Multiscale Approximation as a Bias-Reducing Strategy with Applications to Manifold-Valued Functions

TL;DR

This paper introduces a multiscale approximation framework that reduces bias in scattered-data approximation, especially for manifold-valued functions, by employing a new bias measure and an error-correction scheme.

Contribution

It develops a novel bias measure called the bias ratio and demonstrates multiscale approximation as an effective bias-reduction strategy for quasi-interpolation operators.

Findings

Multiscale approximation effectively reduces bias in scattered-data approximation.

The bias ratio provides a quantitative measure of bias reduction.

Applicable to manifold-valued functions and general quasi-interpolation operators.

Abstract

We study the bias-variance tradeoff within a multiscale approximation framework. Our approach uses a given quasi-interpolation operator, which is repeatedly applied within an error-correction scheme over a hierarchical data structure. We introduce a new bias measure, the bias ratio, to quantitatively assess the improvements afforded by multiscale approximations and demonstrate that this strategy effectively reduces the bias component of the approximation error, thereby providing an operator-level bias reduction framework for addressing scattered-data approximation problems. Our findings establish multiscale approximation as a bias-reduction methodology applicable to general quasi-interpolation operators, including applications to manifold-valued functions.