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MonoKAN: Certified Monotonic Kolmogorov-Arnold Network

Alejandro Polo-Molina, David Alfaya, Jose Portela

TL;DR

This work tackles the need for trustworthy AI by enforcing domain-aligned monotonic constraints within neural models. It introduces MonoKAN, a certified partial monotonic Kolmogorov-Arnold Network that integrates monotone cubic Hermite splines into a KAN framework to guarantee monotonicity across chosen inputs while preserving interpretability. A mathematical certification theorem and a lightweight training algorithm with projection ensure that the network remains monotone across the entire input space, not just training data. Empirical results on a 2D fairness case study and multiple real-world benchmarks show MonoKAN often surpasses state-of-the-art monotonic MLPs, with competitive accuracy and improved transparency, supported by accessible code.

Abstract

Artificial Neural Networks (ANNs) have significantly advanced various fields by effectively recognizing patterns and solving complex problems. Despite these advancements, their interpretability remains a critical challenge, especially in applications where transparency and accountability are essential. To address this, explainable AI (XAI) has made progress in demystifying ANNs, yet interpretability alone is often insufficient. In certain applications, model predictions must align with expert-imposed requirements, sometimes exemplified by partial monotonicity constraints. While monotonic approaches are found in the literature for traditional Multi-layer Perceptrons (MLPs), they still face difficulties in achieving both interpretability and certified partial monotonicity. Recently, the Kolmogorov-Arnold Network (KAN) architecture, based on learnable activation functions parametrized as splines, has been proposed as a more interpretable alternative to MLPs. Building on this, we introduce a novel ANN architecture called MonoKAN, which is based on the KAN architecture and achieves certified partial monotonicity while enhancing interpretability. To achieve this, we employ cubic Hermite splines, which guarantee monotonicity through a set of straightforward conditions. Additionally, by using positive weights in the linear combinations of these splines, we ensure that the network preserves the monotonic relationships between input and output. Our experiments demonstrate that MonoKAN not only enhances interpretability but also improves predictive performance across the majority of benchmarks, outperforming state-of-the-art monotonic MLP approaches.

MonoKAN: Certified Monotonic Kolmogorov-Arnold Network

TL;DR

This work tackles the need for trustworthy AI by enforcing domain-aligned monotonic constraints within neural models. It introduces MonoKAN, a certified partial monotonic Kolmogorov-Arnold Network that integrates monotone cubic Hermite splines into a KAN framework to guarantee monotonicity across chosen inputs while preserving interpretability. A mathematical certification theorem and a lightweight training algorithm with projection ensure that the network remains monotone across the entire input space, not just training data. Empirical results on a 2D fairness case study and multiple real-world benchmarks show MonoKAN often surpasses state-of-the-art monotonic MLPs, with competitive accuracy and improved transparency, supported by accessible code.

Abstract

Artificial Neural Networks (ANNs) have significantly advanced various fields by effectively recognizing patterns and solving complex problems. Despite these advancements, their interpretability remains a critical challenge, especially in applications where transparency and accountability are essential. To address this, explainable AI (XAI) has made progress in demystifying ANNs, yet interpretability alone is often insufficient. In certain applications, model predictions must align with expert-imposed requirements, sometimes exemplified by partial monotonicity constraints. While monotonic approaches are found in the literature for traditional Multi-layer Perceptrons (MLPs), they still face difficulties in achieving both interpretability and certified partial monotonicity. Recently, the Kolmogorov-Arnold Network (KAN) architecture, based on learnable activation functions parametrized as splines, has been proposed as a more interpretable alternative to MLPs. Building on this, we introduce a novel ANN architecture called MonoKAN, which is based on the KAN architecture and achieves certified partial monotonicity while enhancing interpretability. To achieve this, we employ cubic Hermite splines, which guarantee monotonicity through a set of straightforward conditions. Additionally, by using positive weights in the linear combinations of these splines, we ensure that the network preserves the monotonic relationships between input and output. Our experiments demonstrate that MonoKAN not only enhances interpretability but also improves predictive performance across the majority of benchmarks, outperforming state-of-the-art monotonic MLP approaches.
Paper Structure (16 sections, 5 theorems, 20 equations, 8 figures, 6 tables, 2 algorithms)

This paper contains 16 sections, 5 theorems, 20 equations, 8 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Kolmogorov-Arnold Networks
  3. MonoKAN
  4. Partial Monotonicity
  5. Monotonic Cubic Hermite Splines
  6. Mathematical Certification of Partial Monotonicity of a KAN
  7. MonoKAN Algorithm
  8. Experiments
  9. Case Study: Visualizing Monotonicity and Fairness in 2D Predictions
  10. Benchmarking MonoKAN Against State-of-the-Art Algorithms
  11. Results
  12. Computational Implementation Details
  13. Conclusion
  14. Symbol Glossary for Theorem \ref{['teo:mono_conditions']}
  15. Proof of Theorem \ref{['teo:mono_conditions']}
  16. ...and 1 more sections

Key Result

Lemma 3.1

Let $p_k$ be an increasing (resp. decreasing) monotone cubic Hermite spline of the data $\mathcal{X} = \{(x_k, y_k, m_k), (x_{k+1}, y_{k+1}, m_{k+1})\}$ such that the control points verify that $y_k \leq y_{k+1}$ (resp. $y_k \geq y_{k+1}$). Then Moreover, if $d_k = 0$ then $p_k$ is monotone (in fact, constant) if and only if $m_k = m_{k+1} = 0$.

Figures (8)

  • Figure 1: Illustration of a cubic Hermite spline. The spline smoothly interpolates the function values while respecting the slope (derivative) constraints at each knot.
  • Figure 2: Scheme of monotonic and non-monotonic connections of a partial monotonic KAN w.r.t the first input with layers $[n,2,3,1]$
  • Figure 3: Illustration of a simplified MonoKAN, partially monotonic w.r.t. the first input, with two inputs and one output. Each connection between neuron $(l,i)$ and $(l+1,j)$ includes a cubic Hermite spline $\varphi_{l,j,i}$ and a base activation $\mathbf{b}$, weighted respectively by $\omega^\varphi_{l,j,i}$ and $\omega^\mathbf{b}_{l,j,i}$. The outputs of these functions are summed to produce the final activation.
  • Figure 4: Contour plots of model outputs over the 2D input space (psychometric test 1 vs. psychometric test 2) for (a) unconstrained cubic Hermite KAN, (b) unconstrained B-spline KAN, (c) partially constrained MonoKAN (monotonic with respect to the first psychometric test), and (d) fully constrained MonoKAN. In (a) and (b), the model exhibits several violations of monotonicity, as highlighted by candidate comparisons (e.g., A vs. B, and A vs. C), where improved input scores lead to lower predicted suitability. In (c), monotonicity is enforced only along test 1, improving consistency along that axis but still allowing irregularities along test 2 (G vs H). In (d), full monotonicity constraints result in a smooth, fair decision surface aligned with ethical expectations.
  • Figure 5: One-dimensional slices of the model outputs across psychometric test 1 and test 2, with the complementary variable fixed at 0.4, to visualize local monotonicity violations. Each row corresponds to a different model architecture: (a, b) unconstrained cubic Hermite KAN, (c, d) unconstrained B-spline KAN, (e, f) partially constrained MonoKAN (monotonic with respect to the first psychometric test), and (g, h) fully constrained MonoKAN. Red dots highlight specific counterexamples (e.g., A vs. B, D vs. E, G vs. H) where increasing an input leads to a decrease in the predicted output, violating monotonicity. These examples are consistent with the patterns observed in the contour plots of Figure \ref{['fig:contour_2d']}, providing further evidence of undesirable behaviors in unconstrained models. Notably, all violations are eliminated in the fully constrained MonoKAN (g, h), confirming its certified monotonic behavior across both input dimensions.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Lemma 3.1: Necessary conditions for monotonicity, Arandiga2022MonotoneGradient
  • Lemma 3.2: Sufficient conditions for monotonicity, Fritsch1980MonotoneInterpolation
  • Theorem 3.3: Certified Partial Monotonicity Conditions
  • Proposition B.1
  • proof
  • Theorem B.2
  • proof