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The minimal spherical dispersion
No full textWe prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy [Spherical dispersion with an application to polygonal approximation of curves, Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. 132 (1995), 3--10]. In particular, we see that the inverse of the minimal spherical dispersion is, for fixed , linear in the dimension of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse , our bounds are optimal with respect to the dependence on
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New Perspectives and Computational Challenges in High Dimensions
Full text linkHigh-dimensional systems are frequent in mathematics and applied sciences, and the understanding of
high-dimensional phenomena has become increasingly important. The mathematical subdisciplines most
strongly related to such phenomena are functional analysis, convex geometry, and probability theory.
In fact, a new area emerged, called asymptotic geometric analysis, which is at the very core of these
disciplines and bears a number of deep connections to mathematical physics, numerical analysis, and
theoretical computer science. The last two decades have seen a tremendous growth in this area. Far
reaching results were obtained and various powerful techniques have been developed, which rather
often have a probabilistic flavor. The purpose of this workshop was to explored these new perspectives, to reach out to other areas concerned with high-dimensional problems, and to bring together researchers having different angles on high-dimensional phenomena
Combinatorial Inequalities and Sub-spaces of L1
No full textNon UBCUnreviewedAuthor affiliation: University of AlbertaPostdoctora
Musielak-Orlicz spaces that are isomorphic to subspaces of L1
No full textWe prove that 1/n! ∑π∈Gn (∑ni=1 |xiai,π(i)|2)1/2 is equivalent to a Musielak-Orlicz norm ||x||∑Mi. We also provide the converse result, i.e., given the Orlicz functions, we provide a formula for the choice of the matrix that generates the corresponding Musielak-Orlicz norm. As a consequence, we obtain the embedding of 2-concave Musielak-Orlicz spaces into L1
New Perspectives and Computational Challenges in High Dimensions
No full textHigh-dimensional systems are frequent in mathematics and applied sciences, and the understanding of
high-dimensional phenomena has become increasingly important. The mathematical subdisciplines most
strongly related to such phenomena are functional analysis, convex geometry, and probability theory.
In fact, a new area emerged, called asymptotic geometric analysis, which is at the very core of these
disciplines and bears a number of deep connections to mathematical physics, numerical analysis, and
theoretical computer science. The last two decades have seen a tremendous growth in this area. Far
reaching results were obtained and various powerful techniques have been developed, which rather
often have a probabilistic flavor. The purpose of this workshop was to explored these new perspectives, to reach out to other areas concerned with high-dimensional problems, and to bring together researchers having different angles on high-dimensional phenomena
Teilräume von L1 und kombinatorische Ungleichungen in der Banachraumtheorie
Full text linkDie Arbeit beschäftigt sich mit endlich-dimensionalen, symmetrischen Teilräumen von L1. Diese werden mit Hilfe kombinatorischer Ungleichungen studiert und es wird eine neue Klasse endlich-dimensionaler, symmetrischer Teilräume von L1 angeben. Außerdem werden in der Arbeit Musielak-Orlicz Räume studiert und in zwei Kapiteln werden kombinatorische Ungleichungen bewiesen
The large and moderate deviations approach in geometric functional analysis
Full text linkThe work of Gantert, Kim, and Ramanan [Large deviations for random
projections of balls, Ann. Probab. 45 (6B), 2017] has initiated and
inspired a new direction of research in the asymptotic theory of geometric
functional analysis. The moderate deviations perspective, describing the
asymptotic behavior between the scale of a central limit theorem and a large
deviations principle, was later added by Kabluchko, Prochno, and Th\"ale in
[High-dimensional limit theorems for random vectors in balls. II,
Commun. Contemp. Math. 23(3), 2021]. These two approaches nicely complement the
classical study of central limit phenomena or non-asymptotic concentration
bounds for high-dimensional random geometric quantities. Beyond studying large
and moderate deviations principles for random geometric quantities that appear
in geometric functional analysis, other ideas emerged from the theory of large
deviations and the closely related field of statistical mechanics, and have
provided new insight and become the origin for new developments. Within less
than a decade, a variety of results have appeared and formed this direction of
research. Recently, a connection to the famous Kannan-Lov\'asz-Simonovits
conjecture and the study of moderate and large deviations for isotropic
log-concave random vectors was discovered. In this manuscript, we introduce the
basic principles, survey the work that has been done, and aim to manifest this
direction of research, at the same time making it more accessible to a wider
community of researchers.Comment: 57 pages, 2 figure
Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions
Full text linkWe prove large deviation principles (LDPs) for random matrices in the
orthogonal group and Stiefel manifold, determining both the speed and good
convex rate functions that are explicitly given in terms of certain
log-determinants of trace-class operators and are finite on the set of
Hilbert-Schmidt operators satisfying . As an application of
those LDPs, we determine the precise large deviation behavior of
-dimensional random projections of high-dimensional product distributions
using an appropriate interpretation in terms of point processes, also
characterizing the space of all possible deviations. The case of uniform
distributions on -balls, , is then considered and
reduced to appropriate product measures. Those applications generalize
considerably the recent work [Johnston, Kabluchko, Prochno: Projections of the
uniform distribution on the cube - a large deviation perspective, Studia
Mathematica 264 (2022), 103-119].Comment: 41 page
Sharp concentration phenomena in high-dimensional Orlicz balls
Full text linkIn this article, we present a precise deviation formula for the intersection of two Orlicz balls generated by Orlicz functions and . Additionally, we establish a (quantitative) central limit theorem in the critical case and a strong law of large numbers for the -norm of the uniform distribution on . Our techniques also enable us to derive a precise formula for the thin-shell concentration of uniformly distributed random vectors in high-dimensional Orlicz balls. In our approach we establish an Edgeworth-expansion using methods from harmonic analysis together with an exponential change of measure argument.28 pages, 4 figure
The minimal spherical dispersion
No full textWe prove upper and lower bounds on the minimal spherical dispersion, improving upon previous estimates obtained by Rote and Tichy in (Anz Österreich Akad Wiss Math Nat Kl 132:3–10, 1995). In particular, we see that the inverse N(ε,d)of the minimal spherical dispersion is, for fixed ε>0, linear in the dimension d of the ambient space. We also derive upper and lower bounds on the expected dispersion for points chosen independently and uniformly at random from the Euclidean unit sphere. In terms of the corresponding inverse N~(ε,d), our bounds are optimal with respect to the dependence on ε
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