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

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.