In this research, we introduce a novel methodology for the index tracking problem with sparse portfolios by leveraging topological data analysis (TDA). Utilizing persistence homology to measure the riskiness of assets, we introduce a topological method for data-driven learning of the parameters for regularization terms. Specifically, the Vietoris-Rips filtration method is utilized to capture the intricate topological features of asset movements, providing a robust framework for portfolio tracking. Our approach has the advantage of accommodating both $\ell_1$ and $\ell_2$ penalty terms without the requirement for expensive estimation procedures. We empirically validate the performance of our methodology against state-of-the-art sparse index tracking techniques, such as Elastic-Net and SLOPE, using a dataset that covers 23 years of S&P500 index and its constituent data. Our out-of-sample results show that this computationally efficient technique surpasses conventional methods across risk metrics, risk-adjusted performance, and trading expenses in varied market conditions. Furthermore, in turbulent markets, it not only maintains but also enhances tracking performance.

翻译：本研究引入了一种利用拓扑数据分析解决稀疏投资组合指数追踪问题的新方法。通过使用持续同调来度量资产的风险性，我们提出了一种拓扑方法，用于数据驱动地学习正则化项的参数。具体而言，利用Vietoris-Rips过滤方法捕捉资产运动的复杂拓扑特征，为投资组合追踪提供了稳健的框架。本方法的优势在于能够同时容纳$\ell_1$和$\ell_2$惩罚项，而无需昂贵的估计过程。我们使用涵盖23年标普500指数及其成分股数据的数据集，将所提方法的性能与Elastic-Net和SLOPE等先进稀疏指数追踪技术进行了实证验证。样本外结果表明，这种计算高效的方法在不同市场条件下的风险指标、风险调整绩效和交易费用方面均优于传统方法。此外，在市场动荡时期，该方法不仅能保持而且能提升追踪表现。