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Kolmogorov--Arnold against bounded translations
Sviatoslav V. Dzhenzher
TL;DR
The paper asks whether KART remains stable under bounded adversarial translations of its hidden layer. It gives a self-contained constructive proof using fixed piecewise linear inner functions and one outer function independent of the specific translation when its bound is known. The resulting representation is approximate, with error controlled by the target’s modulus of continuity and a term depending on n and m.
Problem
KART’s stability under continuous adversarial perturbations of the hidden layer remains an open question, including the single-outer-function case for translation adversaries.
Method
The paper constructs a piecewise linear inner matrix and a single piecewise linear outer function for bounded translations.
Results
Theorem 2.3 yields an approximate representation whose error is bounded by ωf(γ(m−1)) + n∥f∥/(m−n), uniformly for translations t ∈ [−C,C]^m.
Takeaways & Limitations
The same outer function works for all m summands and does not depend on the particular bounded translation, provided its maximum bound is known in advance.
Takeaways & Limitations
The construction requires advance knowledge of the adversary’s maximum translation scope, and its practical performance is strongest for m ≥ 2n + 1.
Abstract
from arXiv · showhide
Historically originating from Hilbert's 13th problem, the Kolmogorov-Arnold representation theorem (KART) has recently experienced a major revitalisation through its applications to neural networks, specifically Kolmogorov-Arnold Networks (KANs). While the exact representation is well established, its stability under continuous adversarial perturbations of the hidden layer remains a critical open question. In this paper, we investigate the robustness of KART against bounded adversarial translations. We provide an explicit, self-contained, and constructive proof of an approximate representation using fixed, piecewise linear inner functions. Crucially, our construction employs a single outer function that remains invariant for all summands and is independent of the specific adversarial translation, provided its maximum bound is known a priori.
1. History and introduction
KART emerged from Hilbert’s 13th problem and established continuous-function superposition through Kolmogorov and Arnold’s 1956–57 work. This paper studies its stability when known continuous perturbations act on the hidden layer.
- Origins: Hilbert asked whether solutions of seventh-degree equations could be expressed as superpositions of functions of two variables.The question extended the representation of lower-degree equation solutions through arithmetic operations.
- Origins: Kolmogorov and Arnold resolved the continuous-function version affirmatively in 1956–57, producing what is now called KART.The theorem is also known in computer science as the Kolmogorov Superposition Theorem.
- Prior work: Early criticism focused on KART’s non-constructive character and the limited smoothness of its inner and outer functions.Prior work established strict limitations on differentiability, while later research also extended KART topologically and to discontinuous functions.
- Stability question: Stability research examines continuous reparameterisations of the hidden layer while allowing the output layer to adjust against a known perturbation.The setup models a known adversary attempting to perturb hidden-layer representations.
- Stability question: Earlier work established stability under countable families of adversarial homeomorphisms and posed further questions addressed in this paper.The paper’s organization includes background, a main theorem, its proof, discussion, and conclusions.
2. Background and main results
The paper reviews KART formulations and prior stability results before stating a constructive theorem for bounded translations. Its theorem fixes piecewise linear inner functions and one outer function that works across adversarial translations within a known bound.
- KART background: KART uses a fixed continuous inner matrix while target-dependent uniformly continuous outer functions represent continuous functions on [0,1]^n.The classical construction has 2n+1 inner columns and permits multiple outer functions.
- KART background: Some KART variants use rationally independent scalings and equal outer functions, while others permit discontinuous targets with discontinuous outer functions.These are alternative formulations with different inner and outer-function regularity.
- Prior stability results: Earlier stability results treated countable families of adversarial homeomorphisms, while continuous translation groups created equicontinuity difficulties.Using different outer functions makes translation approximation trivial; the theorem here addresses the single-outer-function case.
- Main theorem: Theorem 2.3 constructs a piecewise linear inner matrix ϕ: [0,1] → R^n×m for integers n > 1 and m ≥ n+1.The theorem applies to every target continuous function f: [0,1]^n → R.
- Main theorem: The theorem supplies one piecewise linear outer function g that is 2∥f∥/[δ0(m−n)]-Lipschitz and independent of each translation t ∈ [−C,C]^m.It also bounds the outer function by ∥g∥ ≤ ∥f∥/[m−n].
- Main theorem: The approximation error is bounded by ωf(γ(m−1)) + n∥f∥/(m−n) for x ∈ [0,1]^n.The first term reflects the modulus of continuity, while the second decreases as m grows.
3. Proof of Theorem 2.3
The proof constructs piecewise linear inner functions from red segments and defines one outer function on separated intervals. Coverage and separation lemmas then yield a uniform approximation bound for bounded translations.
- Inner construction: The inner matrix is constructed from red segments of ranks i ∈[m], with each ϕj,i piecewise linear, constant on those segments, and linearly interpolated between them.The construction numbers same-rank red segments from left to right and assigns forced values before linear extension.
- Inner construction: Red n-solids are Cartesian products of n red segments of the same rank, and each such solid is assigned a representative point p(S).These solids provide the domains on which the outer function receives forced values.
- Outer construction: The forced-value intervals [Φ(S)−C, Φ(S)+C] are disjoint because solids of equal and different ranks have separated Φ-values.The separation follows from the two rank-based lemmas and makes the piecewise linear definition of g meaningful.
- Outer construction: The outer function g is 2∥f∥δ0(m−n)-Lipschitz because distinct forced-value intervals remain at least δ0 apart.The gap is δ−2C=δ0.
- Approximation bound: Each selected summand contributes at most ωf(γ(m−1)), while the remaining contribution is bounded by n m−n∥f∥.Together these estimates produce the final upper bound for arbitrary t ∈[−C,C]^m.
- Approximation bound: Every x ∈[0,1]^n lies in red n-solids of at least m−n different ranks, enabling the proof to select m−n solids containing x.The selected solids are indexed by a set I with exactly m−n elements.
4. Discussion of the result
The discussion explains why the construction is approximate and identifies trade-offs in choosing m. It emphasizes that translation-independent outer functions require advance knowledge of the translation bound.
- Approximation scope: The approximation cannot generally be made exact because the construction does not control the modulus of continuity of f.If f were known to be L-Lipschitz, choosing γ sufficiently small would yield a corresponding error bound.
- Approximation scope: Classical recursive and Baire-category arguments cannot be applied here because bounded-Lipschitz function spaces are not separable.The discussion also notes that the required matrices cannot thereby be shown open and dense.
- Parameter trade-offs: The theorem is formulated for m≥n+1 but works well only for m≥2n+1, since smaller m can leave the error n m−n∥f∥ no better than using g=0.The choice of m trades increasing ωf(γ(m−1)) against decreasing n m−n∥f∥.
- Parameter trade-offs: Unlike the earlier result, the outer function is independent of the adversarial translation, but constructing it requires knowing the translation bound C in advance.This is the stated cost of translation-independent reuse.
5. Conclusion and further research
The paper gives a constructive, self-contained stability result for bounded adversarial translations, using one translation-independent outer function and fixed piecewise linear inner functions. The construction has structural limitations: it requires a known translation bound and lacks sharp control because the modulus of continuity of f is uncontrolled.
- 5. Conclusion and further research: Theorem 2.3 constructs an explicit piecewise linear inner matrix whose outer function is shared by all m summands and independent of the specific translation when its bound C is known.This independence requires inner functions whose ranges expand as the allowed adversarial translation scope increases.
- 5. Conclusion and further research: The proof of Theorem 2.3 is constructive and self-contained.The paper compares its proof with earlier constructive proofs of KART.
- 5. Conclusion and further research: The construction is non-adaptive because the upper translation bound must be known before the inner function is built.The paper leaves overcoming this obstruction with a more sophisticated choice of inner function as an open question.
- 5. Conclusion and further research: Uncontrolled modulus of continuity for f prevents the result from being sharp.The paper identifies overcoming this obstruction as an open question.
- 5. Conclusion and further research: The introduced techniques are proposed as a starting point for studying stability under larger classes of adversarial actions.The authors identify this as a direction for future work.