Metrics on diagrams and persistent homology Peter Bubenik - PowerPoint PPT Presentation

Background Categorical ph Relative ph More structure Metrics on diagrams and persistent homology Peter Bubenik Department of Mathematics Cleveland State University p.bubenik@csuohio.edu http://academic.csuohio.edu/bubenik_p/ July 18, 2013 joint work with Vin de Silva and Jonathan A. Scott funded by AFOSR FA9550-13-1-0115 Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Topological data analysis From data to topology: 1 Start with a finite set of points in some metric space. 2 Apply a geometric construction (e.g. ˇ Cech, Rips) to obtain a nested sequence of simplicial complexes. Peter Bubenik Metrics on diagrams and persistent homology

� � � � � � R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Persistent homology We have a nested sequence of simplicial complexes, · · · ( ∗ ) K 0 K 1 K n . Apply simplicial homology, · · · H ( K 0 ) H ( K 1 ) H ( K n ) . ( H ∗ ) Peter Bubenik Metrics on diagrams and persistent homology

� � � � � � � � � R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Persistent homology We have a nested sequence of simplicial complexes, · · · ( ∗ ) K 0 K 1 K n . Apply simplicial homology, · · · H ( K 0 ) H ( K 1 ) H ( K n ) . ( H ∗ ) The shape of these diagrams is given by the category n , · · · n . 0 1 Then ( ∗ ) is equivalent to n K − → Simp , and ( H ∗ ) is equivalent to n K → Simp H − − → Vect F . Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Persistent homology Another paradigm: 1 Start with a function f : X → R . For each a ∈ R , consider f − 1 (( −∞ , a ]). 2 This gives us a diagram F : ( R , ≤ ) → Top . 3 Composing with singular homology we have, ( R , ≤ ) F → Top H − − → Vect F . Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Multidimensional persistent homology 1 Start with a function f : X → R n . For each a ∈ R n , consider f − 1 ( R n ≤ a ). 2 This gives us a diagram F : ( R n , ≤ ) → Top . 3 Composing with singular homology we have, ( R n , ≤ ) F → Top H − − → Vect F . Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Levelset persistent homology 1 Start with a function f : X → R . For each interval I ⊆ R , consider f − 1 ( I ). 2 This gives us a diagram F : Intervals → Top . 3 Composing with singular homology we have, Intervals F → Top H − − → Vect F . Peter Bubenik Metrics on diagrams and persistent homology

R n S 1 Background Categorical ph Relative ph More structure TDA R levelset Angle-valued persistent homology 1 Start with a function f : X → S 1 . For each arc A ⊆ S 1 , consider f − 1 ( A ). 2 This gives us a diagram F : Arcs → Top . 3 Composing with singular homology we have, Arcs F → Top H − − → Vect F . Peter Bubenik Metrics on diagrams and persistent homology

Background Categorical ph Relative ph More structure Results Interleaving Payoff Goals We will use category theory to give a unified treatment of each of the above flavors of persistent homology. Why? Give simpler, common proofs to some basic persistence results. Remove assumptions. Apply persistence to functions, f : X → ( M , d ). Allow homology to be replaced with other functors. Provide a framework for new applications. Specific goal: Interpret and prove stability in this setting. Peter Bubenik Metrics on diagrams and persistent homology

Background Categorical ph Relative ph More structure Results Interleaving Payoff Terminology In this talk a metric will be allowed to have d ( x , y ) = ∞ for x � = y , and have d ( x , y ) = 0 for x � = y . That is, it is an extended pseudometric. Example: The Hausdorff distance on the set of all subspaces of R . Peter Bubenik Metrics on diagrams and persistent homology

Background Categorical ph Relative ph More structure Results Interleaving Payoff Unified framework Generalized persistence module, P F → C H − − → A . Here, The indexing category P is a poset together with some notion of distance; C is some category; A is some abelian category (e.g. Vect F , R - mod ); F and H are arbitrary functors. Peter Bubenik Metrics on diagrams and persistent homology