Pairwise KAN Matrix (PKAN)

• PKAN is a matrix-based tool for quantifying and visualizing nonlinear associations between variable pairs using learnable, spline-based mappings derived from Kolmogorov–Arnold theory.
• It computes a normalized association strength score and extracts explicit functional forms via a streamlined, single-layer KAN approach for each input–target pair.
• PKAN overcomes the limitations of linear and mutual information measures by revealing asymmetric, explicit nonlinear relationships for better feature selection and model interpretation.

The Pairwise KAN Matrix (PKAN) is a matrix-based tool for quantifying and visualizing nonlinear associations between ordered pairs of variables in a dataset. Derived from the theoretical foundations of the Kolmogorov–Arnold representation theorem and implemented using the Kolmogorov–Arnold Network (KAN) architecture, PKAN provides a color-coded summary of both the strength and the explicit functional form of each pairwise relationship, thereby addressing the limitations of classical linear and information-theoretic association measures (Fuente et al., 12 Dec 2025).

1. Theoretical Foundation

PKAN is grounded in the Kolmogorov–Arnold representation theorem, which asserts that any continuous multivariate function $f(x_1,\ldots,x_n)$ can be written as a finite superposition of continuous univariate functions:

$$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$

where $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$ and $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$ are continuous functions. The KAN architecture provides a parametric, learnable analogue to these $\phi_{ij}$ and $\Phi_i$ functions, typically using spline-based single-layer neural networks and training via standard prediction losses such as mean squared error.

PKAN specifically utilizes a simplified KAN, omitting the outer $\Phi_i$ layer and learning only a single univariate mapping for each input–target pair. The learned mapping $P_{ij}(\cdot)\approx \phi_{ij}(\cdot)$ characterizes the predictive relationship and the form of nonlinearity between $x_j$ and $x_i$.

2. Formal Definition

Given a set of $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$0 variables $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$1, PKAN constructs an $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$2 matrix where each entry $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$3 encodes the fit of a single-input single-output network predicting $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$4 from $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$5:

$$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$6

Two outputs are obtained from each fit:

• Functional form: The learned univariate $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$7, representing the explicit relationship.
• Association strength: A normalized scalar $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$8 quantifying the degree to which $$f(x_1,\ldots,x_n) = \sum_{i=1}^{2n+1} \Phi_i \left( \sum_{j=1}^n \phi_{ij}(x_j) \right)$$9 predicts $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$0.

Strength calculation involves:

• Computing prediction loss $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$1 (e.g., MSE) on held-out data.
• Converting $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$2 to a predictive strength score using the Nash–Sutcliffe Efficiency (NSE) or similar:

$\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$3

• Estimating the KAN attribute score $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$4 as the ratio of standard deviation of activation through $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$5 to the total at the node.
• Calculating raw strength $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$6.
• Rescaling $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$7 for all pairs to $\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$8:

$\phi_{ij}:\mathbb{R}\rightarrow\mathbb{R}$9

with $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$0 for diagonal elements.

3. Algorithmic Construction

PKAN computation follows these steps:

1. Preprocessing: Optionally normalize each variable to $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$1.
2. Model fitting: For each ordered pair $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$2 with $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$3, fit a univariate KAN mapping $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$4 via mean squared error minimization.
3. Output computation:
   • Evaluate $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$5 on held-out data.
   • Compute $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$6 as per Liu et al.
   • Obtain $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$7.
4. Normalization: Transform $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$8 to $\Phi_i:\mathbb{R}\rightarrow\mathbb{R}$9 as above.
5. Storage: Record $\phi_{ij}$0 and the learned curve $\phi_{ij}$1 (sampled over a grid) for visualization.

The diagonal $\phi_{ij}$2 is set to $\phi_{ij}$3 and the identity function is stored for reference.

Algorithm Pseudocode (from (Fuente et al., 12 Dec 2025))

Step	Operation	Output
Preprocess	Normalize $\phi_{ij}$4 (optional)	Normalized variables
KAN Fit	Train $\phi_{ij}$5 for $\phi_{ij}$6	Parameters $\phi_{ij}$7
Evaluate	Compute $\phi_{ij}$8	Raw strength $\phi_{ij}$9
Normalize	Convert $\Phi_i$0 to $\Phi_i$1	PKAN matrix $\Phi_i$2
Visualize	Store $\Phi_i$3 curves	Heatmap overlay

4. Visualization and Interpretation

PKAN is rendered as an $\Phi_i$4 color-coded matrix:

• Rows $\Phi_i$5: target variables $\Phi_i$6
• Columns $\Phi_i$7: input variables $\Phi_i$8
• Cell $\Phi_i$9: color encodes $\Phi_i$0 (white for $\Phi_i$1, dark red for $\Phi_i$2); a curve is overlaid to show $\Phi_i$3

Interpreting a cell:

• Color intensity (magnitude) indicates how strongly $\Phi_i$4 alone predicts $\Phi_i$5
• Curve shape (form) reveals monotonicity, nonlinearity, symmetry, or injectivity
• Directional asymmetry is manifest as $\Phi_i$6 except for mutually injective mappings

5. Comparative Evaluation with Classical Metrics

PKAN provides a richer characterization compared to standard association measures:

• Pearson correlation $\Phi_i$7: Captures only linear, symmetric dependence; fails for nonlinear monotonic relationships (e.g., $\Phi_i$8 on symmetric domains with $\Phi_i$9).
• Mutual information $P_{ij}(\cdot)\approx \phi_{ij}(\cdot)$0: Detects nonlinearity and is symmetric, but unable to represent directionality or explicit function shape, and becomes computationally challenging as $P_{ij}(\cdot)\approx \phi_{ij}(\cdot)$1 increases.
• PKAN: Encodes strength (asymmetrically), explicit function form, and directionality. It remains robust to collinearity by isolating each predictor.

Empirical demonstrations include:

• Quadratic/cubic: PKAN captures $P_{ij}(\cdot)\approx \phi_{ij}(\cdot)$2, $P_{ij}(\cdot)\approx \phi_{ij}(\cdot)$3 accurately in strength and learned curve; Pearson and MI miss essential features.
• Heteroscedastic noise: PKAN scores track Pearson but with curve shape reflecting the true underlying relation.
• Lagged sinusoids: PKAN recognizes cyclic structure and directionality, which Pearson and MI cannot fully recover.

PKAN thus provides both an interpretable score of association and direct visualization of the relationship, serving as a tool for pre-processing (feature selection, redundancy analysis), post-processing (model explanation), and discovery of physical patterning in data (Fuente et al., 12 Dec 2025).

6. Applications and Broader Significance

PKAN supports multiple stages in the model development workflow:

• Exploratory data analysis: Rapid assessment of all pairwise relationships in high-dimensional data.
• Feature selection and redundancy analysis: Identification of inputs with non-trivial, possibly nonlinear predictive value.
• Physical insight: Visualization of explicit functional forms to guide scientific hypothesis formation.
• Model explanation: Post hoc interrogation of black-box models via analogous association patterns.

Case studies, such as analysis on the CAMELS hydrology dataset and large-eddy simulations of river flow, corroborate the robustness, informativeness, and interpretability advantages of PKAN over prior approaches (Fuente et al., 12 Dec 2025).

Plausible implications: The combined strength-and-form rendering aids in domain-informed modeling and hypothesis-driven discovery, particularly under high-dimensional and collinear regimes where conventional statistics may obscure essential structure.