FSMLP: Modelling Channel Dependencies With Simplex Theory Based Multi-Layer Perceptions In Frequency Domain

Time series forecasting (TSF) plays a crucial role in various domains, including web data analysis, energy consumption prediction, and weather forecasting. While Multi-Layer Perceptrons (MLPs) are lightweight and effective for capturing temporal dependencies, they are prone to overfitting when used to model inter-channel dependencies. In this paper, we investigate the overfitting problem in channel-wise MLPs using Rademacher complexity theory, revealing that extreme values in time series data exacerbate this issue. To mitigate this issue, we introduce a novel Simplex-MLP layer, where the weights are constrained within a standard simplex. This strategy encourages the model to learn simpler patterns and thereby reducing overfitting to extreme values. Based on the Simplex-MLP layer, we propose a novel \textbf{F}requency \textbf{S}implex \textbf{MLP} (FSMLP) framework for time series forecasting, comprising of two kinds of modules: \textbf{S}implex \textbf{C}hannel-\textbf{W}ise MLP (SCWM) and \textbf{F}requency \textbf{T}emporal \textbf{M}LP (FTM). The SCWM effectively leverages the Simplex-MLP to capture inter-channel dependencies, while the FTM is a simple yet efficient temporal MLP designed to extract temporal information from the data. Our theoretical analysis shows that the upper bound of the Rademacher Complexity for Simplex-MLP is lower than that for standard MLPs. Moreover, we validate our proposed method on seven benchmark datasets, demonstrating significant improvements in forecasting accuracy and efficiency, while also showcasing superior scalability. Additionally, we demonstrate that Simplex-MLP can improve other methods that use channel-wise MLP to achieve less overfitting and improved performance. Code are available \href{https://github.com/FMLYD/FSMLP}{\textcolor{red}{here}}.