Statistical Analysis of Karcher Means for Random Restricted PSD Matrices

@InProceedings{pmlr-v206-chen23a,
  title = 	 {Statistical Analysis of Karcher Means for Random Restricted PSD Matrices},
  author =       {Chen, Hengchao and Li, Xiang and Sun, Qiang},
  booktitle = 	 {Proceedings of The 26th International Conference on Artificial Intelligence and Statistics},
  pages = 	 {1437--1456},
  year = 	 {2023},
  editor = 	 {Ruiz, Francisco and Dy, Jennifer and van de Meent, Jan-Willem},
  volume = 	 {206},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {25--27 Apr},
  publisher =    {PMLR},
  pdf = 	 {https://proceedings.mlr.press/v206/chen23a/chen23a.pdf},
  url = 	 {https://proceedings.mlr.press/v206/chen23a.html},
  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.}
}