Representation theorem · learnable splines on edges · multivariate function decomposition
Kolmogorov-Arnold theorem (1957): Any continuous multivariate function f: [0,1]ⁿ → ℝ can be written as:
f(x₁,...,xₙ) = Σ_{q=0}^{2n} Φ_q( Σ_{p=1}^{n} φ_{q,p}(x_p) )
where φ_{q,p} and Φ_q are continuous univariate functions. KAN (Liu et al. 2024) turns this into a neural network where learnable activation functions live on edges (not nodes), parameterized as B-splines. This contrasts with standard MLPs where fixed activations live on nodes.
Here we demonstrate approximating f(x,y) = sin(x²+y²) using a KAN-style decomposition with 1D splines.