Classification and Reconstruction of High-Dimensional Signals from Low-Dimensional Noisy Features in the Presence of Side Information

This paper offers a characterization of fundamental limits in the classification and reconstruction of high-dimensional signals from low-dimensional features, in the presence of side information. In particular, we consider a scenario where a decoder has access both to noisy linear features of the signal of interest and to noisy linear features of the side information signal; while the side information may be in a compressed form, the objective is recovery or classification of the primary signal, not the side information. We assume the signal of interest and the side information signal are drawn from a correlated mixture of distributions/components, where each component associated with a specific class label follows a Gaussian mixture model (GMM). By considering bounds to the misclassification probability associated with the recovery of the underlying class label of the signal of interest, and bounds to the reconstruction error associated with the recovery of the signal of interest itself, we then provide sharp sufficient and/or necessary conditions for the phase transition of these quantities in the low-noise regime. These conditions, which are reminiscent of the well-known Slepian-Wolf and Wyner-Ziv conditions, are a function of the number of linear features extracted from the signal of interest, the number of linear features extracted from the side information signal, and the geometry of these signals and their interplay. Our framework, which also offers a principled mechanism to integrate side information in high-dimensional data problems, is also tested in the context of imaging applications. In particular, we report state-of-the-art results in compressive hyperspectral imaging applications, where the accompanying side information is a conventional digital photograph.