Bias Reduction by Multiscale Quasi-Interpolation for Scalar- and Manifold-Valued Functions

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 use the bias ratio, the fraction of mean squared error attributable to squared bias, to assess multiscale estimators. We also provide a theoretical mechanism for the observed bias reduction: for scalar-valued linear quasi-interpolation with zero-mean additive noise, the statistical bias of the multiscale estimator is exactly the deterministic residual produced by the multiscale error-correction scheme. A residual-contraction condition then yields bias reduction, and a spectral-filter model gives a monotone nonincreasing integrated bias ratio with respect to the number of levels. Within the same model, we obtain an exact per-level error increment whose bias--variance crossover gives a criterion for when an additional level is beneficial. Under a local consistency assumption, a tangent-space analysis extends the residual-bias mechanism to manifold-valued constructions and yields a leading-order bias--variance decomposition. Our findings establish multiscale approximation as a bias-reduction methodology applicable to general quasi-interpolation operators, including applications to manifold-valued functions.