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

arXiv · AI, language, vision and robotics · article · Aug 26, 2026 · UTC

Kolmogorov-Arnold Networks (KANs) replace fixed activations in deep architectures with learnable univariate edge functions, making the choice of edge parametrisation central. Existing variants rely on fixed bases such as splines, polynomials, or Fourier features, which impose a function-space geometry before data are observed. We introduce geometry-constrained KANs, a family of edge activations derived from Banach duality maps in which the geometry itself is learned through a scalar exponent $p > 1$ per edge. This exponent controls the qualitative response: sub-Euclidean values produce sharp,

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