随机约束PSD矩阵Karcher均值的统计分析
Statistical Analysis of Karcher Means for Random Restricted PSD Matrices
摘要：
由于可能复杂的非线性流形结构，现代几何感知机器学习算法通常缺少非渐近统计分析。本文研究了限制半正定矩阵流形上的一个内在平均模型，给出了Karcher平均的一个非渐近统计分析。我们还考虑了一个一般的外在信号加噪声模型，根据该模型提供了一个确定性的误差界的Karcher平均。作为一个应用程序，我们表明，分布式主成分分析算法，LRC-dPCA，实现了相同的性能作为全样本PCA算法。数值实验为我们的理论提供了强有力的支持。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
