Covering number

Covering number

The covering number refers to the minimum number of balls or hyperspheres with a given radius needed to cover a set or data set in a metric space. It captures the size of the metric space and is used to determine the level of coverage required to encompass all observations in the set.From: Handbook of Big Data [2019], Quantitative stability of full random two-stage problems with quadratic recourse [2019], Exploratory Data Analysis with MATLAB [2019]

Dimensionality Reduction — Linear Methods

View Chapter

Purchase Book

Published in Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka, Exploratory Data Analysis with MATLAB®, 2017

Wendy L. Martinez, Angel R. Martinez, Jeffrey L. Solka

Given a metric space, the r-covering number N(r) of a set S={x1,...,xn} $ S = \{ x_{1} , ..., x_{n} \} $ in this space is the smallest number of hyperspheres with radius r that is needed to cover all observations xi in the data set S. The r-covering number N(r) of a d-dimensional data set is proportional to r-d, and we can define the capacity dimension as d=-limr→0logN(r)logr. $$ d = - \mathop {\lim }\limits_{{r \to 0}} \frac{{{{\log~}}N(r)}}{{{{\log~}}r}} . $$

Quantitative stability of full random two-stage problems with quadratic recourse

View Article

Journal Information

Published in Optimization, 2019

Sainan Zhang, Shaoyan Guo, Hongwei Zhang, Liwei Zhang

[2] Let be a subset of the normed space (for some ) equipped with the usual norm . The covering number is the minimal number of balls of radius ϵ needed to cover the set . The centres of the balls need not belong to , but they should have finite norms.Given two functions and from , the bracket for P-almost all is called an ϵ-bracket if . Then the bracketing number is the minimum number of ϵ-brackets needed to cover .

Support vector machines regression with unbounded sampling

View Article

Journal Information

Published in Applicable Analysis, 2019

Hongzhi Tong, Di-Rong Chen, Fenghong Yang

For a subset of C(X), the covering number is defined to be the minimal integer such that there exist l balls with radius covering .

Explore chapters and articles related to this topic

Dimensionality Reduction — Linear Methods

Quantitative stability of full random two-stage problems with quadratic recourse

Support vector machines regression with unbounded sampling