Statistical Analysis of Karcher Means for Random Restricted PSD Matrices

• URL: http://arxiv.org/abs/2302.12426v3
• Date: Tue, 21 Mar 2023 01:49:46 GMT
• Title: Statistical Analysis of Karcher Means for Random Restricted PSD Matrices
• Authors: Hengchao Chen, Xiang Li, Qiang Sun
• Abstract summary: 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. As an application, we show that the distributed principal component analysis algorithm, LRC-dPCA, achieves the same performance as the full sample PCA algorithm.
• Score: 5.867823829398135
• License: http://creativecommons.org/licenses/by/4.0/
• 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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