Kolmogorov-Arnold Representation Theorem

Updated 18 August 2025

Kolmogorov-Arnold Representation Theorem is a fundamental result that expresses any continuous multivariate function as a finite sum of compositions of univariate functions and addition.
It underpins theoretical analysis and enables the design of neural networks that approximate high-dimensional mappings with efficiency and structural guarantees.
Architectural implementations, such as Kolmogorov-Arnold Networks, employ spline-based univariate functions to enhance interpretability, robustness, and computational efficiency.

The Kolmogorov-Arnold Representation Theorem provides a foundational result in multivariate function theory by establishing that any continuous function of several real variables can be decomposed exactly as a finite sum of compositions of continuous, univariate functions and addition. This theorem underlies a wide range of developments in theoretical analysis, computational mathematics, and modern machine learning, especially in the design and training of neural networks that aim to approximate or represent high-dimensional mappings with rigorous structural guarantees and efficiency.

1. Formal Statement and Mathematical Structure

For a continuous function $f$ defined on a compact subset of $\mathbb{R}^d$ , the Kolmogorov-Arnold theorem guarantees the existence of continuous univariate functions such that: $f(x_1, ..., x_d) = \sum_{q=1}^{2d+1} \Phi_q\left( \sum_{p=1}^d \psi_{q,p}(x_p) \right)$ where:

$\psi_{q,p}: [0,1] \to \mathbb{R}$ are the "inner" functions,
$\Phi_q: \mathbb{R} \to \mathbb{R}$ are the "outer" functions.

This canonical construction demonstrates that any multivariate continuous function can be written as a finite sum of composed univariate functions, with the summations performed first over the input variables and then over the summands.

Key features of the theorem:

The number of required summands is always $2d+1$ for functions of $d$ variables.
Inner functions $\psi_{q,p}$ can typically be chosen independently of $f$ , but outer functions $\Phi_q$ generally depend on the specific target function.

2. Hierarchical Representation, Discrete Operators, and Implementation

The theorem's hierarchical structure implies a deep, layered evaluation: each input $x_j$ is first processed through many inner functions $f^{(kj)}$ , yielding branch activations, which are summed: $\varphi_k = \sum_{j=1}^m f^{(kj)}(x_j).$ The root layer then processes each $\varphi_k$ via an outer function $\Phi^k$ ; the final output is the sum over $k$ .

This structure can be interpreted as a tree of discrete Urysohn operators, where each addend encodes a linear mixture of univariate transformations. Algorithms for constructing such representations often:

Approximate each $f^{(kj)}$ and $\Phi^k$ by sparse, piecewise-linear functions characterized by nodal values and local interpolation (as in $g(x) = (1 - \psi) G[q] + \psi G[q+1]$ ).
Use projection descent or similar iterative update methods to adjust only the affected nodal points for sparse, efficient and computationally stable training.
Handle quantized or discrete inputs seamlessly by aligning nodal positions to data levels, which further reduces computational requirements (Polar et al., 2020).

3. Smoothness Transfer, Cantor Set Embeddings, and Deep Networks

Modifications of the traditional theorem (notably in "The Kolmogorov-Arnold representation theorem revisited" (Schmidt-Hieber, 2020)) address the irregularity of the outer functions by reparameterizing the inner composition using a map into the Cantor set. The interior mapping

$\phi(x) = \sum_{j=1}^\infty \frac{2a_j^x}{3^{1+d(j-1)}}, ~~~ x = [0.a_1^x a_2^x ...]_2$

enables control over the regularity of the outer function $g$ : $f(x_1, ..., x_d) = g\left(3 \sum_{p=1}^d 3^{-p} \phi(x_p)\right).$ If the target $f$ is Hölder continuous with exponent $\beta$ , then $g$ inherits a related smoothness (with exponent rescaled by $({\beta \log 2})/{(d \log 3)}$ ), making $g$ much more amenable to approximation by piecewise-linear or ReLU neural networks.

This insight yields a direct correspondence between the Kolmogorov-Arnold decomposition and deep neural architectures: hidden layers are required to efficiently extract and encode the binary structure of input variables, with the outermost layer handling a smooth univariate function. This leads to architectures of depth approximately $2K+3$, where $K$ is the bit truncation level, and width governed by the input dimension and approximation accuracy.

4. Theorem in Neural Network Design: Kolmogorov-Arnold Networks

Kolmogorov-Arnold Networks (KANs) directly operationalize the theorem for machine learning:

Linear transformations at each layer are replaced by spline-based or learnable univariate functions along the edges, giving the architecture interpretability and theoretical universality.
Both inner ( $\phi_{q,p}$ ) and outer ( $\Phi_q$ ) functions are parameterized as basis expansions (commonly B-splines, rational functions, or—more recently—sinusoidal units (Gleyzer et al., 1 Aug 2025)), tuned from data via backpropagation.
KANs can often approximate complicated nonlinear relationships with fewer parameters than Multilayer Perceptrons (MLPs), as these functions are matched to the polynomial decomposition the theorem suggests.
Function classes with quantized inputs, hybrid quantized-continuous domains, and varying degrees of regularity can be handled by aligning discretization and interpolation strategies.
Empirical studies report state-of-the-art or competitive results in regression, classification, time series forecasting, operator learning, and physics-informed modeling (Peng et al., 13 May 2024, Toscano et al., 21 Dec 2024, Bhattacharya et al., 19 Dec 2024).

The separation of edges and node activations allows for more modular control and analysis of model inductive bias and interpretation (Moradi et al., 2 Oct 2024).

5. Extensions: Geometric, p-adic, and Probabilistic Variants

Recent research generalizes the theorem and the associated network structures:

Geometric invariance and equivariance: The superposition representation is extended to encode symmetry properties (e.g., invariance under rotations $O(n)$ , permutations $S_n$ , and general linear groups) by expressing the function in terms of invariant quantities, such as pairwise inner products, and adapting the addend structure to respect group symmetries. This enables KANs to model physical systems in molecular dynamics and particle physics with correct geometric properties (Alesiani et al., 23 Feb 2025).
$p$ -adic analogs: For continuous functions over multiple $p$ -adic variables ( $\mathbb{Z}_p^n$ ), every such function can be represented as a single-variable composition using digit-interleaving homeomorphisms specific to the $p$ -adic canonical expansion (Zubarev, 11 Mar 2025).
Probabilistic and generative models: The inner functions are interpreted as inverse cumulative distribution functions, allowing KANs to serve as flow-based or probabilistic generative models. Each latent dimension is sampled by inverse transform sampling from learnable energy-based priors, and mixture or Langevin methods are used to relax independence or enhance posterior-prior matching (Raj, 17 Jun 2025).

6. Practical Impact and Architectural Trends

The theorem’s operationalization via KANs and its modern variants has yielded:

Improved parameter and computational efficiency relative to comparable-width MLPs, especially for problems where target structure matches the hierarchical decomposition (e.g., structured data, scientific and engineering modeling, operator learning) (Basina et al., 15 Nov 2024).
Robustness to the curse of dimensionality, as KANs distribute approximation complexity across 1D bases rather than directly over the exponentially scaling multivariate input space. Error bounds in B-spline KANs scale with grid resolution or smoothness, not explicitly with dimension (Basina et al., 15 Nov 2024).
Enhanced interpretability, as the building blocks correspond to observable, contextually meaningful 1D relationships.

Recent innovations—incorporating active subspaces (asKAN (Zhou et al., 7 Apr 2025)), self-scaled attention (KKAN (Toscano et al., 21 Dec 2024)), kernel-based perspectives (Liu et al., 29 Mar 2025), quantum circuit implementations (Ivashkov et al., 6 Oct 2024), and convolutional or transformer extensions (Ferdaus et al., 22 Oct 2024, Bodner et al., 19 Jun 2024)—illustrate the breadth of algorithmic development inspired by the theorem.

7. Limitations and Ongoing Research Directions

Despite its universality, the Kolmogorov-Arnold representation faces practical challenges:

The precise functional forms of the outer functions may be highly irregular in the original construction, necessitating careful reparameterizations or regularizations for tractable numerical implementation (Schmidt-Hieber, 2020).
Classical KANs may be inflexible for functions whose main dependencies are ridge-type (i.e., depend on linear combinations of inputs rather than coordinate-wise structure), prompting hybrid approaches, such as embedding active subspace projections (Zhou et al., 7 Apr 2025).
Efficiently learning the number and type of basis functions for each univariate component can be addressed through variational or adaptive regularization schemes (Alesiani et al., 3 Jul 2025).
Extending the superposition architecture to model equivariance under general transformation groups requires new theoretical and computational frameworks (Alesiani et al., 23 Feb 2025).

Ongoing research addresses optimization of basis selection (e.g., through infinite-basis variational schemes), geometric extension to encode invariance and equivariance, and integrating the architecture into generative modeling pipelines.

In summary, the Kolmogorov-Arnold Representation Theorem delivers a rigorous decomposition of continuous multivariate functions into finite sums of univariate functions and underpins a class of scalable, interpretable, and universal neural architectures. Its influence is manifest in contemporary function approximators, operator learning frameworks, generative models, and invariant deep learning systems, where the structural and theoretical clarity of the theorem is leveraged for both practical machine learning tasks and the analysis of physical systems.