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：现代几何感知机器学习算法往往因可能复杂的非线性流形结构而缺乏非渐近统计分析。本文研究受限正半定矩阵流形上的内蕴均值模型，并提供Karcher均值的非渐近统计分析。我们还考虑一般性外蕴信号加噪声模型，在此模型下给出Karcher均值的确定性误差界。作为应用，我们证明分布式主成分分析算法LRC-dPCA能达到与全样本PCA算法相同的性能。数值实验有力支持了我们的理论。