Non-asymptotic statistical analysis is often missing for modern geometry-aware machine learning algorithms due to the possibly intricate non-linear manifold structure. This paper studies an intrinsic mean model on the manifold of restricted positive semi-definite matrices and provides a non-asymptotic statistical analysis of the Karcher mean. We also consider a general extrinsic signal-plus-noise model, under which a deterministic error bound of the Karcher mean is provided. As an application, we show that the distributed principal component analysis algorithm, LRC-dPCA, achieves the same performance as the full sample PCA algorithm. Numerical experiments lend strong support to our theories.

翻译：由于可能复杂的非线性多元结构,现代几何学机器学习算法往往缺少非线性统计分析。本文研究了限制正半无限制矩阵的内在平均值模型,并对Karcher的平均值进行了非线性统计分析。我们还考虑了一种一般的外在信号加氮模型,根据这一模型,提供了Karcher含义的确定性错误。作为一个应用程序,我们展示了分布式主要组成部分分析算法(LRC-dPCA)的性能与完全样本的五氯苯甲醚算法相同。数字实验为我们的理论提供了强有力的支持。</s>