Skip to main content Accessibility help
×
Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T14:08:58.849Z Has data issue: false hasContentIssue false

Appendix B - An Update on Some Problems in High-Dimensional Convex Geometry and Related Probabilistic Results

Published online by Cambridge University Press:  26 October 2017

Daniel Li
Affiliation:
Université d'Artois, France
Hervé Queffélec
Affiliation:
Université de Lille I
Get access

Summary

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Adamczak, R., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles, J. Amer. Math. Soc. 23, 2 2010, 535–561.Google Scholar
[2] Adamczak, R., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. Sharp bounds on the rate of convergence of the empirical covariance matrix, C. R. Math. Acad.Sci. Paris 349, 3-4 2011, 195–200.Google Scholar
[3] Alonso-Guttiérez, D. & Bastero, J. The variance conjecture on hyperplane projections of _np balls, https://arxiv.org/abs/1610.04023 (preprint).
[4] Anttila,M., Ball, K. & Perissinaki, I. The central limit problem for convex bodies, Trans. Amer. Math. Soc. 355, 12 2003, 4723–4735.(electronic).Google Scholar
[5] Bandeira, A. S. & van Handel, R. Sharp nonasymptotic bounds on the norm of random matrices with independent entries, Ann. Probab. 44, 4 2016, 2479– 2506.Google Scholar
[6] Berinde, R., Gilbert, A., Indyk, P. H. K. & Strauss, M. Combining geometry and combinatorics: a unified approach to sparse signal recovery, in Communication, Control, and Computing, 2008 46th Annual Allerton Conference on, IEEE 2008, 798–805.Google Scholar
[7] Bourgain, J. Bounded orthogonal systems and the (p)-set problem, Acta Math. 162, 3-4 1989, 227–245.Google Scholar
[8] Bourgain, J. Random points in isotropic convex sets, in Convex Geometric Analysis (Berkeley, CA, 1996), Mathematical Sciences Research Institute Publications 34, Cambridge University Press 1999, 53–58.
[9] Bourgain, J. An improved estimate in the restricted isometry problem, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics 2116, Springer 2014, 65–70.
[10] Buser, P. A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Super. (4) 15, 2 1982, 213–230.Google Scholar
[11] CandèsE. J, Romberg, J. K. & Tao, T. Stable signal recovery from incomplete and inaccurate measurements, Comm. Pure Appl. Math. 59, 8 2006, 1207–1223.Google Scholar
[12] Candès, E. J. & Tao, T. Decoding by linear programming, IEEE Trans. Inform.Theory 51, 12 2005, 4203–4215.Google Scholar
[13] Chafaï, D., Guédon, O., Lecué, G. & Pajor, A. Interactions Between Compressed Sensing Random Matrices and High Dimensional Geometry, Panoramas et Synthèses 37, Société Mathématique de France 2012.
[14] Dirksen, S., Lecué, G. & Rauhut, H. On the gap between restricted isometry properties and sparse recovery conditions, https:doi.org/10.1109/TIT.2016.2570244, to appear in IEEE Trans. Inform. Theory.
[15] Donoho, D. L., Compressed sensing, IEEE Trans. Inform. Theory 52, 4 2006, 1289–1306.Google Scholar
[16] Eldan, R. Thin shell implies spectral gap up to polylog via a stochastic localization scheme, Geom. Funct. Anal. 23, 2 2013, 532–569.Google Scholar
[17] Fleury, B. Concentration in a thin Euclidean shell for log-concave measures, J.Funct. Anal. 259, 4 2010, 832–841.Google Scholar
[18] Fleury, B. Poincaré inequality in mean value for Gaussian polytopes, Probab.Theory Related Fields 152, 1-2 2012, 141–178.Google Scholar
[19] Fleury, B., Guédon, O. & Paouris, G. A stability result for mean width of Lp-centroid bodies, Adv. Math. 214, 2 2007, 865–877.Google Scholar
[20] Foucart, S. & Rauhut, H. A Mathematical Introduction to Compressive Sensing, Applied and Numerical Harmonic Analysis. Birkhäuser/Springer 2013.
[21] Friedland, O. & Guédon, O. Sparsity and non-Euclidean embeddings, Israel J.Math. 197, 1 2013, 329–345.Google Scholar
[22] Giannopoulos, A. A. & Milman, V. D., Concentration property on probability spaces, Adv. Math. 156, 1 2000, 77–106.Google Scholar
[23] Gozlan, N., Roberto, C. & Samson, P.-M., From dimension free concentration to the Poincaré inequality, Calc. Var. Partial Differential Equations 52, 3-4 2015, 899–925.Google Scholar
[24] Guédon, O. Kahane–Khinchine type inequalities for negative exponent, Mathematika 46, 1 1999, 165–173.Google Scholar
[25] Guédon, O., Litvak, A. E., Pajor, A. & Tomczak-Jaegermann, N. On the interval of fluctuation of the singular values of random matrices, https://arxiv.org/ abs/1509.02322 (preprint), to appear in J. Eur. Math. Soc. (JEMS).
[26] Guédon, O. & Milman, E. Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures, Geom. Funct. Anal. 21, 5 2011, 1043–1068.Google Scholar
[27] Haviv, I. & Regev, O. The restricted isometry property of subsampled Fourier matrices, https://arxiv.org/abs/1507.01768 (preprint).
[28] Kannan, R., Lovász, L. & Simonovits, M. Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13, 3-4 1995, 541–559.Google Scholar
[29] Kannan, R., Lovász, L. & Simonovits, M. Random walks and an O∗ (n5) volume algorithm for convex bodies, Random Structures Algorithms 11, 1 1997, 1–50.Google Scholar
[30] Klartag, B. On convex perturbations with a bounded isotropic constant, Geom.Funct. Anal. 16, 6 2006, 1274–1290.Google Scholar
[31] Klartag, B. A central limit theorem for convex sets, Invent. Math. 168, 1 2007, 91–131.Google Scholar
[32] Klartag, B. Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245, 1 2007, 284–310.Google Scholar
[33] Kolesnikov, A. V. & Milman, E. The KLS isoperimetric conjecture for generalized Orlicz balls, https://arxiv.org/abs/1610.06336 (preprint).
[34] Latała, R. Some estimates of norms of random matrices, Proc. Amer. Math. Soc. 133, 5 2005, 1273–1282.Google Scholar
[35] Latała, R. & Wojtaszczyk, J. O. On the infimum convolution inequality, Studia Math. 189, 2 2008, 147–187.Google Scholar
[36] Ledoux, M. A simple analytic proof of an inequality by P.Buser, Proc. Amer.Math. Soc. 121, 3 1994, 951–959.Google Scholar
[37] Lee, Y. T. & Vempala, S. S., Eldan's stochastic localization and the KLS hyperplane conjecture: an improved lower bound for expansion, https://arxiv.org/abs/1612.01507 (preprint).
[38] Mendelson, S. & Paouris, G. On the singular values of random matrices, J. Eur.Math. Soc. (JEMS) 16, 4 2014, 823–834.Google Scholar
[39] Milman, E. On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math. 177, 1 2009, 1–43.Google Scholar
[40] Milman, V. D. & Schechtman, G. Asymptotic Theory of Finite-Dimensional Normed Spaces , Lecture Notes in Mathematics 1200, Springer (1986), With an appendix by M. Gromov.
[41] Oymak, S. & Tropp, J. A., Universality laws for randomized dimension reduction, with applications, https://arxiv.org/abs/1511.09433 (preprint).
[42] Paouris, G. Concentration of mass on convex bodies, Geom. Funct. Anal. 16, 5 2006, 1021–1049.Google Scholar
[43] Rudelson, M. Random vectors in the isotropic position, J. Funct. Anal. 164, 1 1999, 60–72.Google Scholar
[44] Rudelson, M. & Vershynin, R. The Littlewood–Offord problem and invertibility of random matrices, Adv. Math. 218, 2 2008, 600–633.Google Scholar
[45] Rudelson, M. & Vershynin, R. On sparse reconstruction from Fourier and Gaussian measurements, Comm. Pure Appl. Math. 61, 8 2008, 1025–1045.Google Scholar
[46] Rudelson, M. & Vershynin, R. Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math. 62, 12 2009, 1707–1739.Google Scholar
[47] Seginer, Y. The expected norm of random matrices, Combin. Probab. Comput. 9, 2 2000, 149–166.Google Scholar
[48] Sodin, S. An isoperimetric inequality on the lp balls, Ann. Inst. Henri Poincaré Probab. Stat. 44, 2 2008, 362–373.Google Scholar
[49] Tikhomirov, K. Sample covariance matrices of heavy-tailed distributions, https://arxiv.org/abs/1606.03557 (preprint), to appear in Int. Math. Res. Not.IMRN.
[50] Tikhomirov, K. & Youssef, P. When does a discrete-time random walk in Rn absorb the origin into its convex hull? https://arxiv.org/abs/1410.0458 (preprint), to appear in Ann. Probab.
[51] van Handel, R. On the spectral norm of Gaussian random matrices, https://arxiv.org/abs/1502.05003 (preprint), to appear in Trans. Amer. Math. Soc.
[52] Vempala, S. S., Recent progress and open problems in algorithmic convex geometry, in 30th International Conference on Foundations of Software Technology and Theoretical Computer Science, LIPIcs: Leibniz International Proceedings in Informatics 8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2010), pp. 42–64.

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×