼尼方块上通常具有计算成本。 Fr\'echet 表示对通用度空间,特别是对里曼尼方块,以普通度空间为平均值的欧几里德方块进行概括化。 Fr\'echet 表示对里曼尼方块进行数字化评估,需要计算每个抽样点的大地学特征。当闭式表达方式不存在时,将采用基于优化的方法。在几何深学习中,特别是里曼尼共进神经网络中,加权Fr\'echet 表示进入网络的每一层,可能要求每个层进行优化。加权扩散平均值为里曼方块提供了替代的加权平均样本测量器,而不需要计算地德性。相反,我们提出了一个模拟方案,用于在预定的时间对产品多层进行导出扩散桥的取样。这种调节是非三角的,因为一般来说,指数图无法覆盖多个元件,因此,调整可以与Ecl 设置类似。