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

arXiv: https://arxiv.org/abs/2606.17886 · HTML: https://mlsanity.com/paper/2606.17886/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 39 references that are in the map.
- First posted on arXiv 16 Jun 2026.

## Builds on (most cited first, 25 of 25)

- [KAN: Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2404.19756/) (2024)
- [Gradient Descent Finds Global Minima of Deep Neural Networks](https://mlsanity.com/paper/1811.03804/) (2018)
- [Interpretable Machine Learning: Fundamental Principles and 10 Grand Challenges](https://mlsanity.com/paper/2103.11251/) (2021)
- [Variational Inference of Disentangled Latent Concepts from Unlabeled Observations](https://mlsanity.com/paper/1711.00848/) (2017)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Unconstrained Monotonic Neural Networks](https://mlsanity.com/paper/1908.05164/) (2019)
- [Disentanglement by Nonlinear ICA with General Incompressible-flow Networks (GIN)](https://mlsanity.com/paper/2001.04872/) (2020)
- [A Survey on Kolmogorov-Arnold Network](https://mlsanity.com/paper/2411.06078/) (2024)
- [Deep Lattice Networks and Partial Monotonic Functions](https://mlsanity.com/paper/1709.06680/) (2017)
- [Block Neural Autoregressive Flow](https://mlsanity.com/paper/1904.04676/) (2019)
- [Certified Monotonic Neural Networks](https://mlsanity.com/paper/2011.10219/) (2020)
- [Monotonic Calibrated Interpolated Look-Up Tables](https://mlsanity.com/paper/1505.06378/) (2015)
- [Constrained Monotonic Neural Networks](https://mlsanity.com/paper/2205.11775/) (2022)
- [Convergence of gradient descent for deep neural networks](https://mlsanity.com/paper/2203.16462/) (2022)
- [Monotonic classification: an overview on algorithms, performance measures and data sets](https://mlsanity.com/paper/1811.07155/) (2018)
- [Deontological Ethics By Monotonicity Shape Constraints](https://mlsanity.com/paper/2001.11990/) (2020)
- [Principled Weight Initialisation for Input-Convex Neural Networks](https://mlsanity.com/paper/2312.12474/) (2023)
- [Monotone, Bi-Lipschitz, and Polyak-Lojasiewicz Networks](https://mlsanity.com/paper/2402.01344/) (2024)
- [Advancing Constrained Monotonic Neural Networks: Achieving Universal Approximation Beyond Bounded Activations](https://mlsanity.com/paper/2505.02537/) (2025)
- [Semantic Neighborhood-Aware Deep Facial Expression Recognition](https://mlsanity.com/paper/2004.12725/) (2020)
- [Smooth Min-Max Monotonic Networks](https://mlsanity.com/paper/2306.01147/) (2023)
- [Avoiding Resentment Via Monotonic Fairness](https://mlsanity.com/paper/1909.01251/) (2019)
- [MonoKAN: Certified Monotonic Kolmogorov-Arnold Network](https://mlsanity.com/paper/2409.11078/) (2024)
- [Monotone Cubic B-Splines with a Neural-Network Generator](https://mlsanity.com/paper/2307.01748/) (2023)

## Built on by (most cited first, 1 of 1)

- [Interpretable Feature Learning for RF Fingerprinting via Polar MKANs](https://mlsanity.com/paper/2608.19881/) (2026)

## Abstract (arXiv)

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 MKAN, a KAN with hard monotonicity guaranteed for 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 representation-cost theorem: any CK, K >0 feature extractor inducing a ball-shaped semantic-neighborhood partition admits a monotone realization of the equivalent neighborhood structure at N' = N^* + k ≤ 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.