Multivariate KAN Contribution Matrix

• MKAN is a nonlinear feature ranking tool using Kolmogorov–Arnold Networks to map individual input contributions in multivariate prediction tasks.
• It computes smooth univariate transforms and normalized scores to reveal both the functional form and magnitude of each feature's influence.
• Empirical applications, including hydrological studies, demonstrate MKAN’s ability to enhance model parsimony and interpretability over linear methods.

The Multivariate KAN Contribution Matrix (MKAN) is a nonlinear feature‐ranking and interpretability tool constructed using Kolmogorov–Arnold Networks (KANs). MKAN enables researchers to quantify, visualize, and interpret the relative contributions and functional entry forms of multiple input variables in predicting target variables in high-dimensional, potentially collinear datasets. Unlike marginal or pairwise analysis, MKAN quantifies each input’s unique predictive utility in the joint multivariate context, supporting both more robust feature selection and deeper physical insight in model development workflows (Fuente et al., 12 Dec 2025).

1. Definition and Role

MKAN is designed to summarize both the shape of each input-to-target transformation and the magnitude of its contribution in a multivariate nonlinear prediction setting. For a set of $n$ variables, each MKAN row is associated with a chosen target $y_i$, and each column with an input $x_j$ (excluding $x_i$ as an input to its own prediction). Each cell encodes:

• The univariate function $\phi_{ij}(x_j)$, as learned by the KAN for predicting $y_i$, which expresses how $x_j$ is mapped to its effect on the target under the influence of all other variables.
• A scalar contribution score $C_{ij}\in[0,1]$ quantifying the importance of $x_j$ in predicting $y_i$ given all other inputs.

By summarizing these properties, MKAN enables automated, nonlinear feature ranking, redundancy detection, and parsimony-driven selection—tasks beyond the capacity of linear correlation or information-theoretic metrics.

2. Mathematical Construction

MKAN employs the simplest KAN architecture, in which the predictive mapping from non-target inputs to a selected target $y_i$0 takes the form:

$y_i$1

where each $y_i$2 is typically a cubic spline or other smooth basis function, jointly optimized to minimize a regularized empirical loss (e.g., mean-square error).

The relative contribution of each $y_i$3 to $y_i$4 is quantified by an “attribute score” $y_i$5, defined as:

$y_i$6

where $y_i$7 is an effective connectivity weight accounting for KAN network topology (see Liu et al. 2024 for details). Within each matrix row, $y_i$8 values are then linearly rescaled to $y_i$9 (min–max normalization). To reflect actual predictive skill, these are multiplied by a held-out performance metric (e.g., row-wise Nash–Sutcliffe or Kling–Gupta efficiency $x_j$0), yielding the final contribution score:

$x_j$1

3. Computational Algorithm

The MKAN is constructed through the following algorithmic workflow:

Step	Description	Notes
1	For each target $x_j$2, define input set $x_j$3	All variables except $x_j$4 considered as inputs
2	Train a one-layer KAN with input-to-target univariate transforms $x_j$5	Regularized to prevent overfitting
3	For each input $x_j$6, $x_j$7
3a	Compute $x_j$8 (edge-contribution score)	Uses std-ratio and connectivity $x_j$9
3b	Normalize all $x_i$0 in the row to $x_i$1	Min–max normalization
3c	Calculate $x_i$2, the prediction skill for $x_i$3	Held-out data
3d	Set $x_i$4
4	Store $x_i$5 in cell $x_i$6	Visualization included
5	Repeat for all $x_i$7

This results in an $x_i$8 matrix where each cell provides both a quantitative ranking and an interpretable, learned transformation.

4. Visualization and Interpretation

Each entry $x_i$9 of the MKAN contains:

• A color-coded background whose intensity reflects $\phi_{ij}(x_j)$0 (nearly black for $\phi_{ij}(x_j)$1, near-white for $\phi_{ij}(x_j)$2).
• An overlaid sketch of $\phi_{ij}(x_j)$3, conveying both sign and nonlinearity (e.g., monotonic, threshold, or complex behavior).

Key interpretive principles:

• Sign: The direction of $\phi_{ij}(x_j)$4 determines in which regions increases in $\phi_{ij}(x_j)$5 drive $\phi_{ij}(x_j)$6 up or down.
• Magnitude: Steepness in $\phi_{ij}(x_j)$7 denotes local sensitivity, but overall $\phi_{ij}(x_j)$8 encodes relative multivariate predictive value.
• Column comparison (within-row): Reveals which predictors most matter for a given target.
• Row comparison (within-column): Indicates which targets are most sensitive to a specific input.

5. Practical Considerations

• Computational Scaling: Training $\phi_{ij}(x_j)$9 independent KANs scales as $y_i$0, where $y_i$1 is the cost of fitting a single shallow network. Owing to exclusively univariate transforms, $y_i$2 is modest.
• Hyperparameters: The number of spline knots or basis functions must be chosen. Overparametrization can cause overfitting, while underparametrization can mask subtleties. Cross-validation on prediction skill $y_i$3 is advised.
• Collinearity: MKAN downweights redundant features naturally—when two inputs are collinear, the most informative or simplest (in terms of $y_i$4 complexity) dominates, and the other’s $y_i$5 approaches zero.
• Comparison to Linear Methods: MKAN captures both nonlinear effect strength and detailed functional form. Empirical benchmarks indicate that KAN-based rankings achieve parsimony and model skill unattainable via Pearson correlation or mutual information: for instance, hydrological prediction experiments attain the same model accuracy with 2–6 fewer features than Pearson‐based rankings.

6. Empirical Examples

Two illustrative cases from (Fuente et al., 12 Dec 2025) demonstrate MKAN’s utility:

• Synthetic Nonlinear Trio: For data $y_i$6, $y_i$7, $y_i$8 with $y_i$9, the MKAN row for $x_j$0 (target) yields $x_j$1 (with $x_j$2) and $x_j$3, indicating $x_j$4 supplies no independent information beyond $x_j$5 for $x_j$6.
• CAMELS Hydrology Dataset: Predicting 5th-percentile streamflow, MKAN ranks “frequency of dry days” and “mean slope” highest ($x_j$7 and $x_j$8, respectively), whereas Pearson correlation misranks these monotonic effects. A random forest trained on the top 12 MKAN-ranked features achieves $x_j$9, exceeding the corresponding Pearson-based subset.

7. Significance and Research Context

MKAN enables fully nonlinear, interpretable, and multivariate attribution of feature influence in scientific datasets. The matrix structure and integrated curve visualizations support both human‐guided and automated scientific workflows, including physical insight extraction, redundancy analysis, and model parsimony validation. Empirical findings show improved robustness and interpretability over traditional correlation and mutual information tools, particularly in domains with hidden nonlinear mechanisms or substantial collinearity (Fuente et al., 12 Dec 2025).