The maximum entropy principle and volumetric properties of Orlicz balls

Kabluchko, Zakhar; Prochno, Joscha

doi:10.1016/j.jmaa.2020.124687

Cited by 19 publications

(36 citation statements)

References 21 publications

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“…As the authors point out, this means that conditioning on a sufficiently small ℓ q -norm induces a probabilistic change that admits a nice geometric interpretation. The strategy of proof is based on ideas from statistical mechanics and large deviations theory (see also [12] for more in this direction). The authors first prove a level-2 large deviation principle for the empirical measure of the coordinates of a random point on a properly scaled ℓ n p -sphere, where the random choice is made with respect to the cone probability measure on the boundary.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Key to our generalization, which follows the general theme of first proving a level-2 large deviation principle for the sequence of empirical measures of the coordinates of random points chosen uniformly from a suitably scaled Orlicz ball and then using Gibbs conditioning, is the statistical mechanics point of view. Indeed, here conditional distributions appear in connection to the study of non-interacting particle systems (micro-canonical and canonical ensembles), where one is seeking to describe the most likely state of the system under an energy constraint (see, e.g., [9], [26], and [12,Subsection 1.2]). This point of view directly links such questions to certain Gibbs distributions.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Considering such Gibbs measures at suitable critical temperatures with potentials given by an Orlicz function allows us to work around the problem arising from the lack of a probabilistic representation. This idea has recently been put forward by Kabluchko and Prochno [12] (see particularly Subsection 1.2 there) in their probabilistic approach to the geometry of Orlicz balls and has also successfully been used in [1], [2], and [15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

Fruehwirth¹,

Prochno²

2021

Preprint

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In this paper, we prove a Sanov-type large deviation principle for the sequence of empirical measures of vectors chosen uniformly at random from an Orlicz ball. From this level-2 large deviation result, in a combination with Gibbs conditioning, entropy maximization and an Orlicz version of the Poincaré-Maxwell-Borel lemma, we deduce a conditional limit theorem for highdimensional Orlicz balls. Roughly speaking, the latter shows that if V 1 and V 2 are Orlicz functions, then random points in the V 1 -Orlicz ball, conditioned on having a small V 2 -Orlicz radius, look like an appropriately scaled V 2 -Orlicz ball. In fact, we show that the limiting distribution in our Poincaré-Maxwell-Borel lemma, and thus the geometric interpretation, undergoes a phase transition depending on the magnitude of the V 2 -Orlicz radius.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

Fruehwirth¹,

Prochno²

2021

Preprint

Self Cite

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“…They gave sharp large deviation (SLD) results in the spirit of Bahadur and Ranga Rao [7] and Petrov [39] for the projections of random points in n p -balls with distributions C n,p and U n,p onto a fixed one-dimensional subspace. Other works in asymptotic geometric analysis have also employed methods from sharp large deviations theory as well, such as Kabluchko and Prochno [27], who derived asymptotic volumes for generalizations of n p -balls, known as Orlicz balls, and showed a Schechtman and Schmuckenschläger-type result by considering intersection volumes of Orlicz balls. Their results on Orlicz balls were then expanded upon by Alonso-Guiterréz and Prochno in [2], who gave the exact asymptotic volume of Orlicz balls and provided thin-shell concentrations for them, augmenting their results into sharp asymptotics under certain conditions.While LDPs only give tail asymptotics on a logarithmic scale, the sharp asymptotics provided by sharp large deviations theory can give tail estimates for concrete values of n ∈ N, which makes them significantly more useful for practical applications.…”

Section: Introductionmentioning

confidence: 99%

Sharp asymptotics for q-norms of random vectors in high-dimensional ℓpn-balls

Kaufmann¹

2021

Modern Stochastics: Theory and Applications

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Sharp large deviation results of Bahadur-Ranga Rao type are provided for the qnorm of random vectors distributed on the n p -ball B n p according to the cone probability measure or the uniform distribution for 1 ≤ q < p < ∞, thereby furthering previous large deviation results by Kabluchko, Prochno and Thäle in the same setting. These results are then applied to deduce sharp asymptotics for intersection volumes of different n p -balls in the spirit of Schechtman and Schmuckenschläger, and for the length of the projection of an n p -ball onto a line with uniform random direction. The sharp large deviation results are proven by providing convenient probabilistic representations of the q-norms, employing local limit theorems to approximate their densities, and then using geometric results for asymptotic expansions of Laplace integrals to integrate these densities and derive concrete probability estimates.

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“…The maximum entropy principle is to maintain maximum uncertainty to ensure minimum risk. Applications of the maximum entropy principle are everywhere [27]. For example, it is often said, 'Do not put all your eggs in one basket'.…”

Section: Introductionmentioning

confidence: 99%

BW-MaxEnt: A Novel MCDM Method for Limited Knowledge

et al. 2021

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With the development of the social economy and an enlarged volume of information, the application of multiple-criteria decision making (MCDM) has become increasingly wide and deep. As a brilliant MCDM technique, the best–worst method (BWM) has attracted many scholars’ attention because it can determine the weights of criteria with less comparison time and higher consistency between judgments than analytic hierarchy process. However, the effectiveness of the BWM is based on complete comparison information among criteria. Considering the fact that the decision makers may have limited time and energy to study all criteria, they cannot construct a complete comparison system. In this paper, we propose a novel MCDM method named BW-MaxEnt that combines BWM and the maximum entropy method (MaxEnt) to identify the weights of unfamiliar criteria with incomplete decision information. The model can be translated into a convex optimization problem that can be solved effectively and has an overall optimal solution. Finally, a practical application concerning the procurement of GPU workstations illustrates the feasibility of the proposed BW-MaxEnt method.

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The maximum entropy principle and volumetric properties of Orlicz balls

Cited by 19 publications

References 21 publications

Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

Sanov-type large deviations and conditional limit theorems for high-dimensional Orlicz balls

Sharp asymptotics for q-norms of random vectors in high-dimensional ℓpn-balls

BW-MaxEnt: A Novel MCDM Method for Limited Knowledge

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