Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networks

The Kolmogorov-Arnold representation theorem states that any continuous multivariable function can be exactly represented as a finite superposition of continuous single variable functions. Subsequent simplifications of this representation involve expressing these functions as parameterized sums of a smaller number of unique monotonic functions. These developments led to the proof of the universal approximation capabilities of multilayer perceptron networks with sigmoidal activations, forming the alternative theoretical direction of most modern neural networks. Kolmogorov-Arnold Networks (KANs) have been recently proposed as an alternative to multilayer perceptrons. KANs feature learnable nonlinear activations applied directly to input values, modeled as weighted sums of basis spline functions. This approach replaces the linear transformations and sigmoidal post-activations used in traditional perceptrons. Subsequent works have explored alternatives to spline-based activations. In this work, we propose a novel KAN variant by replacing both the inner and outer functions in the Kolmogorov-Arnold representation with weighted sinusoidal functions of learnable frequencies. Inspired by simplifications introduced by Lorentz and Sprecher, we fix the phases of the sinusoidal activations to linearly spaced constant values and provide a proof of its theoretical validity. We also conduct numerical experiments to evaluate its performance on a range of multivariable functions, comparing it with fixed-frequency Fourier transform methods and multilayer perceptrons (MLPs). We show that it outperforms the fixed-frequency Fourier transform and achieves comparable performance to MLPs.

SineKAN: Kolmogorov-Arnold Networks Using Sinusoidal Activation Functions

Recent work has established an alternative to traditional multi-layer perceptron neural networks in the form of Kolmogorov-Arnold Networks (KAN). The general KAN framework uses learnable activation functions on the edges of the computational graph followed by summation on nodes. The learnable edge activation functions in the original implementation are basis spline functions (B-Spline). Here, we present a model in which learnable grids of B-Spline activation functions can be replaced by grids of re-weighted sine functions. We show that this leads to better or comparable numerical performance to B-Spline KAN models on the MNIST benchmark, while also providing a substantial speed increase on the order of 4-9 times.