The persistence space in multidimensional persistent homology A. - - PowerPoint PPT Presentation

The persistence space in multidimensional persistent homology

• A. Cerri1,2
• C. Landi2,3

1CNR – IMATI, Genova 2DISMI – Universit`

a di Modena e Reggio Emilia

3ARCES – Universit`

a di Bologna

DGCI 2013, March 20-22, Sevilla, Spain

Overview

• Persistent homology is a geometrical/topological approach to the

analysis of data;

2 of 4

Overview

• Persistent homology is a geometrical/topological approach to the

analysis of data;

• Persistence diagrams provide a qualitative, multi-scale

description of data w.r.t. properties modeled by scalar functions;

2 of 4

Overview

• Persistent homology is a geometrical/topological approach to the

analysis of data;

• Persistence diagrams provide a qualitative, multi-scale

description of data w.r.t. properties modeled by scalar functions;

• We introduce the persistence space of a vector-valued function

to generalize the concept of persistence diagram.

2 of 4

Overview: Persistence diagrams

• Modeling data as a pair (X,f ), with f : X → R...

0.4

• 0.2

0.4 0.6

• 0.6
• 0.4
• 0.2

0.2 0.2

• 0.6
• 0.8
• 0.8

0.4 0.6 0.2 0.4 0.8

• 0.2

0.2 0.6 0.6

• 0.2
• 0.4
• 0.4

0.4 0.8

(X,f ) Dgm(f ) Dgm(g) (X,g)

• Persistence diagrams provide a compact representation of data...

3 of 4

Overview: Persistence diagrams

• Modeling data as a pair (X,f ), with f : X → R...

0.4

• 0.2

0.4 0.6

• 0.6
• 0.4
• 0.2

0.2 0.2

• 0.6
• 0.8
• 0.8

0.4 0.6 0.4

• 0.2

0.4 0.6

• 0.6
• 0.4
• 0.2

0.2 0.2

• 0.6
• 0.8
• 0.8

0.4 0.6

(X,f ) Dgm(f ) Dgm(f ′) (X,f ′)

• ... which can be stably compared.

3 of 4

Motivations and contributions

• Using functions valued in Rn → multi-parameter information

4 of 4

Motivations and contributions

• Using functions valued in Rn → multi-parameter information
• Our contribution is threefold:

4 of 4

Motivations and contributions

• Using functions valued in Rn → multi-parameter information
• Our contribution is threefold:
• 1. Theoretical foundations of persistence spaces;

4 of 4

Motivations and contributions

• Using functions valued in Rn → multi-parameter information
• Our contribution is threefold:
• 1. Theoretical foundations of persistence spaces;
• 2. Method to visualize persistence spaces (they live in R2n);

−20 −10 10 −10 10 20 30 40 −15 −10 −5 5 10

4 of 4

Motivations and contributions

• Using functions valued in Rn → multi-parameter information
• Our contribution is threefold:
• 1. Theoretical foundations of persistence spaces;
• 2. Method to visualize persistence spaces (they live in R2n);
• 3. We show that persistence spaces can be stably compared.

(X,f1) (X,f2) Spc(f ) Spc(g) (X,g1) (X,g2) f = (f1,f2) : X → R2 g = (g1,g2) : X → R2

4 of 4