Low-Dimensional Models for Dimensionality Reduction and Signal Recovery: A Geometric Perspective There are many signal techniques that can be used to perform data acquisition, analysis or processing more efficiently and accurately than can be done by most random signal processors.

TLDR: This work compares and contrast from a geometric perspective a number of low-dimensional signal models that support stable information-preserving dimensionality reduction, and points out a common misconception related to probabilistic compressible signal models.

Abstract: 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 deter- ministic 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 to be stably preserved via a simple linear and nonadaptive projection to a much lower dimensional space; in each case the projection dimension is independent of the signal's ambient dimension at best or grows logarithmically with it at worst. As abonus, we point out a common misconception related to probabilistic compressible signal models, namely, by showing that the oft-used generalized Gaussian and Laplacian models do not support stable linear dimensionality reduction.