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Many applications, such as photon-limited imaging and genomics, involve largedatasets with noisy entries from exponential family distributions. It is ofinterest to estimate the covariance structure and principal components of thenoiseless distribution. Principal Component Analysis (PCA), the standard methodfor this setting, can be inefficient when the noise is non-Gaussian.  We develop $e$PCA (exponential family PCA), a new methodology for PCA onexponential family distributions. $e$PCA can be used for dimensionalityreduction and denoising of large data matrices. $e$PCA involves theeigendecomposition of a new covariance matrix estimator, constructed in asimple and deterministic way using moment calculations, shrinkage, and randommatrix theory.  We provide several theoretical justifications for our estimator, includingthe finite-sample convergence rate, and the Marchenko-Pastur law in highdimensions. $e$PCA compares favorably to PCA and various PCA alternatives forexponential families, in simulations as well as in XFEL and SNP data analysis.An open-source implementation is available.

$e$PCA: High Dimensional Exponential Family PCA