On computational algorithms for real-valued continuous functions of several variables

• Published in: Neural networks Vol. 59; pp. 16 - 22
• Main Author: Sprecher, David
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Ltd 01.11.2014; Elsevier
• Subjects: Algorithm; Algorithms; Applied sciences; Artificial intelligence; Computer science; control theory; systems; Connectionism. Neural networks; Exact sciences and technology; Hilbert curve; Humans; Kolmogorov superpositions; Mathematics; Models, Neurological; Neural Networks (Computer); Sciences and techniques of general use; Space-filling curves; Superpositions; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Hilbert curve; Space-filling curves; Kolmogorov superpositions; Algorithm; Superpositions; Prototype; Characteristic curve; Continuous function; Neural network; Complexity; Feedforward neural nets; Coding; Real function; space-filling curve; Feedforward; Kolmogorov equation
• ISSN: 0893-6080; 1879-2782; 1879-2782
• DOI: 10.1016/j.neunet.2014.05.015

The subject of this paper is algorithms for computing superpositions of real-valued continuous functions of several variables based on space-filling curves. The prototypes of these algorithms were based on Kolmogorov’s dimension-reducing superpositions (Kolmogorov, 1957). Interest in these grew significantly with the discovery of Hecht-Nielsen that a version of Kolmogorov’s formula has an interpretation as a feedforward neural network (Hecht-Nielse, 1987). These superpositions were constructed with devil’s staircase-type functions to answer a question in functional complexity, rather than become computational algorithms, and their utility as an efficient computational tool turned out to be limited by the characteristics of space-filling curves that they determined. After discussing the link between the algorithms and these curves, this paper presents two algorithms for the case of two variables: one based on space-filling curves with worked out coding, and the Hilbert curve (Hilbert, 1891).