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Tagged barcodes for the topological analysis of gradient-like vector fields

Clemens Bannwart, Claudia Landi

TL;DR

It is shown how to apply this pipeline to the weighted and based Morse chain complex of a gradient-like Morse-Smale vector field on a compact Riemannian manifold in both the smooth and discrete settings.

Abstract

Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a factored chain complex, and encodes it as a barcode of tagged intervals (briefly, a tagged barcode). We show how to apply this pipeline to the weighted and based Morse chain complex of a gradient-like Morse-Smale vector field on a compact Riemannian manifold in both the smooth and discrete settings. Interestingly for computations, it turns out that there is an isometry between factored chain complexes endowed with the interleaving distance and their tagged barcodes endowed with the bottleneck distance. Concerning stability, we show that the map taking a generic enough gradient-like vector field to its barcode of tagged intervals is continuous. Finally, we prove that the tagged barcode of any such vector field can be approximated by the tagged barcode of a combinatorial version of it with arbitrary precision.

Tagged barcodes for the topological analysis of gradient-like vector fields

TL;DR

It is shown how to apply this pipeline to the weighted and based Morse chain complex of a gradient-like Morse-Smale vector field on a compact Riemannian manifold in both the smooth and discrete settings.

Abstract

Intending to introduce a method for the topological analysis of fields, we present a pipeline that takes as an input a weighted and based chain complex, produces a factored chain complex, and encodes it as a barcode of tagged intervals (briefly, a tagged barcode). We show how to apply this pipeline to the weighted and based Morse chain complex of a gradient-like Morse-Smale vector field on a compact Riemannian manifold in both the smooth and discrete settings. Interestingly for computations, it turns out that there is an isometry between factored chain complexes endowed with the interleaving distance and their tagged barcodes endowed with the bottleneck distance. Concerning stability, we show that the map taking a generic enough gradient-like vector field to its barcode of tagged intervals is continuous. Finally, we prove that the tagged barcode of any such vector field can be approximated by the tagged barcode of a combinatorial version of it with arbitrary precision.
Paper Structure (29 sections, 43 theorems, 79 equations, 3 figures, 1 algorithm)

This paper contains 29 sections, 43 theorems, 79 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Preliminaries
  2. Parametrized objects
  3. The generalized interleaving distance.
  4. The generalized bottleneck distance.
  5. Parametrized vector spaces.
  6. Decomposition of chain complexes.
  7. Dynamical systems
  8. Structural stability and Morse-Smale vector fields
  9. The smooth Morse complex
  10. Combinatorial vector fields and the combinatorial Morse complex
  11. Relating the smooth and combinatorial Morse complexes
  12. Barycentric subdivision.
  13. Factored chain complexes
  14. Interval functors in $\mathop{\mathrm{TEPCh}}\nolimits$
  15. Parametrized vector spaces induced from parametrized chain complexes
  16. ...and 14 more sections

Key Result

Theorem 1.4

Any tame parametrized vector space $V$ is isomorphic to a finite direct sum of interval functors in $\mathop{\mathrm{TPVect}}\nolimits$, i.e. there exists a unique finite multiset $\mathop{\mathrm{Bar}}\nolimits =\mathop{\mathrm{Bar}}\nolimits(V)\in \mathop{\mathrm{Mult}}\nolimits(\mathcal{I})$ of i The multiset $\mathop{\mathrm{Bar}}\nolimits$ is called the persistence barcode of $V$.

Figures (3)

  • Figure 1: Left: Gradient-like Morse-Smale vector field $v$ on the 2-sphere with two singular points $p,q$ of index 2, one singular point $s$ of index 1, and one singular point $x$ of index 0. The point $x$ is represented in the image by the whole boundary. Right: Visualization of the resulting tagged barcode $\mathop{\mathrm{tBar}}\nolimits(Y(v))$, as explained in \ref{['ex:tagged-barcode']}.
  • Figure 2: Left: Gradient-like Morse-Smale vector field on the 2-sphere with five sources, seven saddles, and four sinks. Right: The corresponding tagged barcode is shown in a persistence diagram like fashion as explained in Example \ref{['ex:tagged-barcode-complicated']}.
  • Figure 3: Different orderings of the pairs $(p,r)$ and $(p,s)$ yield different tagged barcodes for this vector field, as explained in \ref{['ex:different-barcodes']}.

Theorems & Definitions (108)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4: Structure theorem in $\mathop{\mathrm{TPVect}}\nolimits$
  • Theorem 1.5: Isometry Theorem in $\mathop{\mathrm{TPVect}}\nolimits$
  • Lemma 1.6
  • Lemma 1.7
  • proof
  • Theorem 1.8
  • Definition 1.9
  • ...and 98 more