The field of topological data analysis attempts to use homologies that persist in multiple scales (="Persistent Homologies") as learnable features, but frankly

The field of topological data analysis attempts to use homologies that persist in multiple scales (="Persistent Homologies") as learnable features, but frankly nothing I saw from this field seemed to work particularly well. The one practical algorithm I've seen which uses topological arguments for justification is UMAP for dimensionality reduction, the core idea IMO* is very much like density estimation and I'm not sure the mathematical justification gives more insight. * The core idea IMO: while in high-dimensional space, absolute distances lose meaning, relative distances between neighboring points are good yard-sticks. This insight is also what powers DBSCAN and other such density-based algorithms. I don't think using the language of simplical complexes when explaining the algorithm simplifies or adds insight.