Geometry-Constrained Kolmogorov-Arnold Networks: Learning Edge Geometry via Banach Duality

arXiv:2608.25807v2 Announce Type: replace Abstract: Kolmogorov-Arnold Networks (KANs) learn univariate functions on edges, making the choice of edge response central to their design. Existing variants use basis expansions such as splines or polynomials, whose functional form is chosen before training. We introduce geometry-constrained KANs, which instead learn the shape of power-law edge responses through an exponent inspired by Banach duality maps. Banach-KAN combines this adaptive power-law response with a bounded hyperbolic tangent (tanh) component. We relate these responses to convex potentials and bound edge sensitivity on bounded input ranges. Across symbolic regression benchmarks, Banach-KAN is competitive with fixed-basis models, particularly with limited clean data. Learned power-law responses remain competitive under measurement noise, while splines are stronger with plentiful clean data. Fixed-exponent and branch-removal controls on a core subset support learning the exponent and retaining both Banach components. Learned exponents describe the shapes of the fitted edge responses.