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arXiv · 2302.12426

Statistical Analysis of Karcher Means for Random Restricted PSD Matrices

Hengchao Chen·Xiang Li·Qiang Sun
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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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Hengchao Chen, Xiang Li, Qiang Sun. 2023-02-24. Statistical Analysis of Karcher Means for Random Restricted PSD Matrices. https://arxiv.org/abs/2302.12426

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