Geometric Variational Inference.

Area of Science:

• Statistics
• Computational Mathematics
• Machine Learning

Background:

• Variational Inference (VI) and Markov-Chain Monte-Carlo (MCMC) are traditional methods for estimating probability distributions.
• VI methods often neglect the geometric properties of probability distributions, unlike MCMC.
• Efficiently accessing information in high-dimensional, non-linear distributions is a significant statistical challenge.

Purpose of the Study:

• To introduce a novel method, geometric Variational Inference (geoVI), that incorporates Riemannian geometry into VI.
• To address the underutilization of geometric properties in existing VI techniques.
• To develop an efficient and accurate method for approximating complex probability distributions.

Main Methods:

• geoVI utilizes Riemannian geometry and the Fisher information metric to construct coordinate transformations.
• This transformation maps the probability distribution's Riemannian manifold to Euclidean space.
• The transformed distribution simplifies, allowing for accurate variational approximation using a normal distribution.

Main Results:

• geoVI enables accurate variational approximation of complex distributions by simplifying their form.
• The method demonstrates efficient implementation across various dimensions, from low to thousands.
• Successful application to non-linear, hierarchical Bayesian inverse problems is shown.

Conclusions:

• Geometric Variational Inference (geoVI) offers a powerful new approach to statistical estimation.
• The method enhances the efficiency and accuracy of Variational Inference by incorporating geometric principles.
• geoVI provides a computationally efficient solution for complex, high-dimensional probability distributions.