Semi-supervised Classification for Noisy Functional Data with Application to Astronomical Spectra
Despite its extensive development for multivariate data, semi-supervised learning remains underdeveloped for functional data, especially under discrete and noisy observations. We develop a density-sensitive semi-supervised framework for functional data supported on a low-dimensional manifold by adapting the Fermat distance to reconstructed trajectories. The resulting pairwise distances are used to construct a weighted $k$-nearest-neighbor classifier and multidimensional-scaling-based 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 that incorporating unlabeled data may not lead to improved classification accuracy without a sufficiently fast-growing individual sampling rate, precisely due to discrete and noisy observations. In most simulation settings satisfying the manifold and cluster assumptions, the proposed classifiers outperform the supervised benchmarks considered; in the Gaia spectra analysis, they attain higher agreement with high-confidence proxy labels.