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On multiplier processes under weak moment assumptions. (English) Zbl 1366.60044
Klartag, Bo’az (ed.) et al., Geometric aspects of functional analysis. Israel seminar (GAFA) 2014–2016. Cham: Springer (ISBN 978-3-319-45281-4/pbk; 978-3-319-45282-1/ebook). Lecture Notes in Mathematics 2169, 301-318 (2017).
Summary: We show that if \(V \subset \mathbb{R}^{n}\) satisfies a certain symmetry condition that is closely related to unconditionality, and if \(X\) is an isotropic random vector for which \(\|\langle X, t\rangle\| _{L_{p}} \leq L\sqrt{p}\) for every \(t\in S^{n-1}\) and every \(1 \leq p\lesssim \log n\), then the suprema of the corresponding empirical and multiplier processes indexed by \(V\) behave as if \(X\) were \(L\)-subgaussian.
For the entire collection see [Zbl 1369.46001].

60E05 Probability distributions: general theory
60G99 Stochastic processes
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[1] F. Albiac, N.J. Kalton, Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233 (Springer, New York, 2006) · Zbl 1094.46002
[2] S. Artstein-Avidan, A. Giannopoulos, V.D. Milman, Asymptotic Geometric Analysis. Part I. Mathematical Surveys and Monographs, vol. 202 (American Mathematical Society, Providence, RI, 2015) · Zbl 1337.52001
[3] P. Bühlmann, S. van de Geer, Statistics for High-Dimensional Data. Methods, Theory and Applications. Springer Series in Statistics (Springer, Heidelberg, 2011) · Zbl 1273.62015
[4] S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2013) · Zbl 1315.94002
[5] V. Koltchinskii, Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems. Lecture Notes in Mathematics, vol. 2033 (Springer, Heidelberg, 2011). Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School] · Zbl 1223.91002
[6] G. Lecué, S. Mendelson, Sparse recovery under weak moment assumptions. Technical report, CNRS, Ecole Polytechnique and Technion (2014). J. Eur. Math. Soc. 19 (3), 881–904 (2017) · Zbl 1414.62135
[7] G. Lecué, S. Mendelson, Regularization and the small-ball method I: sparse recovery. Technical report, CNRS, ENSAE and Technion, I.I.T. (2015). Ann. Stati. (to appear) · Zbl 1318.62178
[8] M. Ledoux, M. Talagrand, Probability in Banach Spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23 (Springer, Berlin, 1991) · Zbl 0748.60004
[9] S. Mendelson, Learning without concentration for general loss function. Technical report, Technion, I.I.T. (2013). arXiv:1410.3192 · Zbl 1393.62038
[10] S. Mendelson, A remark on the diameter of random sections of convex bodies, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, vol. 2116, pp. 395–404 (Springer, Cham, 2014) · Zbl 1317.52011
[11] S. Mendelson, Upper bounds on product and multiplier empirical processes. Stoch. Process. Appl. 126 (12), 3652–3680 (2016) · Zbl 1386.60077
[12] S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17 (4), 1248–1282 (2007) · Zbl 1163.46008
[13] S. Mendelson, G. Paouris, On generic chaining and the smallest singular value of random matrices with heavy tails. J. Funct. Anal. 262 (9), 3775–3811 (2012) · Zbl 1242.60008
[14] V.D. Milman, Random subspaces of proportional dimension of finite-dimensional normed spaces: approach through the isoperimetric inequality, in Banach Spaces (Columbia, MO, 1984). Lecture Notes in Mathematics, vol. 1166, pp. 106–115 (Springer, Berlin, 1985)
[15] A. Pajor, N. Tomczak-Jaegermann, Nombres de Gel’ fand et sections euclidiennes de grande dimension, in Séminaire d’Analyse Fonctionelle 1984/1985. Publ. Math. Univ. Paris VII, vol. 26, pp. 37–47 (University of Paris VII, Paris, 1986)
[16] A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Am. Math. Soc. 97 (4), 637–642 (1986) · Zbl 0623.46008
[17] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, vol. 94 (Cambridge University Press, Cambridge, 1989) · Zbl 0698.46008
[18] M. Talagrand, Regularity of Gaussian processes. Acta Math. 159 (1–2), 99–149 (1987) · Zbl 0712.60044
[19] M. Talagrand, Upper and Lower Bounds for Stochastic Processes. Modern Methods and Classical Problems. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 60. (Springer, Heidelberg, 2014)
[20] A.W. van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics (Springer, New York, 1996) · Zbl 0862.60002
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