Optimal projection of observations in a Bayesian setting

TL;DR

This paper introduces three optimal projection methods for Bayesian inference in Gaussian linear models, leveraging information theory to improve dimensionality reduction over standard PCA, with proven theoretical properties and numerical validation.

Contribution

It proposes novel information-theoretic optimal projections for Bayesian models, including solutions via Riemannian optimization and eigenvalue problems, with theoretical error bounds and efficiency analysis.

Findings

The proposed methods outperform PCA in preserving information.

Optimal subspaces can be computed efficiently using eigenvalue problems.

The methods are validated on linear and nonlinear models.

Abstract

Optimal dimensionality reduction methods are proposed for the Bayesian inference of a Gaussian linear model with additive noise in presence of overabundant data. Three different optimal projections of the observations are proposed based on information theory: the projection that minimizes the Kullback-Leibler divergence between the posterior distributions of the original and the projected models, the one that minimizes the expected Kullback-Leibler divergence between the same distributions, and the one that maximizes the mutual information between the parameter of interest and the projected observations. The first two optimization problems are formulated as the determination of an optimal subspace and therefore the solution is computed using Riemannian optimization algorithms on the Grassmann manifold. Regarding the maximization of the mutual information, it is shown that there exists an optimal subspace that minimizes the entropy of the posterior distribution of the reduced model; a basis of the subspace can be computed as the solution to a generalized eigenvalue problem; an a priori error estimate on the mutual information is available for this particular solution; and that the dimensionality of the subspace to exactly conserve the mutual information between the input and the output of the models is less than the number of parameters to be inferred. Numerical applications to linear and nonlinear models are used to assess the efficiency of the proposed approaches, and to highlight their advantages compared to standard approaches based on the principal component analysis of the observations.