# zbMATH — the first resource for mathematics

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].

##### MSC:
 60E05 Probability distributions: general theory 60G99 Stochastic processes
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.