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.

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