Kolmogorov-Arnold Representation Theorem

• KART is a foundational theorem that decomposes any continuous multivariate function into a finite sum of univariate continuous functions, enabling simpler analysis.
• It employs a hierarchical structure of Urysohn trees using piecewise-linear interpolation and projection descent to achieve efficient and stable function approximation.
• This methodology yields rapid convergence and enhanced interpretability, proving effective in applications such as regression, classification, and dynamic system modeling.

The Kolmogorov-Arnold Representation Theorem (KART) is a foundational result in multivariate function theory that ensures any continuous function on a compact domain can be expressed as a finite superposition of univariate continuous functions and the addition operation. It has had a transformative impact on the analysis, approximation, and implementation of high-dimensional mappings, particularly within modern machine learning, system identification, and scientific computing.

1. Classical Formulation and Theoretical Structure

KART states that for any continuous function $f : [0, 1]^m \to \mathbb{R}$, there exist continuous univariate functions $f^{(k j)}(x_j)$ and $\Phi^k$ such that

$$F(x_1, \ldots, x_m) = \sum_{k=1}^{2m+1} \Phi^k \left( \sum_{j=1}^m f^{(k j)}(x_j) \right)$$

This decomposition reduces multivariate function approximation to a composition-summation hierarchy, where each branch of the tree acts on a single variable and the collection is combined via outer functions $\Phi^k$.

The paper (Polar et al., 2020) refines this by casting the KA representation as a tree of discrete Urysohn operators, which are themselves defined as the sum of univariate component functions

$U(x_1, ..., x_m) = \sum_{j=1}^m g^j(x_j)$

and reveals that the full Kolmogorov-Arnold structure is a hierarchical, multi-level tree where each node, or operator, acts on a reduced set of features formed from the lower-level outputs.

2. Algorithmic Construction: Hierarchical Urysohn Trees

Practical KA representation construction is achieved through identification and iterative training of each node/operator in the Urysohn tree. Each univariate function, both within the inner branches and at the root level, is parameterized as a piecewise-linear map with a finite set of nodal points, allowing efficient interpolation and parameter updates. The main identification workflow is as follows:

1. Input Rescaling: Each input $x_{j,i}$ is mapped into a normalized coordinate $b_{j,i}$ over a grid of $n_j$ nodes; $q_{j,i}$ and $\psi_{j,i}$ denote the integer and fractional parts respectively.
2. Piecewise-Linear Interpolation: The function value is given by linear interpolation of nodal values $G^j$:

$$g^j(x_{j,i}) = (1 - \psi_{j,i}) G^j[q_{j,i}] + \psi_{j,i} G^j[q_{j,i}+1]$$

1. Projection Descent Update: For each training record $i$ (with prediction $\hat{z}_i$ and true $z_i$), the residual $D_i = z_i - \hat{z}_i$ is projected onto the affected nodes and updated:

$$G^j[q_{j,i}] \gets G^j[q_{j,i}] + \alpha D_i (1-\psi_{j,i})/\chi_i, \quad G^j[r_{j,i}] \gets G^j[r_{j,i}] + \alpha D_i \psi_{j,i}/\chi_i$$

with normalization $$\chi_i = \sum_{j=1}^m [(1-\psi_{j,i})^2 + \psi_{j,i}^2]$$ and step size $\alpha \in (0,2)$.

1. Extension to Full Tree: At the composite (tree) level, two hierarchies are managed:
   • Inner (branch) functions $f^{(k j)}$ form auxiliary variables $\varphi_k = \sum_{j=1}^m f^{(k j)}(x_j)$.
   • Outer functions $\Phi^k$ combine these variables into the output.

Since auxiliary variables $\varphi_k$ are hidden, an outer loop incrementally adjusts them via

$$\Delta \varphi_{k,i} = \frac{\mu R_i(\varphi_{k,i})}{(2m+1) T((\Phi^k)'(\varphi_{k,i}))}$$

where $T(\zeta)$ is a safeguarded update (clipped by $\delta$), ensuring numerical stability when $(\Phi^k)'$ is near zero.

1. Recordwise Loop and Convergence: The identification proceeds by updating each component (branches, root, nodal domains) record-by-record and iterating until the prescribed error or stability threshold is attained.

3. Computational Efficiency and Stability

Several computational advantages distinguish the presented algorithm:

• Sparse, Localized Updates: Each input update touches only two nodal values per variable, and the full set of univariate functions are updated simultaneously, greatly expediting convergence compared to sequential or full-matrix updates.
• Normalization and Step Control: The inclusion of scale factors $\chi_i$ and adjustable step sizes $\alpha$, $\mu$ with the clipped $T(\cdot)$ function guarantees stable descent, mitigating overshooting and divergence even when function derivatives degenerate.
• Mixed Input Handling: Discrete/quantized variables are naturally incorporated by grid alignment, and combinations of quantized and continuous features are supported without restrictive assumptions.

4. Empirical Evaluation

The proposed identification framework is validated across a suite of benchmark data sets, both synthetic and real:

Dataset	Model Structure	Normalized RMSE	Correlation (P)
Synthetic nonlinear function	Full KA: 11 addends, 5 inputs	0.0203	0.9935
Airfoil self-noise (UCI)	Standard implementation	-	0.9506
Mushroom classification	1 quantized Urysohn operator	3–4 errors/fold	-

In mushroom classification, the single quantized operator achieved a computational runtime of only 2 seconds and outperformed alternative classifiers in both accuracy and efficiency. Additional testing on dynamic system and social data (e.g., churn prediction) further corroborates the competitive error rates and fast convergence of the deep tree architecture.

5. Theoretical and Practical Significance of Urysohn Trees

The Urysohn tree framework generalizes classical KA representation, bringing together hierarchical decomposition and operator theory with modern machine learning:

• Hierarchical Structure: Each node processes a one-dimensional projection, forming an interpretable tree or deep network of univariate operators culminating in the final output.
• Efficiency and Interpretability: The sparse, node-wise updates allow for clear visualization, understanding, and control of the model's behavior—a sharp contrast with many substantially less interpretable black-box architectures.
• Adaptability: The modular structure accommodates various types of features (quantized, continuous, hybrid) and is extensible to dynamic and high-dimensional data domains.

6. Relationship to Deep Learning and Network Architecture

The layered (tree) structure of Urysohn operators encapsulates both universal approximation and the hierarchical processing central to deep learning:

• Classical KA decomposition—originally interpreted as a two-layer structure—can be seen more generally as a deep tree or network where most layers encode inner function transformations, and the root layer acts as a final aggregator.
• Efficient update and identification strategies naturally parallel advances in deep learning optimization, including simultaneous weight updates and normalization schemes.

The Urysohn tree perspective links representation theory with current data-driven learning architectures, opening new directions for interpretable, robust, and scalable model identification across scientific, engineering, and data-centric disciplines.

7. Limitations and Scope

While the presented method is computationally efficient and widely applicable, the following considerations inform its deployment:

• The piecewise-linear univariate parameterization presumes sufficient coverage of nodal points; in extremely high-frequency or highly non-smooth regions the density of nodes may require adjustment.
• The method's convergence and accuracy rest on the decay and adaptation of the residual-based iterative loop; tuning of the $\alpha$ and $\mu$ parameters is necessary for stability in noisy or imbalanced data.
• As with all tree-structured, hierarchical models, the interpretability benefit is preserved best when the number of layers and functions is not excessive relative to the scale of the problem.

Summary

The Kolmogorov-Arnold Representation Theorem provides both the mathematical and structural basis for efficient and stable identification of continuous multivariate mappings via hierarchical superpositions of univariate functions. The algorithm described in (Polar et al., 2020) operationalizes this principle as a Urysohn tree, controlled via projection descent and piecewise-linear nodal updates, yielding rapid convergence, interpretability, and broad applicability. Its local update rules, efficient handling of mixed-type inputs, and robust empirical validation substantiate it as an effective foundation for machine learning models in regression, classification, and system modeling tasks.