Function Approximation via The Subsampled Poincar\' e Inequality

TL;DR

This paper introduces a generalized subsampled Poincaré inequality for function approximation, analyzing its optimality and impact on approximation accuracy, especially under limited regularity and small lengthscales.

Contribution

It develops a new subsampled Poincaré inequality, explores its optimality, and applies it to improve understanding of function approximation errors under various regularity conditions.

Findings

The approximation error increases as the subsampled lengthscale approaches zero.

A weighted Poincaré inequality is proposed to mitigate error blow-up.

The inequality's optimality is established in relation to the subsampled lengthscale.

Abstract

Function approximation and recovery via some sampled data have long been studied in a wide array of applied mathematics and statistics fields. Analytic tools, such as the Poincar\'e inequality, have been handy for estimating the approximation errors in different scales. The purpose of this paper is to study a generalized Poincar\' e inequality, where the measurement function is of subsampled type, with a small but non-zero lengthscale that will be made precise. Our analysis identifies this inequality as a basic tool for function recovery problems. We discuss and demonstrate the optimality of the inequality concerning the subsampled lengthscale, connecting it to existing results in the literature. In application to function approximation problems, the approximation accuracy using different basis functions and under different regularity assumptions is established by using the subsampled Poincar\'e inequality. We observe that the error bound blows up as the subsampled lengthscale approaches zero, due to the fact that the underlying function is not regular enough to have well-defined pointwise values. A weighted version of the Poincar\' e inequality is proposed to address this problem; its optimality is also discussed.