This paper addresses the problem of computing the Riemannian center of mass of a collection of symmetric positive definite matrices. We show in detail that the condition number of the Riemannian Hessian of the underlying optimization problem is never very ill conditioned in practice, which explains why the Riemannian steepest descent approach has been observed to perform well. We also show theoretically and empirically that this property is not shared by the Euclidean Hessian. We then present a limited‐memory Riemannian BFGS method to handle this computational task. We also provide methods to produce efficient numerical representations of geometric objects that are required for Riemannian optimization methods on the manifold of symmetric positive definite matrices. Through empirical results and a computational complexity analysis, we demonstrate the robust behavior of the limited‐memory Riemannian BFGS method and the efficiency of our implementation when compared to state‐of‐the‐art algorithms.

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计算矩阵的几何均值：黎曼与欧几里得条件，实现技术和黎曼BFGS方法

本文解决了计算对称正定矩阵集合的黎曼质量中心的问题。我们详细显示了基本优化问题的黎曼Hessian的条件编号在实践中绝不会病得很厉害，这解释了为什么观察到黎曼最速下降法的效果很好。我们还从理论和经验上证明，欧几里得黑森州不共享此属性。然后，我们提出了一种有限内存的黎曼BFGS方法来处理此计算任务。我们还提供在对称正定矩阵的流形上产生黎曼优化方法所需的几何对象的有效数值表示的方法。通过经验结果和计算复杂性分析，