Convergence Analysis of a Greedy Algorithm for Conditioning Gaussian Random Variables

TL;DR

This paper analyzes the convergence of greedy algorithms used for conditioning Gaussian random variables, introducing an operator to transfer convergence rates and applying approximation theory to establish bounds.

Contribution

It introduces an operator that links convergence rates of observed and conditional Gaussian variables and applies approximation theory to analyze greedy algorithms.

Findings

Established an upper bound on convergence rates for the conditional covariance operator

Transferred convergence rate analysis from observed to conditional Gaussian variables

Demonstrated the optimality of greedy methods in this context

Abstract

In the context of Gaussian conditioning, greedy algorithms iteratively select the most informative measurements, given an observed Gaussian random variable. However, the convergence analysis for conditioning Gaussian random variables remains an open problem. We adress this by introducing an operator $M$ that allows us to transfer convergence rates of the observed Gaussian random variable approximation onto the conditional Gaussian random variable. Furthermore we apply greedy methods from approximation theory to obtain convergence rates. These greedy methods have already demonstrated optimal convergence rates within the setting of kernel based function approximation. In this paper, we establish an upper bound on the convergence rates concerning the norm of the approximation error of the conditional covariance operator.