用非线性分解方法提升股市收益分析，危机时期表现更优
Nonlinear Factor Decomposition via Kolmogorov-Arnold Networks: A Spectral Approach to Asset Return Analysis
- 用KAN构建非线性编码器，替代传统PCA的线性投影
- 在20只标普500股票上，重构R²达66.57%（传统PCA为62.99%）
- 适用于金融时间序列建模，尤其适合市场剧烈波动场景
KAN-PCA是一种自编码器，采用KAN作为编码器、线性映射作为解码器。它通过在每条边使用学习的B样条函数，将经典PCA推广为非线性形式。其动机在于：当市场危机导致资产间相关性剧烈变化时，线性假设失效，经典PCA效率下降。我们证明，若强制样条激活函数为线性，则KAN-PCA结果与经典PCA完全一致，表明后者是前者特例。在2015至2024年20只标普500成分股数据上的实验表明，使用相同3个因子时，KAN-PCA的重构决定系数R²达到66.57%，高于经典PCA的62.99%；且在修正训练过程中的数据泄漏问题后，其预测性能与经典PCA相当。
原文摘要 · Abstract (English)
KAN-PCA is an autoencoder that uses a KAN as encoder and a linear map as decoder. It generalizes classical PCA by replacing linear projections with learned B-spline functions on each edge. The motivation is to capture more variance than classical PCA, which becomes inefficient during market crises when the linear assumption breaks down and correlations between assets change dramatically. We prove that if the spline activations are forced to be linear, KAN-PCA yields exactly the same results as classical PCA, establishing PCA as a special case. Experiments on 20 S&P 500 stocks (2015-2024) show that KAN-PCA achieves a reconstruction R^2 of 66.57%, compared to 62.99% for classical PCA with the same 3 factors, while matching PCA out-of-sample after correcting for data leakage in the training procedure.
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