A machine-learning model that lets each network edge learn its own exponent recorded a median normalized root mean squared error (NRMSE) of 0.030 on a clean regression benchmark, with a mean rank of 2.32 across all 50 targets. But the picture changed with measurement noise: when σ rose from 0 to 1, Banach-KAN’s aggregated median error was 8.8 times its starting level, compared with 3.7 times for ℓp-KAN and 3.2 times for Tanh-KAN.
What the models changed
The findings come from an arXiv version-1 preprint dated 26 August 2026. Its main synthetic test covered 50 symbolic-regression targets: 40 equations from AI Feynman and 10 synthetic stress tests. The first 18 equations were reused as a core set for noise, small-sample and ablation experiments.
The models change the function applied along each edge of the network. In ℓp-KAN, each edge has five learnable parameters, with the exponent parameterized as p=1+exp(θ). Banach-KAN combines tanh and Jp branches and uses eight learnable parameters per edge. The clean control comparison also tested mixture and rational/Padé activations.
Clean results, then a noise penalty
On the clean benchmark, using 500 training samples and σ=0, Banach-KAN had median NRMSE 0.030. Its mean rank was 2.00 on the Core 18 and 2.32 on the full set, compared with 2.34 for Spline G=3 on the full set. No confidence interval was reported for the median or rank comparisons.
The noise comparison exposed a trade-off within the family. From σ=0 to σ=1, the corresponding degradation ratios were 3.7-fold for ℓp-KAN, 3.2-fold for Tanh-KAN, 8.8-fold for Banach-KAN, 11.2-fold for Spline CV and 21.6-fold for Spline G=3. The ratios aggregate median NRMSE, and the test concerned measurement noise rather than worst-case perturbations.
In the clean small-sample sweep, Banach-KAN won 8 of 18 equations at n=50 and 7 of 18 at n=100; across the full sweep it won 28 of 90 model-condition cells. Those counts indicate a recurring lead in limited-data conditions, but not a universal one.
Learning a different geometry
The exponent itself also moved substantially during fitting. Across 4,465 Banach-KAN edges, 18.7% learned p<2, only 1.2% remained within ±0.01 of the p=2.5 initialization, and 77% of runs had within-run p standard deviation above 0.5. The authors describe this signal as aggregate, relative and initialization-dependent, with sensitivity to the random seed at the edge and equation levels.
That pattern varied with input dimension: the share of edges with p<2 rose monotonically from 11% at two dimensions to 34% at six dimensions. The paper treats the learned exponents as a possible readout of target structure, not as an absolute, seed-independent measure.
Against flexible controls
Against generic flexible controls, Banach-KAN had NRMSE 0.029, 15 wins out of 18 equations and mean rank 1.17 in the clean Core-18 comparison, averaged over five seeds. The mixture control scored 0.052 and rational/Padé 0.066. A paired equation-bootstrap estimated the Banach–mixture gap at −0.023, with a 95% confidence interval of −0.039 to −0.001.
The study also documented two controls around the ablation. The Spline–Adam schedule was selected before the 18-equation ablation using two held-out equations and five seeds each. In the Banach ablation, the median degradation ratio was 1.91-fold; full Banach-KAN beat the ablated variant on all 18 equations, with the largest single-equation gap reaching 5.00-fold.
Beyond the core equations
Beyond the core benchmark, Banach-KAN had the lowest reported extrapolation NRMSE in an out-of-range test: 0.35, with seven wins out of 18. ℓp-KAN recorded 0.42 and six wins, while spline recorded 0.73 and no wins. Across 12 PMLB regression datasets, the geometry-constrained family won 10 of 12; Tanh-KAN had five wins and mean rank 1.67, while Spline G=3 had no wins and mean rank 4.83.
The missing depth test
The clearest limit is depth. An analytic proposition says the relevant order parameter is finite if and only if p>3/2, with a unique critical weight scale above that threshold but none at or below it. The faithful deep inner-layer variant did not complete within the available budget, however, so the study did not directly test whether that condition predicts trainability in a genuinely deep network.
The authors interpret the results as evidence that learned function-space geometry, rather than fixed-basis choice or generic parametric flexibility, is the operative advantage of this KAN family in low-to-mid-dimensional regression. The reported experiments show a regime split: Banach-KAN led several clean and limited-sample comparisons, while the noise-degradation ratios were smaller for ℓp-KAN and Tanh-KAN. Those conclusions remain tied to the computational benchmarks, and the deep trainability claim awaits direct testing.
Paper data and sources
Original title: Geometry-Constrained Kolmogorov-Arnold Networks: Learning Edge Geometry via Banach Duality
Authors: K S Sesh Kumar
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text