We present information-theoretic results on signal classification and reconstruction from compressive measurements. We particularize to data lying near low-dimensional subspaces, which we represent with a nearly low-rank Gaussian mixture model. The low-rank components model the signal subspaces, and additional full-rank noise captures deviation from the subspace structure. For subspace classification, we present TRAIT, a method for choosing compressive measurements that encourages separation between subspaces while preserving intra-class features. Applying TRAIT to the YaleB face database, we demonstrate superior performance over standard methods such as LDA. For signal reconstruction, we present bounds on the MMSE of the estimated signal when we have access to side information that conforms to a subspace structure. In general, the subspace geometry, as well as the degree to which the signals deviate from the subspace model, govern performance.
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