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

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).