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

The persistence space in multidimensional persistent homology A. Cerri 1 , 2 C. Landi 2 , 3 1 CNR – IMATI, Genova 2 DISMI – Universit` a di Modena e Reggio Emilia 3 ARCES – 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 ... 0.6 0.8 0.4 0.6 0.2 0.4 0 0.2 -0.2 0 0.4 -0.2 -0.6 -0.4 -0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 Dgm ( f ) Dgm ( g ) ( X , f ) ( 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 ... 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -0.2 -0.2 0.4 0.4 -0.6 -0.6 -0.8 -0.8 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Dgm ( f ) Dgm ( f ′ ) ( X , f ) ( X , f ′ ) • ... which can be stably compared. 3 of 4

Motivations and contributions • Using functions valued in R n → multi-parameter information 4 of 4

Motivations and contributions • Using functions valued in R n → multi-parameter information • Our contribution is threefold: 4 of 4

Motivations and contributions • Using functions valued in R n → multi-parameter information • Our contribution is threefold: 1. Theoretical foundations of persistence spaces; 4 of 4

Motivations and contributions • Using functions valued in R n → multi-parameter information • Our contribution is threefold: 1. Theoretical foundations of persistence spaces; 2. Method to visualize persistence spaces (they live in R 2 n ); 10 5 0 −5 −10 −20 −15 −10 −10 0 10 0 20 30 10 40 4 of 4

Motivations and contributions • Using functions valued in R n → multi-parameter information • Our contribution is threefold: 1. Theoretical foundations of persistence spaces; 2. Method to visualize persistence spaces (they live in R 2 n ); 3. We show that persistence spaces can be stably compared . Spc ( f ) Spc ( g ) ( X , f 1 ) ( X , f 2 ) ( X , g 1 ) ( X , g 2 ) f = ( f 1 , f 2 ) : X → R 2 g = ( g 1 , g 2 ) : X → R 2 4 of 4