David Loiseaux (10/16/24): Multiparameter Persistence for Machine Learning

The main tool of Topological Data Analysis is Persistent Homology, which captures the persistence of topological features as a filtration parameter changes---typically, a geometrical scale. For many applications however, it is beneficial to simultaneously vary multiple filtration parameters, such as, scales, the dataset's sampling density, or simply take into account other intrinsic properties of the data. To that end, a generalization of this construction, called Multiparameter Persistent Homology, enables the use of multiple filters at once.

In this talk, I will present the recent advancements in multiparameter persistence aiming at making it practical for machine learning. To that end, I will introduce several descriptors; one is obtained by gluing one-dimensional slices, and takes the form of an interval decomposable module called an MMA decomposition, and the other is computed from the Möbius inversion of classical invariants called the signed barcode. Finally, I will discuss corresponding vectorization for these descriptors, and their differentiability properties.