Statistical Analysis of Karcher Means for Random Restricted PSD Matrices
This work addresses a theoretical gap for researchers in manifold-based ML, offering incremental analysis with specific applications like distributed PCA.
The paper tackles the lack of non-asymptotic statistical analysis for geometry-aware ML algorithms on manifolds by studying the Karcher mean on restricted PSD matrices, providing error bounds and showing that the distributed PCA algorithm LRC-dPCA matches full sample PCA performance.
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.