Reduced Effectiveness of Kolmogorov-Arnold Networks on Functions with Noise

Reduced Effectiveness of Kolmogorov-Arnold Networks on Functions with Noise

20 Jul 2024 | Haoran Shen, Chen Zeng, Jiahui Wang, and Qiao Wang
This paper investigates the reduced effectiveness of Kolmogorov-Arnold Networks (KANs) when subjected to noise. The authors propose two strategies to mitigate the impact of noise: kernel filtering and oversampling. Kernel filtering involves using diffusion maps or Gaussian-like kernels to pre-filter noisy data before training KANs. Oversampling increases the training dataset size, which helps KANs better approximate the original function in noisy environments. The study shows that adding independent and identically distributed (i.i.d.) noise with a fixed signal-to-noise ratio (SNR) significantly degrades KAN performance. However, increasing the training data size by a factor of $ r_s $ leads to a test loss (RMSE) that asymptotically decreases as $ r^{-\frac{1}{2}} $ as $ r \to +\infty $. This indicates that oversampling can improve KAN performance in noisy conditions. Kernel filtering is effective in low SNR scenarios but becomes less effective as SNR increases. The optimal filter parameter $ \sigma $, which controls the kernel width, is nonlinearly dependent on SNR, making it challenging to determine. Increasing the training dataset size is more effective than filtering in real-world situations, as it reduces test loss and enhances filtering performance. Combining oversampling and kernel filtering does not consistently improve performance, as kernel filtering can disrupt the effectiveness of oversampling. The study concludes that while both strategies can reduce noise effects, the overall performance of KANs remains limited due to the high data requirements of oversampling. Therefore, KANs must overcome the challenges posed by noise interference to be more robust in noisy environments.This paper investigates the reduced effectiveness of Kolmogorov-Arnold Networks (KANs) when subjected to noise. The authors propose two strategies to mitigate the impact of noise: kernel filtering and oversampling. Kernel filtering involves using diffusion maps or Gaussian-like kernels to pre-filter noisy data before training KANs. Oversampling increases the training dataset size, which helps KANs better approximate the original function in noisy environments. The study shows that adding independent and identically distributed (i.i.d.) noise with a fixed signal-to-noise ratio (SNR) significantly degrades KAN performance. However, increasing the training data size by a factor of $ r_s $ leads to a test loss (RMSE) that asymptotically decreases as $ r^{-\frac{1}{2}} $ as $ r \to +\infty $. This indicates that oversampling can improve KAN performance in noisy conditions. Kernel filtering is effective in low SNR scenarios but becomes less effective as SNR increases. The optimal filter parameter $ \sigma $, which controls the kernel width, is nonlinearly dependent on SNR, making it challenging to determine. Increasing the training dataset size is more effective than filtering in real-world situations, as it reduces test loss and enhances filtering performance. Combining oversampling and kernel filtering does not consistently improve performance, as kernel filtering can disrupt the effectiveness of oversampling. The study concludes that while both strategies can reduce noise effects, the overall performance of KANs remains limited due to the high data requirements of oversampling. Therefore, KANs must overcome the challenges posed by noise interference to be more robust in noisy environments.
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