I find the topic intriguing, but can someone who is well versed with <i>both</i> topology and machine learning comment on what is the key innovation here?<p>On first glance, the methods here seem a lot like the toolbox of dimensionality reduction techniques (PCA, spectral embedding, or more general manifold learning, etc.) from machine learning literature.<p>What specific insights from the field of topology have helped further our understanding of data that we missed earlier?