Bernstein-type approximation of set-valued functions in the symmetric difference metric\@textsuperscript\safe@setrefT1thanksT1\@nil,\safe@setrefT1thanks\@nil\@@T1,\safe@setrefT1thanks
We study the approximation of univariate and multivariate set-valued functions (SVFs) by the adaptation to SVFs of positive sample-based approximation operators for real-valued functions. To this end, we introduce a new weighted average of several sets and study its properties. The approximation results are obtained in the space of Lebesgue measurable sets with the symmetric difference metric.
In particular, we apply the new average of sets to adapt to SVFs the classical Bernstein approximation operators, and show that these operators approximate continuous SVFs. The rate of approximation of Hölder continuous SVFs by the adapted Bernstein operators is studied and shown to be asymptotically equal to the one for real-valued functions. Finally, the results obtained in the metric space of sets are generalized to metric spaces endowed with an average satisfying certain properties.
School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv,
T1thanksThis is a preprint version of the paper.\@finalstrut\@textsuperscript\safe@setref\r@t1thankst1thankscorresponding author\@finalstrut
Set-valued functions (SVFs) have various applications in optimization, control theory, mathematical economics and other areas. The approximation of SVFs from a finite number of samples has been the subject of several recent research works ([baier2011set],[dyn2007approximations],[dyn2006approximations],[kels2011subdivision]) and reviews ([dynapproximation],[muresanset]).
In order to adapt to SVFs sample-based approximation methods known for real-valued functions, it is required to define linear combinations of two or more sets. For most approximation methods it is sufficient to consider linear combinations with weights summing up to one, while for positive approximation operators only convex combinations (non-negative weights summing up to one) are considered. We term convex linear combinations as weighted averages.
In case of data sampled from a SVF mapping real-numbers to convex sets, methods based on the classical Minkowski sum of sets can be used for the approximation [dyn2000spline, vitale1979approximation]. In this approach, sums of numbers in positive operators for real-valued approximation are replaced by Minkowski sums of sets. A generalization to sets which are either convex or differences of convex sets is done in [baier2001differences], where convex sets are embedded into the Banach space of directed sets. This approach allows to apply existing methods for the approximation in Banach spaces [baier2011set].
Approximation of set-valued functions mapping real-numbers to general sets is a more challenging task. In this case, methods based on Minkowski sum of sets fail to approximate the sampled function [vitale1979approximation, dyn2005set], and other weighted averages of sets are needed.
Artstein [artstein1989pla] introduced a weighted average of two sets with the property that the Hausdorff metric between the average and any of the averaged sets changes linearly with the weight of the average. This average was later termed as the metric average of sets. An extension of the metric average to the weighted average of several sets, named the metric linear combination, is given in[dyn2007approximations]. The metric linear combinations were used in [dyn2007approximations] to adapt to sets positive and non-positive approximation operators, with the approximation error measured in the Hausdorff metric. However, the metric linear combination is applicable only to ordered sequences of sets, which limits its usage to the approximation of univariate SVFs.
As it is noticed in [artstein1989pla], the particular choice of a metric is crucial to the construction and analysis of set-valued approximation methods. While previous works develop and analyze set-valued approximation methods in the metric space of compact sets endowed with the Hausdorff metric, we consider here the approximation problem in the metric space of Lebesgue measurable sets with the symmetric difference metric111The measure of the symmetric difference is only a pseudo-metric on Lebesgue measurable sets. The metric space is obtained in a standard way as described in Section Bernstein-type approximation of set-valued functions in the symmetric difference metric\@textsuperscript\safe@setrefT1thanksT1\@nil,\safe@setrefT1thanks\@nil\@@T1,\safe@setrefT1thanks. The symmetric difference metric allows to obtain approximation results for a wider class of functions, as is demonstrated in [kels2011subdivision], where set-valued subdivision techniques are investigated.
In this work, we consider the adaptation of positive sample-based approximation operators to univariate and multivariate SVFs . The adaptation is based on a new weighted average of several sets, termed the partition average, which is studied in details.
As it is very well known, the concept of the weighted average of numbers is closely related to that of the mathematical expectation of a discrete random variable. Similarly, a weighted average of several sets may be interpreted as the expectation of a random set [molchanov2005theory]. We use tools from the theory of random sets to prove properties of the partition average of sets.
First, we adapt to SVFs the classical Bernstein operators, and show that these operators approximate continuous SVFs. Furthermore, we consider the rate of approximation of Hölder continuous SVFs by set-valued Bernstein operators, and obtain a result for SVFs analogous to that of Kac [kac1938remarque, kac1939reconnaissance] for real-valued functions. Moreover, we show that the adaptation to SVFs of the classical de Casteljau’s algorithm (see, e.g. [farin2002curves], Chapter 4) yields another sequence of adapted operators having the same rate of approximation as that for the adapted Bernstein operators.
The results for Bernstein operators are then extended to general positive sample-based operators. Moreover, we study the application of positive sample-based operators to monotone SVFs, and show that the adapted operator is monotonicity preserving if and only if the corresponding operator for real-valued functions is monotonicity preserving.
Due to the commutativity of the partition average of sets, the results are easily generalized to approximation operators for multivariate SVFs. Finally, we generalize the approximation results to functions with values in metric spaces endowed with a weighted average, with properties similar to those of the partition average of sets.
The structure of this work is as follows. In Section Bernstein-type approximation of set-valued functions in the symmetric difference metric\@textsuperscript\safe@setrefT1thanksT1\@nil,\safe@setrefT1thanks\@nil\@@T1,\safe@setrefT1thanks, we survey definitions and results relevant to our work. In Section Bernstein-type approximation of set-valued functions in the symmetric difference metric\@textsuperscript\safe@setrefT1thanksT1\@nil,\safe@setrefT1thanks\@nil\@@T1,\safe@setrefT1thanks, we introduce the partition average of sets and study its properties. In Section LABEL:sectionBernstein, we adapt to sets the Bernstein approximation operators. In section LABEL:sectionCastelio, we study another type of set-valued Bernstein operators, obtained by adapting to sets of the de Casteljau’s algorithm . In Section LABEL:sectionOperators, we consider the adaptation to SVFs of positive sample-based operators. In Section LABEL:sectionMonotone, we discuss the approximation of monotone SVFs. The approximation of multi-variate SVFs is the subject of Section LABEL:sectionMulti. Finally in Section LABEL:sectionGeneral, we generalize the results to functions with values in general metric spaces.
We denote by the -dimensional Lebesgue measure and by the collection of Lebesgue measurable subsets of having finite measure. The set difference of two sets is
and the symmetric difference is defined by
The measure of the symmetric difference of ,
induces a pseudo-metric on , and is a complete metric space by regarding any two sets such that as equal ([halmos1974measure], Chapter 8). For , such that , it is easy to observe that
We use the notation for the closure of the interior of . A bounded set , such that is called regular compact. Regular compact sets are closed under finite unions, but not under finite intersections, yet for regular compact sets such that ,
We recall that a set is Jordan measurable if and only if its boundary has zero Lebesgue measure. Jordan measurable sets are denoted by . We recall that is closed under finite unions and finite intersections. Note that for ,
Moreover for ,
We denote by the subset of consisting of regular compact sets. Notice that for any , implies , therefore is a metric on . In particular, the empty set is in , and it is the only set in having zero measure. Note that by its definition is closed under finite unions.
For a function , the Bernstein polynomial of degree is
The mapping is called the Bernstein operator. An extensive exposition of Bernstein polynomials is given in [devore1993constructive].
Obviously one can interpret (2.5) as the weighted arithmetic average of the values . The probabilistic nature of the Bernstein polynomials is also well known. It can be recognized by interpreting the weights,
as point probabilities of a binomial distribution with parameters and .
The polynomials are the basis of Bernstein’s proof of the Weierstrass Approximation Theorem ([bernstein1912demonstration], see [levasseur1984probabilistic] for a modern presentation). Using Bernstein polynomials the theorem can be formulated as
Let be a continuous function, then for any there exists , such that for all and all ,
A stronger version of the above theorem for Hölder continuous functions is due to Mark Kac ([kac1938remarque, kac1939reconnaissance], see [mathe1999approximation] for a modern presentation). We denote by the class of Hölder continuous functions with exponent and constant , defined on , namely functions satisfying,
Let , then
Our adaptation of Bernstein operators to SVFs is based on the new average of sets introduced in Section Bernstein-type approximation of set-valued functions in the symmetric difference metric\@textsuperscript\safe@setrefT1thanksT1\@nil,\safe@setrefT1thanks\@nil\@@T1,\safe@setrefT1thanks. To obtain the relevant properties of the new average of sets, we give it a probabilistic interpretation using the notion of a random closed set, which is discussed together with basic relevant results in the next subsection.
We proceed with a few definitions regarding random sets. The following definitions and results are adapted from [molchanov2005theory], which provides a thorough account of random sets theory.
Here we denote by the collection of closed subsets of .
Let be a probability space. A map is called a random closed set, if for every compact set ,
In the sequel we assume that is discretely distributed, namely, , with and . Moreover, we assume that , . Note that for any and any random set , defines a real-valued random variable.
Random closed sets are said to be independent if,
for all . Here is generated by all collections of closed sets of the form with running through all compact subsets of ([molchanov2005theory], Section 1.2).
The coverage function of the closed random set is ([molchanov2005theory], Section 2.2)
Notice that for discretely distributed random set ,
The following relation is useful,
where denotes the expectation of a real-valued random variable. Clearly, for a discretely distributed accepting values the integral in (2.12) can be taken over .
Let , setting , one obtains from (2.9) that for independent ,
The construction of our average of sets is built upon several definitions. We begin with
Let be such that
For , .
The function is called the subset-generating function.
Note that since and ,
For any collection of sets in , we consider a special partition of their union to mutually disjoint sets,
Let . For any subset of the indices , we define the set
For a fixed collection of sets we use the shorthand notation . The collection of sets
where denotes all subsets of the set of integers , is termed the partition of the union of . The sets are termed elements of the partition.
An example of the partition of the union of three subsets on is given in Figure 1.
Here we state several properties of the partition of the union, that follow easily from Definition 3.2.
, , .
For a fixed , , .
Let be a random set, , . The coverage function, is constant over each element of the partition of the union of , and
We are now in a position to define a new weighted average of sets in , which is based on the partition of the union of the averaged sets.
Let and , . The partition average of with the weights is
where is a subset-generating function in Definition 3.1.
Using the partition average, we can define expectation of a discretely distributed random set as
Let be a random set, , . The partition expectation of is
In view of Lemma 3.4, the partition expectation is related to the coverage function through
Next we state relevant properties of the partition average of sets.
In the notation of Definition 3.5,
For any permutation of ,
If for some , , then with and . In particular,
If for some , , then
and thus . The other part of Property 4, follows from the observation that .
Property 5 is an immediate consequence of Property 4. Next we prove Property 6. From the definition of the partition average, from the fact that the sets are pairwise disjoint and from the properties of the subset-generating function, we obtain that
Finally, we apply (2.12) to obtain that
The above properties of the partition average are analogous to those of weighted averages between non-negative numbers. In this analogy, the measure of a set replaces the absolute value of a number, the measure of the symmetric difference of two sets replaces the absolute value of the difference between two numbers. Moreover, the intersection and union of sets replace the minimum and the maximum of numbers, and finally the relation between sets replaces the relation between numbers.
The next theorem treats the distance between the partition expectations of two independent random sets distributed over the same collection of sets .
Let , be independent random sets, , , , with . Then
where is the real-valued random variable , namely