Towards a Theoretical Analysis of PCA for Heteroscedastic Data

TL;DR

This paper offers a theoretical framework for understanding PCA performance on heteroscedastic data, providing asymptotic predictions that help quantify and interpret the impact of non-uniform noise variances.

Contribution

It introduces a simple asymptotic prediction model for PCA recovery of a one-dimensional subspace with heteroscedastic noise, enhancing understanding of PCA's behavior under non-uniform noise conditions.

Findings

Asymptotic prediction of PCA recovery performance with heteroscedastic noise

Efficient calculation method for PCA performance metrics

Qualitative insights into PCA's sensitivity to outliers and noise variance

Abstract

Principal Component Analysis (PCA) is a method for estimating a subspace given noisy samples. It is useful in a variety of problems ranging from dimensionality reduction to anomaly detection and the visualization of high dimensional data. PCA performs well in the presence of moderate noise and even with missing data, but is also sensitive to outliers. PCA is also known to have a phase transition when noise is independent and identically distributed; recovery of the subspace sharply declines at a threshold noise variance. Effective use of PCA requires a rigorous understanding of these behaviors. This paper provides a step towards an analysis of PCA for samples with heteroscedastic noise, that is, samples that have non-uniform noise variances and so are no longer identically distributed. In particular, we provide a simple asymptotic prediction of the recovery of a one-dimensional subspace from noisy heteroscedastic samples. The prediction enables: a) easy and efficient calculation of the asymptotic performance, and b) qualitative reasoning to understand how PCA is impacted by heteroscedasticity (such as outliers).