Semi-supervised Classification for Functional Data with Application to Astronomical Spectra Analysis

TL;DR

This paper extends semi-supervised classification methods to functional data, introducing new classifiers based on the Fermat distance, with theoretical guarantees and application to astronomical spectra.

Contribution

It develops novel semi-supervised classifiers for functional data using the Fermat distance, with efficient estimation procedures and theoretical convergence guarantees.

Findings

Unlabeled data improves accuracy only with sufficiently high sampling rate.

Proposed classifiers outperform existing supervised methods on simulated and real astronomical data.

Established exponential convergence rates for the $k$-NN classifier and Fermat distance consistency.

Abstract

Despite its extensive development for multivariate data, semi-supervised learning remains underdeveloped for functional data. To address this challenge, we extend the Fermat distance, a density-sensitive metric aligning with the semi-supervised setting, to the functional domain. Leveraging the Fermat distance, we propose novel semi-supervised classifiers, including the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. To accommodate massive datasets commonly seen in semi-supervised applications, we design a computationally efficient estimation procedure tailored for discrete and noisy functional observations. Theoretically, we establish exponentially decaying convergence rates of the $k$-NN classifier and the consistency of the estimated Fermat distance. Crucially, our results reveal a phenomenon unique to error-contaminated functional data: Incorporating unlabeled data leads to improved classification accuracy only when the individual sampling rate grows sufficiently fast. Applying our framework to simulated data and a large-scale dataset of Gaia astronomical spectra, we demonstrate that our proposed semi-supervised classifiers uniformly outperform existing supervised benchmarks.