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

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 978-1-4666-3942-3.ch011.m11 variables with superpositions of continuous functions of 978-1-4666-3942-3.ch011.m12 variables. The function

978-1-4666-3942-3.ch011.m13
of two variables can offers a simple example of a representation with superpositions of functions of one variable:
978-1-4666-3942-3.ch011.m14
where
978-1-4666-3942-3.ch011.m15
978-1-4666-3942-3.ch011.m16
978-1-4666-3942-3.ch011.m17
and
978-1-4666-3942-3.ch011.m18
i.e.,

In general, we are interested in representations

978-1-4666-3942-3.ch011.m20
for some number r of summands, in which
978-1-4666-3942-3.ch011.m21
are fixed continuous functions that are independent of

978-1-4666-3942-3.ch011.m22.

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 978-1-4666-3942-3.ch011.m23 (Figure 1): The problem is to find continuous mappings 978-1-4666-3942-3.ch011.m24 and 978-1-4666-3942-3.ch011.m25 such that f can be replaced with the superposition 978-1-4666-3942-3.ch011.m26, where 978-1-4666-3942-3.ch011.m27 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 978-1-4666-3942-3.ch011.m28, in which

978-1-4666-3942-3.ch011.m29
Figure 1.

Superpositions978-1-4666-3942-3.ch011.m33 as a commuting diagram

978-1-4666-3942-3.ch011.f01
is the n-dimensional Euclidean unit cube, 978-1-4666-3942-3.ch011.m30,
978-1-4666-3942-3.ch011.m31
is the r-dimensional Euclidean space, and 978-1-4666-3942-3.ch011.m32.

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