Conditioning of trained hybrid KAN layers

Determine the conditioning of trained layers in the hybrid BesselKAN architecture to establish whether the severe ill-conditioning observed in the synthetic design-matrix analysis persists after network training.

Background

The paper combines B-spline and Bessel bases and reports substantial reductions in in-sample fitting error, but the resulting design matrices are extremely ill conditioned, with Gram-matrix condition numbers of approximately 1014–1016. Because the reported fits are evaluated on the same points used for fitting and the hybrid basis contains more columns, those results do not establish improved generalization or approximation quality.

The authors explicitly note that they did not evaluate the conditioning of trained layers. Measuring this quantity would clarify whether the numerical instability identified in the prescribed basis construction remains a practical problem during optimization and inference.

References

Combining the bases lowered the in-sample fit error but made the design matrix extremely ill conditioned ($\kappa(G)\approx10{14}$--$10{16}$); because the fit is evaluated on the fitting points and the hybrid has more columns, this does not show better approximation, and we did not measure the conditioning of trained layers.

— Hybrid Parallel Bessel KAN Model: a Kolmogorov Arnold Network: integrated framework for developing a novel architecture to reconstruct the fused-basis approach via a parameter free quadratic interaction  (2610.17779 - Pathak et al., 13 Oct 2026) in Section 6.4, “Conditioning and edge functions”