Tractability of approximation in the weighted Korobov space in the worst-case setting

In this paper we consider L_p-approximation, p ∈{2,∞}, of periodic functions from weighted Korobov spaces. In particular, we discuss tractability properties of such problems, which means that we aim to relate the dependence of the information complexity on the error demand ε and the dimension d to the decay rate of the weight sequence (γ_j)_j ≥ 1 assigned to the Korobov space. Some results have been well known since the beginning of this millennium, others have been proven quite recently. We give a survey of these findings and will add some new results on the L_∞-approximation problem. To conclude, we give a concise overview of results and collect a number of interesting open problems.

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1 Introduction

In this paper we consider -approximation, where , of periodic functions from a weighted Korobov space with smoothness parameter from the viewpoint of Information-Based Complexity. In particular, we study the information complexity of these problems, which is the minimal number of information evaluations required to push the approximation error below a certain error demand for problems in dimension . The information classes considered are the class consisting of arbitrary continuous linear functionals and the class consisting of point evaluations only. Furthermore, we will distinguish between the absolute and the normalized error criterion in the worst-case setting.

If the information complexity grows exponentially in for

tending to infinity, the problem is said to suffer from the curse of dimensionality. Otherwise, for sub-exponential growth rates, the problem is said to be tractable. Initially, only the notions of polynomial and strong polynomial tractability were introduced and studied in the literature. An extensive overview of tractability of multivariate problems can be found in the trilogy

[10, 11, 12].

For weighted function classes, one assigns real numbers (weights) to the coordinates in order to model varying influence of the single variables on the approximation problem, and one is interested in (matching) necessary and sufficient conditions on the weights which guarantee tractability. In the particular case of -approximation for the weighted Korobov space, matching conditions can be found in the paper [14] by Wasilkowski and Woźniakowski for the information class and in the paper [9] by Novak, Sloan, and Woźniakowski for . For -approximation, results on (strong) polynomial tractability are due to Kuo, Wasilkowski, and Woźniakowski; see [5] for and [6] for .

After (strong) polynomial tractability, more and finer notions of tractability of tractability have been introduced with the aim of obtaining a more detailed and clearer picture of the tractability of multivariate problems. Nowadays, there is a variety of finer notions of tractability comprising quasi-polynomial tractability, weak tractability, and uniform weak tractability. The exact definitions will be given in Definition 1

. Based on this development, many multivariate problems need to be reconsidered in order to classify them further with respect to the newer notions of tractability. This has been done recently in

[2] for the problem of -approximation for weighted Korobov spaces. These results will be summarized in Section 3. In the present paper we shall also study the -case. We derive necessary and sufficient conditions for several notions of tractability (see Section 4). The presented conditions are tight, but unfortunately do not match exactly. Here, some problems remain open.

In Section 5 we give a concise survey of the current state of research in tractability theory of approximation in weighted Korobov spaces and formulate some interesting open questions.

Notation and basic definitions will be introduced in the following section.

2 Basic definitions

Function space setting

The Korobov space with weight sequence in is a reproducing kernel Hilbert space with kernel function given by

where by “” we denote the usual dot product. The corresponding inner product and norm are given by

Here, the Fourier coefficients of a function are given by

and the decay function equals, for , , with (the so-called smoothness parameter of the space), and

The kernel is well-defined for and for all , since

where is the Riemann zeta function (note that since ).

Furthermore, we assume here that the weights are ordered and satisfy

The weighted Korobov space is a popular reference space for quasi-Monte Carlo rules, in particular for lattice rules. See, e.g., [7, Chapter 4] or [10, Appendix A] and the references therein.

Approximation in

In this paper we consider -approximation of functions from the weighted Korobov space for . We consider the operator with for all . The operator is the embedding from the weighted Korobov space to the space .

In order to approximate with respect to the -norm over , , it suffices to employ linear algorithms that use information evaluations and are of the form

(1)

with functions and bounded linear functionals for ; see [1] and also [8, 10]. We will assume that the functionals belong to some permissible class of information . In particular, we study the class consisting of the entire dual space and the class , which consists only of point evaluation functionals. Recall that is a reproducing kernel Hilbert space, which means that point evaluations are continuous linear functionals and therefore is a subclass of . With some abuse of notation we will write if is a linear algorithm of the form (1) using information from the class .

We remark that in both cases and , the embedding operator is continuous for all , which can be seen as follows.

  • For , we have for all that

    By considering the choice , it follows that the above inequality is sharp such that the operator norm of is given by

  • For , we have for all that

    By considering the choice , it follows that the above inequality is sharp such that the operator norm of is given by

The worst-case setting

The worst-case error of an algorithm as in (1) is defined as

and the -th minimal worst-case error with respect to the information class is given by

where the infimum is extended over all linear algorithms of the form (1) with information from the class . In the case the essential supremum is used in the calculation of .

The initial error, i.e., the error obtained by approximating by zero, equals

Note that for the initial error may be exponential in if it is not properly normalized. In the following analysis, we will therefore consider the normalized as well as the absolute error criterion.

We are interested in how the approximation error of algorithms depends on the number of information evaluations used and how it depends on the problem dimension . To this end, we define the so-called information complexity as

with and , and where either for the absolute error criterion (we then write ) and for the normalized error criterion (then, we write ).

Useful relations

In the case of -approximation we have and hence the absolute and the normalized error criteria coincide. This means that

and we just write for .

In the case of -approximation the situation is different, since . Hence we only have

(2)

Furthermore, it is well known, see, e.g., [3], that -approximation is not harder than -approximation for the absolute error criterion, which means that for we have

Thus, necessary conditions for tractability of -approximation in the weighted space are also necessary conditions for tractability of -approximation in for the absolute error criterion.

For the information class , -approximation for

can be fully characterized in terms of the eigenvalues of the self-adjoint, compact operator

The following well-known lemma (see, e.g., [10, p. 215]) provides information on the eigenpairs of the operator .

Lemma 1.

The eigenpairs of the operator are with , where for we set

Furthermore, denote the ordered eigenvalues of by , where

Note that , since and for all .

We then have the following relations (see, for example, [10, 13] for and [5, Theorem 2] for ) for the -th minimal error with respect to ,

Consequently,

for , and

(3)

for .

Relations to the average-case setting

Note that (3) is exactly the same as the information complexity for -approximation in the average-case setting for certain spaces (see [12, p. 190] for a general introduction to the average-case setting). Indeed, following the outline in [4], assume that we are given a sequence of spaces , , and study the operator with for . Furthermore, we assume that

is equipped with a Gaussian probability measure

, which has mean zero and a covariance function that coincides with the reproducing kernel of the Korobov space , with all parameters as above. I.e.,

Again, it is of interest to study approximation of by linear algorithms of the form (1). The average-case error of such an algorithm is given by

and the initial error by

We can also define the -th minimal average-case error of -approximation in for an information class by

Now define, for any Borel set in , the inverse image under by and let . Then, is a Gaussian measure on , again with mean zero, and a covariance operator given by

For more detailed information we refer to [4] and the references therein.

Using the notation just introduced, there are several relations to be observed between the worst-case setting and the average-case setting. Indeed, it is known that the eigenvalues of the covariance operator coincide with the eigenvalues of the operator introduced above. Furthermore, by making use of the relation between the covariance function of and the kernel , it can easily be shown that

Hence the initial error of average-case -approximation in is exactly the same as the initial error of worst-case -approximation in . What is more, if one allows information from , we have

for the -th minimal error, i.e., the -th minimal error of average-case -approximation in equals the -th minimal error of worst-case -approximation in . For the derivation of these results and further details, we refer to [13, Chapter 6], see also [10].

These observations (which have been pointed out in the literature before) imply that the results on -approximation in presented here can also be interpreted as results on average-case -approximation in . Indeed some of the theorems presented on -approximation below recover some of the results in [4] and the references therein, formulated for the average-case setting there.

Notions of tractability

An important goal of tractability theory is to analyze which problems suffer from the curse of dimensionality, i.e., whether there exist such that for infinitely many , and which do not. In the latter case it is then an important task to classify the growth rate of the information complexity with respect to the dimension  tending to infinity () and the error threshold  tending to zero (). Different growth rates are characterized by means of various notions of tractability which are given in the following definition.

Definition 1.

Consider the approximation problem for the information class . We say that for this problem we have:

  1. [label=()]

  2. Strong polynomial tractability (SPT) if there exist non-negative numbers such that

    (4)

    The infimum of all exponents such that (4) holds for some is called the exponent of strong polynomial tractability and is denoted by .

  3. Polynomial tractability (PT) if there exist non-negative numbers such that

  4. Quasi-polynomial tractability (QPT) if there exist non-negative numbers such that

    (5)

    The infimum of all exponents such that (5) holds for some is called the exponent of quasi-polynomial tractability and is denoted by .

  5. Weak tractability (WT) if

  6. -weak tractability (-WT) for positive numbers if

  7. Uniform weak tractability (UWT) if -weak tractability holds for all .

We obviously have the following hierarchy of tractability notions:

Furthermore, WT coincides with -WT for .

The characterization of the applicable tractability classes will be done with respect to decay conditions on the weight sequence . To this end, we introduce the following notation.

  • The infimum of the sequence is denoted by .

  • The sum exponent is defined as

  • The exponent is defined as

  • The exponent , for , is defined as

In the definitions of , , and we use the convention that .

3 The results for

A complete overview of necessary and sufficient conditions for tractability of -approximation in the weighted Korobov space has recently been published in [2].

Theorem 2.

Consider the approximation problem for the information class and let . Then we have the following results.

  1. (Cf. [14]) Strong polynomial tractability for the class holds if and only if . In this case the exponent of strong polynomial tractability is

  2. (Cf. [14]) Strong polynomial tractability and polynomial tractability for the class are equivalent.

  3. Quasi-polynomial tractability, uniform weak tractability, and weak tractability for the class are equivalent and hold if and only if .

  4. If we have quasi-polynomial tractability, then the exponent of quasi-polynomial tractability satisfies

    In particular, if , we set and we have that .

  5. For , -weak tractability for the class holds for all weights .

Theorem 3.

Consider multivariate approximation for the information class and . Then we have the following results.

  1. (Cf. [9]) Strong polynomial tractability for the class holds if and only if

    which implies . In this case the exponent of strong polynomial tractability satisfies

  2. (Cf. [9]) Polynomial tractability for the class holds if and only if

  3. Polynomial and quasi-polynomial tractability for the class are equivalent.

  4. Weak tractability for the class holds if and only if

  5. For , -weak tractability for the class holds if and only if

    For , -weak tractability for the class holds for all weights .

  6. Uniform weak tractability for the class holds if and only if

Theorems 2 and 3 imply that in the case of -approximation no open questions remain, at least for the currently most common tractability classes.

4 The results for

We have the following result for -approximation in the space .

Theorem 4.

Consider multivariate approximation for the information classes and for the normalized and absolute error criterion and . Then we have the following results.

  1. (Cf. [5] for and [6] for ) The approximation problem is strongly polynomially tractable if and only if . If this holds, then for any we have

    where in both cases the implied factor is independent of and .

  2. (Cf. [5] for and [6] for ) The approximation problem is polynomially tractable if and only if . If this holds, then for any and any we have

    where in both cases the implied factor is independent of and .

  3. A necessary condition for quasi-polynomial tractability is

    which implies .

  4. A necessary condition for weak tractability is

    which implies , and a sufficient condition for weak tractability is .

  5. A necessary condition for -weak tractability for is

    which implies , and a sufficient condition for -weak tractability is .

    For , -weak tractability holds for all weights .

  6. A necessary condition for uniform weak tractability is

    which implies for all , and a sufficient condition for uniform weak tractability is

Remark 5.

Some remarks on Theorem 4 are in order.

  1. So far we only have a necessary condition for QPT, which is

    (6)

    and which in turn implies . However, this condition is very close to the “if and only if”-condition for PT, which is . It is an interesting question whether (6) is already strong enough to imply QPT or whether is really necessary. The latter case would imply that PT and QPT are equivalent.

  2. The necessary and sufficient conditions for the notions of weak tractability in Items 4–6 are very tight, although not matching exactly. How to close these gaps is another interesting problem. Regarding Item 4, we also refer to [10, Section 6.3], where a corresponding result for -approximation in the average-case setting is shown, and this is—as pointed out in our remarks above—equivalent to our result for -approximation in the worst-case setting. There, the same gap is observed, but the authors of [10] point out that at least for general weights the condition is not sufficient for weak tractability. Whether a similar observation also holds for the special case of product weights, which are considered in the present paper, remains open.

of Theorem 4.

Proofs of the results on (strong) polynomial tractability in Items 1 and 2 can be found in [5, Theorem 11] for the class and in [6, Theorem 11] for .

Now we consider QPT. From (3) and the fact that for all , we have for that

Hence,

(7)

Assume that we have QPT for -approximation for and the normalized error criterion. Then there exist positive and such that

for all and all .

Fixing , e.g., choosing and taking the logarithm implies the condition

for all . This implies . Since for , this then implies

(8)

Thus we have shown that (8) is a necessary condition for QPT for and the normalized error criterion. Since QPT for implies QPT for , we find that (8) is also a necessary condition for QPT for and the normalized error criterion.

Assume that we have QPT for -approximation for and the absolute error criterion. Then, according to (2), we have QPT for -approximation for and the normalized error criterion, and hence (8) holds. Thus the proof of Item 3 is complete.

We now discuss -WT and first consider the necessary conditions. Assume that we have -WT for for -approximation for and the normalized error criterion. Then, according to (7),

For fixed this implies

which in turn implies that

(9)

So (9) is a necessary condition for -WT for for and the normalized error criterion. In the same way as for QPT we see that (9) is a necessary condition for -WT for for and the normalized and the absolute error criterion. Note that (9) implies . This finishes the proof of the necessary conditions in Items 4–6.

Next, we discuss sufficient conditions for -WT. In [15] Zeng, Kritzer, and Hickernell constructed a spline algorithm based on lattice rules with a prime number of nodes, for which for arbitrary

(10)

Assume that . Then there exists a such that