BiKAN: Restoring Collapsed Basis of Binary Kolmogorov--Arnold Networks

Binarizing a polynomial Kolmogorov--Arnold Network (KAN) not only changes parameter precision, but also alters the function space available to each layer. When activations are restricted to {-1,+1}, all even powers reduce to 1 and all odd powers reduce to x, causing the elementwise polynomial basis to collapse to constant and first-order responses. We refer to this structural failure as Spatial Orthogonality Collapse. Our proposed BiKAN addresses this critical issue by augmenting each binary KAN layer with selected degree-2 Walsh characters. Fixed circular channel rolls generate pairwise parities, and learned binary projections mix them using the same XNOR--popcount operations as the remaining W1A1 paths. This restores explicit pairwise coordinates without learned routing or multiplier-based feature generation. Experiments on CIFAR-10 confirms that removing parity reduces accuracy by 1.23 points over five paired seeds (p=0.003), the gain increases as width decreases, and accuracy improves monotonically as more parity planes are added. At an equal \sim11.9M-parameter budget, parity outperforms conventional widening by 3.09 points (p<10^{-4}). At W1A1, BiKAN reaches 99.48%, 84.38%, and 55.81% on MNIST, CIFAR-10, and CIFAR-100, respectively. Post-route Zynq-7020 FPGA results show that the repair remains hardware-efficient; the convolutional design cuts DSP usage from 164 to 72 and estimated compute-core latency from 401 to 54.8 ms, while the power-of-two-aware dense design achieves zero-DSP inference with a 0.03-point accuracy loss. The BiKAN implementation is available at https://github.com/OSU-STARLAB/BiKAN.