The representation theorem of persistence revisited and generalized

Corbet, René; Kerber, Michael

doi:10.1007/s41468-018-0015-3

Cited by 11 publications

(9 citation statements)

References 33 publications

(49 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A1 the direct sums of vector spaces are endowed with additional structure that comes from identifications encoded by the structure morphisms: these direct sums are graded modules over certain monoid rings, and the finiteness condition is understood as being a condition on these modules. We refer the reader to [72] and references therein for more details.…”

Section: Conclusion and Further Directionsmentioning

confidence: 99%

A Topological Perspective on Weather Regimes

Strommen

Chantry

Dorrington

et al. 2021

Preprint

View full text Add to dashboard Cite

It has long been suggested that the mid-latitude atmospheric circulation possesses what has come to be known as `weather regimes', loosely categorised as regions of phase space with above-average density and/or extended persistence. Their existence and behaviour has been extensively studied in meteorology and climate science, due to their potential for drastically simplifying the complex and chaotic mid-latitude dynamics. Several well-known, simple non-linear dynamical systems have been used as toy-models of the atmosphere in order to understand and exemplify such regime behaviour. Nevertheless, no agreed-upon and clear-cut definition of a `regime' exists in the literature. We argue here for an approach which equates the existence of regimes in a dynamical system with the existence of non-trivial topological structure of the system's attractor. We show using persistent homology, an algorithmic tool in topological data analysis, that this approach is computationally tractable, practically informative, and identifies the relevant regime structure across a range of examples.

show abstract

Section: Conclusion and Further Directionsmentioning

confidence: 99%

A Topological Perspective on Weather Regimes

Strommen

Chantry

Dorrington

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…But, in fact, the module has more structure than its component vector spaces. Especially in our case, under suitable finiteness condition [22], a persistence (homology) module is indeed a finitely presented graded module over multivariate polynomial ring R = k[t 1 , • • • , t d ], which was first recognized by Carlsson et al [16,17] and Knudson [37] and further studied by Lesnick et al [38,40]. The graded module structure studied in algebraic geometry and commutative algebra [29,42] encodes a lot of information and compresses the space leading to an improved time complexity.…”

Section: Introductionmentioning

confidence: 84%

Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules

Dey¹,

Xin²

2019

Preprint

View full text Add to dashboard Cite

The classical persistence algorithm virtually computes the unique decomposition of a persistence module implicitly given by an input simplicial filtration. Based on matrix reduction, this algorithm is a cornerstone of the emergent area of topological data analysis. Its input is a simplicial filtration defined over the integers Z giving rise to a 1-parameter persistence module. It has been recognized that multi-parameter version of persistence modules given by simplicial filtrations over d-dimensional integer grids Z d is equally or perhaps more important in data science applications. However, in the multi-parameter setting, one of the main challenges is that topological summaries based on algebraic structure such as decompositions and bottleneck distances cannot be as efficiently computed as in the 1-parameter case because there is no known extension of the persistence algorithm to multi-parameter persistence modules. The Meataxe algorithm, a popular one known for computing such a decomposition runs in Õ(n 6(d+1) ) time where n is the size of the input filtration. We present for the first time a generalization of the persistence algorithm based on a generalized matrix reduction technique that runs in O(n 2ω+1 ) time for d = 2 and in O(n d(2ω+1) ) time for d > 2 where ω < 2.373 is the exponent for matrix multiplication. Various structural and computational results connecting the graded modules from commutative algebra to matrix reductions are established through the course.

show abstract

“…Persistence modules. A persistence module is a family (V p ) p∈Z 2 of K-vector spaces for each grade, together with maps f p→q : V p → V q for every pair of grades with p ≤ q which are functorial, that means, f p→p is the identity and f q→r • f p→q = f p→r for p ≤ q ≤ r. The term "module" comes from the fact that this object carries the algebraic structure of a Z 2graded module over the polynomial ring K[x, y] [10,15].…”

Section: Preliminariesmentioning

confidence: 99%

Fast Minimal Presentations of Bi-graded Persistence Modules

Kerber¹,

Rolle²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

Multi-parameter persistent homology is a recent branch of topological data analysis. In this area, data sets are investigated through the lens of homology with respect to two or more scale parameters. The high computational cost of many algorithms calls for a preprocessing step to reduce the input size. In general, a minimal presentation is the smallest possible representation of a persistence module. Lesnick and Wright [28] proposed recently an algorithm (the LW-algorithm) for computing minimal presentations based on matrix reduction.In this work, we propose, implement and benchmark several improvements over the LW-algorithm. Most notably, we propose the use of priority queues to avoid extensive scanning of the matrix columns, which constitutes the computational bottleneck in the LWalgorithm, and we combine their algorithm with ideas from the multi-parameter chunk algorithm by Fugacci and Kerber [21]. Our extensive experiments show that our algorithm outperforms the LW-algorithm and computes the minimal presentation for data sets with millions of simplices within a few seconds. Our software is publicly available.

show abstract

The representation theorem of persistence revisited and generalized

Cited by 11 publications

References 33 publications

A Topological Perspective on Weather Regimes

A Topological Perspective on Weather Regimes

Generalized Persistence Algorithm for Decomposing Multi-parameter Persistence Modules

Fast Minimal Presentations of Bi-graded Persistence Modules

Contact Info

Product

Resources

About