Statistical Analysis of Karcher Means for Random Restricted PSD Matrices
Abstract
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.
Cite
Text
Chen et al. "Statistical Analysis of Karcher Means for Random Restricted PSD Matrices." Artificial Intelligence and Statistics, 2023.Markdown
[Chen et al. "Statistical Analysis of Karcher Means for Random Restricted PSD Matrices." Artificial Intelligence and Statistics, 2023.](https://mlanthology.org/aistats/2023/chen2023aistats-statistical/)BibTeX
@inproceedings{chen2023aistats-statistical,
title = {{Statistical Analysis of Karcher Means for Random Restricted PSD Matrices}},
author = {Chen, Hengchao and Li, Xiang and Sun, Qiang},
booktitle = {Artificial Intelligence and Statistics},
year = {2023},
pages = {1437-1456},
volume = {206},
url = {https://mlanthology.org/aistats/2023/chen2023aistats-statistical/}
}