# Robust Streaming PCA

## TL;DR

This paper studies robust streaming PCA under model perturbations, establishing fundamental convergence limits and analyzing the optimality of existing algorithms like the noisy power method and Oja's algorithm.

## Contribution

It introduces a robust framework for streaming PCA with uncertain covariance, providing convergence bounds and demonstrating the optimality of the noisy power method.

## Key findings

- Noisy power method is rate-optimal under perturbations.
- Convergence limits are established for algorithms in uncertain covariance settings.
- Numerical experiments validate theoretical results.

## Abstract

We consider streaming principal component analysis when the stochastic data-generating model is subject to perturbations. While existing models assume a fixed covariance, we adopt a robust perspective where the covariance matrix belongs to a temporal uncertainty set. Under this setting, we provide fundamental limits on convergence of any algorithm recovering principal components. We analyze the convergence of the noisy power method and Oja's algorithm, both studied for the stationary data generating model, and argue that the noisy power method is rate-optimal in our setting. Finally, we demonstrate the validity of our analysis through numerical experiments on synthetic and real-world dataset.