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Convergence Analysis of a Greedy Algorithm for Conditioning Gaussian Random Variables

Daniel Winkle, Ingo Steinwart, Bernard Haasdonk

TL;DR

This work addresses the difficult problem of conditioning Gaussian random variables in Banach spaces by introducing an operator $M$ that maps the observed variable $Y$ to the conditional variable $Z=\mathbb{E}(X|Y)$, enabling transfer of convergence rates from $cov(Y)$ to $cov(X|Y)$. It shows that conditioning corresponds to orthogonal projections across Gaussian spaces and derives a kernel-based representation for the conditioned process via RKHS theory. By applying the $P$-greedy algorithm to the conditioning measurements, the authors establish polynomial convergence rates for the conditional covariance and provide explicit bounds in terms of Kolmogorov widths. The theoretical framework is complemented by several examples, including Brownian motion with partial observations and noisy measurement models, and by proofs detailing the extension of the operator $M_W$ and the projection structure. This integrates Gaussian conditioning with kernel methods and greedy approximation to yield practical convergence guarantees for high-dimensional conditioning problems.

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.

Convergence Analysis of a Greedy Algorithm for Conditioning Gaussian Random Variables

TL;DR

This work addresses the difficult problem of conditioning Gaussian random variables in Banach spaces by introducing an operator that maps the observed variable to the conditional variable , enabling transfer of convergence rates from to . It shows that conditioning corresponds to orthogonal projections across Gaussian spaces and derives a kernel-based representation for the conditioned process via RKHS theory. By applying the -greedy algorithm to the conditioning measurements, the authors establish polynomial convergence rates for the conditional covariance and provide explicit bounds in terms of Kolmogorov widths. The theoretical framework is complemented by several examples, including Brownian motion with partial observations and noisy measurement models, and by proofs detailing the extension of the operator and the projection structure. This integrates Gaussian conditioning with kernel methods and greedy approximation to yield practical convergence guarantees for high-dimensional conditioning problems.

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 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.

Paper Structure

This paper contains 10 sections, 35 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Gaussian Hilbert Spaces
  4. Abstract Wiener Spaces
  5. Reproducing Covariance Space
  6. P-Greedy
  7. Main Results
  8. Examples
  9. Proofs
  10. Additional Examples

Key Result

Corollary 2

Given an RKHS $H$ with kernel $k$ selecting points via the weak $P$-greedy one obtains with $d_m(\mathcal{F}) := \min_{V} \sup_{f \in \mathcal{F}} \Vert f - \Pi_{V} f \Vert_H$ denoting the Kolmogorov width of the set $\mathcal{F}$, where the minimum is taken over all $m$-dimensional subspaces $V\subseteq H$ and $\Pi_{V}$ denotes the orthogonal projection in $H$ onto $V$.

Theorems & Definitions (76)

  • Definition 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Corollary 5
  • Lemma 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Corollary 10
  • ...and 66 more