[Paper Review] On optimal recovery in $L_2$

Vladimir Temlyakov|arXiv (Cornell University)|Oct 7, 2020

Mathematical Approximation and Integration17 references6 citations

TL;DR

This paper establishes a fundamental inequality showing that the optimal recovery error in the $L_2$ norm is bounded above by the Kolmogorov width in the uniform norm, leveraging recent discretization results in $L_2$ for finite-dimensional subspaces. The key contribution is a general bound linking recovery error to uniform norm widths, enabling powerful error estimates for classes with mixed smoothness via weighted least squares algorithms.

ABSTRACT

We prove that the optimal error of recovery in the $L_2$ norm of functions from a class $\bF$ can be bounded above by the value of the Kolmogorov width of $\bF$ in the uniform norm. We demonstrate on a number of examples of $\bF$ from classes of functions with mixed smoothness that the obtained inequality provides a powerful tool for estimating errors of optimal recovery.

Motivation & Objective

• To establish a general upper bound for the optimal recovery error in $L_2$ using Kolmogorov widths in the uniform norm.
• To demonstrate the effectiveness of weighted least squares algorithms for optimal recovery in $L_2$.
• To apply the main inequality to function classes with mixed smoothness, providing concrete error estimates.
• To connect recovery error bounds with deep discretization results in $L_2$ for finite-dimensional subspaces.

Proposed method

• Uses the discretization of $L_2$ norms on finite-dimensional subspaces via weighted sampling, relying on recent results from [16], [7], and [12].
• Applies the weighted least squares algorithm $\ell p\mathbf{w}(\xi)$, which minimizes the weighted $\ell_p$-norm of sampling errors.
• Employs the Chebyshev projection $P_{X_N,p}(f)$ as the best approximation operator in $L_p$-norm.
• Establishes a link between the recovery error $\varrho_{m}(\mathbf{F},L_2)$ and the Kolmogorov width $d_n(\mathbf{F},L_\infty)$ via a key inequality involving absolute constants.
• Relies on Condition E(t) for orthonormal systems to ensure uniform sampling discretization with uniform weights.
• Uses Theorem 4.2 (discretization with uniform weights) to derive a variant of the main result for classical least squares algorithms.

Experimental results

Research questions

• RQ1 Can the optimal recovery error in $L_2$ be bounded using the Kolmogorov width in the uniform norm?
• RQ2 How do weighted least squares algorithms perform in terms of error bounds for $L_2$ recovery?
• RQ3 What is the role of discretization results in connecting $L_2$ recovery to uniform norm widths?
• RQ4 Can the main inequality be extended to $L_p$ norms for $p \neq 2$?
• RQ5 What are the implications of the result for function classes with mixed smoothness?

Key findings

• The optimal recovery error in $L_2$ satisfies $\varrho_{bn}(\mathbf{F},L_2) \leq B d_n(\mathbf{F},L_\infty)$ for some absolute constants $b, B > 0$, linking $L_2$ recovery to uniform norm widths.
• The result provides a powerful tool for estimating recovery errors in classes of functions with mixed smoothness.
• For classical least squares with uniform weights, the bound $\varrho_{bn}^{ls}(\mathbf{F},L_2) \leq B d_n^{E(t)}(\mathbf{F},L_\infty)$ holds under Condition E(t).
• The discretization result in Theorem 4.1 allows extension to $L_p$ for $1 \leq p \leq 2$, but no such analog is known for $p > 2$.
• The inequality is sharp in the sense that it captures the correct order of magnitude for known recovery rates in $L_\infty$ for Sobolev-type classes.
• The method is robust and applies to arbitrary compact, centrally symmetric function classes in $\mathcal{C}(\Omega)$.