  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.
