Expository Talk by Prof. Hideto Asashiba (Interval replacements of persistence modules)

Speaker: Prof. Hideto Asashiba (Shizuoka University/Kyoto University KUIAS, Japan)

Title: Interval replacements of persistence modules

Abstract: We define two notions. The first one is a compression system ξ for a finite poset P , which assigns to each interval subposet I an order-preserving map ξ_I : Q_I → P satisfying some conditions, where Q_I is a connected finite poset. An example is given by the total compression system that assigns to each I the inclusion of I into P . The second one is an I-rank of a persistence module M under ξ, the family of which is called the interval rank invariant of M under ξ. A compression system ξ makes it possible to define the interval replacement (also called the interval-decomposable approximation) not only for 2D persistence modules but also for any persistence modules over any finite poset. We will show that the forming of the interval replacement preserves the interval rank invariant, which is a stronger property than the preservation of the usual rank invariant. Moreover, to know explicitly what is preserved, we will give a formula of the I-rank of M under ξ in terms of the structure linear maps of M for any compression system ξ. This makes it possible to define an essential cover property for a compression system, and by using this notion we give a sufficient condition for the I-rank of M under ξ to coincide with that under the total compression system, the value of which is equal to the generalized rank invariant introduced by Kim–Memoli.
This is a joint work with Etienne Gauthier and Enhao Liu.

Time: 14:00-15:30, July 2, 2025

Location: L5D23: OIST campus, Lab 5, Floor D, Room L5D23

Zoom Link: Will be posted closer to the event!

This talk is part of the Thematic Program TDA PARTI: Topological Data Analysis, Persistence And Representation Theory Intertwined.