Multiscale Approximation as a Bias-Reducing Strategy for Scalar and Manifold-Valued Functions

Abstract: We study the bias-variance tradeoff within a multiscale approximation framework. Our approach utilizes a given quasi-approximation operator, repeatedly applied in an error-correction scheme over a hierarchical data structure. We introduce a new bias measurement, the bias ratio, to quantitatively assess the improvements made by multiscale approximations and demonstrate that this multiscale strategy effectively reduces the bias component of the approximation error, thereby providing a more flexible and robust framework for addressing scattered data approximation problems. Our findings exhibit consistent bias decay across various scenarios, including applications to manifold-valued functions.