Persistent Local Systems

Transcription

1 Persistent Local Systems Amit Patel Department of Mathematics Colorado State University joint work with Robert MacPherson School of Mathematics Institute for Advanced Study Abel Symposium on Topological Data Analysis June 5-7, 2018

2 Lays foundation for a theory of persistence in the setting of constructible maps to manifolds X M f Real Algebraic Maps Real Analytic Maps Proper Piecewise Linear Maps Structurally Stable Smooth Maps Involves new (co)sheaf theoretic methods

3 Point Cloud P R n What is the shape of this point cloud at all scales?

4 Grow Balls X(r) [ p2p B p (r)

5 Grow Bigger Balls X(r) [ p2p B p (r)

6 Filtration X(0),! X(r 1 ),! X(r 2 ),!,! X(1) =R n HX(0)! HX(r 1 )! HX(r 2 )!! HX(1) Using coefficients in a field k F :(R, apple)! Vect k

7 What Persists? F :(R, apple)! Vect k Given two values r apple s what persists along the entire interval [r, s]? Answer: the persistent homology group which is the image of the map F(r apple s) Edelsbrunner, Letcher, Zomorodian. Topological persistence and simplification. 2002

8 Property I of the PH Group F :(R, apple)! Vect k Consider two intervals of the real line: [p, s] [q, r] F(p) F(q) F(s) F(r) rank F(p apple s) apple rank F(q apple r)

9 Property I of the PH Group F :(R, apple)! Vect k Consider two intervals of the real line: [p, s] [q, r] Contravariant F(p) F(q) Covariant F(s) F(r) rank F(p apple s) apple rank F(q apple r)

10 Property II of the PH Group F, G :(R, apple)! Vect k - interleaved For every long enough interval [p, q] F(p) G(p + ) F(q) G(q ) rank F(p apple q) apple rank G(p + apple q )

11 Constructible Persistence Modules The persistence diagram can be defined solely from persistent homology groups Let S = {s 1,...,s n } be a finite set of real numbers A persistence module F is S-constructible if F(p) =0, for all p<s 1 for s i apple p apple q<s i+1, F(p apple q) is an isomorphism for s n apple p, F(p apple q) is an isomorphism s 1 s 2 s 3 s 4

12 Poset of Intervals Dgm Dgm poset of half-open intervals of the form [r, s) [r, 1) and

13 Persistence Diagram A persistence diagram is a map that is nonzero on finitely many intervals P : Dgm! Z

14 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

15 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

16 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

17 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

18 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

19 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

20 Construction of P. Diagram F :(R, apple)! Vect Möbius Inversion df : Dgm! Z P F : Dgm! Z

21 Persistence and the Möbius Inversion The Inclusion-Exclusion construction of the persistence diagram is by Cohen-Steiner, Edelsbrunner, and Harer. Stability of persistence diagrams Recognized as a Möbius Inversion leading to persistence diagrams for constructible persistence modules valued in small symmetric monoidal categories A. Patel. Generalized Persistence Diagrams There is even bottleneck stability in the case of an abelian category A. McCleary, A. Patel. Bottleneck Stability for Generalized Persistence Diagrams. arxiv See also T. Leinster. Notions of Möbius Inversion. 2012

22 X f Maps to Manifolds We don t have sublevel sets, but we have level sets or fibers of the map. M Leray invented sheaf theory to study the fibers of a map We believe the persistent homology group is the fundamental unit of persistence. For each open set of the map over U perturbations? U M we ask, what is the fiberwise homology that is stable to infinitesimally small Our answer is the persistent local system. It behaves like the persistent homology group.

23 What is a Local System? A local system of abelian groups over a manifold is: A representation of the fundamental groupoid L : Fund(M)! Ab A locally constant cosheaf L : Open(M)! Ab A locally constant sheaf L : Open op (M)! Ab

24 Examples of Local Systems f g Take fiberwise one-dimensional homology L f is the trivial local system with fiber Z L g is the Möbius local system with fiber Z

25 Fiberwise Stable Homology R 2 B f R 2

26 Cap Product Given a constructible map f : X! M Suppose the target is oriented For each point U M of p p 2 M and a small enough open neighborhood H +m X, X f 1 (U) H m X, X f 1 _ (U)! H f 1 (U) f? H m (M,M U) = H m (M) = Z

27 Fiberwise Stable Homology R 2 B H 2 X, X f 1 (U) f _ f? (O u ) R 2 H 0 f 1 (U) image is 0

28 Fiberwise Stable Homology H 2 X, X f 1 (U) f _ f? (O u ) H 0 f 1 (U) Image has rank 1

29 Bisheaf

30 Persistence Stack X Given a map M f to an oriented manifold we construct some structure F d where for each open set U M F d+m (U) Episheaf over U (some kind of sheaf over U) F d (U) Monocosheaf under U (some kind of cosheaf under U) The image is the persistent local system over U

31 Property I For each inclusion of open sets i : V U we have i? F d+m (U) F d+m (V ) i? F(U) F(V ) i? F d (U) F d (V ) rank i? F(U) apple rank F(V )

32 Property II For an open set U M fix any > 0 and shrink U by U U Shrink

33 Property II X then there is > 0 such that for any second constructible map where Dist(f,g) < M g i? F d+m (U) G d+m (U ) i? F(U) i : U U G(U ) i? F d (U) G m (U ) rank i? F(U) apple rank G(U )

34 Ongoing Work Given a constructible map, we create a persistence stack. For each open set of the target space, we have a map from an episheaf to a monocosheaf. The image of this map is the persistent local system over this open set. The persistent local system behaves like the persistent homology group What is the persistence diagram of a persistence stack?

35 Thank You This research was supported by NSF grant CCF