메뉴 건너뛰기




Volumn 57, Issue 10, 2011, Pages 6894-6919

Precise stability phase transitions for ℓ1 minimization: A unified geometric framework

Author keywords

1 minimization; Compressed sensing; convex geometry; Grassmann angle; phase transition; robustness; sparse recovery; stability

Indexed keywords

BALANCEDNESS; COMPRESSED SENSING; CONVEX GEOMETRY; FUNDAMENTAL PROPERTIES; GEOMETRIC FRAMEWORK; GRASSMANN ANGLE; LINEAR SUBSPACE; NOISY MEASUREMENTS; NULL SPACE; PERFORMANCE BOUNDS; POLYTOPES; SHARP BOUNDS; SIGNAL RECOVERY; SIGNAL VECTORS; SPARSE RECOVERY; SPARSE SIGNALS; SUFFICIENT CONDITIONS; UNDER-DETERMINED; UNIFIED FRAMEWORK;

EID: 80053958456     PISSN: 00189448     EISSN: None     Source Type: Journal    
DOI: 10.1109/TIT.2011.2165825     Document Type: Article
Times cited : (28)

References (45)
  • 1
    • 0039127194 scopus 로고
    • Random projections of regular simplices
    • F. Affentranger and R. Schneider, "Random projections of regular simplices," Discrete Comput. Geom., vol. 7, no. 3, pp. 219-226, 1992.
    • (1992) Discrete Comput. Geom. , vol.7 , Issue.3 , pp. 219-226
    • Affentranger, F.1    Schneider, R.2
  • 2
    • 79957444557 scopus 로고    scopus 로고
    • Compressed sensing: How sharp is the restricted isometry property?
    • J. D. Blanchard, C. Cartis, and J. Tanner, "Compressed sensing: How sharp is the restricted isometry property?," SIAM Rev., vol. 53, no. 1, pp. 105-125, 2011.
    • (2011) SIAM Rev. , vol.53 , Issue.1 , pp. 105-125
    • Blanchard, J.D.1    Cartis, C.2    Tanner, J.3
  • 3
    • 55649115527 scopus 로고    scopus 로고
    • A simple proof of the restricted isometry property for random matrices
    • R. Baraniuk, M.Davenport, R. DeVore, and M.Wakin, "A simple proof of the restricted isometry property for random matrices," Construct. Approx., vol. 28, no. 3, pp. 253-263, 2008.
    • (2008) Construct. Approx. , vol.28 , Issue.3 , pp. 253-263
    • Baraniuk, R.1    Davenport, M.2    DeVore, R.3    Wakin, M.4
  • 4
    • 0040816160 scopus 로고
    • Intrinsic volumes and lattice points of crosspolytopes
    • U. Betke and M. Henk, "Intrinsic volumes and lattice points of crosspolytopes," Monatshefte fr Mathematik, vol. 115, no. 1-2, pp. 27-33, 1993.
    • (1993) Monatshefte fr Mathematik , vol.115 , Issue.1-2 , pp. 27-33
    • Betke, U.1    Henk, M.2
  • 6
    • 84878104490 scopus 로고    scopus 로고
    • Compressive sampling
    • Eur. Math. Soc., Zürich
    • E. J. Candès, "Compressive sampling," in International Congress of Mathematicians, 2006, vol. III, pp. 1433-1452, Eur. Math. Soc., Zürich.
    • (2006) International Congress of Mathematicians , vol.3 , pp. 1433-1452
    • Candès, E.J.1
  • 7
    • 57349181932 scopus 로고    scopus 로고
    • Compressed sensing and best -term approximation
    • A. Cohen,W. Dahmen, and R. DeVore, "Compressed sensing and best -term approximation," J. Amer. Math. Soc., vol. 22, no. 2009, pp. 211-231, 2008.
    • (2008) J. Amer. Math. Soc. , vol.22 , Issue.2009 , pp. 211-231
    • Cohen, A.1    Dahmen, W.2    DeVore, R.3
  • 8
    • 0000364765 scopus 로고
    • Robust modeling with erratic data
    • J. F. Claerbout and F.Muir, "Robust modeling with erratic data," Geophysics, vol. 38, no. 5, pp. 826-844, 1973.
    • (1973) Geophysics , vol.38 , Issue.5 , pp. 826-844
    • Claerbout, J.F.1    Muir, F.2
  • 9
    • 33745604236 scopus 로고    scopus 로고
    • Stable signal recovery from incomplete and inaccuratemeasurements
    • E. J. Candès, J. Romberg, and T. Tao, "Stable signal recovery from incomplete and inaccuratemeasurements," Commun. of Pure and Appl. Math., vol. 59, pp. 1207-1223, 2005.
    • (2005) Commun. of Pure and Appl. Math. , vol.59 , pp. 1207-1223
    • Candès, E.J.1    Romberg, J.2    Tao, T.3
  • 10
    • 31744440684 scopus 로고    scopus 로고
    • Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
    • Feb
    • E. J. Candès, J. Romberg, and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information," IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489-509, Feb. 2006.
    • (2006) IEEE Trans. Inf. Theory , vol.52 , Issue.2 , pp. 489-509
    • Candès, E.J.1    Romberg, J.2    Tao, T.3
  • 11
    • 29144439194 scopus 로고    scopus 로고
    • Decoding by linear programming
    • Dec
    • E. J. Candès and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory , vol. 51, no. 12, pp. 4203-4215, Dec. 2005.
    • (2005) IEEE Trans. Inf. Theory , vol.51 , Issue.12 , pp. 4203-4215
    • Candès, E.J.1    Tao, T.2
  • 13
    • 33645712892 scopus 로고    scopus 로고
    • Compressed sensing
    • Apr
    • D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289-1306, Apr. 2006.
    • (2006) IEEE Trans. Inf. Theory , vol.52 , Issue.4 , pp. 1289-1306
    • Donoho, D.L.1
  • 14
    • 33646479497 scopus 로고    scopus 로고
    • High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension
    • D. Donoho, "High-dimensional centrally symmetric polytopes with neighborliness proportional to dimension," Discrete and Comput. Geom., vol. 35, no. 4, pp. 617-652, 2006.
    • (2006) Discrete and Comput. Geom. , vol.35 , Issue.4 , pp. 617-652
    • Donoho, D.1
  • 15
    • 22144471532 scopus 로고    scopus 로고
    • Neighborliness of randomly projected simplices in high dimensions
    • D. L. Donoho and J. Tanner, "Neighborliness of randomly projected simplices in high dimensions," in Proc. Natl. Acad. Sci. USA, 2005, vol. 102, no. 27, pp. 9452-9457.
    • (2005) Proc. Natl. Acad. Sci. USA , vol.102 , Issue.27 , pp. 9452-9457
    • Donoho, D.L.1    Tanner, J.2
  • 16
    • 73149095252 scopus 로고    scopus 로고
    • Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing
    • D. Donoho and J. Tanner, "Observed universality of phase transitions in high-dimensional geometry, with implications for modern data analysis and signal processing," Philosoph. Trans. of the Roy. Soc. A, vol. 367, pp. 4273-4293, 2009.
    • (2009) Philosoph. Trans. of the Roy. Soc. A , vol.367 , pp. 4273-4293
    • Donoho, D.1    Tanner, J.2
  • 17
    • 0038718564 scopus 로고    scopus 로고
    • On sparse representation in pairs of bases
    • Jun
    • A. Feuer and A. Nemirovski, "On sparse representation in pairs of bases," IEEE Trans. Inf. Theory, vol. 49, no. 6, pp. 1579-1581, Jun. 2003.
    • (2003) IEEE Trans. Inf. Theory , vol.49 , Issue.6 , pp. 1579-1581
    • Feuer, A.1    Nemirovski, A.2
  • 18
    • 0001436319 scopus 로고
    • A limit theorem for the norm of random matrices
    • S. Geman, "A limit theorem for the norm of random matrices," Ann. Probabil., vol. 8, no. 2, pp. 252-261, 1980.
    • (1980) Ann. Probabil. , vol.8 , Issue.2 , pp. 252-261
    • Geman, S.1
  • 19
    • 22144445039 scopus 로고
    • Grassmann angles of convex polytopes
    • B. Grünbaum, "Grassmann angles of convex polytopes," Acta Math., vol. 121, pp. 293-302, 1968.
    • (1968) Acta Math. , vol.121 , pp. 293-302
    • Grünbaum, B.1
  • 20
    • 38849143845 scopus 로고    scopus 로고
    • 2nd ed. New York: Springer-Verlag, Prepared and with a preface by Volker Kaibel, Victor Klee and Gnterü M. Ziegler
    • B. Grönbaum, Convex Polytopes Volume 221 of Graduate Texts in Mathematics, 2nd ed. New York: Springer-Verlag, 2003, Prepared and with a preface by Volker Kaibel, Victor Klee and Gnterü M. Ziegler.
    • (2003) Convex Polytopes Volume 221 of Graduate Texts in Mathematics
    • Grönbaum, B.1
  • 21
    • 0040221640 scopus 로고
    • Gitterpunktanzahl im simplex und willssche vermutung"
    • H. Hadwiger, "Gitterpunktanzahl im simplex und willssche vermutung," Math. Ann., vol. 239, pp. 271-288, 1979.
    • (1979) Math. Ann. , vol.239 , pp. 271-288
    • Hadwiger, H.1
  • 22
    • 33748575920 scopus 로고    scopus 로고
    • Signal reconstruction from noisy random projections
    • J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Inf. Theory.
    • IEEE Trans. Inf. Theory
    • Haupt, J.1    Nowak, R.2
  • 23
    • 0033273787 scopus 로고    scopus 로고
    • Random projections of regular polytopes
    • K. Böröczky, Jr. and M. Henk, "Random projections of regular polytopes," Arch. Math. (Basel), vol. 73, no. 6, pp. 465-473, 1999.
    • (1999) Arch. Math. (Basel) , vol.73 , Issue.6 , pp. 465-473
    • Böröczky Jr., K.1    Henk, M.2
  • 24
    • 38349077426 scopus 로고    scopus 로고
    • A remark on compressed sensing
    • Nov
    • B. S. Kashin andV.N. Temlyakov, "A remark on compressed sensing," Mathemat. Notes, vol. 82, no. 5, pp. 748-755, Nov. 2007.
    • (2007) Mathemat. Notes , vol.82 , Issue.5 , pp. 748-755
  • 26
    • 33746923055 scopus 로고    scopus 로고
    • How neighborly can a centrally symmetric polytope be?
    • N. Linial and I. Novik, "How neighborly can a centrally symmetric polytope be?," Discrete and Comput. Geom., vol. 36, no. 6, pp. 273-281, 2006.
    • (2006) Discrete and Comput. Geom. , vol.36 , Issue.6 , pp. 273-281
    • Linial, N.1    Novik, I.2
  • 27
    • 84976038751 scopus 로고
    • Non-linear angle-sum relations for polyhedral cones and polytopes
    • P. McMullen, "Non-linear angle-sum relations for polyhedral cones and polytopes," Math. Proc. Cambridge Philos. Soc., vol. 78, no. 2, pp. 247-261, 1975.
    • (1975) Math. Proc. Cambridge Philos. Soc. , vol.78 , Issue.2 , pp. 247-261
    • McMullen, P.1
  • 28
    • 0000263234 scopus 로고
    • Distributions of eigenvalues for some sets of random matrices
    • V. A. Marcenko and L. A. Pastur, "Distributions of eigenvalues for some sets of random matrices," Math. USSR-Sbornik, vol. 1, pp. 457-483, 1967.
    • (1967) Math. USSR-Sbornik , vol.1 , pp. 457-483
    • Marcenko, V.A.1    Pastur, L.A.2
  • 30
    • 80053998645 scopus 로고    scopus 로고
    • [Online]. Available:
    • [Online]. Available: http://www.dsp.ece.rice.edu/cs/
  • 33
    • 0001629248 scopus 로고
    • The smallest eigenvalue of a large dimensional wishart matrix
    • J. W. Silverstein, "The smallest eigenvalue of a large dimensional wishart matrix," The Annals of Probability, vol. 13, pp. 1364-1368, 1985.
    • (1985) The Annals of Probability , vol.13 , pp. 1364-1368
    • Silverstein, J.W.1
  • 39
    • 0038860738 scopus 로고
    • Asymptotic behavior of the number of faces of randompolyhedra and the neighborliness problem
    • A. M. Vershik and P. V. Sporyshev, "Asymptotic behavior of the number of faces of randompolyhedra and the neighborliness problem," Selecta Mathemat. Soviet., vol. 11, no. 2, pp. 181-201, 1992.
    • (1992) Selecta Mathemat. Soviet. , vol.11 , Issue.2 , pp. 181-201
    • Vershik, A.M.1    Sporyshev, P.V.2
  • 42
    • 70449484735 scopus 로고    scopus 로고
    • On sharp performance bounds for robust sparse signal recoveries
    • W. Xu and B. Hassibi, "On sharp performance bounds for robust sparse signal recoveries," in Proc. Int. Symp. Information Theory, 2009.
    • (2009) Proc. Int. Symp. Information Theory
    • Xu, W.1    Hassibi, B.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.