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Principal component analysis (PCA) is often central to dimensionality reduction and compression in many applications, yet its data-dependent nature as a transform computed via expensive eigendecomposition often hinders its use in severely resource-constrained settings such as satellite-borne sensors. A process is presented that effectively shifts the computational burden of PCA from the resource-constrained encoder to a presumably more capable base-station decoder. The proposed approach, compressive-projection PCA (CPPCA), is driven by projections at the sensor onto lower-dimensional subspaces chosen at random, while the CPPCA decoder, given only these random projections, recovers not only the coefficients associated with the PCA transform, but also an approximation to the PCA transform basis itself. An analysis is presented that extends existing Rayleigh-Ritz theory to the special case of highly eccentric distributions; this analysis in turn motivates a reconstruction process at the CPPCA decoder that consists of a novel eigenvector reconstruction based on a convex-set optimization driven by Ritz vectors within the projected subspaces. As such, CPPCA constitutes a fundamental departure from traditional PCA in that it permits its excellent dimensionality-reduction and compression performance to be realized in an light-encoder/heavy-decoder system architecture. In experimental results, CPPCA outperforms a multiple-vector variant of compressed sensing for the reconstruction of hyperspectral data.