Applied & Computational Mathematics seminar: Automated Parameter Selection for Mapper through Metric Geometry

Speaker:  Enrique Alvarado, University of Texas at El Paso

Title:  Automated Parameter Selection for Mapper through Metric Geometry

Abstract:   Mapper is a construction from topological data analysis that turns a dataset hat_X into a graph, or more generally, a (simplicial) complex intended to summarize the large-scale structure of the dataset. Its construction is done with three additional ingredients: a lens function from hat_X to a lens space Y, a cover of Y, and a clustering algorithm on X. Mapper was originally introduced as a data-driven approximation to the Reeb graph (more generally, space) of , the space that our data hat_X was sampled from. Depending on the choices we pick for mapper's parameters, the resulting graph can vary wildly. Finding the "best" parameter's is often a matter of trial and error. In this talk we will look at a automated way of picking mapper's cover set, called G-mapper.

In the second half of the talk, we present recent results from an ongoing project with Bala Krishnamoorthy, where we use tools from metric geometry to study automated ways in which one can pick mapper's lens function. Our framework applies when the the underlying space X comes equipped with a pseudometric and the lens function maps takes values in an arbitrary metric space. Although metrics on mapper graphs have been studied, the distinguishing feature of our metric is that it is derived from a natural metric on the Reeb space. Under mild assumptions on the underlying metric spaces and lens function, the topology induced by this natural metric agrees with the quotient topology on the Reeb space. We will discuss additional advantages of this Reeb space metric and show how it can be approximated from a finite sample hat_X of X. We describe how it can be used to equip the Mapper complex with a metric and then show how they Gromov-Hausdorff convergence of the metric mapper complex to the metric Reeb space as the diameters of the covers uniformly go to zero.
 

Time:  Friday, October 2, 1:30pm – 2:30pm
Place:  Exploratory Hall, room 4106 or Zoom