The Shift-Dimension of Multipersistence Modules

TL;DR

This paper introduces the shift-dimension as a new algebraic invariant for multipersistence modules, providing computational tools and applications in machine learning through multipersistence contours.

Contribution

It defines the shift-dimension, explores its properties, and develops algorithms for its computation, especially in the bivariate case, with applications to feature mapping.

Findings

Defined the shift-dimension for multipersistence modules

Developed a fast algorithm for bivariate interval modules

Constructed multipersistence contours for machine learning

Abstract

We present the shift-dimension of multipersistence modules and investigate its algebraic properties. This gives rise to a new invariant of multigraded modules over the multivariate polynomial ring arising from the hierarchical stabilization of the zeroth total multigraded Betti number. We give a fast algorithm for the computation of the shift-dimension of interval modules in the bivariate case. We construct multipersistence contours that are parameterized by multivariate functions and hence provide a large class of feature maps for machine learning tasks.