Sobolev-Regularized Objective Functions for Robust Pairwise Alignment of Functional Data

2026-04-13

AI Summary

This work addresses the challenges in functional data registration—specifically, the sensitivity to noise due to reliance on derivatives and the difficulty in disentangling phase and amplitude variability—by proposing a robust, derivative-free registration framework. The method formulates an objective function in the original function space using second-order Sobolev regularization, parameterizes warping functions via the centered log-ratio (CLR) transformation, and enforces strict monotonicity and non-degeneracy of diffeomorphic maps by jointly penalizing velocity and acceleration. It accommodates four L²-type mismatch functionals and finite-dimensional approximations, enabling unconstrained, efficient optimization. Theoretically, the existence of an optimal warping function and the asymptotic consistency of the estimator are established, achieving markedly improved noise robustness while preserving computational scalability.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Functional data registration is a critical challenge in modern statistics, essential for separating phase variability from amplitude variability. While derivative-based frameworks offer mathematically elegant solutions, their dependence on signal velocities renders them susceptible to additive noise. This study proposes and evaluates a family of robust, Sobolev-regularized objective functions for the pairwise alignment of functional data, operating entirely within the original function space to avoid the need for numerical differentiation of the data. We define our optimization over a second-order Sobolev space and utilize the Centered Log-Ratio (CLR) transform to represent the warping functions. By penalizing both the velocity and acceleration of the centered log-derivative, this geometric approach preempts degenerate "pinching" artifacts and ensures the resulting warps are strictly monotonic, valid diffeomorphisms. In practice, this allows for highly efficient, unconstrained optimization within a finite-dimensional space. We systematically investigate four distinct pairwise data mismatch formulations: a Standard L2 baseline, a Symmetric L2 formulation, an Isometry (L2-preserving) mapping, and a Jacobian-weighted L2 functional. We establish robust theoretical foundations for these methods, proving the existence of optimal warps and the asymptotic consistency of the finite-dimensional estimators. Our results demonstrate that this CLR-regularized framework offers a powerful, computationally scalable, and noise-robust alternative to traditional derivative-based registration.

Problem

• functional data registration
• phase variability
• amplitude variability
• noise robustness
• Sobolev regularization

Innovation

• Sobolev regularization
• functional data registration
• Centered Log-Ratio transform
• diffeomorphic warping
• noise robustness