Complex-Valued KANs

Kolmogorov-Arnold Networks (KANs) replace the fixed scalar weights of an MLP with learnable univariate functions on the edges. complextorch.nn.CVKANLayer is a complex adaptation that places a learnable radial-basis expansion over the complex plane, and complextorch.models.CVKAN stacks those layers into a network.

The complex-plane RBF edge function

Each edge response is a sum of Gaussian bumps centred on a grid of complex points \(c_1,\dots,c_G\), combined with complex coefficients, plus a complex linear base (residual):

\[ \psi_g(z) = \exp\!\Big(-\frac{|z - c_g|^2}{2 h^2}\Big), \qquad y_o = \sum_{i}\sum_{g} W_{o,i,g}\,\psi_g(z_i) + (\mathbf{B} z)_o . \]

The Gaussian acts on the full complex value (both quadratures), so the learned edge function is genuinely two-dimensional rather than a real function applied per part. The grid uses num_grid**2 centres laid out on a square in the complex plane (optionally learnable).

import torch
import complextorch as ctorch

# A 2-layer complex KAN: 4 -> 8 -> 3 complex features.
model = ctorch.models.CVKAN([4, 8, 3], num_grid=8)
x = torch.randn(16, 4, dtype=torch.cfloat)
y = model(x)
print(y.shape, y.dtype)

# A single edge-function layer can fit a complex map such as z -> z**2.
layer = ctorch.nn.CVKANLayer(1, 1, num_grid=8)

See CVKAN (arXiv:2502.02417).