Clustering is a fundamental operation in many data science workflows. This talk will approach clustering from the mathematical point of view.
First we will review and compare different clustering methods (e.g. k-means and spectral clustering), with an emphasis on deciding when methods succeed or fail based on understanding their mathematical properties. We will analyze several examples.
This leads naturally to a more theoretical discussion about clustering algorithms. To this end, we will discuss Kleinberg’s impossibility theorem, namely his axioms and a sketch of the proof. After explaining some ideas from category theory, we will examine the functorial approach to clustering developed by Carlsson-Memoli. We will compare the functorial approach to Kleinberg’s axioms, and the role of density in the functorial approach will emerge.
One of the insights of Carlsson-Memoli is that clustering is the statistical counterpart to taking the connected components of a topological space. We conclude by discussing generalizations of clustering (persistent homology) suggested by this motto.
This talk will be mostly expository. The main hope is that practitioners will be able to better apply and reason about clustering algorithms.