# The Shift-Dimension of Multipersistence Modules

@inproceedings{Chacholski2021TheSO, title={The Shift-Dimension of Multipersistence Modules}, author={Wojciech Chach'olski and Ren{\'e} Corbet and Anna-Laura Sattelberger}, year={2021} }

We introduce 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 present 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…

## References

SHOWING 1-10 OF 29 REFERENCES

The theory of multidimensional persistence

- MathematicsSCG '07
- 2007

This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.

The representation theorem of persistence revisited and generalized

- MathematicsJ. Appl. Comput. Topol.
- 2018

This work gives a more accurate statement of the original Representation Theorem and provides a complete and self-contained proof and generalizes the statement from the case of linear sequences of R- modules to R-modules indexed over more general monoids.

Multidimensional Persistence and Noise

- Mathematics, Computer ScienceFound. Comput. Math.
- 2017

The feature counting invariant is introduced and it is proved that assigning this invariant to compact tame functors is a 1-Lipschitz operation.

Metrics for Generalized Persistence Modules

- MathematicsFound. Comput. Math.
- 2015

This work considers the question of defining interleaving metrics on generalized persistence modules over arbitrary preordered sets, and introduces a distinction between ‘soft’ and ‘hard’ stability theorems.

Stratifying Multiparameter Persistent Homology

- MathematicsSIAM J. Appl. Algebra Geom.
- 2019

This work proposes multigrade Hilbert series, multigraded associated primes and local cohomology as invariants for studying multiparameter persistent homology, which generalize in a suitable sense the invariant for the one-parameter case.

Decomposition of persistence modules

- Mathematics
- 2018

We show that a pointwise finite-dimensional persistence module indexed over a small category decomposes into a direct sum of indecomposables with local endomorphism rings. As an application of this…

Persistence Modules on Commutative Ladders of Finite Type

- MathematicsDiscret. Comput. Geom.
- 2016

It is proved that the commutative ladders of length less than 5 are representation-finite and explicitly show their Auslander–Reiten quivers.

Fast Minimal Presentations of Bi-graded Persistence Modules

- Computer Science, MathematicsALENEX
- 2021

This work proposes the use of priority queues to avoid extensive scanning of the matrix columns, which constitutes the computational bottleneck in the LW-algorithm, and combines their algorithm with ideas from the multi-parameter chunk algorithm by Fugacci and Kerber.