Kolmogorov Superpositions: A New Computational Algorithm

Kolmogorov Superpositions: A New Computational Algorithm

David Sprecher (University of California at Santa Barbara, USA)
Copyright: © 2013 |Pages: 27
DOI: 10.4018/978-1-4666-3942-3.ch011
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Abstract

Kolmogorov’s superpositions enable the representation of every real-valued continuous function f defined on the Euclidean n-cube in the form , with continuous functions that compute f, and fixed continuous functions dependent only on n. The functions specify space-filling curves that determine characteristics that are not suitable for efficient computational algorithms. Reversing the process, we specify suitable space-filling curves that enable new functions that give a computational algorithm better adaptable to applications. Detailed numerical constructions are worked out for the case n = 2.
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Introduction

This chapter concerns the computation of continuous functions of variables with superpositions of continuous functions of variables. The function

of two variables can offers a simple example of a representation with superpositions of functions of one variable:
where
and
i.e.,

In general, we are interested in representations

for some number r of summands, in which
are fixed continuous functions that are independent of

.

In their most general form, such superpositions belong to a class of problems conveniently described by means of a commuting diagram with metric spaces X, Y, and T, and a given mapping (Figure 1): The problem is to find continuous mappings and such that f can be replaced with the superposition , where is a fixed embedding of X into T, and when this representation is valid for a sufficiently large class of functions f. We are not going to pursue here this aspect of generality, and instead will adhere to the setting , in which

Figure 1.

Superpositions as a commuting diagram

is the n-dimensional Euclidean unit cube, ,
is the r-dimensional Euclidean space, and .

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