Covering number

Not to be confused with Winding number or degree of a continuous mapping, sometimes called "covering number" or "engulfing 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:

${\displaystyle K\subseteq \bigcup _{x\in C}B_{r}(x)}$.

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

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

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

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

Examples

1. The metric space is the real line ${\displaystyle \mathbb {R} }$. ${\displaystyle K\subset \mathbb {R} }$ is a set of real numbers whose absolute value is at most ${\displaystyle k}$. Then, there is an external covering of ${\textstyle \left\lceil {\frac {2k}{r}}\right\rceil }$ intervals of length ${\displaystyle r}$, covering the interval ${\displaystyle [-k,k]}$. Hence:

   ${\displaystyle N_{r}^{\text{ext}}(K)\leq {\frac {2k}{r}}}$

2. The metric space is the Euclidean space ${\displaystyle \mathbb {R} ^{m}}$ with the Euclidean metric. ${\displaystyle K\subset \mathbb {R} ^{m}}$ is a set of vectors whose length (norm) is at most ${\displaystyle k}$. If ${\displaystyle K}$ lies in a d-dimensional subspace of ${\displaystyle \mathbb {R} ^{m}}$, then: ^{[1] : 337}

   ${\displaystyle N_{r}^{\text{ext}}(K)\leq \left({\frac {2k{\sqrt {d}}}{r}}\right)^{d}}$.

3. The metric space is the space of real-valued functions, with the l-infinity metric. The covering number ${\displaystyle N_{r}^{\text{int}}(K)}$ is the smallest number ${\displaystyle k}$ such that, there exist ${\displaystyle h_{1},\ldots ,h_{k}\in K}$ such that, for all ${\displaystyle h\in K}$ there exists ${\displaystyle i\in \{1,\ldots ,k\}}$ such that the supremum distance between ${\displaystyle h}$ and ${\displaystyle h_{i}}$ is at most ${\displaystyle r}$. The above bound is not relevant since the space is ${\displaystyle \infty }$-dimensional. However, when ${\displaystyle K}$ is a compact set, every covering of it has a finite sub-covering, so ${\displaystyle N_{r}^{\text{int}}(K)}$ is finite. ^{[2] : 61}

Properties

1. The internal and external covering numbers, the packing number, and the metric entropy are all closely related. The following chain of inequalities holds for any subset K of a metric space and any positive real number r. ^[3]

   ${\displaystyle N_{2r}^{\text{met}}(K)\leq N_{r}^{\text{pack}}(K)\leq N_{r}^{\text{ext}}(K)\leq N_{r}^{\text{int}}(K)\leq N_{r}^{\text{met}}(K)}$

2. Each function except the internal covering number is non-increasing in r and non-decreasing in K. The internal covering number is monotone in r but not necessarily in K.

The following properties relate to covering numbers in the standard Euclidean space, ${\displaystyle \mathbb {R} ^{m}}$: ^{[1] : 338}

3. If all vectors in ${\displaystyle K}$ are translated by a constant vector ${\displaystyle k_{0}\in \mathbb {R} ^{m}}$, then the covering number does not change.
4. If all vectors in ${\displaystyle K}$ are multiplied by a scalar ${\displaystyle k\in \mathbb {R} }$, then:

   for all ${\displaystyle r}$: ${\displaystyle N_{|k|\cdot r}^{\text{ext}}(k\cdot K)=N_{r}^{\text{ext}}(K)}$

5. If all vectors in ${\displaystyle K}$ are operated by a Lipschitz function ${\displaystyle \phi }$ with Lipschitz constant ${\displaystyle k}$, then:

   for all ${\displaystyle r}$: ${\displaystyle N_{|k|\cdot r}^{\text{ext}}(\phi \circ K)\leq N_{r}^{\text{ext}}(K)}$

Application to machine learning

Let ${\displaystyle K}$ be a space of real-valued functions, with the l-infinity metric (see example 3 above). Suppose all functions in ${\displaystyle K}$ are bounded by a real constant ${\displaystyle M}$. Then, the covering number can be used to bound the generalization error of learning functions from ${\displaystyle K}$, relative to the squared loss: ^{[2] : 61}

${\displaystyle \operatorname {Prob} \left[\sup _{h\in K}{\big \vert }{\text{GeneralizationError}}(h)-{\text{EmpiricalError}}(h){\big \vert }\geq \epsilon \right]\leq N_{r}^{\text{int}}(K)\,2\exp {-m\epsilon ^{2} \over 2M^{4}}}$

where ${\displaystyle r={\epsilon \over 8M}}$ and ${\displaystyle m}$ is the number of samples.

See also

• Polygon covering
• Kissing number

References

1. Shalev-Shwartz, Shai; Ben-David, Shai (2014). Understanding Machine Learning – from Theory to Algorithms. Cambridge University Press. ISBN   9781107057135.
2. Mohri, Mehryar; Rostamizadeh, Afshin; Talwalkar, Ameet (2012). Foundations of Machine Learning. US, Massachusetts: MIT Press. ISBN   9780262018258.
3. Tao, Terence. "Metric entropy analogues of sum set theory" . Retrieved 2 June 2014.