Persistent 1-Cycles: Definition, Computation, and Its Application

Dey, Tamal K.; Hou, Tao; Mandal, Sujit

doi:10.1007/978-3-030-10828-1_10

Cited by 15 publications

(16 citation statements)

References 27 publications

(36 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, we are interested in representatives that satisfy some minimality condition: for holes we compute optimal representatives [54] using the software Persloop [55], while for components, we use representatives to find all the points in a component. We note that finding optimal representatives for holes is a challenging problem; the software Persloop implements an algorithm that gives a heuristic approximation for 1-cycles in 3D, but which might fail to give meaningful 1-cycles on higher dimensional data sets.…”

Section: Optimal Representatives Of Cyclesmentioning

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: Optimal Representatives Of Cyclesmentioning

confidence: 99%

A Topological Perspective on Weather Regimes

Strommen

Chantry

Dorrington

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The homology classes that correspond to each of these representative cycles are highlighted in Figure 10C . We remark that instead of the representative cycles produced by Dionysus one may want to use (approximate) shortest cycle representatives ( Jeff Erickson, 2012 ; Dey et al, 2018 ; Obayashi, 2018 ; Day et al, 2019 ).…”

Section: Resultsmentioning

confidence: 99%

Topological Data Analysis of C. elegans Locomotion and Behavior

Thomas¹,

Bates²,

Elchesen³

et al. 2021

Front. Artif. Intell.

View full text Add to dashboard Cite

We apply topological data analysis to the behavior of C. elegans, a widely studied model organism in biology. In particular, we use topology to produce a quantitative summary of complex behavior which may be applied to high-throughput data. Our methods allow us to distinguish and classify videos from various environmental conditions and we analyze the trade-off between accuracy and interpretability. Furthermore, we present a novel technique for visualizing the outputs of our analysis in terms of the input. Specifically, we use representative cycles of persistent homology to produce synthetic videos of stereotypical behaviors.

show abstract

“…Dey et al (2010) propose a polynomial-time algorithm that computes a set of loops from a VR complex of the given data whose lengths approximate those of a shortest basis of the one dimensional homology group H 1 . In Dey et al (2019), show that finding optimal (minimal) persistent 1-cycles is NP-hard and then propose a polynomial time algorithm to find an alternative set of meaningful cycle representatives. This alternative set of representatives is not always optimal but still meaningful because each persistent 1-cycle is a sum of shortest cycles born at different indices.…”

Section: Frontiers Inmentioning

confidence: 99%

“…finding a minimal length or bounding area/ volume using an appropriate metric. The algorithmic problem of selecting such optimal representatives is currently an active area of research (Chen and Freedman, 2010a;Dey et al, 2011;Wu et al, 2017;Obayashi, 2018;Dey et al, 2019).…”

Section: Introductionmentioning

confidence: 99%

Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User’s Guide

Thompson

Henselman-Petrusek

et al. 2021

Front. Artif. Intell.

View full text Add to dashboard Cite

Cycle representatives of persistent homology classes can be used to provide descriptions of topological features in data. However, the non-uniqueness of these representatives creates ambiguity and can lead to many different interpretations of the same set of classes. One approach to solving this problem is to optimize the choice of representative against some measure that is meaningful in the context of the data. In this work, we provide a study of the effectiveness and computational cost of several ℓ1 minimization optimization procedures for constructing homological cycle bases for persistent homology with rational coefficients in dimension one, including uniform-weighted and length-weighted edge-loss algorithms as well as uniform-weighted and area-weighted triangle-loss algorithms. We conduct these optimizations via standard linear programming methods, applying general-purpose solvers to optimize over column bases of simplicial boundary matrices. Our key findings are: 1) optimization is effective in reducing the size of cycle representatives, though the extent of the reduction varies according to the dimension and distribution of the underlying data, 2) the computational cost of optimizing a basis of cycle representatives exceeds the cost of computing such a basis, in most data sets we consider, 3) the choice of linear solvers matters a lot to the computation time of optimizing cycles, 4) the computation time of solving an integer program is not significantly longer than the computation time of solving a linear program for most of the cycle representatives, using the Gurobi linear solver, 5) strikingly, whether requiring integer solutions or not, we almost always obtain a solution with the same cost and almost all solutions found have entries in {‐1,0,1} and therefore, are also solutions to a restricted ℓ0 optimization problem, and 6) we obtain qualitatively different results for generators in Erdős-Rényi random clique complexes than in real-world and synthetic point cloud data.

show abstract

Persistent 1-Cycles: Definition, Computation, and Its Application

Cited by 15 publications

References 27 publications

A Topological Perspective on Weather Regimes

A Topological Perspective on Weather Regimes

Topological Data Analysis of C. elegans Locomotion and Behavior

Minimal Cycle Representatives in Persistent Homology Using Linear Programming: An Empirical Study With User’s Guide

Contact Info

Product

Resources

About