UNIVERSITY
OF WROC£AW
 
Main Page
Contents
Online First
General Information
Instructions for authors


VOLUMES
44.1 43.2 43.1 42.2 42.1 41.2 41.1
40.2 40.1 39.2 39.1 38.2 38.1 37.2
37.1 36.2 36.1 35.2 35.1 34.2 34.1
33.2 33.1 32.2 32.1 31.2 31.1 30.2
30.1 29.2 29.1 28.2 28.1 27.2 27.1
26.2 26.1 25.2 25.1 24.2 24.1 23.2
23.1 22.2 22.1 21.2 21.1 20.2 20.1
19.2 19.1 18.2 18.1 17.2 17.1 16.2
16.1 15 14.2 14.1 13.2 13.1 12.2
12.1 11.2 11.1 10.2 10.1 9.2 9.1
8 7.2 7.1 6.2 6.1 5.2 5.1
4.2 4.1 3.2 3.1 2.2 2.1 1.2
1.1
 
 
WROC£AW UNIVERSITY
OF SCIENCE AND
TECHNOLOGY

Contents of PMS, Vol. 42, Fasc. 2,
pages 303 - 317
DOI: 10.37190/0208-4147.00084
Published online 1.12.2022
 

Large deviations for uniform projections of p-radial distributionon on lpn-Balls

T. Kaufmann
H. Sambale
C. Thäle

Abstract: We consider products of uniform random variables from the Stiefel manifold of orthonormal k-frames in Rn, k ≤ n, and random vectors from the n-dimensional lpn-balls Bpn with certain p-radial distributions, p ∈ [1,∞ ). The distribution of this product geometrically corresponds to the projection of the p-radial distribution on Bpn onto a random k-dimensional subspace. We derive large deviation principles (LDPs) on the space of probability measures on Rk for sequences of such projections.

2010 AMS Mathematics Subject Classification: Primary 52A23; Secondary 60F10.

Keywords and phrases: large deviation principle, lpn-ball, random projection, Stiefel manifold.

D. Alonso-Gutiérrez, J. Prochno, and C. Thäle, Large deviations for high-dimensional random projections of lpn-balls, Adv. Appl. Math. 99 (2018), 1-35.

F. Barthe, O. Guédon, S. Mendelson, and A. Naor, A probabilistic approach to the geometry of the lpn-ball, Ann. Probab. 33 (2005), 480-513.

D. Chafaï ˆ, O. Guédon, G. Lecué, and A. Pajor, Interactions between Compressed Sensing Random Matrices and High Dimensional Geometry, Soc. Math. France, 2012.

A. Dembo and O. Zeitouni, Large Deviations: Techniques and Applications, Stochastic Modelling Appl. Probab. 38, Springer, Berlin, 2010 (corrected reprint of the second (1998) edition).

S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Appl. Numer. Harmonic Anal., Birkhäuser/Springer, New York, 2013.

N. Gantert, S. S. Kim, and K. Ramanan, Large deviations for random projections of lp balls, Ann. Probab. 45 (2017), 4419-4476.

A. Hinrichs, D. Krieg, E. Novak, J. Prochno, and M. Ullrich, Random sections of ellipsoids and the power of random information, Trans. Amer. Math. Soc. 374 (2021), 8691-8713.

A. Hinrichs, J. Prochno, and M. Sonnleitner, Random sections of lp-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators, arXiv:2109.14504 (2021).

G. J. O. Jameson, Inequalities for gamma function ratios, Amer. Math. Monthly 120 (2013), 936-940.

Z. Kabluchko and J. Prochno, Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions, arXiv:2110.12977 (2021).

Z. Kabluchko, J. Prochno, and C. Thäle, High-dimensional limit theorems for random vectors in lpn-balls, Commun. Contemp. Math. 21 (2019), art. 1750092, 30 pp.

Z. Kabluchko, J. Prochno, and C. Thäle, Sanov-type large deviations in Schatten classes, Ann. Inst. H. Poincaré Probab. Statist. 56 (2019), 928-953.

Z. Kabluchko, J. Prochno, and C. Thäle, High-dimensional limit theorems for random vectors in lpn-balls. II, Commun. Contemp. Math. 23 (2021), art. 1950073, 35 pp.

T. Kaufmann and C. Thäle, Weighted p-radial distributions on Euclidean and matrix p-balls with applications to large deviations, J. Math. Anal. Appl. 515 (2022), no. 1, art. 126377, 41 pp.

S. S. Kim and K. Ramanan, Large deviation principles induced by the Stiefel manifold, and random multi-dimensional projections, Adv. in Appl. Math. 134 (2022), art. 102306, 64 pp.

B. Klartag, A central limit theorem for convex sets, Invent. Math. 168 (2007), 91-131.

B. Klartag, Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245 (2007), 284-310.

D. Krieg and M. Sonnleitner, Random points are optimal for the approximation of Sobolev functions, arXiv:2009.11275 (2020).

J. Prochno, C. Thäle, and N. Turchi, Geometry of lpn-balls: Classical results and recent developments, in: High Dimensional Probability VIII, Progr. Probab. 74, Birkhäuser, 2019, 121-150.

Download:    Abstract    Full text