  This paper uses topological data analysis (TDA) tools and introduces a
data-driven clustering-based stock selection strategy tailored for sparse
portfolio construction. Our asset selection strategy exploits the topological
features of stock price movements to select a subset of topologically similar
(different) assets for a sparse index tracking (Markowitz) portfolio. We
introduce new distance measures, which serve as an input to the clustering
algorithm, on the space of persistence diagrams and landscapes that consider
the time component of a time series. We conduct an empirical analysis on the
S\&P index from 2009 to 2020, including a study on the COVID-19 data to
validate the robustness of our methodology. Our strategy to integrate TDA with
the clustering algorithm significantly enhanced the performance of sparse
portfolios across various performance measures in diverse market scenarios.
