Topological data analysis (TDA) is a novel, and relatively new approach to analyzing high-dimensional data sets. The underlying assumption is that the real life data has shape and knowing this shape offers an insight different from more conventional techniques. TDA uses results from algebraic topology and Morse theory to understand qualitative features of the data such as coarse-scale, global geometric features. These topology-inspired techniques analyze the data in a manner which is independent of choice of embedding and coordinates.
In this talk I will give a gentle introduction to two popular TDA techniques: persistent homology and the mapper algorithm. Persistent homology is a "hole-detection" formalism based on classical simplicial homology. It describes the topology using bar codes (an incarnation of quiver representation) which serves as a natural input to various machine learning algorithms. On the other hand the mapper algorithm uses Morse theory to describe complex relationship among various features of the data. I will explain these techniques using a lot of real-life examples.
