Kolmogorov-Arnold网络的正弦近似定理
Sinusoidal Approximation Theorem for Kolmogorov-Arnold Networks
AI总结：
本研究提出一种以可学习频率加权正弦函数替代Kolmogorov-Arnold表示中内外函数的新型KAN变体，固定相位为线性间隔常数值并证明其理论有效性，实验显示其性能优于固定频率傅里叶变换、与MLPs相当。
摘要：
Kolmogorov-Arnold表示定理指出，任意连续多变量函数都可精确表示为有限个连续单变量函数的叠加。该表示的后续简化工作涉及将这些函数表达为少量唯一单调函数的参数化和。这些研究进展推动了带sigmoid激活的多层感知器网络的通用逼近能力证明，构成了大多数现代神经网络的另一理论方向。\nKolmogorov-Arnold网络（KANs，柯尔莫哥洛夫-阿诺德网络）是近期被提出的多层感知器替代方案。KANs的特点是可学习的非线性激活直接作用于输入值，这些激活被建模为基样条函数的加权和。该方法替代了传统感知器中使用的线性变换和sigmoid后置激活。后续研究探索了基于样条的激活的替代方案。在本工作中，我们提出了一种新型KAN变体，将Kolmogorov-Arnold表示中的内函数和外函数都替换为带可学习频率的加权正弦函数。受Lorentz和Sprecher提出的简化方法启发，我们将正弦激活的相位固定为线性间隔的常数值，并给出了其理论有效性的证明。我们还开展了数值实验，在一系列多变量函数上评估其性能，将其与固定频率傅里叶变换方法以及多层感知器（MLPs）进行对比。结果表明，该方法的性能优于固定频率傅里叶变换，且能达到与MLPs相当的性能。
英文摘要：
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.
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