Covering number

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.

Definition

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

$K\subseteq \bigcup _{x\in C}{B}_r(x)$.

In other words, for every $y\in K$ there exists $x\in C$ such that $d(x,y)\le r$.

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

The external covering number of K, denoted $N_r^{\text{ext}}(K)$, is the minimum cardinality of any external covering of K. The internal covering number, denoted $N_r^{\text{int}}(K)$, is the minimum cardinality of any internal covering.

A subset P of K is a packing if $P\subseteq K$ and the set $\{B_r(x){\}}_{x\in P}$ is pairwise disjoint. The packing number of K, denoted $N_r^{\text{pack}}(K)$, 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, denoted $N_r^{\text{met}}(K)$, is the maximum cardinality of any r-separated subset of K.

Properties

The following properties relate to covering numbers in the standard Euclidean space, ${ℝ}^m$:Template:Rp Template:Ordered list

Application to machine learning

Let $K$ be a space of real-valued functions, with the l-infinity metric (see example 3 above). Suppose all functions in $K$ are bounded by a real constant $M$. Then, the covering number can be used to bound the generalization error of learning functions from $K$, relative to the squared loss:Template:Rp

$\mathrm{Prob}\left[\underset{h\in K}{\sup }\right|\text{GeneralizationError}(h)-\text{EmpiricalError}(h)|\ge ϵ]\le N_r^{\text{int}}(K)2\exp \frac{-m{ϵ}^2}{2M^4}$

where $r=\frac{ϵ}{8M}$ and $m$ is the number of samples.

See also

• Polygon covering
• Kissing number