# Statistical Analysis of Karcher Means for Random Restricted PSD Matrices

## TL;DR

This paper provides a non-asymptotic statistical analysis of the Karcher mean on the manifold of restricted positive semi-definite matrices, with applications to distributed PCA algorithms and strong numerical validation.

## Contribution

It offers the first non-asymptotic analysis of the Karcher mean on restricted PSD matrices and applies it to distributed PCA performance guarantees.

## Key findings

- Deterministic error bounds for the Karcher mean under noise.
- Distributed PCA matches full sample PCA performance.
- Numerical experiments support theoretical results.

## 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.