Dataset comparison using persistent homology morphisms

Abstract: Persistent homology summarizes geometrical information of data by means of a barcode. Given a pair of datasets, $X$ and $Y$, one might obtain their respective barcodes $B(X)$ and $B(Y)$. Thanks to stability results, if $X$ and $Y$ are similar enough one deduces that the barcodes $B(X)$ and $B(Y)$ are also close enough; however, the converse is not true in general. In this talk we consider the case when there is a known relation between $X$ and $Y$ encoded by a morphism between persistence modules. For example, this is the case when $Y$ is a finite subset of euclidean space and $X$ is a sample taken from $Y$. As in linear algebra, a morphism between persistence modules is understood by a choice of a pair of bases together with the associated matrix. I will explain how to use this matrix to get barcodes for images, kernels and cokernels. Additionally, I will explain how to compute an induced block function that relates the barcodes $B(X)$ and $B(Y)$. I will finish the talk revising some applications of this theory as well as future research directions.

machine learningmathematical physicscommutative algebraalgebraic geometryalgebraic topologycombinatoricsdifferential geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Machine Learning Seminar

Series comments: Online machine learning in pure mathematics seminar, typically held on Wednesday. This seminar takes place online via Zoom.

For recordings of past talks and copies of the speaker's slides, please visit the seminar homepage at: kasprzyk.work/seminars/ml.html