When Lie Groups Meet Hyperspectral Images: Equivariant Manifold Network for Few-Shot HSI Classification

Hyperspectral imagery (HSI) offers rich spectral signatures and fine-grained spatial structures for remote sensing, but practical HSI classification is often constrained by scarce labels and complex geometric disturbances, including translation, rotation, scaling, and shear. Existing deep models are typically developed under Euclidean assumptions and rely on data-hungry training pipelines, which makes them brittle in the few-shot regime. To address this challenge, we propose EMNet, a Lie-group-based Equivariant Manifold Network for few-shot HSI classification that explicitly encodes geometric invariance and improves discriminative accuracy. EMNet couples an SE(2)-based Equivariance-Guided Module (EGM) to enforce equivariance to translations and rotations with an affine Lie-group-based Characteristic Filtering Convolution (CFC) that models scaling and shearing on the feature manifold while adaptively suppressing redundant responses. Extensive experiments on WHU-Hi-HongHu, Houston2013, and Indian Pines demonstrate state-of-the-art performance with competitive complexity, achieving OAs of 95.77% (50 samples/class), 97.37% (50 samples/class), and 96.09% (5% labeled samples), respectively, and yielding up to +3.34% OA, +6.01% AA, and +4.14% Kappa over the strong DGPF-RENet baseline. Under a stricter 25-samples-per-class protocol with 10 repeated random hold-out splits, EMNet consistently improves the mean accuracy while exhibiting lower variance, indicating better stability to sampling uncertainty. On the city-scale Xiongan New Area dataset with extreme long-tail imbalance (1580 × 3750 pixels, 256 bands, and 5.925 M labeled pixels), EMNet further boosts OA from 85.89% to 93.77% under the 1% labeled-sample protocol, highlighting robust generalization for large-area mapping. Beyond point estimates, we report mean ± SD/SE across repeated splits and provide rigorous statistical validation by computing Yule's Q statistic for class-wise behavior similarity, performing the Friedman test with Nemenyi post hoc comparisons for multi-method ranking significance, and presenting 95% confidence intervals together with Cohen's d effect sizes to quantify practical improvement.