Compressed sensing over the Grassmann manifold: A unified analytical framework

46th Annual Allerton Conference on Communication, Control, and Computing

Volumn , Issue , 2008, Pages 562-567

  Xu, Weiyu ^a   Hassibi, Babak ^a  

^a California Institute of Technology   (United States)

Author keywords

balancedness; Basis pursuit; Compressed sensing; Convex polytopes; Geometric probability; Grassmann angle; Grassmann manifold; High dimensional integral geometry; L1 optimization; Neighborly polytopes; Random linear subspaces

Indexed keywords

BASIS PURSUIT; COMPRESSED SENSING; CONVEX POLYTOPES; GEOMETRIC PROBABILITY; GRASSMANN ANGLE; GRASSMANN MANIFOLD; HIGH-DIMENSIONAL INTEGRAL GEOMETRY; L1-OPTIMIZATION; NEIGHBORLY POLYTOPES; RANDOM LINEAR SUBSPACES;

COMPUTATIONAL GEOMETRY; LINEAR PROGRAMMING; LINEARIZATION; OPTIMIZATION; SIGNAL RECONSTRUCTION; TRELLIS CODES;

TOPOLOGY;

EID: 64549099406     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/ALLERTON.2008.4797608     Document Type: Conference Paper

References (27)

1. David Donoho, "High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension", Discrete and Computational Geometry, 102 (27), pp. 617-652, 2006, Springer.
2. Emmanuel Candes and Terence Tao, "Decoding by linear programming", IEEE Trans. on Information Theory, 51 (12), pp. 4203-4215, December 2005.
3. David Donoho and Jared Tanner, "Neighborliness of randomly-projected simplices in high dimensions", Proc. National Academy of Sciences, 102 (27), pp. 9452-9457, 2005.
4. Richard Baraniuk, Mark Davenport, Ronald DeVore, and Michael Wakin, "A simple proof of the restricted isometry property for random matrices", To appear in Constructive Approximation, available online at http://www.dsp. ece.rice.edu/cs/.
5. Mark Rudelson and Roman Vershynin, "Geometric approach to error correcting codes and reconstruction of signals", International Mathematical Research Notices, 64, pp. 4019-4041, 2005.
6. A. M. Vershik and P. V. Sporyshev, "Asymptotic Behavior of the Number of Faces of Random Polyhedra and the Neighborliness Problem", Selecta Mathematica Sovietica vol. 11, No. 2 (1992).
7. W. M. Boothby, An Introduction to Differential Manifolds and Riemannian Geometry, 2nd ed. San Diego, CA: Academic, 1986.
8. A. Feuer and A. Nemirovski, "On sparse representation in pairs of bases", IEEE Transactions on Information Theory 49 (6) : 1579-1581 (2003).
9. N. Linial and I. Novik, "How neighborly can a centrally symmetric polytope be?", Discrete and Computational Geometry, 36 (2006) 273-281.
10. Y. Zhang, "When is missing data recoverable", available online at http://www.dsp. ece.rice.edu/cs/.
11. J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections", IEEE Trans. on Information Theory, 52 (9), pp. 4036-4048, September 2006.
12. M. J. Wainwright, "Sharp thresholds for high-dimensional and noisy recovery of sparsity", Proc. Allerton Conference on Communication, Control, and Computing, Monticello, IL, September 2006.
13. A. Cohen, W. Dahmen, and R. DeVore, "Compressed sensing and best k-term approximation". (Preprint, 2006), available online at http://www.dsp. ece.rice.edu/cs/.
14. J. Tropp, M. Wakin, M. Duarte, D. Baron, and R. Baraniuk, "Random filters for compressive sampling and reconstruction". Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), Toulouse, France, May 2006.
15. Branko Grünbaum, Grassmann angles of convex polytopes. Acta Math., 121:pp. 293-302, 1968.
16. Branko Grünbaum, Convex polytopes, volume 221 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 2003. Prepared and with a preface by Volker Kaibel, Victor Klee and GnterüM. Ziegler.
17. Fernando Affentranger and Rolf Schneider. "Random projections of regular simplices". Discrete Comput. Geom., 7 (3) :pp. 219-226, 1992.
18. Károly Böröczky, Jr. and Martin Henk."Random projections of regular polytopes". Arch. Math. (Basel), 73 (6) :pp. 465-473, 1999.
19. L. A. Santaló, Geometría integral en espacios de curvatura constante, Rep. Argetina Publ. Com. Nac. Energí Atómica, Ser. Mat 1, No. 1 (1952)
20. Peter McMullen. "Non-linear angle-sum relations for polyhedral cones and polytopes". Math. Proc. Cambridge Philos. Soc., 78 (2) :pp. 247-261, 1975.
21. Emmanuel Candes, "Compressive sampling", Proc. International Congress of Mathematics, 3, pp. 1433-1452, Madrid, Spain, 2006.
22. Emmanuel Cands, Justin Romberg, and Terence Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information.", IEEE Trans. on Information Theory, 52 (2) pp. 489-509, February 2006.
23. David Donoho, "Compressed sensing", IEEE Trans. on Information Theory, 52 (4), pp. 1289-1306, April 2006
24. M. Stojnic, W. Xu, and B. Hassibi, "Compressed sensing - probabilistic analysis of a null-space characterization", IEEE International Conference on Acoustics, Speech and Signal Processing, Pages:3377-3380, March 31 2008-April 4 2008.
25. M. Stojnic, W. Xu, and B. Hassibi, "Compressed sensing of approximately sparse signals", IEEE International Symposium on Information Theory, July 2008.
26. W. Xu, and B. Hassibi, "Compressed sensing of approximately sparse signals - A Grassman angle approach", under preparation.
27. Boris S. Kashin and Vladimir N. Temlyakov, "A remark on compressed sensing," Mathematical Notes, 82 (5-6), pp. 748-755, Nov. 2007.