  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.
