# Statistical Analysis of Karcher Means for Random Restricted PSD Matrices

arXiv: https://arxiv.org/abs/2302.12426 · HTML: https://mlsanity.com/paper/2302.12426/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 42 references that are in the map.
- First posted on arXiv 24 Feb 2023.

## Builds on (most cited first, 5 of 5)

- [Spectral Methods for Data Science: A Statistical Perspective](https://mlsanity.com/paper/2012.08496/) (2020)
- [Randomized Sketches of Convex Programs with Sharp Guarantees](https://mlsanity.com/paper/1404.7203/) (2014)
- [Communication-Efficient Distributed SVD via Local Power Iterations](https://mlsanity.com/paper/2002.08014/) (2020)
- [Communication-efficient Algorithms for Distributed Stochastic Principal Component Analysis](https://mlsanity.com/paper/1702.08169/) (2017)
- [Communication-efficient distributed eigenspace estimation](https://mlsanity.com/paper/2009.02436/) (2020)

## Abstract (arXiv)

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.