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Mat. Zametki, 2012, Volume 91, Issue 6, Pages 853–860 (Mi mz9388)  

Bracketing Entropy and VC-Dimension

Yu. V. Malykhin

Steklov Mathematical Institute of the Russian Academy of Sciences

Abstract: We study the relationship between two characteristics of functional classes, pseudodimension and bracketing entropy. (Pseudodimension is a generalization of VC-dimension to classes of functions. Bracket entropy characterizes the $L_1$-error of one-sided approximation of a class by finite sets.) It is shown that classes of continuous functions with finite pseudodimension possess a finite bracketing $\varepsilon$-entropy for any $\varepsilon>0$. We establish a general result concerning the relationship between the VC-dimension of classes of sets and their bracketing entropy.

Keywords: bracketing entropy, pseudodimension, VC-dimension, measurable function, bracketing compactness of sets, random variable, probability space

Funding Agency Grant Number
Russian Foundation for Basic Research 09-01-12173-офи-м


DOI: https://doi.org/10.4213/mzm9388

Full text: PDF file (478 kB)
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English version:
Mathematical Notes, 2012, 91:6, 800–807

Bibliographic databases:

UDC: 517.518.8
Received: 18.11.2010

Citation: Yu. V. Malykhin, “Bracketing Entropy and VC-Dimension”, Mat. Zametki, 91:6 (2012), 853–860; Math. Notes, 91:6 (2012), 800–807

Citation in format AMSBIB
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