Sheaf-Theoretic Stratification Learning

Transcription

1 Symposium on Computational Geometry 2018 Sheaf-Theoretic Stratification Learning Adam Brown and Bei Wang University of Utah 1 / 27

2 Motivation Manifold Learning Infer structure from data 2 / 27

3 Motivation Manifold Learning Infer structure from data manifold learning 2 / 27

4 Motivation Manifold Learning Infer structure from data manifold learning Study the structure of singularities in data stratification learning 2 / 27

5 Motivation Topological Stratifications = 3 / 27

6 Motivation Topological Stratifications =

7 Motivation Topological Stratifications = =

8 Motivation Topological Stratifications = = =

9 Motivation Topological Stratifications = = = = 4 / 27

10 Stratifications Topological Stratifications Definition A topological stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that X i X i 1 is an i-dimensional (topological) manifold. 5 / 27

11 Stratifications Topological Stratifications Definition A topological stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that X i X i 1 is an i-dimensional (topological) manifold. + conditions on how the strata fit together. 5 / 27

12 Stratifications Topological Stratifications Definition A topological stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that X i X i 1 is an i-dimensional (topological) manifold. + conditions on how the strata fit together. Notation: We call X i X i 1 a stratum, and denote it by S i. 5 / 27

13 Stratifications Topological Stratifications Definition A topological stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that X i X i 1 is an i-dimensional (topological) manifold. + conditions on how the strata fit together. Notation: We call X i X i 1 a stratum, and denote it by S i. + visualize the stratification by X = S n S n 1 S 0 5 / 27

14 Computation Computing Topological Stratifications Problem: Compute (reasonable) topological stratifications of triangulated topological spaces. 6 / 27

15 Computation Computing Topological Stratifications Problem: Compute (reasonable) topological stratifications of triangulated topological spaces. I need to check if X i X i 1 is a manifold for each i. 6 / 27

16 Computation Computing Topological Stratifications Problem: Compute (reasonable) topological stratifications of triangulated topological spaces. I need to check if X i X i 1 is a manifold for each i. In practice, this is computationally difficult... 6 / 27

17 Computation Computing Topological Stratifications Problem: Compute (reasonable) topological stratifications of triangulated topological spaces. I need to check if X i X i 1 is a manifold for each i. In practice, this is computationally difficult... Revise our definition of stratification 6 / 27

18 Computation Computing Topological Stratifications Problem: Compute (reasonable) topological stratifications of triangulated topological spaces. I need to check if X i X i 1 is a manifold for each i. In practice, this is computationally difficult... Revise our definition of stratification Rourke-Sanderson [4], Bendich-Wang-Mukherjee [1], Nanda [3] 6 / 27

19 Results Results Our contributions are four-fold: 7 / 27

20 Results Results Our contributions are four-fold: 1 Definition of F-stratification 7 / 27

21 Results Results Our contributions are four-fold: 1 Definition of F-stratification 2 Existence and uniqueness results for certain F-stratifications 7 / 27

22 Results Results Our contributions are four-fold: 1 Definition of F-stratification 2 Existence and uniqueness results for certain F-stratifications 3 Algorithm for computation of F-stratifications 7 / 27

23 Results Results Our contributions are four-fold: 1 Definition of F-stratification 2 Existence and uniqueness results for certain F-stratifications 3 Algorithm for computation of F-stratifications 4 Application to local homology stratifications 7 / 27

24 Results Results Our contributions are four-fold: 1 Definition of F-stratification 2 Existence and uniqueness results for certain F-stratifications 3 Algorithm for computation of F-stratifications 4 Application to local homology stratifications We envision that our abstraction could give rise to a larger class of computable stratifications beyond homological stratification. 7 / 27

25 Sheaves Sheaves Suppose X is a topological space. 8 / 27

26 Sheaves Sheaves Suppose X is a topological space. A sheaf F on X associates each open set U X to a vector space denoted F(U), 8 / 27

27 Sheaves Sheaves Suppose X is a topological space. A sheaf F on X associates each open set U X to a vector space denoted F(U), and to each inclusion U V a linear map F(U V ) : F(V ) F(U) 8 / 27

28 Sheaves Sheaves Suppose X is a topological space. A sheaf F on X associates each open set U X to a vector space denoted F(U), and to each inclusion U V a linear map such that 1 F( ) = 0; 2 F(U U) = id U. F(U V ) : F(V ) F(U) 3 If U V W, then F(U W ) = F(U V ) F(V W ). 4 If {V i } is an open cover of U, and s i F(V i ) has the property that i, j, F((V i V j ) V i )(s i ) = F((V j V i ) V j )(s j ), then there exists a unique s F(U) such that i, F(V i U)(s) = s i. 8 / 27

29 Stratifications Topological Stratifications Definition A topological stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that X i X i 1 is an i-dimensional (topological) manifold. 9 / 27

30 Stratifications Topological Stratifications Definition A topological stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that X i X i 1 is an i-dimensional (topological) manifold. Definition An F-stratification of a topological space X is a filtration by closed subsets = X 1 X 0 X 1 X n = X such that F is locally constant when restricted to X i X i 1, for each i. 9 / 27

31 Stratifications Local Homology Sheaf Suppose X is an (abstract) simplicial complex 10 / 27

32 Stratifications Local Homology Sheaf Suppose X is an (abstract) simplicial complex, viewed as a topological space with the up-set topology: U X is open if σ U implies that τ U for all τ σ. 10 / 27

33 Stratifications Local Homology Sheaf Suppose X is an (abstract) simplicial complex, viewed as a topological space with the up-set topology: U X is open if σ U implies that τ U for all τ σ. Local homology can be viewed as a sheaf on X defined by: L(U) = H (X, X U) X U

34 Stratifications Local Homology Sheaf Suppose X is an (abstract) simplicial complex, viewed as a topological space with the up-set topology: U X is open if σ U implies that τ U for all τ σ. Local homology can be viewed as a sheaf on X defined by: L(U) = H (X, X U) X U

35 Stratifications Local Homology Sheaf Suppose X is an (abstract) simplicial complex, viewed as a topological space with the up-set topology: U X is open if σ U implies that τ U for all τ σ. Local homology can be viewed as a sheaf on X defined by: L(U) = H (X, X U) X U H ( ) 10 / 27

36 Example Triangulation of the Sundial / 27

37 Example Hasse Diagram [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [1, 3] [0, 3] [1, 2] [0, 1] [2, 3] [0, 2] [1, 4] [0, 4] [3] [2] [0] [1] [4] / 27

38 Example Labeled Hasse Diagram [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [1, 3] [0, 3] [1, 2] [0, 1] [2, 3] [0, 2] [1, 4] [0, 4] [3] [2] [0] [1] [4] / 27

39 Example Inductively defined stratification Following Goresky-MacPherson [2], Rourke-Sanderson [4], Nanda [3], we define a stratification of X inductively: 14 / 27

40 Example Inductively defined stratification Following Goresky-MacPherson [2], Rourke-Sanderson [4], Nanda [3], we define a stratification of X inductively: S n = {σ X : L Stσ is constant} 14 / 27

41 Example Labeled Hasse Diagram, St[4] [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [1, 3] [0, 3] [1, 2] [0, 1] [2, 3] [0, 2] [1, 4] [0, 4] [3] [2] [0] [1] [4]

42 Example Labeled Hasse Diagram, St[4] [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [1, 3] [0, 3] [1, 2] [0, 1] [2, 3] [0, 2] [1, 4] [0, 4] [3] [2] [0] [1] [4] / 27

43 Example Labeled Hasse Diagram, St[0, 2] [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [1, 3] [0, 3] [1, 2] [0, 1] [2, 3] [0, 2] [1, 4] [0, 4] [3] [2] [0] [1] [4]

44 Example Labeled Hasse Diagram, St[0, 2] [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [1, 3] [0, 3] [1, 2] [0, 1] [2, 3] [0, 2] [1, 4] [0, 4] [3] [2] [0] [1] [4] / 27

45 Example Labeled Hasse Diagram of S 2 [0, 1, 3] [0, 1, 2] [0, 2, 3] [0, 1, 4] [0, 3] [0, 2] / 27

46 Example Inductively defined stratification Inductive step: Restrict the local homology sheaf from X to the complement of S n and repeat: S n 1 = {σ X S n : L St(X Sn) σ is constant} 18 / 27

47 Example Example [1, 3] [1, 2] [0, 1] [2, 3] [1, 4] [0, 4] [3] [2] [0] [1] [4] / 27

48 Example Example = / 27

49 Example Example

50 Example Example / 27

51 Example Example [1, 2, 3] [1, 2, 4] [1, 3, 4] [1, 4, 5] [1, 5, 6] [1, 5, 7] [1, 6, 7] [1, 2] [1, 3] [1, 4] [1, 5] [1, 6] [1, 7] [1] 22 / 27

52 Example Example [1, 4] [1, 5] [1] 23 / 27

53 Example Example [1, 4] [1, 5] [1] 1 23 / 27

54 Example Example = 24 / 27

55 Conclusion computational topology computational geometry statistics

56 Conclusion sheaf theory computational topology computational geometry statistics

57 Conclusion sheaf theory computational topology computational geometry statistics computable stratification theory 25 / 27

58 Citations Paul Bendich, Bei Wang, and Sayan Mukherjee. Local homology transfer and stratification learning. ACM-SIAM Symposium on Discrete Algorithms (SODA), pages , Mark Goresky and Robert MacPherson. Intersection homology II. Inventiones Mathematicae, 71:77 129, Vidit Nanda. Local cohomology and stratification. ArXiv: , Colin Rourke and Brian Sanderson. Homology stratifications and intersection homology. Geometry and Topology Monographs, 2: , / 27

59 Citations Thank you for listening. 27 / 27