Girosi and Poggio's 1989 critique argued Kolmogorov's exact representation is unsuited to neural networks for mathematical reasons, not for lack of tooling

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Responding to Robert Hecht-Nielsen's 1987 proposal that Kolmogorov's representation theorem grounds a trainable neural network — a proposal Hecht-Nielsen himself hedged, writing "the direct usefulness of this result is doubtful, at least in the near term, because no constructive method for developing the g_i functions is known" — Federico Girosi and Tomaso Poggio gave two specific mathematical objections to the exact two-hidden-layer Kolmogorov construction, independent of any missing infrastructure. First, smoothness: citing results of Vituškin and Henkin, they note the theorem's inner functions are "highly not smooth," and smoothness is what a representation needs to generalize and resist noise. Second, and more fundamental for a trainable network: Kolmogorov's construction is not a parametrized representation of fixed units with modifiable weights — the outer functions g_q depend on the specific function being represented, and are "at least as complex... as f" itself, defeating the point of learning a compact model. Their conclusion: an exact Kolmogorov network "seems hopeless," and the theorem is better read as "a 'pathology' of the continuous functions" than as a blueprint. Nothing in this argument turns on backpropagation or compute; it targets the shape of the mathematical object itself.