Large deviations for uniform projections of p-radial distributions on lnp - balls

• Autorzy: Kaufmann Tom , Sambale Holger , Thäle Christoph

Identyfikatory

DOI

10.37190/0208-4147.00084

• Języki publikacji: EN

Abstrakty

EN

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 ℓⁿ_p-ball Bⁿ_p with certain p-radial distributions, p ∈ [1, ∞). The distribution of this product geometrically corresponds to the projection of the p-radial distribution on Bⁿ_p onto a random k-dimensional subspace. We derive large deviation principles (LDPs) on the space of probability measures on R^k for sequences of such projections.

Słowa kluczowe

EN

large deviation principle; lnp -ball; random projection; Stiefel manifold

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2022
• Tom: Vol. 42, Fasc. 2
• Strony: 303--317
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Kaufmann Tom

• Ruhr University Bochum, Faculty of Mathematics, Universitätsstr. 150, 44780 Bochum, Germany

autor

Sambale Holger

• Ruhr University Bochum, Faculty of Mathematics, Universitätsstr. 150, 44780 Bochum, Germany

autor

Thäle Christoph

• Ruhr University Bochum, Faculty of Mathematics, Universitätsstr. 150, 44780 Bochum, Germany

Bibliografia

• [1] D. Alonso-Gutiérrez, J. Prochno, and C. Thäle, Large deviations for high-dimensional random projections of ℓnp -balls, Adv. Appl. Math. 99 (2018), 1-35.
• [2] F. Barthe, O. Guédon, S. Mendelson, and A. Naor, A probabilistic approach to the geometry of the ℓnp -ball, Ann. Probab. 33 (2005), 480-513.
• [3] D. Chafaï, O. Guédon, G. Lecué, and A. Pajor, Interactions between Compressed Sensing Random Matrices and High Dimensional Geometry, Soc. Math. France, 2012.
• [4] 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).
• [5] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing, Appl. Numer. Harmonic Anal., Birkhäuser/Springer, New York, 2013.
• [6] N. Gantert, S. S. Kim, and K. Ramanan, Large deviations for random projections of ℓp balls, Ann. Probab. 45 (2017), 4419-4476.
• [7] 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.
• [8] A. Hinrichs, J. Prochno, and M. Sonnleitner, Random sections of ℓp-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators, arXiv:2109.14504 (2021).
• [9] G. J. O. Jameson, Inequalities for gamma function ratios, Amer. Math. Monthly 120 (2013), 936-940.
• [10] 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).
• [11] Z. Kabluchko, J. Prochno, and C. Thäle, High-dimensional limit theorems for random vectors in ℓnp -balls, Commun. Contemp. Math. 21 (2019), art. 1750092, 30 pp.
• [12] Z. Kabluchko, J. Prochno, and C. Thäle, Sanov-type large deviations in Schatten classes, Ann. Inst. H. Poincaré Probab. Statist. 56 (2019), 928-953.
• [13] Z. Kabluchko, J. Prochno, and C. Thäle, High-dimensional limit theorems for random vectors in ℓnp -balls. II, Commun. Contemp. Math. 23 (2021), art. 1950073, 35 pp.
• [14] 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.
• [15] 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.
• [16] B. Klartag, A central limit theorem for convex sets, Invent. Math. 168 (2007), 91-131.
• [17] B. Klartag, Power-law estimates for the central limit theorem for convex sets, J. Funct. Anal. 245 (2007), 284-310.
• [18] D. Krieg and M. Sonnleitner, Random points are optimal for the approximation of Sobolev functions, arXiv:2009.11275 (2020).
• [19] J. Prochno, C. Thäle, and N. Turchi, Geometry of ℓnp -balls: Classical results and recent developments, in: High Dimensional Probability VIII, Progr. Probab. 74, Birkhäuser, 2019, 121-150.

Uwagi

Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).