Fermat Active Laplace Learning for Semi-Supervised Hyperspectral Image Classification

Hyperspectral image classification is complicated by mixed pixels, spectral ambiguity, class imbalance, and limited annotations. Most current classifiers encode a pixel or patch as a deterministic vector and apply a linear or multilayer softmax head. Although effective for discrimination, this representation does not directly expose how mixed or uncertain a sample is. This paper presents \method, a classical density-matrix representation learning framework for hyperspectral images. The spectral bands are divided into groups and each group is mapped to a positive semi-definite, Hermitian, trace-normalized matrix state. A composable stack of spectral, spatial, and inter-group transitions then updates the states while repeatedly projecting them back to the valid state set. Instead of flattening the final features, \method\ aggregates the group states and compares them with learnable class-prototype density matrices through Uhlmann fidelity. The normalized eigenspectrum, von Neumann entropy, purity, and prototype fidelity provide sample-level diagnostics that are unavailable from a conventional vector head. On Indian Pines, ten runs yield an overall accuracy of $96.20\pm0.70\%$, an average accuracy of $95.57\pm1.29\%$, and a kappa coefficient of $95.66\pm0.80\%$. On WHU-Hi-LongKou, the best of ten runs reaches $97.52\%$ overall accuracy. Classification maps and feature projections show that the transition stack produces compact and better separated class structures. The results support constrained matrix-state learning as a practical alternative to vector-only hyperspectral classification without requiring quantum hardware.

Hyperspectral imaging enables fine-grained recognition of materials by capturing rich spectral signatures, but learning robust classifiers is challenging due to high dimensionality, spectral redundancy, limited labeled data, and strong domain shifts. Beyond earth observation, labeled HSI data is often scarce and imbalanced, motivating compact models for generic hyperspectral classification across diverse acquisition regimes. We propose the lightweight Spectral-Spatial Fusion Transformer (SSFT), which factorizes representation learning into spectral and spatial pathways and integrates them via cross-attention to capture complementary wavelength-dependent and structural information. We evaluate our SSFT on the challenging HSI-Benchmark, a heterogeneous multi-dataset benchmark covering earth observation, fruit condition assessment, and fine-grained material recognition. SSFT achieves state-of-the-art overall performance, ranking first while using less than 2% of the parameters of the previous leading method. We further evaluate transfer to the substantially larger SpectralEarth benchmark under the official protocol, where SSFT remains competitive despite its compact size. Ablation studies show that both spectral and spatial pathways are crucial, with spatial modeling contributing most, and that SSFT remains robust without data augmentation.

Semi-supervised classification, where unlabeled data are massive but labeled data are limited, often arises in machine learning applications. We address this challenge under high-dimensional data by leveraging the manifold and cluster assumptions. Based on the Fermat distance, a density-sensitive metric that naturally encodes the cluster assumption, we propose the weighted $k$-nearest neighbors (NN) classifier and multidimensional scaling (MDS)-induced classifiers. The use of MDS with a large target dimension allows the effective application of linear classifiers to complex manifold data. Theoretically, we derive a sharp lower bound for the expected excess risk within clusters and prove that the weighted $k$-NN classifier utilizing the true Fermat distance is minimax optimal. Furthermore, we explicitly quantify the utility of unlabeled data by showing that the error arising from estimating the Fermat distance decays exponentially with the pooled sample size. Such a rate is much faster than the related rates in the literature. Extensive experiments on synthetic and real datasets demonstrate competitive or superior performance of our approaches compared to state-of-the-art graph-based semi-supervised classifiers.