arXiv:2408.07906v1 Announce Type: cross Abstract: In this paper, we compare the performance of Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) networks on irregular or noisy functions. We control the number of parameters and the size of the training samples to ensure a fair comparison. For clarity, we categorize the functions into six types: regular functions, continuous functions with local non-differentiable points, functions with jump discontinuities, functions with singularities, functions with coherent oscillations, and noisy functions. Our experimental results indicate that KAN does not always perform best. For some types of functions, MLP outperforms or performs comparably to KAN. Furthermore, increasing the size of training samples can improve performance to some extent. When noise is added to functions, the irregular features are often obscured by the noise, making it challenging for both MLP and KAN to extract these features effectively. We hope these experiments provide valuable insights for future neural network research and encourage further investigations to overcome these challenges.
Introduction:
The performance of Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) networks on irregular or noisy functions is compared in this paper. By controlling the number of parameters and the size of training samples, a fair comparison is ensured. The functions are categorized into six types, including regular functions, continuous functions with local non-differentiable points, functions with jump discontinuities, functions with singularities, functions with coherent oscillations, and noisy functions. The experimental results reveal that KAN does not always outperform MLP, as in some cases, MLP performs better or comparably to KAN. Additionally, increasing the size of training samples shows some improvement in performance. However, when noise is introduced to functions, both MLP and KAN struggle to effectively extract the irregular features that are often obscured by the noise. These findings provide valuable insights for future neural network research and call for further investigations to overcome these challenges.
Exploring the Performance of Kolmogorov-Arnold Networks and Multi-Layer Perceptron Networks on Irregular or Noisy Functions
Neural networks have shown immense potential in solving complex problems across various domains. However, understanding their performance on irregular or noisy functions is crucial to improving their effectiveness. In this paper, we compare the performance of two popular neural network architectures – Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) networks – on different types of functions, exploring their strengths and weaknesses.
Categorizing Functions for Comparison
To ensure a fair comparison, we categorize the functions into six types:
- Regular functions: These are well-behaved functions without any irregular features.
- Continuous functions with local non-differentiable points: Functions that are continuous but have points where the derivative does not exist.
- Functions with jump discontinuities: Functions with abrupt changes or jumps in values.
- Functions with singularities: Functions that have points where they become infinitely large or undefined.
- Functions with coherent oscillations: These functions exhibit regular oscillations or periodic patterns.
- Noisy functions: Functions affected by random noise, making it challenging to extract underlying patterns.
Comparing KAN and MLP Performance
Our experimental results reveal that KAN does not always outperform MLP. The performance of both architectures varies depending on the type of function being analyzed.
Regular functions: Since regular functions do not have any irregularities, both KAN and MLP perform equally well.
Continuous functions with local non-differentiable points: MLP, with its ability to handle non-differentiability, often outperforms KAN in capturing these irregularities.
Functions with jump discontinuities: KAN excels in capturing sudden changes in functions, making it perform better than MLP in such cases.
Functions with singularities: KAN, designed to handle singularities, surpasses MLP in accurately representing these functions.
Functions with coherent oscillations: Both KAN and MLP excel at capturing periodic patterns, yielding comparable results.
Noisy functions: The presence of noise poses challenges for both architectures, making it difficult to effectively extract underlying features. Increasing the size of training samples can improve performance to some extent, but further investigations are required to overcome this limitation.
Insights for Future Research
These experiments shed light on the performance of KAN and MLP networks on different types of functions. While KAN proves advantageous in certain scenarios, MLP also demonstrates its effectiveness in others. Understanding these strengths and weaknesses can guide researchers in choosing the appropriate architecture for their specific problem domains.
Furthermore, the challenges posed by noisy functions call for innovative approaches to enhance the robustness of neural networks. Exploring techniques such as denoising autoencoders or incorporating noise reduction mechanisms during training could prove promising in overcoming these challenges.
By delving into the performance of neural networks on irregular or noisy functions, this study paves the way for future research in refining existing architectures and developing novel solutions to extract hidden patterns effectively.
The paper titled “Comparing Kolmogorov-Arnold Networks and Multi-Layer Perceptron Networks on Irregular or Noisy Functions” presents an analysis of the performance of two types of neural networks, namely Kolmogorov-Arnold Networks (KAN) and Multi-Layer Perceptron (MLP) networks, on various types of functions. The authors aim to provide insights into how these networks perform on irregular or noisy functions and highlight the challenges associated with extracting features from such functions.
To ensure a fair comparison, the authors control the number of parameters and the size of the training samples. They categorize the functions into six types: regular functions, continuous functions with local non-differentiable points, functions with jump discontinuities, functions with singularities, functions with coherent oscillations, and noisy functions.
The experimental results reveal that KAN does not always outperform MLP. In fact, for certain types of functions, MLP either outperforms or performs comparably to KAN. This finding suggests that the choice of network architecture should be carefully considered depending on the nature of the function being analyzed.
Additionally, the authors find that increasing the size of training samples can lead to improved performance to some extent. This observation highlights the importance of having sufficient data for training neural networks, especially when dealing with irregular or noisy functions.
One interesting challenge discussed in the paper is the effect of noise on the performance of both KAN and MLP. When noise is added to functions, the irregular features that the networks are meant to extract are often obscured, making it difficult for the networks to effectively capture these features. This limitation indicates the need for further research and development to overcome the challenges posed by noisy data.
Overall, this paper provides valuable insights into the performance of KAN and MLP networks on irregular or noisy functions. It emphasizes the importance of considering the specific characteristics of the function being analyzed and the potential impact of noise on the network’s performance. Future research should focus on developing techniques to improve the extraction of features from noisy functions and explore alternative network architectures that may better handle irregularities in data.
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