Sprecher's 1996 constructive Kolmogorov-network algorithm rested on unproven properties that turned out false, and the fix wasn't proven correct until Braun and Griebel's 2009 paper, 13 years later
A separate line of attempts tried to make Kolmogorov's representation directly computable — an explicit, numerically constructible inner function — rather than merely approximable, as in Kůrková's approach. David Sprecher's 1996 paper (Neural Networks 9) gave a convergent algorithm defining the theorem's inner functions via a single function ψ, claiming (per Braun & Griebel's 2009 account) that ψ was monotonic and continuous. Those properties "were not explicitly proved and turned out to be not valid" — a real mathematical error, not a missing-infrastructure gap. Mario Köppen's 2002 paper (ICANN) proposed a corrected definition of ψ and claimed, again without proof, that it was monotonic and continuous. Braun and Griebel's 2009 paper is the first to actually prove Köppen's corrected construction has these properties — thirteen years after Sprecher's original claimed algorithm, and independent of any change in available training tools. The obstacle on this branch of the research program was mathematical correctness, established by direct proof, not the arrival of backpropagation or better hardware.
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“Sprecher gave in [27, 28] a constructive proof of Kolmogorov's superposition theorem in form of a convergent algorithm which defines the inner functions explicitly via one inner function ψ... Basic features of this function as monotonicity and continuity were supposed to be true, but were not explicitly proved and turned out to be not valid. Köppen suggested in [16] a corrected definition of the inner function ψ and claimed, without proof, its continuity and monotonicity. In this paper we now show that these properties indeed hold for Köppen's ψ.”
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