Probabilistic Unrolling: Scalable, Inverse-Free Maximum Likelihood Estimation for Latent Gaussian Models

TL;DR

Probabilistic unrolling offers a scalable, inverse-free approach for maximum likelihood estimation in latent Gaussian models, significantly reducing computation time while maintaining accuracy.

Contribution

The paper introduces probabilistic unrolling, a novel method combining Monte Carlo sampling and iterative solvers to avoid matrix inversion in high-dimensional latent Gaussian models.

Findings

Learns latent Gaussian models up to 10 times faster than gradient EM.

Maintains comparable model performance with reduced computational cost.

Applicable to both simulated and real datasets.

Abstract

Latent Gaussian models have a rich history in statistics and machine learning, with applications ranging from factor analysis to compressed sensing to time series analysis. The classical method for maximizing the likelihood of these models is the expectation-maximization (EM) algorithm. For problems with high-dimensional latent variables and large datasets, EM scales poorly because it needs to invert as many large covariance matrices as the number of data points. We introduce probabilistic unrolling, a method that combines Monte Carlo sampling with iterative linear solvers to circumvent matrix inversion. Our theoretical analyses reveal that unrolling and backpropagation through the iterations of the solver can accelerate gradient estimation for maximum likelihood estimation. In experiments on simulated and real data, we demonstrate that probabilistic unrolling learns latent Gaussian models up to an order of magnitude faster than gradient EM, with minimal losses in model performance.