Title: Quantifying the Estimation Error of Principal Components,
Abstract: Principal component analysis is an important pattern recognition and
dimensionality reduction tool in many applications. Principal components are
computed as eigenvectors of a maximum likelihood covariance $\widehat{\Sigma}$
that approximates a population covariance $\Sigma$, and these eigenvectors are
often used to extract structural information about the variables (or
attributes) of the studied population. Since PCA is based on the
eigendecomposition of the proxy covariance $\widehat{\Sigma}$ rather than the
ground-truth $\Sigma$, it is important to understand the approximation error in
each individual eigenvector as a function of the number of available samples.
The recent results of Kolchinskii and Lounici yield such bounds. In the present
paper we sharpen these bounds and show that eigenvectors can often be
reconstructed to a required accuracy from a sample of strictly smaller size
order.