Topology is concerned with studying properties of spaces that are preserved under continuous transformation. In particular, using notions from topology we can classify spaces according to certain global properties that are invariant under such transformations. While commonly considered a branch of pure mathematics, the use of topological ideas for data analysis has recently seen a surge of interest under the name "Topological Data Analysis".

Topological Data Analysis (TDA) is typically concerned with high-dimensional point cloud data. TDA aims to extract the "global shape" of this point cloud using computational tools, such as persistent homology, which aim to extract a global topological description of the point cloud. Stated differently, the whole dataset is treated as a single object, which we aim to characterize. This view contrasts somewhat with the standard perspective of unsupervised learning in which the objects of interest are the points (feature vectors of different objects) themselves, and we are interested in characterizing these objects relative to each other.

In this talk, we will provide a brief introduction to topological data analysis and its relation to unsupervised machine learning. We will showcase a few methods that aim to bridge the gap between these two seemingly different viewpoints, and discuss open directions and challenges in this context.