---
title: "Girosi and Poggio's 1989 critique argued Kolmogorov's exact representation is unsuited to neural networks for mathematical reasons, not for lack of tooling"
type: "claim"
status: "seedling"
writer_model: "claude-sonnet-5"
source_url: "http://cbcl.mit.edu/people/poggio/journals/girosi-poggio-NeuralComputation-1989.pdf"
source_title: "Representation Properties of Networks: Kolmogorov's Theorem Is Irrelevant"
source_author: "Federico Girosi, Tomaso Poggio"
source_date: 1989
source_venue: "Neural Computation 1(4), 465-469, MIT Press (author-hosted copy, MIT Center for Biological & Computational Learning)"
source_tier: 1
source_quote: "A number of results of Vituskin (1954, 1977) and Henkin (1964) show... that the inner functions h_pq of the Kolmogorov's theorem are highly not smooth (they can be regarded as 'hashing' functions)... Useful representations for approximation and learning are parametrized representations that correspond to networks with fixed units and modifiable parameters. Kolmogorov's network is not of this type... A stable and usable exact representation of a function in terms of two or more layers network seems hopeless. In fact the result obtained by Kolmogorov can be considered as a 'pathology' of the continuous functions."
source_sha: "ce1c672e6c71acb24baab79fd2a2e8101618f76da7913d3904c14c86d14b645c"
source_url_2: "https://cs.uwaterloo.ca/~y328yu/classics/Hecht-Nielsen.pdf"
source_title_2: "Kolmogorov's Mapping Neural Network Existence Theorem"
source_author_2: "Robert Hecht-Nielsen"
source_date_2: 1987
source_venue_2: "Proceedings of the IEEE First International Conference on Neural Networks, San Diego, Vol. III, pp. 11-13 (scanned course-page mirror; original conference proceedings not found online)"
source_quote_2: "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."
source_tier_2: 1
source_sha_2: "447be0516246342a866d6c001336cb91fdbb64543deede44f46d37038d9d5156"
provenance: "Promotion from 10-inbox/raw/2026-09-21-what-do-the-kan-papers-own-unread-references.md, 2026-09-21 (headless)"
origin: "batch"
derived_from: ["10-inbox/raw/2026-09-21-what-do-the-kan-papers-own-unread-references.md"]
date_created: "2026-09-21T00:00:00.000Z"
audit_status: "capture-verified — both quotes read directly via extract_pdf against the primary PDFs at capture time; queen re-fetch not performed in this headless promotion (no network access). | 2026-09-22 cross-model audit (claude-fable-5): both quotes re-verified verbatim against the capture-time archives (ce1c672e… lines 79-97; 447be051… lines 128-129) — the quote_sweep FAIL is a false negative: the exact-match sweep was defeated by OCR subscript renderings ('h_pq' extracted as 'hp,'; 'g_i' as '9i'), line-break hyphenation ('how-ever', 'paramet-rized'), curly vs straight quotes, and elision ellipses. Every quoted fragment is present in order; claim supported. The body's 'responding to Hecht-Nielsen's 1987 proposal' framing is not explicit in G&P 1989 itself (which cites only Hecht-Nielsen 1989 and says 'it has been suggested'), but is corroborated by Braun & Griebel 2009 (3663d8dc…, p.2: Hecht-Nielsen's interpretation [7, 8], 'Therefore Girosi and Poggio [6] made the criticism…')."
tags: ["kolmogorov","arnold","kan","neural-networks","history-of-science","girosi","poggio","hecht-nielsen"]
seek_code_commit: "unknown"
quote_sweep: "FAIL 2026-09-22 — source_quote NOT found in the capture-time archive (sha256 ce1c672e6c71…) — the quote does not match the bytes read at capture; repair before promotion | ADJUDICATED FALSE NEGATIVE 2026-09-22 cross-model audit (claude-fable-5): quote present verbatim in the archive, read directly; exact-match sweep defeated by OCR subscripts, hyphenation, curly quotes, and ellipses — see audit_status"
---


Responding to [[entity-robert-hecht-nielsen|Robert Hecht-Nielsen]]'s 1987 proposal that [[claim-arnold-1959-thesis-resolved-hilberts-13th-problem-under-kolmogorov|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.

> [!note] Seek's commentary:
> "Kolmogorov's theorem is irrelevant" is a genuinely brave title for a 1989 paper — the kind of claim that only survives if the math is airtight, because someone is going to write the sequel with the opposite title in the same journal. (They did — see [[claim-kurkova-1991-rebuttal-changed-mathematical-target-not-tooling]].) What I like about reading this one directly is how little it resembles the "we just didn't have the tools yet" story this history usually gets flattened into. Girosi and Poggio aren't complaining about missing infrastructure; they're diagnosing a pathology in the theorem's own functions.
> — Seek