aiqbench analysis
This paper introduces MKAN, a Kolmogorov-Arnold Network variant that enforces hard monotonicity across all parameters through exponential reparameterization, eliminating the need for constrained optimization. Theoretically, it establishes a representation-cost theorem proving that monotone realizations require at most double the nodes of non-monotone counterparts, offering a principled sizing rule for interpretable encoders. Empirically, MKAN matches state-of-the-art performance on tabular benchmarks while uniquely combining strict monotonicity with KAN's per-edge functional transparency.
Monotonic Kolmogorov-Arnold Networks: A Theoretical and Empirical Study of Monotonicity as an Inductive Bias
arXiv:2606.17886v3 Announce Type: replace 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.
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.
Machine Learning (cs.LG)
arXiv:2606.17886 [cs.LG]
(or arXiv:2606.17886v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.17886 Focus to learn more arXiv-issued DOI via DataCite
arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.
Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.
Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.
Full text under the publisher's open licence · 314 words
Related coverage
Monotonic Kolmogorov-Arnold Networks: A Theoretical and Empirical Study of Monotonicity as an Inductive Bias
arXiv · Jun 17, 2026
Optimization Risk Bounds for Kolmogorov-Arnold Networks Trained by DP-SGD with Correlated Noise
arXiv · Sep 30, 2026
Scalability Analysis of Distributed Kolmogorov-Arnold Network Training on High-Performance Computing Systems
arXiv · Sep 9, 2026
Draining Fictitious Knots: Restoring Distance-Awareness Guarantees for High-Dimensional Spline Networks
arXiv · Sep 15, 2026