Geometric variational inference
This work addresses the problem of improving variational inference efficiency for statisticians and machine learning practitioners by incorporating geometric properties, representing an incremental advancement over traditional methods.
The paper tackles the challenge of efficiently accessing information in non-linear, high-dimensional probability distributions by proposing geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric, which constructs a coordinate transformation to simplify distributions for accurate variational approximation by a normal distribution, demonstrated on examples from low-dimensional to hierarchical Bayesian inverse problems in thousands of dimensions.
Efficiently accessing the information contained in non-linear and high dimensional probability distributions remains a core challenge in modern statistics. Traditionally, estimators that go beyond point estimates are either categorized as Variational Inference (VI) or Markov-Chain Monte-Carlo (MCMC) techniques. While MCMC methods that utilize the geometric properties of continuous probability distributions to increase their efficiency have been proposed, VI methods rarely use the geometry. This work aims to fill this gap and proposes geometric Variational Inference (geoVI), a method based on Riemannian geometry and the Fisher information metric. It is used to construct a coordinate transformation that relates the Riemannian manifold associated with the metric to Euclidean space. The distribution, expressed in the coordinate system induced by the transformation, takes a particularly simple form that allows for an accurate variational approximation by a normal distribution. Furthermore, the algorithmic structure allows for an efficient implementation of geoVI which is demonstrated on multiple examples, ranging from low-dimensional illustrative ones to non-linear, hierarchical Bayesian inverse problems in thousands of dimensions.