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Activity Number: 126 - Topics at the Frontier of Statistical Computing and Machine Learning
Type: Invited
Date/Time: Monday, August 8, 2022 : 10:30 AM to 12:20 PM
Sponsor: Section on Bayesian Statistical Science
Abstract #320331
Title: Variational Bayes on Manifolds
Author(s): Minh-Ngoc Tran* and Dang Nguyen and Duy Nguyen
Companies: University of Sydney and University of Alabama and Marist College
Keywords: natural gradient; stochastic approximation; Riemannian manifold

Variational Bayes (VB) has become a widely-used tool for Bayesian inference in statistics and machine learning. Nonetheless, the development of the existing VB algorithms is so far generally restricted to the case where the variational parameter space is Euclidean, which hinders the potential broad application of VB methods. This paper extends the scope of VB to the case where the variational parameter space is a Riemannian manifold. We develop an efficient manifold-based VB algorithm that exploits both the geometric structure of the constraint parameter space and the information geometry of the manifold of VB approximating probability distributions. Our algorithm is provably convergent and achieves a convergence rate of order $\mathcal O(1/\sqrt{T})$ and $\mathcal O(1/T^{2-2\epsilon})$ for a non-convex evidence lower bound function and a strongly retraction-convex evidence lower bound function, respectively. We develop in particular two manifold VB algorithms, Manifold Gaussian VB and Manifold Wishart VB, and demonstrate through numerical experiments that the proposed algorithms are stable, less sensitive to initialization and compares favourably to existing VB methods.

Authors who are presenting talks have a * after their name.

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