$e$PCA: High Dimensional Exponential Family PCA

TL;DR

The paper introduces ePCA, a novel method for principal component analysis tailored to exponential family distributions, improving covariance estimation and denoising in high-dimensional noisy data.

Contribution

ePCA provides a new covariance estimator and PCA methodology specifically designed for exponential family data, with theoretical guarantees and practical advantages.

Findings

ePCA outperforms standard PCA in simulations.

ePCA effectively denoises large exponential family datasets.

Theoretical analysis confirms convergence and spectral properties.

Abstract

Many applications, such as photon-limited imaging and genomics, involve large datasets with noisy entries from exponential family distributions. It is of interest to estimate the covariance structure and principal components of the noiseless distribution. Principal Component Analysis (PCA), the standard method for this setting, can be inefficient when the noise is non-Gaussian. We develop $e$PCA (exponential family PCA), a new methodology for PCA on exponential family distributions. $e$PCA can be used for dimensionality reduction and denoising of large data matrices. $e$PCA involves the eigendecomposition of a new covariance matrix estimator, constructed in a simple and deterministic way using moment calculations, shrinkage, and random matrix theory. We provide several theoretical justifications for our estimator, including the finite-sample convergence rate, and the Marchenko-Pastur law in high dimensions. $e$PCA compares favorably to PCA and various PCA alternatives for exponential families, in simulations as well as in XFEL and SNP data analysis. An open-source implementation is available.