Space-filling curves and Kolmogorov superposition-based neural networks

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 lambda(k), k = 1,2,3,..., each with 10(2k) rational nodal-points whose order is determined for each k by the linear order of their image-points under a nomographic function y = alpha1psi(x1) + alpha2psi(x2) that is the basis of a computable version of the Kolmogorov superpositions in two dimensions. The function psi:[0,1] --> [0,1] is continuous and monotonic increasing, and alpha1, alpha2 are suitable constants. The curves lambda(k) are composed of families of disjoint closed squares of diminishing diameters and connecting joins of diminishing lengths as k --> infinity.