Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Asaf Abas, Nir Sharon

TL;DR

This work tackles the bias dominance in quasi-approximation of scattered data on manifolds by introducing a multiscale error-correction framework. It formalizes nested-scale operators, defines a bias-variance diagnostic (the bias ratio Br), and extends the analysis to manifold-valued targets using geodesic distances. Through extensive numerical experiments on scalar, SO(3), and SPD manifolds, the authors show consistent bias reduction and improved accuracy, often equivalent to increasing the approximation degree by one. The bias-ratio diagnostic offers a practical tool for tuning and deploying bias-aware multiscale approximations in geometric data analysis and manifold learning.

Abstract

We study the bias-variance tradeoff within a multiscale approximation framework. Our approach utilizes a given quasi-approximation operator that is repeatedly applied in 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 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.

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

TL;DR

This work tackles the bias dominance in quasi-approximation of scattered data on manifolds by introducing a multiscale error-correction framework. It formalizes nested-scale operators, defines a bias-variance diagnostic (the bias ratio Br), and extends the analysis to manifold-valued targets using geodesic distances. Through extensive numerical experiments on scalar, SO(3), and SPD manifolds, the authors show consistent bias reduction and improved accuracy, often equivalent to increasing the approximation degree by one. The bias-ratio diagnostic offers a practical tool for tuning and deploying bias-aware multiscale approximations in geometric data analysis and manifold learning.

Abstract

We study the bias-variance tradeoff within a multiscale approximation framework. Our approach utilizes a given quasi-approximation operator that is repeatedly applied in 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 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.

Paper Structure

This paper contains 11 sections, 14 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The multiscale approximation and the bias-variance tradeoff
  3. The multiscale approximation
  4. Bias-Variance Tradeoff for scalar and manifold-valued functions
  5. Bias ratio
  6. Numerical Evaluation
  7. Settings of the Numerical Framework
  8. Multiscale based on linear moving least squares over scattered data
  9. Approximation function with values on SO(3) manifold
  10. Approximating function with values on a SPD manifold
  11. Conclusion

Figures (4)

  • Figure 1: Bias ratio for MLS and multiscale MLS across dataset sizes ($N$) and polynomial degrees (1-3, from left to right, respectively). The MLS graphs are shown in lighter colors, and their multiscale counterparts in darker colors: linear (blue), quadratic (red), and cubic (green). The target function is a smooth scalar-valued function \ref{['Eq-3-3']}.
  • Figure 2: MSE for MLS and multiscale MLS across dataset sizes ($N$) and polynomial degrees. The MLS graphs appear in lighter colors, and their multiscale counterparts appear in darker colors: linear (blue), quadratic (red), and cubic (green). The target function is a smooth scalar-valued function \ref{['Eq-3-3']}.
  • Figure 3: Comparing Shepard (red) and its multiscale (blue) across different noise levels (standard deviation $\sigma$). The target function is a noisy $\operatorname{SO}(3)$-valued function \ref{['Eq-3-4']}.
  • Figure 4: Comparing Shepard (red) and its multiscale (blue) across different signal-to-noise ratios. The target function is a noisy SPD-valued function \ref{['Eq-3-2']}.