Fast and Simple PCA via Convex Optimization

TL;DR

This paper introduces a new, efficient convex optimization-based method for principal component analysis (PCA) that significantly speeds up computation of top eigenvectors and eigenvalues, especially for large datasets.

Contribution

The paper presents a novel PCA algorithm reducing the problem to convex optimization, achieving the fastest known bounds for eigenvector approximation in various regimes.

Findings

Achieves near-optimal runtime for PCA eigenvector computation.

Provides bounds that outperform previous methods in large-scale settings.

Applicable when eigenvalue gaps or approximation errors are small.

Abstract

The problem of principle component analysis (PCA) is traditionally solved by spectral or algebraic methods. We show how computing the leading principal component could be reduced to solving a \textit{small} number of well-conditioned {\it convex} optimization problems. This gives rise to a new efficient method for PCA based on recent advances in stochastic methods for convex optimization. In particular we show that given a $d\times d$ matrix $\X = \frac{1}{n}\sum_{i=1}^n\x_i\x_i^{\top}$ with top eigenvector $\u$ and top eigenvalue $\lambda_1$ it is possible to: \begin{itemize} \item compute a unit vector $\w$ such that $(\w^{\top}\u)^2 \geq 1-\epsilon$ in $\tilde{O}\left({\frac{d}{\delta^2}+N}\right)$ time, where $\delta = \lambda_1 - \lambda_2$ and $N$ is the total number of non-zero entries in $\x_1,...,\x_n$, \item compute a unit vector $\w$ such that $\w^{\top}\X\w \geq \lambda_1-\epsilon$ in $\tilde{O}(d/\epsilon^2)$ time. \end{itemize} To the best of our knowledge, these bounds are the fastest to date for a wide regime of parameters. These results could be further accelerated when $\delta$ (in the first case) and $\epsilon$ (in the second case) are smaller than $\sqrt{d/N}$.