On a Constructive Proof of Kolmogorov's Superposition Theorem

Kolmogorov (Dokl. Akad. Nauk USSR, 14(5): 953-956, 1957) showed that any multivariate continuous function can be represented as a superposition of one-dimensional functions, i.e., f (x(1), ..., x(n)) = Sigma(2n)(q=0) Phi(q) (Sigma(n)(p=1) psi(q,p)(x(p))). The proof of this fact, however, was not constructive, and it was not clear how to choose the outer and inner functions Phi(q) and psi(q,p), respectively. Sprecher (Neural Netw. 9(5): 765-772, 1996; Neural Netw. 10(3): 447-457, 1997) gave a constructive proof of Kolmogorov's superposition theorem in the form of a convergent algorithm which defines the inner functions explicitly via one inner function psi by psi(p,q) :=lambda(p)psi(x(p)+ qa) with appropriate values lambda(p), a is an element of R. Basic features of this function such as monotonicity and continuity were supposed to be true but were not explicitly proved and turned out to be not valid. Koppen (ICANN 2002, Lecture Notes in Computer Science, vol. 2415, pp. 474-479, 2002) suggested a corrected definition of the inner function psi and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Koppen's psi, and we present a correct constructive proof of Kolmogorov's superposition theorem for continuous inner functions psi similar to Sprecher's approach.