title: On computational algorithms for real-valued continuous functions of several variables.
abstract: 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).
authors.affiliation: Department of Mathematics, University of California, Santa Barbara, CA 93106, United States. Electronic address: sprecher@math.ucsb.edu.
authors.forename: David
authors.initials: D
authors.lastname: Sprecher
mesh_terms: D000465:Algorithms; D006801:Humans; D008959:Models, Neurological; D016571:Neural Networks, Computer
keywords: Algorithm; Hilbert curve; Kolmogorov superpositions; Space-filling curves; Superpositions