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 @math PCA (exponential family PCA), a new methodology for PCA on exponential family distributions. @math PCA can be used for dimensionality reduction and denoising of large data matrices. @math 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. @math 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.
The recent work of @cite_13 develops a generalized Stein's Unbiased Risk Estimation (SURE) approach for singular value shrinkage denoising of low-rank matrices in exponential families. However, their shrinkage formulas become numerically intractable for Frobenius norm beyond Gaussian errors, and they instead introduce a heuristic algorithm. Their work is geared towards higher signal-to-noise ratio settings.
@cite_13
2404992943
We consider the problem of estimating a low-rank signal matrix from noisy measurements under the assumption that the distribution of the data matrix belongs to an exponential family. In this setting, we derive generalized Stein's unbiased risk estimation (SURE) formulas that hold for any spectral estimators which shrink or threshold the singular values of the data matrix. This leads to new data-driven spectral estimators, whose optimality is discussed using tools from random matrix theory and through numerical experiments. Under the spiked population model and in the asymptotic setting where the dimensions of the data matrix are let going to infinity, some theoretical properties of our approach are compared to recent results on asymptotically optimal shrinking rules for Gaussian noise. It also leads to new procedures for singular values shrinkage in finite-dimensional matrix denoising for Gamma-distributed and Poisson-distributed measurements.