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：在现代几何感知机器学习算法中，由于可能涉及复杂的非线性流形结构，非渐近统计分析常常缺失。本文研究了受限半正定矩阵流形上的内蕴均值模型，并提供了Karcher均值的非渐近统计分析。我们还考虑了一般的泛化信号加噪声模型，在该模型下给出了Karcher均值的确定性误差界。作为一个应用，我们表明分布式主成分分析算法LRC-dPCA能够达到与全样本PCA算法相同的性能。数值实验有力地支持了我们的理论。