Manifold Optimisation Assisted Gaussian Variational Approximation

Variational approximation methods are a way to approximate the posterior in Bayesian inference especially when the dataset has a large volume or high dimension. Factor covariance structure was introduced in previous work with three restrictions to handle the problem of computational infeasibility in Gaussian approximation. However, the three strong constraints on the covariance matrix could possibly break down during the process of the structure optimization, and the identification issue could still possibly exist within the final approximation. In this paper, we consider two types of manifold parameterization, Stiefel manifold and Grassmann manifold, to address the problems. Moreover, the Riemannian stochastic gradient descent method is applied to solve the resulting optimization problem while maintaining the orthogonal factors. Results from two experiments demonstrate that our model fixes the potential issue of the previous method with comparable accuracy and competitive converge speed even in high-dimensional problems.