Source-linked AI summary
Kolmogorov--Arnold stability for discontinuous functions
Sviatoslav V. Dzhenzher
TL;DR
KART stability under hidden-layer reparameterisations had been studied for continuous targets, while extensions to discontinuous and unbounded targets assumed a static, error-free hidden layer. This paper proves stability for the broader target class under countable collections of homeomorphisms, with the outer function preserving discontinuity or unboundedness when present.
Problem
Earlier KART extensions covered discontinuous and unbounded functions but assumed a static, error-free hidden layer, leaving stability under hidden-layer perturbations to be addressed.
Source: §1 · pp. 1–2Method
The paper extends the stability theorem to discontinuous and unbounded targets, assuming the homeomorphism set already stabilizes KART for continuous targets.
Source: §2 · p. 2Results
Takeaways & Limitations
For KANs, the theorem gives a mathematical guarantee for non-smooth data, with fixed continuous hidden activations and an adjustable outer function.
Source: §2 · p. 3For unbounded f, the final proof step cannot use Hahn–Banach because the functional Ff is unbounded; the authors instead use Zorn’s lemma.
Source: §3 · p. 5
Abstract
from arXiv · showhide
Here we investigate the stability of the Kolmogorov--Arnold representation theorem (KART) under adversarial reparameterisations of the hidden layer for multivariate discontinuous and unbounded functions. Our results provide a rigorous mathematical foundation for the structural robustness of modern deep learning architectures, such as Kolmogorov--Arnold Networks (KANs), under adversarial configurations.
1. Introduction
The introduction traces KART from Hilbert’s question to its representation of continuous multivariate functions, then frames stability under hidden-layer reparameterisations as an open issue for discontinuous and unbounded targets. This work extends stability to that broader function class under countable collections of homeomorphisms.
1. Introduction
- KART establishes that any multivariate continuous function can be represented using superpositions of univariate continuous functions and addition.The result of Kolmogorov and Arnold (1956–57) answered Hilbert’s question about such representations.Source: §1 · p. 1
- The relevance of KART to neural networks includes representing continuous multivariate functions with three-layer feedforward networks.Hecht-Nielsen also showed that computable multivariate functions can be implemented with computable activations and computable real weights.Source: §1 · p. 1
- Earlier work studied KART stability under hidden-layer reparameterisations, assuming the reparameterisation was known and the output layer adjusted to it.That work showed stability for countable collections of homeomorphisms.Source: §1 · pp. 1–2
- Recent extensions generalised KART to discontinuous, unbounded functions, but assumed a static, error-free hidden layer; this paper proves stability under adversarial reparameterisations for that class.The stated result concerns countable collections of homeomorphisms.Source: §1 · p. 2
2. Background and the main result
The section defines stability of KART under hidden-layer homeomorphisms and states that this stability extends to discontinuous and unbounded target functions. It also connects the result to KANs, where the inner tuple is fixed and continuous while the outer function is adjustable.
2. Background and the main result
- The section notes that KART stability under any countable set of homeomorphisms was already stated in [18].It also identifies the identity-only case as KART itself.Source: §2 · p. 2
- Theorem 2.2 extends KART stability to target functions that may be discontinuous or unbounded, provided KART is stable under the chosen homeomorphism set H.For each rationally independent choice of λ values, the theorem provides an inner continuous tuple and, for each target and adversarial h ∈ H, an outer function g.Source: §2 · p. 2
- The theorem describes the target functions on [0, 1]^n and uses an inner continuous tuple ϕ mapping [0, 1] into [0, 1]^(2n+1).The classical form given here concerns continuous targets and an outer uniformly continuous function.Source: §2 · p. 2
- For continuous targets, g can be continuous; for discontinuous targets, g must be discontinuous, and for unbounded targets, g must be unbounded.These properties constrain the regularity or boundedness of the outer function to match the target function.Source: §2 · p. 3
- Theorem 2.2 is presented as a mathematical guarantee for KANs on non-smooth data, with ϕ corresponding to fixed hidden-layer activations and g to learnable output-layer functions.The paper says the distortion from a hidden-layer homeomorphism can be compensated by readjusting g, preserving expressivity under the stated stability condition.Source: §2 · p. 3
- The homeomorphism set H models hidden-layer perturbations associated with data drift, adversarial attacks, and hardware precision degradation or analogue drift.The passage also names coordinate warping as a feature-pipeline distortion.Source: §2 · p. 3
3. Proof of the main result
The proof treats bounded and unbounded target functions separately: dual-operator surjectivity handles the bounded case, while a Hamel-basis extension handles the unbounded case. In both cases it constructs an outer function for the representation.
3. Proof of the main result
- The proof fixes rationally independent λ values and an inner tuple stable under H, then forms the reparameterized maps Φ_i and their compact union of images.The outer function need only be constructed on a subset of R and can then be extended arbitrarily while preserving boundedness.Source: §3 · pp. 3–4
- In the bounded case, T maps C(K) to C([0,1]^n), and stability under h makes T surjective, equivalent to T* being bounded below.Banach’s theorem then yields surjectivity of Z* and the bounded-target representation.Source: §3 · p. 4
- The proof handles bounded discontinuous targets through surjectivity of the dual operator, concluding the representation for bounded target functions.The proof relates surjectivity of T to its adjoint being bounded below, then uses Banach’s theorem to establish surjectivity of Z*.Source: §3 · p. 4
- For unbounded targets, bounded-below Z makes the vectors Zδ_x linearly independent, allowing F_f to extend over a Hamel basis by Zorn’s lemma.Hahn–Banach cannot replace this step because F_f is unbounded when f is unbounded.Source: §3 · pp. 4–5
- Extending F_f and defining the outer function yields the required representation for each x in [0,1]^n.The proof overview identifies dual operator theory for bounded functions and a Zorn-based algebraic extension for unbounded functions.Source: §3 · pp. 3, 5
4. Conclusion and further research
The paper establishes KART stability under hidden-layer reparameterisations for discontinuous and unbounded multivariate target functions. It identifies future work on alternative and shift-invariant formulations, and on constructive stability proofs for KANs.
4. Conclusion and further research
- KART remains expressive under countable hidden-layer homeomorphisms for discontinuous and unbounded multivariate target functions.This extends prior stability results constrained to continuous settings.Source: §4 · p. 5
- The proof establishes existence of the outer function g non-constructively, relying on Zorn’s lemma.The paper identifies constructive proofs accommodating adversarial reparameterisations as a future direction.Source: §4 · p. 5
- Stability remains to be investigated for alternative KART forms, including Sprecher’s formulations that reduce the number of inner and outer functions.The paper also identifies shift-invariant variants with inner functions differing strictly by shifts as a direction for stability proofs.Source: §4 · p. 5
- Constructive proofs for adversarially reparameterised KART could bridge theoretical stability and computational deep learning, potentially enabling provably robust KAN architectures.The paper points to Braun and Griebel’s algorithmic approach as an example to generalise.Source: §4 · p. 5