Low-dimensional models for dimensionality reduction and signal recovery: A geometric perspective

• Published in: Proceedings of the IEEE, vol. 98, num. 6, p. 959–971
• Publication date: 2010

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.

Keywords: Compression ; compressive sensing ; dimensionality reduction ; manifold ; point cloud ; sparsity ; stable embedding ; Hidden Markov-Models ; Simultaneous Sparse Approximation ; Restricted Isometry Property ; Random Projections ; Wavelet-Domain ; Algorithms ; Regression ; Manifolds ; Pursuit ; Subspaces

Reference

• EPFL-ARTICLE-150570
• doi:10.1109/JPROC.2009.2038076
• View record in Web of Science

Record created on 2010-08-26, modified on 2012-03-21