### Atomic Cycles and Finding Holes in Manifolds

It's been a while! Turns out being in a PhD program keeps you pretty busy. I'm gonna try to get back into research blogging, but we'll see how long that lasts. This post is going back to manifold learning , my research focus during my undergrad at the University of Michigan. Machine learning theoreticians and practitioners frequently make use of the manifold hypothesis , which asserts that high-dimensional data generally lie along lower-dimensional latent manifolds. Manifold learning makes heavy use of that assumption: it tries to find a coordinate chart of that manifold, towards producing an explicit low-dimensional representation of the data. One potential shortcoming of this strategy is that not all manifolds can be represented using a single coordinate chart, due to holes in the manifold. One can get around this by making cuts to the manifold or embedding pieces of the manifold as an atlas of charts , but to do so, one must identify the holes. Since the manifold is un