Topological Data Analysis

The methods of topological data analysis (TDA) have been built to reveal “the shape of data.” But, to date, when a structural feature of interest is identified, there is no rigorous statistical foundation to assess its significance through hypothesis testing or other familiar techniques.

``My analysis has revealed a large hole in this point cloud. But is it possible that a hole of this size might arise due to randomness alone?''

— Curious, but observant researcher

In on-going work with Veronica Ciocanel, we argue that, when studying certain structures in point clouds, a framework for hypothesis testing can indeed be constructed. However, unlike classical statistical applications, which rely on distributions like the standard Normal, $t$, chi-squared, and $F$ distributions, we find that the Gumbel distribution is fundamental for assessing significance in TDA.