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Volumn 22, Issue 1, 2009, Pages 1-53

Counting faces of randomly projected polytopes when the projection radically lowers dimension

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EID: 57349138185     PISSN: 08940347     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0894-0347-08-00600-0     Document Type: Article
Times cited : (297)

References (34)
  • 1
    • 0039127194 scopus 로고
    • Random projections of regular simplices
    • MR1149653 92k:52008
    • Fernando Affentranger and Rolf Schneider, Random projections of regular simplices, Discrete Comput. Geom. 7 (1992), no. 3, 219-226. MR1149653 (92k:52008)
    • (1992) Discrete Comput. Geom. , vol.7 , Issue.3 , pp. 219-226
    • Affentranger, F.1    Schneider, R.2
  • 3
    • 0033273787 scopus 로고    scopus 로고
    • Random projections of regular polytopes
    • MR1725183 2001b:52004
    • Károly Böröczky, Jr. and Martin Henk, Random projections of regular polytopes, Arch. Math. (Basel) 73 (1999), no. 6, 465-473. MR1725183 (2001b:52004)
    • (1999) Arch. Math. (Basel) , vol.73 , Issue.6 , pp. 465-473
    • Böröczky, K.J.1    Henk, M.2
  • 4
    • 31744440684 scopus 로고    scopus 로고
    • Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information
    • MR2236170 2007e:94020
    • Emmanuel J. Candès, Justin Romberg, and Terence Tao, Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information, IEEE Trans. Inform. Theory 52 (2006), no. 2, 489-509. MR2236170 (2007e:94020)
    • (2006) IEEE Trans. Inform. Theory , vol.52 , Issue.2 , pp. 489-509
    • Candès, E.J.1    Romberg, J.2    Tao, T.3
  • 5
    • 29144439194 scopus 로고    scopus 로고
    • Decoding via linear programming
    • MR2243152 2007b:94313
    • Emmanuel J. Candès and Terence Tao, Decoding via linear programming, IEEE Trans. Inform. Theory 51 (2005), no. 12, 4203-4215. MR2243152 (2007b:94313)
    • (2005) IEEE Trans. Inform. Theory , vol.51 , Issue.12 , pp. 4203-4215
    • Candès, E.J.1    Tao, T.2
  • 6
    • 33947416035 scopus 로고    scopus 로고
    • Near optimal signal recovery from random projections and universal encoding strategies
    • MR2300700
    • -, Near optimal signal recovery from random projections and universal encoding strategies, IEEE. Trans. Info. Thry. 52 (2006), 5406-5425. MR2300700
    • (2006) IEEE. Trans. Info. Thry , vol.52 , pp. 5406-5425
  • 7
    • 77950566092 scopus 로고
    • Introduction to approximation theory
    • MR1656150 99f:41001
    • E. W. Cheney, Introduction to approximation theory, Chelsea, New York, 1982. MR1656150 (99f:41001)
    • (1982) Chelsea, New York
    • Cheney, E.W.1
  • 8
    • 33645712892 scopus 로고    scopus 로고
    • Compressed sensing
    • MR2241189 2007e:94013
    • David L. Donoho, Compressed sensing, IEEE. Trans. Info. Thry. 52 (2006), no. 4, 1289-1306. MR2241189 (2007e:94013)
    • (2006) IEEE. Trans. Info. Thry , vol.52 , Issue.4 , pp. 1289-1306
    • Donoho, D.L.1
  • 9
    • 33744552752 scopus 로고    scopus 로고
    • 1-norm solution is the sparsest solution
    • MR2222440 2006m:65079
    • 1-norm solution is the sparsest solution, Comm. Pure Appl. Math. 59 (2006), no. 7, 907-934. MR2222440 (2006m:65079)
    • (2006) Comm. Pure Appl. Math. , vol.59 , Issue.7 , pp. 907-934
  • 10
    • 33646479497 scopus 로고    scopus 로고
    • High-dimensional centrally-symmetric polytopes with neighborliness proportional to dimension
    • MR2225676 2007h:52012
    • -, High-dimensional centrally-symmetric polytopes with neighborliness proportional to dimension, Disc. Comput. Geometry 35 (2006), no. 4, 617-652. MR2225676 (2007h:52012)
    • (2006) Disc. Comput. Geometry , vol.35 , Issue.4 , pp. 617-652
  • 12
    • 22144471532 scopus 로고    scopus 로고
    • Neighborliness of randomly-projected simplices in high dimensions
    • MR2168716
    • David L. Donoho and Jared Tanner, Neighborliness of randomly-projected simplices in high dimensions, Proc. Natl. Acad. Sci. USA 102 (2005), no. 27, 9452-9457. MR2168716
    • (2005) Proc. Natl. Acad. Sci. USA , vol.102 , Issue.27 , pp. 9452-9457
    • Donoho, D.L.1    Tanner, J.2
  • 13
    • 22144488449 scopus 로고    scopus 로고
    • Sparse nonnegative solutions of underdetermined linear equations by linear programming
    • MR2168715 2006d:90089
    • -, Sparse nonnegative solutions of underdetermined linear equations by linear programming, Proc. Natl. Acad. Sci. USA 102 (2005), no. 27, 9446-9451. MR2168715 (2006d:90089)
    • (2005) Proc. Natl. Acad. Sci. USA , vol.102 , Issue.27 , pp. 9446-9451
  • 16
    • 0017973512 scopus 로고
    • On the inherent intractability of certain coding problems
    • MR0495180 58:13912
    • E. R. Berlekamp, R. J. McEliece, and H. C. A. van Tilborg, On the inherent intractability of certain coding problems, IEEE Trans. Info. Thry. 24 (1978), 384-386. MR0495180 (58:13912)
    • (1978) IEEE Trans. Info. Thry. , vol.24 , pp. 384-386
    • Berlekamp, E.R.1    McEliece, R.J.2    Van Tilborg, H.C.A.3
  • 17
    • 0002670267 scopus 로고
    • Neighboring vertices on a convex polyhedron
    • Annals of Mathematics Studies, Princeton University Press, Princeton, N. J., MR0085552
    • David Gale, Neighboring vertices on a convex polyhedron, Linear inequalities and related system, Annals of Mathematics Studies, no. 38, Princeton University Press, Princeton, N. J., 1956, pp. 255-263. MR0085552
    • (1956) Linear Inequalities and Related System , Issue.38 , pp. 255-263
    • Gale, D.1
  • 18
    • 0010833530 scopus 로고
    • Neighborly and cyclic polytopes
    • Amer. Math. Soc, Providence, R. I., MR0152944 27:2915
    • -, Neighborly and cyclic polytopes, Proc. Sympos. Pure Math., Vol. VII, Amer. Math. Soc, Providence, R. I., 1963, pp. 225-232. MR0152944 (27:2915)
    • (1963) Proc. Sympos. Pure Math. , vol.7 , pp. 225-232
  • 19
    • 40349097228 scopus 로고    scopus 로고
    • Convex polytopes, second ed.
    • Springer-Verlag, New York, prepared and with a preface by Volker Kaibel, Victor Klee and Gun̈ter M. Ziegler. MR1976856
    • Branko Grünbaum, Convex polytopes, second ed., Graduate Texts in Mathematics, vol. 221, Springer-Verlag, New York, 2003, prepared and with a preface by Volker Kaibel, Victor Klee and Gun̈ter M. Ziegler. MR1976856
    • (2003) Graduate Texts in Mathematics , vol.221
    • Grünbaum, B.1
  • 20
    • 20744451888 scopus 로고    scopus 로고
    • Geometric representation of high dimensional low sample size data
    • MR2155347
    • P. Hall, J. S. Marron, and A. Neeman, Geometric representation of high dimensional low sample size data, J. Roy. Stat. Soc. B 67 (2005), 427-444. MR2155347
    • (2005) J. Roy. Stat. Soc. B , vol.67 , pp. 427-444
    • Hall, P.1    Marron, J.S.2    Neeman, A.3
  • 21
    • 77950591917 scopus 로고    scopus 로고
    • Signal reconstruction from noisy randomized projections with applications to wireless sensing
    • University of Wisconsin
    • Jarvis Haupt and Robert A. Nowak, Signal reconstruction from noisy randomized projections with applications to wireless sensing, Tech. report, Electrical Engineering, University of Wisconsin, 2005.
    • (2005) Tech. Report, Electrical Engineering
    • Haupt, J.1    Nowak, R.A.2
  • 23
    • 22844457314 scopus 로고    scopus 로고
    • Limit theorems for the convex hull of random points in higher dimensions
    • MR1670156 2000a:52008
    • Irene Hueter, Limit theorems for the convex hull of random points in higher dimensions, Trans. Amer. Math. Soc. 351 (1999), no. 11, 4337-4363. MR1670156 (2000a:52008)
    • (1999) Trans. Amer. Math. Soc. , vol.351 , Issue.11 , pp. 4337-4363
    • Hueter, I.1
  • 25
    • 33746923055 scopus 로고    scopus 로고
    • How neighborly can a centrally symmetric polytope be?
    • MR2252105 2007f:52027
    • Nathan Linial and Isabella Novik, How neighborly can a centrally symmetric polytope be?, Disc. Comput. Geometry 36 (2006), no. 2, 273-281. MR2252105 (2007f:52027)
    • (2006) Disc. Comput. Geometry , vol.36 , Issue.2 , pp. 273-281
    • Linial, N.1    Novik, I.2
  • 26
    • 4244094761 scopus 로고    scopus 로고
    • Lectures on discrete geometry
    • Springer-Verlag, New York, MR1899299 2003f:52011
    • Jiri Matousek, Lectures on discrete geometry, Graduate Texts in Mathematics, vol. 212, Springer-Verlag, New York, 2002. MR1899299 (2003f:52011)
    • (2002) Graduate Texts in Mathematics , vol.212
    • Matousek, J.1
  • 27
    • 84974069711 scopus 로고
    • Diagrams for centrally symmetric polytopes
    • MR0238180 38:6456
    • Peter McMullen and Geoffrey C. Shephard, Diagrams for centrally symmetric polytopes, Mathematika 15 (1968), 123-138. MR0238180 (38:6456)
    • (1968) Mathematika , vol.15 , pp. 123-138
    • McMullen, P.1    Shephard, G.C.2
  • 28
    • 51249195097 scopus 로고
    • On the geometrical moments of skew-regular simplices in hyper spherical space, with some applications in geometry and mathematical statistics
    • MR0121713 22:12447
    • Harold Ruben, On the geometrical moments of skew-regular simplices in hyper spherical space, with some applications in geometry and mathematical statistics, Acta Math. 103 (1960), 1-23. MR0121713 (22:12447)
    • (1960) Acta Math. , vol.103 , pp. 1-23
    • Ruben, H.1
  • 29
    • 33746801607 scopus 로고    scopus 로고
    • Geometric approach to error-correcting codes and reconstruction of signals
    • University of California, Davis, MR2206919 2006j:94116
    • M. Rudelson and R. Vershynin, Geometric approach to error-correcting codes and reconstruction of signals, Tech. report, Department of Mathematics, University of California, Davis, 2005. MR2206919 (2006j:94116)
    • (2005) Tech. Report, Department of Mathematics
    • Rudelson, M.1    Vershynin, R.2
  • 30
    • 77950583173 scopus 로고    scopus 로고
    • Sparse reconstruction by convex relaxation: Fourier and gaussian measurements
    • Princeton University, 2006
    • -, Sparse reconstruction by convex relaxation: Fourier and gaussian measurements, Proceedings Conference on Information Science and Systems (CISS) 2006, Princeton University, 2006.
    • (2006) Proceedings Conference on Information Science and Systems (CISS)
  • 31
    • 84971922633 scopus 로고
    • Neighbourliness of centrally symmetric polytopes in high dimensions
    • MR0405244 53:9038
    • Rolf Schneider, Neighbourliness of centrally symmetric polytopes in high dimensions, Mathematika 22 (1975), no. 2, 176-181. MR0405244 (53:9038)
    • (1975) Mathematika , vol.22 , Issue.2 , pp. 176-181
    • Schneider, R.1
  • 32
    • 30844471849 scopus 로고    scopus 로고
    • Signal recovery from partial information by orthogonal matching pursuit
    • University of Michigan
    • J. A. Tropp and Anna Gilbert, Signal recovery from partial information by orthogonal matching pursuit, Tech. report, Mathematics Department, University of Michigan, 2005.
    • (2005) Tech. Report, Mathematics Department
    • Tropp, J.A.1    Gilbert, A.2
  • 34
    • 0038860738 scopus 로고
    • Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem
    • MR1166627 93d:60017
    • A. M. Vershik and P. V. Sporyshev, Asymptotic behavior of the number of faces of random polyhedra and the neighborliness problem, Selecta Math. Soviet. 11 (1992), no. 2, 181-201. MR1166627 (93d:60017)
    • (1992) Selecta Math. Soviet , vol.11 , Issue.2 , pp. 181-201
    • Vershik, A.M.1    Sporyshev, P.V.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.