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A remark on covering

01/10/2013
by   Vladimir Temlyakov, et al.
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We discuss construction of coverings of the unit ball of a finite dimensional Banach space. The well known technique of comparing volumes gives upper and lower bounds on covering numbers. This technique does not provide a construction of good coverings. Here we apply incoherent dictionaries for construction of good coverings. We use the following strategy. First, we build a good covering by balls with a radius close to one. Second, we iterate this construction to obtain a good covering for any radius. We mostly concentrate on the first step of this strategy.

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1 Introduction

Let be a Banach space with a norm and let denote the corresponding closed unit ball:

(1.1)

The open unit ball will be denoted by :

(1.2)

Notation and will be used respectively for closed and open balls with the center and radius . In case we drop it from the notation: . For a compact set and a positive number we define the covering number as follows

The following proposition is well known.

Proposition 1.1.

For any -dimensional Banach space we have

This proposition describes the behavior of when . In this paper we concentrate on the case when is close to . In particular, we discuss the following problem: How many balls are needed for covering ? In other words we are interested in the number

(1.3)

We prove here that if is a uniformly smooth Banach space then . With this result in hands we discuss the problem: How small can be for the relation to hold? The left inequality in Proposition 1.1 gives the lower bound for such : . In Section 3 we prove an upper bound: . This upper bound follows from two different constructions given in Propositions 3.2 and 3.4. In both constructions we use a system

of vectors and built a covering of

in the form with an appropriate . In Section 4 we apply this idea with being an incoherent dictionary for covering in the Hilbert space . We prove the following bound in Corollary 4.1. For , , we have

(1.4)

In Section 5 we use incoherent dictionaries in a smooth Banach space to build a good covering for . Let denote the modulus of smoothness of (see Section 3 below for definition) and be a solution (actually, it is a unique solution) to the equation

We prove the following bound in Corollary 5.1. For , , we have

(1.5)

It is interesting to note (see Section 6) that in the case , , we have as in the case .

In Section 6 we consider several specific examples of and make a conclusion that the technique based on extremal incoherent dictionaries works well and provides either optimal or close to optimal bounds in the sense of order of .

2 Lower bounds

We prove the following bound in this section.

Theorem 2.1.

Let be a Banach space with a norm . Then

Proof.

We prove that any balls , do not cover . Indeed, for a given set consider the linear manifold passing through :

It is clear that is a -dimensional linear manifold. We use Lemma 2.1 below which guarantees that there is , such that for any we have . Then is not covered by the . ∎

Lemma 2.1.

Let be a Banach space with a norm . Then for any -dimensional manifold we have

(2.1)
Proof.

Without loss of generality we can assume that is a subspace. Indeed, by symmetry of we have that where

Let

Define a subspace

Then . Indeed, for any there are and such that

Set . Then

So, we assume that is a subspace. A standard proof of statements like Lemma 2.1 is based on the antipodality theorem of Borsuk (see, for instance, [2], p. 405). We give a proof that is based on ideas from functional analysis. Let be a functional such that and for . Consider a norming functional for . Our space is a reflexive Banach space. So . For any we have

This completes the proof of Lemma 2.1

3 Upper bounds

We begin with the case when the norm is the Euclidean norm. Let denote the standard basis: if and .

Proposition 3.1.

Define , and . Then

Proof.

We begin with describing a set that is not covered by , . Take any point . Then . Setting we obtain

If then . Thus those which are not covered by satisfy the inequality which implies . Therefore,

We now prove that . Indeed, for any we have

The inequality implies and

Using

we obtain

Proposition 3.2.

Define , , and . Then

Proof.

The proof repeats the proof of Proposition 3.1. We only point out the places where we make changes. First, we note that if then . Therefore, in this case . We have

We now prove that . Similar to the above argument we get

For a Banach space we define the modulus of smoothness

The uniformly smooth Banach space is the one with the property

Proposition 3.3.

Let be a uniformly smooth Banach space with norm . Define , and . Then there exists an such that

(3.1)
Proof.

Embedding (3.1) is equivalent to the claim that for each at least one of the following inequalities is satisfied

(3.2)
(3.3)

In the proof that follows parameter is small. We assume that . Then for such that all inequalities (3.2) are satisfied. Therefore, in further argument it is sufficient to consider such that .

For let be a norming functional for : and . Existence of such a functional follows from the Hahn-Banach theorem. We note that from the definition of modulus of smoothness we get the following inequality (see, for instance, [4], p.336).

Lemma 3.1.

Let . Then

where is a norming functional of .

This lemma implies the following inequalities

(3.4)
(3.5)

Here, is the norming functional of .

First, we note that for some the is large enough. Indeed, let . Then

We have

which implies that for some

(3.6)

Set and consider three cases:

(3.7)
(3.8)
(3.9)

In the case (3.7) inequality (3.5) implies (3.3) if is sufficiently small (remind that uniform smoothness assumption implies as ). In the case (3.8) we have for some that and this is sufficient to derive (3.2) with from (3.4) and small .

Consider the case (3.9). Inequality (3.6) guarantees that either or . In case we complete the proof as in case (3.8). In case our assumption (3.9) implies that

and

Therefore, for some

and we complete the proof as in case (3.8). ∎

We now discuss another way of constructing a -covering of the Euclidean ball. It is based on the tight frames construction. We begin with a conditional statement.

Proposition 3.4.

Let be a system of normalized vectors, , , satisfying the condition

Then, there exists an such that

Proof.

In our proof is a small number. Let . Then for any , , and any we have

Thus, it is sufficient to consider such that . For each we have

(3.10)

We now need to estimate

from below. It is easy to check that our assumptions on imply the relations

(3.11)
(3.12)
(3.13)

We now need a simple technical lemma.

Lemma 3.2.

If is such that then there exists satisfying

Proof.

The proof goes by contradiction. Suppose for all . Denote

Then our assumption implies (note that )

and, therefore,

It is a contradiction. ∎

We apply Lemma 3.2 with , . Then the condition follows from (3.12). Thus, by (3.13), taking into account that , we derive from Lemma 3.2 that there exists such that

By (3.10) we obtain for this

Specifying we get

We now discuss a question of existence and construction of systems from Proposition 3.4. We only give one example of such construction which is based on the Hadamard matrices. Hadamard matrices are very useful in both theoretical research and engineering applications. In particular, Hadamard matrices are very popular in error-correction coding theory. A Hadamard matrix of order is an matrix with all entries or , and

where

is the identity matrix. Obviously, any two columns or any two rows of a Hadamard matrix

are mutually orthogonal. This orthogonality is kept if we permute some rows or columns, or multiply some rows or columns by -1. Therefore, given any Hadamard matrix, we can always make a new Hadamard matrix which has all 1’s in the first row by multiplying some columns by -1. Hadamard matrices only exist for special orders . The following lemma and remark are from [5].

Lemma 3.3.

If is a Hadamard matrix of order , then , , or .

Remark 3.1.

One of the famous conjectures in the area of combinatorial designs states that a Hadamard matrix of order exists for every . But we are still very far from a proof of this conjecture. The smallest for which a Hadamard matrix could exist but no example is known presently 428.

There exists a variety of methods to construct Hadamard matrices. We can construct Hadamard matrices from so-called conference matrices (see [5]). We will not discuss this way. For illustration purposes we provide a very simple construction of Hadamard matrices of order . The following lemma provides a recursive method to build Hadamard matrices of order , where .

Lemma 3.4.

For , the matrices generated by

are Hadamard matrices.

Proof.

Clearly, and are Hadamard matrices of order 1 and 2 respectively. Assume is a Hadamard matrix of order , then

We need to show that

Indeed,

We can build higher order Hadamard matrices from the Kronecker product of lower order Hadamard matrices. Let matrix with entries and . Then the Kronecker product of and is a matrix,

The following simple lemma is known.

Lemma 3.5.

If and are Hadamard matrices of order and respectively, then is a Hadamard matrix of order .

This lemma provides a good way to build higher order Hadamard matrices from known lower order ones. We can see that Lemma 3.4 is a corollary of Lemma 3.5, where the recursion is .

The Hadamard matrices were used in [1] for construction systems from Proposition 3.4. Such systems are called absolutely equiangular tight frames in [1].

Theorem 3.1.

Let be a Hadamard matrix with all in the first row and . Then, the columns of the matrix generated by deleting the first row of and dividing by form an absolutely equiangular tight frame.

Proof.

All columns of are mutually orthogonal. In other words, for any , the two columns and of satisfy .

Since the first elements of and are both 1, the corresponding columns and of satisfy

for all . ∎

4 Covering using incoherent dictionaries

Proposition 3.4 demonstrates how special dictionaries can be used for building coverings. In this section we discuss an application of incoherent dictionaries in Euclidean space. Let be a normalized (, ) system of vectors in equipped with the Euclidean norm. We define the coherence parameter of the dictionary as follows

In this section we discuss the following characteristics

The problem of studying is equivalent to a fundamental problem of information theory. It is a problem on optimal spherical codes. A spherical code is a set of points (code words) on the -dimensional unit sphere, such that the absolute values of inner products between any two distinct code words is not greater than . The problem is to find the largest such that the spherical code exists. It is clear that . Denote by a dictionary such that and . We call such an extremal dictionary for a given .

Theorem 4.1.

Let be an extremal dictionary for a given . Then

Thus, .

Proof.

Our assumption that is an extremal dictionary for implies that for any there is such that . Suppose, . The other case is treated exactly the same way. Then

The problem of estimating is well studied (see, for instance, [4], section 5.7, p. 314). It is known (see [4], p. 315) that for a system with we have . Thus, a natural range for is . In particular, the following bound is known (see [4], p. 315)

(4.1)

As a corollary of (4.1) and Theorem 4.1 we obtain the following statement.

Corollary 4.1.

For , , we have

5 Covering in Banach spaces using incoherent dictionaries

We use here a generalization of the concept of -coherent dictionary to the case of Banach spaces. This generalization was published in [3] (see also [4], p. 381).

Let be a dictionary in a Banach space . We define the coherence parameter of this dictionary in the following way

where is a norming functional for . We note that, in general, a norming functional is not unique. This is why we take over all norming functionals of in the definition of . We do not need in the definition of if for each there is a unique norming functional . Then we define and call a dual dictionary to a dictionary . It is known that the uniqueness of the norming functional is equivalent to the property that is a point of Gateaux smoothness:

for any . In particular, if is uniformly smooth then is unique for any .

Let be a normalized system of vectors in , which is equipped with a norm , . Denote by

a matrix formed by column vectors . Suppose for simplicity that for each there is a unique norming functional . Each functional can be associated with a vector in such a way that , . Then

Consider the matrix

which is a matrix formed by column vectors . Consider the transposed matrix that is formed by the row vectors , , or by the column vectors , . Define the coherence matrix of a dictionary as follows

Then the coherence matrix of the system satisfies the following inequality for the rank: . Indeed, the columns of are linear combinations of columns , . It is clear that the coherence matrix , , has on the diagonal and for all off-diagonal elements we have .

In this section we discuss the following characteristics

We now use a fundamental result of Alon (see, for instance, [4], p.317) to derive an upper bound for from the property .

Theorem 5.1.

Let be a square matrix of the form , ; , . Then

(5.1)

with an absolute constant .

We apply this theorem with and . For
(5.1) implies that

For (5.1) implies that