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Efficient quantization and weak covering of high dimensional cubes

Let ℤ_n = {Z_1, ..., Z_n} be a design; that is, a collection of n points Z_j ∈ [-1,1]^d. We study the quality of quantization of [-1,1]^d by the points of ℤ_n and the problem of quality of covering of [-1,1]^d by B_d(ℤ_n,r), the union of balls centred at Z_j ∈ℤ_n. We concentrate on the cases where the dimension d is not small (d≥ 5) and n is not too large, n ≤ 2^d. We define the design 𝔻_n,δ as the maximum-resolution 2^d-1 design defined on vertices of the cube [-δ,δ]^d, 0≤δ≤ 1. For this design, we derive a closed-form expression for the quantization error and very accurate approximations for the coverage area vol([-1,1]^d ∩ B_d(ℤ_n,r)). We conjecture that the design 𝔻_n,δ with optimal δ is the most efficient quantizer of [-1,1]^d under the assumption n ≤ 2^d and it is also makes a very efficient (1-γ)-covering. We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs 𝔻_n,δ.

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1 Introduction

1.1 Main notation

  • : the Euclidean norm;

  • : -dimensional ball of radius centered at ;

  • : a design; that is, a collection of points ;

  • ;

  • vol: the proportion of the cube covered by ;

  • vectors in are row-vectors;

  • for any , .

1.2 Main problems of interest

We will study the following two main characteristics of designs .

1. Quantization error. Let be uniform random vector on . The mean squared quantization error for a design is defined by

(1)

2. Weak coverage. Denote the proportion of the cube covered by the union of balls by

For given radius , the union of balls makes the -coverage of the cube if

(2)

Complete coverage corresponds to . In this paper, the complete coverage of will not be enforced and we will mostly be interested in weak coverage, that is, achieving (2) with some small .

Two -point designs and will be differentiated in terms of performance as follows: (a) dominates for quantization if ; (b) if for a given , and , then the design provides a more efficient -covering than and is therefore preferable. In Section 1.4 we extend these definitions by allowing the two designs to have different number of points and, moreover, to have different dimensions.

1.3 Relation between quantization and weak coverage

The two characteristics, and , are related: , as a function of , is the c.d.f. of the r.v. while

is the second moment of the distribution with this c.d.f.:

(3)

In particular, this yields that if an -point design maximizes, in the set of all -point designs, for all , then it also minimizes . Moreover, if r.v. stochastically dominates , so that for all and the inequality is strict for at least one , then .

The relation (3) can alternatively be written as

(4)

where , considered as a function of , is the c.d.f. of the r.v. and hence is the mean of this r.v. Relation (4) is simply another form of (1).

1.4 Renormalised versions and formulation of optimal design problems

In view of (13), the naturally defined re-normalized version of is From (4) and (3), is the expectation of the r.v. and the second moment of the r.v. respectively. This suggests the following re-normalization of the radius with respect to and :

(5)

We can then define optimal designs as follows. Let be fixed, be the set of all -point designs and be the set of all designs.

Definition 1

The design with some is optimal for quantization in , if

(6)
Definition 2

The design with some is optimal for -coverage of , if

(7)

Here and for a given design ,

(8)

where is defined as the smallest such that .

Importance of the factor in (5) will be seen in Section 3.6 where we shall study the asymptotical behaviour of -coverings for large .

1.5 Thickness of covering

Let in Definition 2. Then is the covering radius associated with so that the union of the balls with makes a covering of . Let us tile up the whole space with the translations of the cube and corresponding translations of the balls . This would make a full covering of the whole space; denote this space covering by . The thickness of any space covering is defined, see (Conway and Sloane, 2013, f-la (1), Ch. 2), as the average number of balls containing a point of the whole space. In our case of , the thickness is

The normalised thickness, , is the thickness divided by , the volume of the unit ball, see (Conway and Sloane, 2013, f-la (2), Ch. 2). In the case of , the normalised thickness is

where we have recalled that and for any .

We can thus define the normalised thickness of the covering of the cube by the same formula and extend it to any :

Definition 3

Let be a -covering of the cube with . Its normalised thickness is defined by

(9)

where , see (5).

In view of (9), we can reformulate the definition (7) of the -covering optimal design by saying that this design minimizes (normalised) thickness in the set of all -covering designs.

1.6 The design of the main interest

We will be mostly interested in the following -point design defined only for :

Design : a maximum-resolution design defined on vertices of the cube , .

The design extends to the lattice (shifted by ) containing points with integer components satisfying , see (Conway and Sloane, 2013, Sect. 7.1, Ch. 4); this lattice is sometimes called ‘checkerboard lattice’. The motivation to theoretically study the design is a consequence of numerical results reported in Noonan and Zhigljavsky (2020), where the authors have shown that for all dimensions , the design with suitable provides the best quantization and covering per point among all other designs considered. In Noonan and Zhigljavsky (2020), we have considered -point designs in -dimensional cubes providing good covering and quantization. Aiming at practical applications in computer experiments, optimization and numerical integration, our aim was to consider the designs with which is not too large and in any case does not exceed . In fact, as a result of extensive numerical comparisons and the analysis we have done in this paper, we state the following conjecture.

Conjecture. For all , the design with optimal determined by (20), is the optimal design for quantization, if we add the restriction in (6).

We do not think that the conjecture is true without a restriction on in (6) as for very large we can construct one of the very efficient lattice space quantizers, see (Conway and Sloane, 2013, Sect. 3, Ch. 2), take the lattice points belonging to a very large cube and scale the cube back to ; this may result in a better quantizer than the design . It is difficult to study (both, numerically and theoretically) properties of such designs since they have to have very large number of points and are expected to have several non-congruent types of Voronoi cells due to boundary conditions. As we are interested in practical applications, we believe that the designs with practically reachable values of are more important. We do not state a conjecture concerning the structure of the best covering designs as formulated in Definition 2, as the structure of such designs depends on both and , see discussions in Section 4.

For theoretical comparison with design , we shall consider the following simple design, which extends to the integer point lattice (shifted by ) in the whole space :

Design : the collection of points , all vertices of the cube .

Without loss of generality, while considering the design we assume that the point is . Similarly, the first point in is . Note also that for numerical comparisons, in Section 4 we shall introduce one more design.

1.7 Structure of the rest of the paper and the main results

In Section 2 we study , the normalized mean squared quantization error for the design . There are two important results, Theorems 2.1 and 2.2. In Theorems 2.1, we derive the explicit form for the Voronoi cells for the points of the design and in Theorem 2.2 we derive a closed-form expression for for any . As a consequence, in Corollary 1 we determine the value of the optimal value of .

The main result of Section 3 is Theorem 3.1, where we derive closed-form expressions (in terms of , the fraction of the cube covered by a ball ) for the coverage area with vol. Then, using accurate approximations for , we derive approximations for vol. In Theorem 3.2 we derive asymptotic expressions for the -coverage radius for the design and show that for any , the ratio of the -coverage radius to the -coverage radius tends to as . Numerical results of Section 3.6 confirm that even for rather small , the -coverage radius is much smaller than the -coverage radius providing the full coverage.

In Section 4 we demonstrate that the approximations developed in Section 3 are very accurate and make a comparative study of selected designs used for quantization and covering.

In Appendices A–C, we provide proofs of the most technical results. In Appendix D, for completeness, we briefly derive an approximation for with arbitrary , and .

The main results of the paper are: a) derivation of the closed-form expression for the quantization error for the design , and b) derivation of accurate approximations for the coverage area vol for the design .

2 Quantization

2.1 Reformulation in terms of the Voronoi cells

Consider any -point design . The Voronoi cell for is defined as

The mean squared quantization error introduced in (1) can be written in terms of the Voronoi cells as follows:

(10)

where and .

This reformulation has significant benefit when the design has certain regular structure. In particular, if all of the Voronoi cells are congruent, then we can simplify (10) to

(11)

In Section 2.4, this formula will be the starting point for derivation of the closed-form expression for for the design .

2.2 Re-normalization of the quantization error

To compare efficiency of -point designs with different values of , one must suitably normalise with respect to . Specialising a classical characteristic for quantization in space, as formulated in (Conway and Sloane, 2013, f-la (86), Ch.2), we obtain

(12)

Note that is re-normalised with respect to dimension too, not only with respect to . Normalization with respect to is very natual in view of the definition of the Euclidean norm.

If all Voronoi cells are congruent, then the volume of these Voronoi cells is and hence (12) simplifies to

In the general case, when some of the Voronoi cells are congruent, there is no direct relation between (10) and (12) so that is not proportional to the mean squared quantization error (1). Also, the quantity is intractable theoretically and very expensive numerically when considering designs with non-congruent Voronoi cells. To address this, we directly re-normalize the mean squared quantization error (1) and introduce

(13)

If all Voronoi cells are congruent then so that both criteria coincide.

The additional rationale for introducing in (13) is as follows. From results of (Niederreiter, 1992, Ch.6), for efficient covering schemes the order of convergence of the covering radius to 0 as is . Therefore, for the mean squared distance (which is the quantization error) we should expect the order as .

2.3 Voronoi cells for

Proposition 1

Consider the design , the collection of points , . The Voronoi cells for this design are all congruent. The Voronoi cell for the point is the cube

(14)

. Consider the Voronoi cells created by the design in the whole space . For the point , the Voronoi cell is clearly . By intersecting this set with the cube we obtain (14).

Theorem 2.1

The Voronoi cells of the design are all congruent. The Voronoi cell for the point is

(15)

where is the cube (14) and

(16)

The volume of is .

. The design is symmetric with respect to all components implying that all Voronoi cells are congruent immediately yielding that their volumes equal 2. Consider with .

Since , where design is introduced in Proposition 1, and is the Voronoi set of for design , for design too.

Consider the cubes adjacent to :

A part of each cube belongs to . This part is exactly the set defined by (16). This can be seen as follows. A part of also belongs to the Voronoi set of the point , where with 1 placed at -th place; all components of are except -th and -th components which are . We have to have , for a point to be closer to than to . Joining all constraints for (, ) we obtain (16) and hence (15).


2.4 Explicit formulae for the quantization error

Theorem 2.2

For the design with , we obtain:

(17)
(18)

. To compute , we use (11), where, in view of Theorem 2.1, . Using the expression (15) for with , we obtain

(19)

Consider the two terms in (19) separately. The first term is easy:

For the second term we have:

Inserting the obtained expressions into (19) we obtain (17). The expression (18) is a consequence of (13), (17) and .

A simple consequence of Theorem 2.2 is the following corollary.

Corollary 1

The optimal value of minimising and is

(20)

for this value,

(21)

From (18), for the design with we get

(22)

which is always slightly larger than (21). Let us make five more remarks.

  1. For the one-point design with the single point 0 and the design with points we have , which coincides with the value of in the case of space quantization by the integer-point lattice , see (Conway and Sloane, 2013, Ch. 2 and 21).

  2. The quantization error (22) for the design have almost exactly the same form as the quantization error for the ‘checkerboard lattice’ in ; the difference is in the factor in the last term in (22), see (Conway and Sloane, 2013, f-la (27), Ch.21). Naturally, the quantization error for in is slightly smaller than for in .

  3. The optimal value of in (20) is smaller than . This is caused by a non-symmetrical shape of the Voronoi cells for designs , which is clearly visible in (15).

  4. The minimal value of is achieved with .

  5. Formulas (20) and (21) are in agreement with numerical results seen in Table 4 of Noonan and Zhigljavsky (2020).

Let us briefly illustrate the results above. In Figure 2, the black circles depict the quantity as a function of . The quantity is shown with the solid red line. We conclude that from dimension seven onwards, the design provides better quantization per points than the design . Moreover for , the quantity slowly increases and converges to . Typical behaviour of as a function of is shown in Figure 2. This figure demonstrates the significance of choosing optimally.

Figure 1: and as functions of and ; .
Figure 2: as a function of and with ;

3 Closed-form expressions for the coverage area with and approximations

In this section, we will derive explicit expressions for the coverage area of the cube by the union of the balls associated with the design introduced in Section 1.2. That is, we will derive expressions for the quantity for all values of . Then, in Section 3.4, we shall obtain approximations for . The accuracy of the approximations will be assessed in Section 4.2.

3.1 Reduction to Voronoi cells

For an -point design , denote the proportion of the Voronoi cell around covered by the ball as:

Then we can state the following simple lemma.

Lemma 1

Consider a design such that all Voronoi cells are congruent. Then for any , .

In view of Theorem 2.1, for design all Voronoi cells are congruent and ; recall that . By then applying Lemma 1 and without loss of generality we have choosen , we have for any

(23)

In order to formulate explicit expressions for , we need an important quantity, proportion of intersection of with one ball.

3.2 Intersection of with one ball

Take the cube and a ball centered at a point ; this point could be outside . The fraction of the cube covered by the ball is denoted by

The following relation will prove useful. Assume that we have the cube of volume , the ball with a center at a point . Denote the fraction of the cube covered by the ball by

Then the change of the coordinates and the radius gives

(24)

3.3 Expressing through

Theorem 3.1

Depending on the values of and , the quantity can be expressed through for suitable as follows.

  • For :

    (25)
  • For :

    (26)
  • For :

    (27)

The proof of Theorem 3.1 is given in Appendix A.

3.4 Approximation for

Accurate approximations for for arbitrary and were developed in Noonan and Zhigljavsky (2020)

. By using the general expansion in the central limit theorem for sums of independent non-identical r.v., the following approximation was developed:

(28)

where

A short derivation of this approximation is included in Appendix D. Using (28), we formulate the following approximation for .

Approximations for . Approximate the values in formulas (25),(26),(27) with corresponding approximations (28).

3.5 Simple bounds for

Lemma 2

For any , and , the quantity can be bounded as follows:

(29)

where