In mathematics, a covering number is the number of balls 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.
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 ball of radius r centered at x.
A subset C of M is an r-external covering of K if: In other words, for every
If furthermore C is a subset of K, then it is an r-internal covering.
The external covering number of K, denoted
, is the minimum cardinality of any external covering of K. The internal covering number, denoted
, is the minimum cardinality of any internal covering.
, 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, denoted
, is the maximum cardinality of any r-separated subset of K. The following properties relate to covering numbers in the standard Euclidean space,
be a space of real-valued functions, with the l-infinity metric (see example 3 above).
Then, the covering number can be used to bound the generalization error of learning functions from