@article{2410dea45b0a433d8059d26b4315f2c8,
title = "Low-dimensional models for dimensionality reduction and signal recovery: A geometric perspective",
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 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 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 a bonus, 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.",
keywords = "Compression, Compressive sensing, Dimensionality reduction, Manifold, Point cloud, Sparsity, Stable embedding",
author = "Baraniuk, {Richard G.} and Volkan Cevher and Wakin, {Michael B.}",
note = "Funding Information: Manuscript received May 7, 2009; accepted October 17, 2009. Date of publication April 29, 2010; date of current version May 19, 2010. This work was supported by the National Science Foundation under Grants CCF-0431150, CCF-0728867, CCF-0830320, and DMS-0603606; by the Defense Advanced Research Projects Agency (DARPA) under Grant HR0011-08-1-0078; by DARPA/Office of Naval Research (ONR) under Grant N66001-08-1-2065; by ONR under Grants N00014-07-1-0936 and N00014-08-1-1112; by the Air Force Office of Scientific Research under Grant FA9550-07-1-0301; by the Army Research Office/Multidisciplinary University Research Initiative (ARO MURI) under Grants W311NF-07-1-0185 and W911NF-09-1-0383; and by the Texas Instruments Leadership University Program. R. G. Baraniuk is with the Department of Electrical and Computer Engineering, Rice University, Houston, TX 77001 USA (e-mail: richb@rice.edu). V. Cevher is with the Department of Electrical and Computer Engineering, Rice University, Houston, TX 77001 USA. He is also with the School of Engineering, Ecole Polytechnique F{\'e}d{\'e}rale de Lausanne (LIONS/IDIAP), 1015 Lausanne, Switzerland (e-mail: volkan@rice.edu; volkan.cevher@epfl.ch). M. B. Wakin is with the Division of Engineering, Colorado School of Mines, Golden, CO 80401 USA (e-mail: mwakin@mines.edu). Funding Information: Dr. Baraniuk received a NATO postdoctoral fellowship from NSERC in 1992, the National Young Investigator award from the National Science Foundation in 1994, a Young Investigator Award from the Office of Naval Research in 1995, the Rosenbaum Fellowship from the Isaac Newton Institute of Cambridge University in 1998, the C. Holmes MacDonald National Outstanding Teaching Award from Eta Kappa Nu in 1999, the Charles Duncan Junior Faculty Achievement Award from Rice in 2000, the University of Illinois ECE Young Alumni Achievement Award in 2000, the George R. Brown Award for Superior Teaching at Rice in 2001, 2003, and 2006, the Tech Museum Laureate Award from the Tech Museum of Innovation in 2006, the Hershel M. Rich Invention Award from Rice in 2007, the Wavelet Pioneer Award from SPIE in 2008, the Internet Pioneer Award from the Berkman Center for Internet and Society at Harvard Law School in 2008, and the World Technology Network Education Award in 2009. In 2007, he was selected as one of Edutopia Magazine{\textquoteright}s Daring Dozen educators, and the Rice single-pixel compressive camera was selected by MIT Technology Review Magazine as a TR10 Top 10 Emerging Technology. He was elected a Fellow of AAAS in 2009.",
year = "2010",
month = jun,
doi = "10.1109/JPROC.2009.2038076",
language = "English (US)",
volume = "98",
pages = "959--971",
journal = "Proceedings of the IEEE",
issn = "0018-9219",
publisher = "Institute of Electrical and Electronics Engineers Inc.",
number = "6",
}