Convergence Rate of Krasulina Estimator

Abstract

Principal component analysis (PCA) is one of the most commonly used statistical procedures with a wide range of applications. Consider the points $X_1,X_2,...,X_n$ are vectors drawn i.i.d. from a distribution with mean zero and covariance $\Sigma$, where $\Sigma$ is unknown. Let $A_n=X_n{X}_n^T$, then $E[A_n]=\Sigma$. This paper consider the problem of finding the least eigenvalue and eigenvector of matrix $\Sigma$. A classical such estimator are due to Krasulina\cite{krasulina_method_1969}. We are going to state the convergence proof of Krasulina for the least eigenvalue and corresponding eigenvector, and then find their convergence rate.

References in corpus (6)

• Sparse PCA: Optimal rates and adaptive estimation
• Normal approximation and concentration of spectral projectors of sample covariance
• Memory Limited, Streaming PCA
• Stochastic Optimization of PCA with Capped MSG
• Accelerated Stochastic Power Iteration
• Concentration Inequalities and Moment Bounds for Sample Covariance Operators