• Corpus ID: 245123797

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 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… 
[PDF] Semantic Reader
2 Citations

Figures from this paper

2 Citations

Multiparameter persistence modules in the large scale

. A persistence module with m discrete parameters is a diagram of vector spaces indexed by the poset N m . If we are only interested in the large scale behavior of such a diagram, then we can…

On the bottleneck stability of rank decompositions of multi-parameter persistence modules

The signed barcode induced by the Betti numbers of the module relative to the so-called rank exact structure is proved to be bottleneck stable under signed matchings and a bottleneck stability result for hook-decomposable modules is proved.

References

SHOWING 1-10 OF 29 REFERENCES

Stable Invariants for Multidimensional Persistence

This paper illustrates the process of converting discrete invariants into stable ones via what is called hierarchical stabilization by constructing stable invariants for multi parameter persistence modules with respect to so called simple noise systems.

EFFECTIVE COMPUTATION OF RELATIVE HOMOLOGICAL INVARIANTS FOR FUNCTORS OVER POSETS

. We study and effectively compute relative Betti diagrams for functors indexed by finite posets, with values in finite dimensional vector spaces. In relative homological algebra, free functors are…

Signed Barcodes for Multi-Parameter Persistence via Rank Decompositions and Rank-Exact Resolutions

The signed barcode is introduced, a new visual representation of the global structure of the rank invariant of a multi-parameter persistence module or of a poset representation, and the theory behind these rank decompositions is developed.

Fast Minimal Presentations of Bi-graded Persistence Modules

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.

A Relative Theory of Interleavings

This paper provides a general theory where one maps to a poset that does admit interesting translations, such as the lattice of down sets, and then defines interleavings relative to this map and provides an approximation condition that in the setting of lattices gives rise to two possible pixelizations.

Persistent Homology for Virtual Screening

This paper introduces PH_VS (Persistent Homology for Virtual Screening), a new system for ligand-based screening using a topological technique known as multi-parameter persistent homology, and shows that this approach can match or exceed a reasonable estimate of current state of the art (including well-funded commercial tools), even with relatively little domain-specific tuning.

Invariants and Metrics for Multiparameter Persistent Homology

Metrics and Stabilization in One Parameter Persistence

This work believes topological persistence is fundamentally not about decomposition theorems but a central role is played by a choice of metrics and chooses a pseudometric between persistent vector spaces leads to stabilization of discrete invariants.

Computing Minimal Presentations and Betti Numbers of 2-Parameter Persistent Homology

An efficient algorithm is given, based on Koszul homology, which computes the bigraded Betti numbers without computing a presentation, with these same complexity bounds, which outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

Every 1D persistence module is a restriction of some indecomposable 2D persistence module

It is given a constructive proof that any 1D persistence module with finite support can be found as a restriction of some indecomposable 2D persistence Module with finiteSupport as line restrictions.