🤖 AI Summary
This study addresses the challenge of latent dimension selection in high-dimensional factor models when the model holds only approximately or the signal-to-noise ratio is low. The authors propose EigenBayes, a novel method that uniquely integrates spectral estimation with adaptive empirical Bayes to construct a structured shrinkage prior, enabling efficient shrinkage and uncertainty quantification in over-parameterized factor models. The approach yields an analytically tractable posterior distribution, obviating the need for MCMC sampling, and adaptively accounts for varying signal-to-noise ratios across both observed variables and latent dimensions. Theoretical analysis establishes favorable asymptotic properties, while numerical experiments and a genomics application demonstrate superior performance over state-of-the-art methods.
📝 Abstract
Factor models are popular approaches for analyzing high-dimensional data to extract low-rank signals and estimate covariances. They decompose the covariance matrix as the sum of low-rank and diagonal components. A key issue is how to choose the latent dimension $k$, which is particularly challenging when the factor model only holds approximately and in low signal-to-noise scenarios. Bayesian overfitted factor models specify an upper bound on $k$ and rely on structured shrinkage priors to effectively remove extra components. Such approaches are popular and effective, but computationally expensive. We propose a much faster \texttt{EigenBayes} approach that provides valid uncertainty quantification, based on spectral estimation of latent factors and adaptive empirical Bayes calibration of key hyperparameters. The resulting posterior distribution factorizes across outcomes and is analytically tractable, bypassing Markov chain Monte Carlo. We show that \texttt{EigenBayes} adapts to the signal-to-noise ratio of each outcome and latent dimension, while shrinking superfluous latent components to zero. We establish favorable asymptotic properties and demonstrate strong empirical performance in numerical experiments and a genomics application, where EigenBayes outperforms state-of-the-art alternatives.