Symbolic regression (SR) poses a significant challenge for randomized search heuristics due to its reliance on the synthesis of expressions for input-output mappings. Although traditional genetic programming (GP) algorithms have achieved success in various domains, they exhibit limited performance when tree-based representations are used for SR. To address these limitations, we introduce a novel SR approach called Fourier Tree Growing (FTG) that draws insights from functional analysis. This new perspective enables us to perform optimization directly in a different space, thus avoiding intricate symbolic expressions. Our proposed algorithm exhibits significant performance improvements over traditional GP methods on a range of classical one-dimensional benchmarking problems. To identify and explain limiting factors of GP and FTG, we perform experiments on a large-scale polynomials benchmark with high-order polynomials up to degree 100. To the best of the authors' knowledge, this work represents the pioneering application of functional analysis in addressing SR problems. The superior performance of the proposed algorithm and insights into the limitations of GP open the way for further advancing GP for SR and related areas of explainable machine learning.

在函数分析的新视角下，我们提出了一种名为傅里叶树生长（FTG）的新颖符号回归方法，以解决遗传编程在树状表示用于符号回归时的性能限制问题，并在经典一维基准问题上取得了显著的性能改进，同时揭示了遗传编程在符号回归问题上的局限，并为进一步推动符号回归和可解释机器学习相关领域的发展指明了方向。

符号回归的函数分析方法