# Stochastic Approximation Algorithms for Principal Component Analysis

## TL;DR

This paper reviews stochastic algorithms for PCA, comparing their convergence and runtime, and explores convex relaxation methods that show promising empirical performance for dimensionality reduction.

## Contribution

It provides a comparative analysis of stochastic PCA algorithms and revisits convex relaxation techniques, highlighting their efficiency and empirical advantages.

## Key findings

- Convex relaxation methods can outperform traditional stochastic algorithms in practice.
- Stochastic approaches to PCA have well-understood convergence properties.
- Convex relaxation offers a promising alternative with comparable or better empirical results.

## Abstract

Principal Component Analysis is a novel way of of dimensionality reduction. This problem essentially boils down to finding the top k eigen vectors of the data covariance matrix. A considerable amount of literature is found on algorithms meant to do so such as an online method be Warmuth and Kuzmin, Matrix Stochastic Gradient by Arora, Oja's method and many others. In this paper we see some of these stochastic approaches to the PCA optimization problem and comment on their convergence and runtime to obtain an epsilon sub-optimal solution. We revisit convex relaxation based methods for stochastic optimization of principal component analysis. While methods that directly solve the non convex problem have been shown to be optimal in terms of statistical and computational efficiency, the methods based on convex relaxation have been shown to enjoy comparable, or even superior, empirical performance. This motivates the need for a deeper formal understanding of the latter.