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We compare and contrast from a geometric perspective a number of low-dimensional signal models that support stable information-preserving dimensionality reduction. We consider sparse and compressible signal models for deterministic and random signals, structured sparse and compressible signal models, point clouds, and manifold signal models. Each model has a particular geometrical structure that enables signal information in to be stably preserved via a simple linear and nonadaptive projection to a much lower dimensional space whose dimension either is independent of the ambient dimension at best or grows logarithmically with it at worst. As a bonus, we point out a common misconception related to probabilistic compressible signal models, that is, that the generalized Gaussian and Laplacian random models do not support stable linear dimensionality reduction.
Alfio Quarteroni, Francesco Regazzoni, Stefano Pagani
Daniel Maria Busiello, Giorgio Nicoletti, Amos Maritan