MatrixKAN: Matrix-Based KAN Innovations

• MatrixKAN is a framework that reformulates Kolmogorov–Arnold Networks by implementing spline computations and basis-function evaluations via efficient matrix operations.
• It transforms traditional recursive spline evaluations into parallelized matrix multiplications, drastically reducing computation time and enhancing scalability.
• MatrixKAN underpins advanced visualization tools like PKAN and MKAN, which quantify nonlinear relationships and support applications in scientific data modeling and cryptography.

MatrixKAN denotes a class of techniques and architectures built upon the Kolmogorov–Arnold Network (KAN) framework in which the core spline computations—both for network inference and specialized data analysis—are efficiently implemented through explicit matrix operations, enabling high-performance parallelized evaluation, streamlined model interpretability, and, in some contexts, cryptographic algorithms relying on matrix action key exchange. The main instantiations of the MatrixKAN paradigm are: (1) parallelized spline computation for scalable and expressive neural networks (Coffman et al., 11 Feb 2025), (2) visual analysis tools for quantifying nonlinear, directional relations in multivariate datasets via Kolmogorov–Arnold superposition (Fuente et al., 12 Dec 2025), (3) ReLU-based, purely matrix- and elementwise-operation KAN variants optimized for modern GPU hardware (Qiu et al., 2024), and (4) in entirely different contexts, key exchange protocols involving matrix semidirect products (Rahman et al., 2020). The following exposition focuses on the central mathematical, algorithmic, and practical advances embodied by MatrixKAN in the context of scientific data modeling and learning.

1. Mathematical and Algorithmic Foundation

MatrixKAN is grounded in the Kolmogorov–Arnold superposition theorem, which asserts that any continuous multivariate function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ admits a decomposition

$$f(x_1, ..., x_n) = \sum_{q=0}^{2n} \Phi_q \Big( \sum_{p=1}^{n} \psi_{q,p}(x_p) \Big)$$

with continuous univariate functions $\psi_{q,p}$ (inner) and $\Phi_q$ (outer). Canonical KAN implements this construction in neural architectures by replacing edge activations or intermediate transformations with learnable univariate splines or basis-function expansions.

MatrixKAN accelerates and structures KAN implementations by recasting spline and basis-function computations into matrix-matrix multiplications, which are inherently parallel and optimized on GPU computing backends. For uniform B-splines, each spline segment's value is evaluated as

$$\mathrm{spline}_i(u) = [1, u, ..., u^{k-1}]\, \Psi^{(k)}\, [c_i, ..., c_{i+k-1}]^\top$$

where $\Psi^{(k)}$ is a precomputed basis matrix encoding all Cox–de Boor recursion coefficients for splines of order $k$ (Coffman et al., 11 Feb 2025). This operation is vectorized across samples, network edges, and layers. Alternatively, in ReLU-KAN (“MatrixKAN”), B-splines are replaced by bell-shaped, compactly supported functions constructed solely with matrix addition, dot multiplication, and squared ReLU segments, further reducing computational complexity and memory requirements (Qiu et al., 2024).

For visualization of nonlinear associations, MatrixKAN builds matrices (PKAN, MKAN) by training ensembles of KAN regressors on all ordered pairs or tuples of variables, then quantifying edge contributions through standardized activation ratios and validation skill metrics (Fuente et al., 12 Dec 2025).

2. Efficient Matrix-Based Spline Computation

Traditional KAN implementations are bottlenecked by the Cox–de Boor recursion, whose $d$ nested levels for degree-$d$ splines inhibit full GPU parallelism. MatrixKAN eliminates this by:

• Precomputing the fixed basis matrix $\Psi^{(k)}$ for each order $k$.
• Representing the input positions in power-basis tensors, enabling batch computation.
• Substituting recursion with batched matrix multiplications and tensor contractions.

Algorithmically, each layer forward-pass is reducible to the following matrix operations (per (Coffman et al., 11 Feb 2025)):

1. Compute normalized positions $u$ in all spline intervals across the batch and all edges.
2. Form the tensor $P$ whose last dimension encodes the powers $u^r$, $r=0,\dots,k-1$.
3. Execute $P \cdot \Psi^{(k)}$ for basis evaluation.
4. Multiply by control-point tensors, sum across basis functions, aggregate across inputs.

For ReLU-KAN, the activation function per basis is $$R_p(x) = [\mathrm{ReLU}(e_p - x)\mathrm{ReLU}(x - s_p)]^2\times \frac{16}{(e_p - s_p)^4}$$, and all matrix computations involve only broadcasted subtractions, elementwise ReLU, pointwise multiplication and summation, fully compatible with high-throughput tensor libraries (Qiu et al., 2024).

3. PKAN and MKAN: Interpretable Nonlinear Data Analysis

MatrixKAN underpins novel analysis tools—Pairwise KAN Matrix (PKAN) and Multivariate KAN Contribution Matrix (MKAN)—for interpretable, color-coded quantification of nonlinear, non-injective, and multivariate relationships in scientific datasets (Fuente et al., 12 Dec 2025):

• PKAN: For each ordered variable pair $(x_j, x_i)$, fits a one-input KAN mapping $x_j \mapsto x_i$. Entry strength $\mathrm{PKAN}_{i,j}$ is the product of normalized edge activation ratio $A_{i,j}$ (standard deviation of edge activation over output variable) and validation predictive strength $\mathrm{Perf}_{i,j}$ (e.g., $R^2$ or Kling-Gupta skill).
• MKAN: For each target $x_i$, fits a multi-input KAN $x_{-i} \mapsto x_i$ and attributes feature contributions via normalized $A_{i,j}$ and overall skill $\mathrm{Perf}_i$.

Visualizations plot color-coded $n \times n$ matrices, overlaying each cell with the learned functional form $\Pi_{i,j}(\cdot)$. PKAN asymmetry ($\mathrm{PKAN}_{i,j} \neq \mathrm{PKAN}_{j,i}$) identifies non-injective mappings, crucial for mechanistic insight.

4. Computational Complexity and Empirical Performance

MatrixKAN yields a dramatic improvement in computational scaling with respect to spline degree $d$:

• KAN: $O(N^2 L (d^2 + dG))$ flops; effective wall time $O(L d)$ per forward pass (sequential recursion) (Coffman et al., 11 Feb 2025).
• MatrixKAN: $O(N^2 L (d^2 + G))$ flops; effective wall time $O(L)$ (fully parallel).
• ReLU-KAN: Further simplifies all per-layer operations to batched matrix addition and multiplications.

Empirical benchmarks demonstrate:

• 20–40$\times$ speedup at high spline degree ($d \gtrsim 20$) and with large datasets.
• Equal or better accuracy (in RMSE, MSE) versus unoptimized KAN; for some Feynman equation tasks, RMSE improves up to 27% with higher $d$ (Coffman et al., 11 Feb 2025).
• ReLU-KAN achieves 8–30$\times$ training speedup and $10^2$–$10^3\times$ lower MSE over standard KAN, with minimal additional GPU memory cost (Qiu et al., 2024).

All matrix-operation optimizations precisely preserve KAN's functional approximation properties.

5. Practical Implementation and Visualization Workflows

A standard MatrixKAN workflow comprises:

1. Data normalization to zero mean and unit variance.
2. For each variable pair $(i,j)$, KAN fitting, $A_{i,j}$ and $\mathrm{Perf}_{i,j}$ computation, and PKAN matrix population (Fuente et al., 12 Dec 2025).
3. For each multivariate target $i$, multi-input KAN fitting, featurewise $A_{i,j}$ computation, and MKAN matrix population.
4. Visualization as color-coded matrices with overlaid learned univariate mappings.

The PKAN and MKAN matrices differentiate between strong, weak, and negligible nonlinear associations with empirical thresholds (e.g., $s < 0.2$ negligible, $s \ge 0.8$ strong). Implementation pseudocode for both PKAN and MKAN construction is fully detailed in (Fuente et al., 12 Dec 2025).

6. Comparative Analysis and Use Cases

Comparative studies against Pearson correlation and Mutual Information establish that:

• PKAN/MKAN correctly reflect mapping directionality and non-injectivity (PKAN zeroes for $x^2\mapsto x$, high for $x\mapsto x^2$).
• PKAN/MKAN maintain stable association strengths in the presence of noise, while Pearson and MI degrade.
• Feature selection using top $k$ features ranked by MKAN scores yields higher $R^2$ on downstream models (e.g., Random Forests) than Pearson or MI feature selection: MKAN needs $\sim$2–4 fewer features to match their predictive performance (Fuente et al., 12 Dec 2025). This is attributed to MatrixKAN's ability to detect both nonlinearity and functional redundancy.

Typical applications span physical sciences, feature selection, model pre- and post-processing, and discovery of hidden latent relationships.

7. Limitations and Future Directions

The principal limitations are:

• Precomputation cost: For B-splines, basis matrix $\Psi^{(k)}$ requires $O(d^4)$ initialization, readily amortized in large networks or datasets (Coffman et al., 11 Feb 2025).
• Memory usage: For very high spline degree, power-basis tensors may become large.
• Spline basis assumptions: Efficiency hinges on uniform spline knots; generalization to non-uniform grids may require interval-specific precomputations.
• Extensibility: MatrixKAN can integrate with other KAN accelerations, e.g., free-knot or radial-basis expansions, and with domain-specific architectures (e.g., convolutional KAN) for additional gains.

A plausible implication is that approaches leveraging elementwise-only architectures (such as ReLU-KAN) may generalize further by exploiting hardware accelerators for even higher model complexity and scale (Qiu et al., 2024).

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References:

• MatrixKAN for parallelized Kolmogorov-Arnold Networks (Coffman et al., 11 Feb 2025).
• CUDA-friendly ReLU-KAN architecture as "MatrixKAN" (Qiu et al., 2024).
• KAN-Matrix: PKAN/MKAN interpretable nonlinear visualization (Fuente et al., 12 Dec 2025).
• MatrixKAN as matrix-action key exchange protocol (Rahman et al., 2020).