Space-filling curves and Kolmogorov superposition-based neural networks

Authors:
David A. Sprecher;Sorin Draghici
Affiliations:
Department of Mathematics, University of California, Santa Barbara, CA;Department of Computer Science, Wayne State University, Detroit, MI
Venue:
Neural Networks
Year:
2002

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Abstract

Kolmogorov superpositions and Hecht-Nielsen's neural network based on them are dimension reducing. This dimension reduction can be understood in terms of space-filling curves that characterize Kolmogorov' s functions, and the subject of this paper is the construction of such a curve. We construct a space-filling curve with Lebesgue measure 1 in the unit square, [0,1]2, with approximating curves Λk, k = 1,2,3 ..... each with 102k rational nodal-points whose order is determined for each k by the linear order of their image-points under a nomographic function y = α1ψ(x1) + α2ψ(x2) that is the basis of a computable version of the Kolmogorov superpositions in two dimensions. The function ψ:[0,1 ] → [0,1] is continuous and monotonic increasing, and α1, α2 are suitable constants. The curves Λk are composed of families of disjoint closed squares of diminishing diameters and connecting joins of diminishing lengths as k → ∞.