In mathematics, a covering number is the number of ballss of a given size needed to completely cover a given space, with possible overlaps between the balls. The covering number quantifies the size of a set and can be applied to general metric spaces. Two related concepts are the packing number, the number of disjoint balls that fit in a space, and the metric entropy, the number of points that fit in a space when constrained to lie at some fixed minimum distance apart.

Definition

Let (M, d) be a metric space, let K be a subset of M, and let r be a positive real number. Let Br(x) denote the closed ball of radius r centered at x. A subset C of M is an r-external covering of K if:

1. .

.

If furthermore C is a subset of K, then it is an r-internal covering.

The external covering number of K, is the minimum cardinality of any external covering of K. The internal covering number, is the minimum cardinality of any internal covering.

A subset P of K is a packing is pairwise disjoint. The packing number of K, is the maximum cardinality of any packing of K.

A subset S of K is r-separated if each pair of points x and y in S satisfies d(x, y) ≥ r. The metric entropy of K, is the maximum cardinality of any r-separated subset of K.

Examples

Properties

The following properties relate to covering numbers in the standard Euclidean space:

Application to machine learning

be a space of real-valued functions, with the ℓ-infinity. Then, the covering number can be used to bound the generalization error, relative to the squared loss:

is the number of samples.

See also

• Polygon covering

• Kissing number

References