Statistical Analysis of Karcher Means for Random Restricted PSD Matrices

Hengchao Chen, Xiang Li, Qiang Sun
2/24/2023
stat.MLcs.LG

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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.

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Cite this paper

@article{chen2023statistical,
  title  = {Statistical Analysis of Karcher Means for Random Restricted PSD Matrices},
  author = {Hengchao Chen and Xiang Li and Qiang Sun},
  year   = {2023},
  eprint = {2302.12426},
  archivePrefix = {arXiv},
  url    = {http://arxiv.org/abs/2302.12426v3}
}

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