A method is proposed for the compression of hyperspectral signature vectors on severely resource-constrained encoding platforms. The proposed technique, compressive-projection principal component analysis, recovers from random projections not only transform coefficients but also an approximation to the principal-component basis, effectively shifting the computational burden of principal component analysis from the encoder to the decoder. In its use of random projections, the proposed method resembles compressed sensing but differs in that simple linear reconstruction suffices for coefficient recovery. Existing results from perturbation theory are invoked to argue for the robustness under quantization of the eigenvector-recovery process central to the proposed technique, and experimental results demonstrate a significant rate-distortion performance advantage over compressed sensing using a variety of popular bases.