Asymptotic Behavior of Entropy Numbers for Finite-Dimensional Lorentz Space Embeddings

Core Concepts

The asymptotic behavior of entropy numbers for natural embeddings between finite-dimensional Lorentz spaces is determined, generalizing classical results for Lebesgue sequence spaces.

Abstract

The paper studies the asymptotic behavior of entropy numbers for natural embeddings between finite-dimensional Lorentz sequence spaces. The main results are as follows: For p ≠ q < ∞, the entropy numbers behave the same as for embeddings between Lebesgue sequence spaces ℓ^n_p → ℓ^n_q. For q < p = ∞, the entropy numbers decay like 2^(-k/n) n^(1/q) (log n)^(-1/u). For p < q = ∞, the entropy numbers exhibit a more complex behavior, with different asymptotics in different regimes of k. For p = q < ∞, the asymptotics depend on the relationship between the Lorentz parameters u and v. For p = q = ∞, the asymptotics again depend on the relationship between u and v. The proofs employ techniques such as interpolation, volume comparison, and sparse approximation, as well as combinatorial arguments. The results generalize classical theorems by Schütt, Edmunds-Triebel, Kühn, and Guédon-Litvak for Lebesgue sequence spaces.

Stats

n^(1/q-1/p)+ n^(1/q)(log n)^(-1/u) 1 (log n)^(1/v-1/u)+

Quotes

None.

Key Insights Distilled From

Entropy numbers of finite-dimensional Lorentz space embeddings

by Josc... at arxiv.org 04-10-2024

https://arxiv.org/pdf/2404.06058.pdf

Deeper Inquiries

How do the entropy numbers of embeddings between Lorentz spaces behave if the spaces are not finite-dimensional

In the context of Lorentz spaces, the behavior of entropy numbers for embeddings between non-finite-dimensional spaces can be quite complex. Unlike in the finite-dimensional case where sharp asymptotic results were obtained, the situation becomes more intricate when dealing with infinite-dimensional Lorentz spaces. In general, when the Lorentz spaces are not finite-dimensional, the entropy numbers of embeddings may exhibit more variability and may not follow the same clear patterns as in the finite-dimensional case. The lack of finite dimensionality can introduce additional challenges in characterizing the compactness of linear operators between Lorentz spaces, leading to more nuanced and intricate results. The study of entropy numbers in infinite-dimensional Lorentz spaces requires more advanced mathematical techniques and a deeper understanding of the properties of these function spaces. The behavior of entropy numbers in this setting may depend on various factors such as the specific properties of the Lorentz spaces involved, the nature of the embeddings, and the characteristics of the operators under consideration.

Can the techniques developed in this paper be extended to study entropy numbers of more general operators between Lorentz spaces, beyond just natural embeddings

The techniques developed in the paper for studying entropy numbers of embeddings between Lorentz spaces can indeed be extended to analyze more general operators between these spaces. By building upon the foundational results and methodologies established in the paper, researchers can explore a wider range of operators and their compactness properties in Lorentz spaces. Extending the techniques to study entropy numbers of more general operators may involve adapting the existing methods to handle different types of operators, considering various function space properties, and exploring the interplay between different function space structures. This extension can lead to a deeper understanding of the compactness of operators in Lorentz spaces and provide insights into the approximation and complexity aspects of these function spaces. By leveraging the insights and techniques developed in the paper, researchers can explore a broader class of operators, investigate more complex scenarios, and further advance the theory of entropy numbers in Lorentz spaces.

Are there applications of the precise asymptotic characterization of entropy numbers for Lorentz space embeddings in areas such as approximation theory, information-based complexity, or machine learning

The precise asymptotic characterization of entropy numbers for Lorentz space embeddings has significant implications and applications in various areas of mathematics and related fields. In the context of approximation theory, the detailed understanding of entropy numbers can provide valuable insights into the compactness and approximation properties of operators between Lorentz spaces. This knowledge can be utilized to optimize approximation algorithms, analyze the convergence rates of approximation methods, and improve the efficiency of numerical computations in function space settings. In information-based complexity, the precise asymptotic characterization of entropy numbers can help in quantifying the complexity of problems and algorithms in terms of information content. By studying the behavior of entropy numbers in Lorentz spaces, researchers can gain a deeper understanding of the computational complexity of tasks involving these function spaces. In machine learning, the insights derived from the asymptotic characterization of entropy numbers can contribute to enhancing the performance of algorithms, improving the generalization capabilities of models, and optimizing the representation and processing of data in high-dimensional spaces. The results can be applied to develop more efficient and accurate machine learning models that leverage the properties of Lorentz spaces for various tasks such as classification, regression, and dimensionality reduction.