Manifold Gaussian Variational Bayes on the Precision Matrix

TL;DR

This paper introduces a novel manifold-based Gaussian Variational Bayes method that efficiently optimizes the precision matrix, providing a practical and computationally advantageous solution for variational inference in complex models.

Contribution

It develops a Riemann manifold-based optimization algorithm for Gaussian VI that ensures positive definiteness and simplifies implementation, especially using the precision matrix parametrization.

Findings

Effective on multiple datasets across different models

Outperforms baseline methods in accuracy and efficiency

Provides a black-box, ready-to-use VI solution

Abstract

We propose an optimization algorithm for Variational Inference (VI) in complex models. Our approach relies on natural gradient updates where the variational space is a Riemann manifold. We develop an efficient algorithm for Gaussian Variational Inference whose updates satisfy the positive definite constraint on the variational covariance matrix. Our Manifold Gaussian Variational Bayes on the Precision matrix (MGVBP) solution provides simple update rules, is straightforward to implement, and the use of the precision matrix parametrization has a significant computational advantage. Due to its black-box nature, MGVBP stands as a ready-to-use solution for VI in complex models. Over five datasets, we empirically validate our feasible approach on different statistical and econometric models, discussing its performance with respect to baseline methods.