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As the important component of the Earth observation system, hyperspectral imaging satellites provide high-fidelity and enriched information for the formulation of related policies due to the powerful spectral measurement capabilities. However, the transmission speed of the satellite downlink has become a major bottleneck in certain applications, such as disaster monitoring and emergency mapping, which demand a fast response ability. We propose an efficient AI-enabled Satellite Edge Computing paradigm for hyperspectral image classification, facilitating the satellites to attain autonomous decision-making. To accommodate the resource constraints of satellite platforms, the proposed method adopts a lightweight, non-deep learning framework integrated with a few-shot learning strategy. Moreover, onboard processing on satellites could be faced with sensor failure and scan pattern errors, which result in degraded image quality with bad/misaligned pixels and mixed noise. To address these challenges, we develop a novel two-stage pixel-wise label propagation scheme that utilizes only intrinsic spectral features at the single pixel level without the necessity to consider spatial structural information as requested by deep neural networks. In the first stage, initial pixel labels are obtained by propagating selected anchor labels through the constructed anchor-pixel affinity matrix. Subsequently, a top-k pruned sparse graph is generated by directly computing pixel-level similarities. In the second stage, a closed-form solution derived from the sparse graph is employed to replace iterative computations. Furthermore, we developed a rank constraint-based graph clustering algorithm to determine the anchor labels.