Kolmogorov‑Arnold Networks (KAN) replace fixed scalar weights with learnable univariate functions on each edge. Existing variants fix the function basis—B‑splines, Chebyshev polynomials, wavelets, or Jacobi polynomials—and only learn linear combination coefficients. RecKAN defines the basis through a second‑order polynomial recurrence
$$ R_{n+1}(x) = (a x^2 + b x + c) Rn(x) + (d x + e) R{n-1}(x) $$
and learns the five coefficients $a,b,c,d,e$ jointly with the network. With particular coefficient settings the recurrence collapses to classic families such as first‑ and second‑kind Chebyshev, Fibonacci, Pell, Jacobsthal, and other polynomial families. We prove that on the sub‑family containing all these cases the degree of $R_n$ grows linearly with $n$, giving a concrete sense in which the learned basis can exceed any fixed classical choice.
Across image, text, biomedical time‑series classification and forecasting benchmarks, RecKAN consistently outperforms parameter‑matched Chebyshev, Jacobi, and spline KAN baselines on all classification tasks and achieves the lowest MSE on the ETTh1 forecasting benchmark. When used as a classifier head on top of a convolutional backbone, RecKAN yields higher accuracy than standard MLP heads on Fashion MNIST, CIFAR‑10, and SVHN. A synthetic function‑fitting test shows RecKAN tracking a sharply oscillatory target that a comparable‑size MLP under‑fits.
Further analysis of the learned recurrence coefficients reveals that on tasks demanding strong local structure, training pushes the basis away from the linear‑degree‑growth regime, matching our theoretical analysis of the structural shift and demonstrating that coefficient changes explicitly modulate the expressive power of the basis.
Review