Monotonic Kolmogorov-Arnold Networks: A Theoretical and Empirical Study of Monotonicity as an Inductive Bias

2026-06-16

AI Summary

This work addresses the challenge of efficiently enforcing global hard monotonicity constraints in neural networks while preserving interpretability and training simplicity. To this end, the authors propose MKAN, a novel architecture based on Kolmogorov–Arnold Networks, which guarantees hard monotonicity across all parameters through exponential reparameterization of B-spline coefficients, positive edge weights, and monotonic basis activation functions, while remaining compatible with standard unconstrained gradient descent optimization. The key contributions include the first realization of full-parameter hard monotonicity within the KAN framework and an architecture-agnostic representation cost theorem that informs the design of monotonic encoder widths. Experiments demonstrate that MKAN achieves state-of-the-art performance on the SMM/ICML-2024 benchmark, validates the predicted 2N* width scaling law across four real-world datasets, and significantly outperforms KAN, MLP, and linear models in Spearman correlation alignment on controllable generation tasks.

Abstract

Monotonicity has been a long-running architectural inductive bias for neural networks, motivated by tabular, scientific, and economic settings where outputs are known to respond monotonically to certain inputs. Existing approaches are MLP- or flow-based and lack per-edge functional transparency; the only Kolmogorov--Arnold Network (KAN) variant with monotonicity, MonoKAN, enforces the constraint only on a restricted parameter subset and requires a projection-style training procedure. We close this gap with \textbf{MKAN}, a KAN with hard monotonicity guaranteed for \emph{all} parameter values via exponential reparameterization of B-spline coefficients, positive edge weights, and a monotone base activation. Training reduces to standard unconstrained gradient descent. Our headline theoretical contribution is a \emph{representation-cost} theorem: any $C^K, K >0$ feature extractor inducing a ball-shaped semantic-neighborhood partition admits a monotone realization of the equivalent neighborhood structure at $N' = N^* + k \le 2N^*$, where $k$ is the number of non-monotone coordinates of the original. The bound is architecture-agnostic and gives a principled sizing rule for monotone encoders. Empirically, MKAN is competitive with state-of-the-art monotone NNs on the SMM/ICML-2024 benchmark while being the only method that combines hard unconstrained monotonicity with KAN's per-edge functional transparency; the $2N^*$ prediction is validated in a self-supervised feature-size sweep on four real datasets, and on a controlled monotone-generative dataset MKAN recovers ground-truth factors with substantially higher Spearman alignment than KAN, MLP, and linear baselines.

Problem

• monotonicity
• Kolmogorov-Arnold Networks
• inductive bias
• functional transparency
• neural networks

Innovation

• Monotonic Neural Networks
• Kolmogorov-Arnold Networks
• Exponential Reparameterization
• Representation-Cost Theorem
• Functional Transparency