Learning the regularity of multivariate functional data

TL;DR

This paper introduces a simple estimator for the local regularity of multivariate functional data surfaces, accounting for measurement errors and randomness, with applications in deformation estimation and surface reconstruction.

Contribution

It proposes a novel estimator for surface regularity in multivariate functional data, including anisotropy detection and non-asymptotic error bounds, with practical applications.

Findings

Derived exponential bounds for regularity estimators.

Developed an anisotropy indicator with risk bounds.

Constructed minimax optimal surface reconstruction estimators.

Abstract

Combining information both within and between sample realizations, we propose a simple estimator for the local regularity of surfaces in the functional data framework. The independently generated surfaces are measured with errors at possibly random discrete times. Non-asymptotic exponential bounds for the concentration of the regularity estimators are derived. An indicator for anisotropy is proposed and an exponential bound of its risk is derived. Two applications are proposed. We first consider the class of multi-fractional, bi-dimensional, Brownian sheets with domain deformation, and study the nonparametric estimation of the deformation. As a second application, we build minimax optimal, bivariate kernel estimators for the reconstruction of the surfaces.