Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

DTM-based Filtrations

Hirokazu Anai, Frédéric Chazal, Marc Glisse, Yuichi Ike, Hiroya Inakoshi, Raphaël Tinarrage, Yuhei Umeda

TL;DR

A new family of filtrations is introduced, built on top of point clouds in the Euclidean space which are more robust to noise and outliers and relies on the notion of distance-to-measure functions.

Abstract

Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.

DTM-based Filtrations

TL;DR

A new family of filtrations is introduced, built on top of point clouds in the Euclidean space which are more robust to noise and outliers and relies on the notion of distance-to-measure functions.

Abstract

Despite strong stability properties, the persistent homology of filtrations classically used in Topological Data Analysis, such as, e.g. the Cech or Vietoris-Rips filtrations, are very sensitive to the presence of outliers in the data from which they are computed. In this paper, we introduce and study a new family of filtrations, the DTM-filtrations, built on top of point clouds in the Euclidean space which are more robust to noise and outliers. The approach adopted in this work relies on the notion of distance-to-measure functions, and extends some previous work on the approximation of such functions.

Paper Structure

This paper contains 24 sections, 20 theorems, 68 equations, 11 figures.

Table of Contents

  1. Abstract.
  2. Numerical experiments.
  3. Introduction
  4. Contributions.
  5. Practical motivations.
  6. Organisation of the paper.
  7. Acknowledgements.
  8. Filtrations and interleaving distance
  9. Filtrations of sets and simplicial complexes.
  10. Persistence modules.
  11. Relation between filtrations and persistence modules.
  12. Weighted Cech filtrations
  13. Definition
  14. Stability
  15. Weighted Vietoris-Rips filtrations
  16. ...and 9 more sections

Key Result

Proposition 3.1

If $X \subseteq E$ is finite and $f$ is any function, then $\mathbb V[X,f]$ is a pointwise finite-dimensional persistence module. More generally, if $X$ is a bounded subset of $E$ and $f$ is any function, then $\mathbb V[X,f]$ is q-tame.

Figures (11)

  • Figure 1: A synthetic example comparing Vietoris-Rips filtration to DTM filtration. The first row represents two time series with very different behavior and their embedding into $\mathbb R^3$ (here a series $(x_1, x_2,\dots, x_{n})$ is converted in the 3D point cloud $\{ (x_1,x_2,x_3), (x_2,x_3,x_4), \dots, (x_{n-2},x_{n-1},x_{n}) \}$). The second row shows the persistence diagrams of the Vietoris-Rips filtration built on top of the two point clouds (red and green points represent respectively the $0$-dimensional $1$-dimensional diagrams); one observes that the diagrams do not clearly 'detect' the different behavior of the time series. The third row shows the persistence diagrams of the DTM filtration built on top of the two point clouds; a red point clearly appears away from the diagonal in the second diagram that highlights the rapid shift occurring in the second time series.
  • Figure 2: Graph of $t \mapsto r_x(t)$ for $f(x) = 1$ and several values of $p$.
  • Figure 3: The sets $V^{t}[X,f]$ for $t=0{,}2$ and several values of $p$.
  • Figure 4: The sets $V^{t}[\Gamma, 0]$, $V^{t}[X, 0]$ and $W^{t}[X]$ for $p=1$, $m=0{,}1$ and $t=0{,}3$.
  • Figure 5: Persistence diagrams of some simplicial filtrations. Points in red (resp. green) represent the persistent homology in dimension $0$ (resp. $1$).
  • ...and 6 more figures

Theorems & Definitions (44)

  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Definition 3.2
  • Proposition 3.4
  • proof
  • ...and 34 more