Overfitted high-dimensional matrix factorizations via adaptive spectral shrinkage

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.

Spectral decomposition-assisted multi-study factor analysis

This article focuses on covariance estimation for multi-study data. Popular approaches employ factor-analytic terms with shared and study-specific loadings that decompose the variance into (i) a shared low-rank component, (ii) study-specific low-rank components, and (iii) a diagonal term capturing idiosyncratic variability. Our proposed methodology estimates the latent factors via spectral decompositions and infers the factor loadings via surrogate regression tasks, avoiding identifiability and computational issues of existing alternatives. Reliably inferring shared vs study-specific components requires novel developments that are of independent interest. The approximation error decreases as the sample size and the data dimension diverge, formalizing a blessing of dimensionality. Conditionally on the factors, loadings and residual error variances are inferred via conjugate normal-inverse gamma priors. The conditional posterior distribution of factor loadings has a simple product form across outcomes, facilitating parallelization. We show favorable asymptotic properties, including central limit theorems for point estimators and posterior contraction, and excellent empirical performance in simulations. The methods are applied to integrate three studies on gene associations among immune cells.

Robust Approximate Sampling via Stochastic Gradient Barker Dynamics

Stochastic Gradient (SG) Markov Chain Monte Carlo algorithms (MCMC) are popular algorithms for Bayesian sampling in the presence of large datasets. However, they come with little theoretical guarantees and assessing their empirical performances is non-trivial. In such context, it is crucial to develop algorithms that are robust to the choice of hyperparameters and to gradients heterogeneity since, in practice, both the choice of step-size and behaviour of target gradients induce hard-to-control biases in the invariant distribution. In this work we introduce the stochastic gradient Barker dynamics (SGBD) algorithm, extending the recently developed Barker MCMC scheme, a robust alternative to Langevin-based sampling algorithms, to the stochastic gradient framework. We characterize the impact of stochastic gradients on the Barker transition mechanism and develop a bias-corrected version that, under suitable assumptions, eliminates the error due to the gradient noise in the proposal. We illustrate the performance on a number of high-dimensional examples, showing that SGBD is more robust to hyperparameter tuning and to irregular behavior of the target gradients compared to the popular stochastic gradient Langevin dynamics algorithm.