Learning from persistence diagrams

Transcription

1 Learning from persistence diagrams Ulrich Bauer TUM July 7, 2015 ACAT final meeting IST Austria Joint work with: Jan Reininghaus, Stefan Huber, Roland Kwitt 1 / 28

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21 Homology inference Problem (Homological reconstruction) Given a finite sample P Ω, construct a shape X that is geometrically close to Ω and satisfies H (X) H (Ω). 5 / 28

22 Homology inference Problem (Homological reconstruction) Given a finite sample P Ω, construct a shape X that is geometrically close to Ω and satisfies H (X) H (Ω). Problem (Homology inference) Determine the homology H (Ω) of a shape Ω R d from a finite sample P Ω. 5 / 28

23 Homology reconstruction using union of balls B δ (P): δ-neighborhood of P Theorem (Niyogi, Smale, Weinberger 2006) Let Ω be a submanifold of R d. Let P Ω be such that Ω B δ (P) and δ satisfies a certain (strong) sampling condition δ < C(Ω). Then H (Ω) H (B δ (P)). 6 / 28

24 Homology reconstruction using union of balls B δ (P): δ-neighborhood of P Theorem (Niyogi, Smale, Weinberger 2006) Let Ω be a submanifold of R d. Let P Ω be such that Ω B δ (P) and δ satisfies a certain (strong) sampling condition δ < C(Ω). Then H (Ω) H (B δ (P)). 6 / 28

25 Homology reconstruction using union of balls B δ (P): δ-neighborhood of P Theorem (Niyogi, Smale, Weinberger 2006) Let Ω be a submanifold of R d. Let P Ω be such that Ω B δ (P) and δ satisfies a certain (strong) sampling condition δ < C(Ω). Then H (Ω) H (B δ (P)). 6 / 28

26 Homology reconstruction using union of balls B δ (P): δ-neighborhood of P Theorem (Niyogi, Smale, Weinberger 2006) Let Ω be a submanifold of R d. Let P Ω be such that Ω B δ (P) and δ satisfies a certain (strong) sampling condition δ < C(Ω). Then H (Ω) H (B δ (P)). 6 / 28

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46 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

47 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

48 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

49 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

50 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

51 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

52 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

53 Homology inference using persistent homology Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) Let Ω R d. Let P Ω be such that Then Ω B δ (P) for some δ > 0 (sampling density), P B є (Ω) for some є > 0 (sampling error), H (Ω B δ+є (Ω)) is an isomorphism, and H (B δ+є (Ω) B 2(δ+є) (Ω)) is a monomorphism. H (Ω) im H (B δ (P) B 2δ+є (P)). 8 / 28

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55 The pipeline of topological data analysis Data point cloud distance Geometry function sublevel sets Topology topological spaces homology Algebra vector spaces barcode Combinatorics intervals 10 / 28

56 Stability of persistence barcodes for functions Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) If two functions f, g K R have distance f g δ then there exists a δ-matching of their barcodes. 11 / 28

57 Stability of persistence barcodes for functions Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) If two functions f, g K R have distance f g δ then there exists a δ-matching of their barcodes. matching of sets X, Y: bijection of subsets X X, Y Y 11 / 28

58 Stability of persistence barcodes for functions Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) If two functions f, g K R have distance f g δ then there exists a δ-matching of their barcodes. matching of sets X, Y: bijection of subsets X X, Y Y δ-matching of barcodes: 11 / 28

59 Stability of persistence barcodes for functions Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) If two functions f, g K R have distance f g δ then there exists a δ-matching of their barcodes. δ matching of sets X, Y: bijection of subsets X X, Y Y δ-matching of barcodes: matched intervals have endpoints within distance δ 11 / 28

60 Stability of persistence barcodes for functions Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) If two functions f, g K R have distance f g δ then there exists a δ-matching of their barcodes. δ 2δ matching of sets X, Y: bijection of subsets X X, Y Y δ-matching of barcodes: matched intervals have endpoints within distance δ unmatched intervals have length 2δ 11 / 28

61 Stability of persistence barcodes for functions Theorem (Cohen-Steiner, Edelsbrunner, Harer 2005) If two functions f, g K R have distance f g δ then there exists a δ-matching of their barcodes. δ 2δ matching of sets X, Y: bijection of subsets X X, Y Y δ-matching of barcodes: matched intervals have endpoints within distance δ unmatched intervals have length 2δ Bottleneck distance d B : infimum δ admitting a δ-matching 11 / 28

62 Persistence of functions Instead of point clouds, we can also study the persistence barcode of functions Example: surface mesh of the corpus callosum Surface is filtered by sublevel sets of the function 12 / 28

63 Persistence barcodes and persistence diagrams / 28

64 Topological machine learning Task: shape retrieval Task: object recognition Task: texture recognition Topological data analysis 14 / 28

65 Topological machine learning Task: shape retrieval SVM PCA Task: object recognition k-means Task: texture recognition Topological data analysis Machine learning 14 / 28

66 Topological machine learning Task: shape retrieval SVM Kernel PCA Task: object recognition k-means Task: texture recognition Topological data analysis Machine learning 14 / 28

67 Linear machine learning methods Many machine learning methods solve linear geometric problems on vector spaces Principal component analysis (PCA): find orthogonal directions of largest variance 15 / 28

68 Linear machine learning methods Many machine learning methods solve linear geometric problems on vector spaces Principal component analysis (PCA): find orthogonal directions of largest variance Support vector machines (SVM): find best-separating hyperplane 15 / 28

69 Kernels and kernel methods Let X be a set, and H a vector space with inner product, H. Consider a map Φ X H 16 / 28

70 Kernels and kernel methods Let X be a set, and H a vector space with inner product, H. Consider a map Φ X H Then k(, ) = Φ( ), Φ( ) H X X R is a kernel (and Φ is the associated feature map) 16 / 28

71 Kernels and kernel methods Let X be a set, and H a vector space with inner product, H. Consider a map Φ X H Then k(, ) = Φ( ), Φ( ) H X X R is a kernel (and Φ is the associated feature map) For many linear methods: Feature map Φ not required explicitly 16 / 28

72 Kernels and kernel methods Let X be a set, and H a vector space with inner product, H. Consider a map Φ X H Then k(, ) = Φ( ), Φ( ) H X X R is a kernel (and Φ is the associated feature map) For many linear methods: Feature map Φ not required explicitly No coordinate basis for H required 16 / 28

73 Kernels and kernel methods Let X be a set, and H a vector space with inner product, H. Consider a map Φ X H Then k(, ) = Φ( ), Φ( ) H X X R is a kernel (and Φ is the associated feature map) For many linear methods: Feature map Φ not required explicitly No coordinate basis for H required Computations can be performed by evaluating k(, ) 16 / 28

74 Kernels and kernel methods Let X be a set, and H a vector space with inner product, H. Consider a map Φ X H Then k(, ) = Φ( ), Φ( ) H X X R is a kernel (and Φ is the associated feature map) For many linear methods: Feature map Φ not required explicitly No coordinate basis for H required Computations can be performed by evaluating k(, ) This is called the kernel trick. 16 / 28

75 A feature map for persistence diagrams 17 / 28

76 A feature map for persistence diagrams 17 / 28

77 A feature map for persistence diagrams 17 / 28

78 A feature map for persistence diagrams 17 / 28

79 Smoothing persistence diagrams Define domain Ω = {(b, d) b d}. 18 / 28

80 Smoothing persistence diagrams Define domain Ω = {(b, d) b d}. For a persistence diagram D, consider the solution u Ω [0, ) R, (x, t) u(x, t) of the heat equation t u = x u u = 0 in Ω [0, ), on Ω [0, ), u = δ y on Ω {0} y D 18 / 28

81 Smoothing persistence diagrams Define domain Ω = {(b, d) b d}. For a persistence diagram D, consider the solution u Ω [0, ) R, (x, t) u(x, t) of the heat equation t u = x u u = 0 in Ω [0, ), on Ω [0, ), u = δ y on Ω {0} y D For σ > 0, define feature map Φ σ (D) = u t=σ. 18 / 28

82 Smoothing persistence diagrams Define domain Ω = {(b, d) b d}. For a persistence diagram D, consider the solution u Ω [0, ) R, (x, t) u(x, t) of the heat equation t u = x u u = 0 in Ω [0, ), on Ω [0, ), u = δ y on Ω {0} y D For σ > 0, define feature map Φ σ (D) = u t=σ. This yields the persistence scale space (PSS) kernel k σ (D f, D g ) = Φ σ (D f ), Φ σ (D g ) L 2 (Ω). 18 / 28

83 Extending the domain Extend domain from Ω to R 2 19 / 28

84 Extending the domain Extend domain from Ω to R 2 Remove boundary condition, change initial condition: t u = x u in Ω [0, ), u = δ y δ y on R 2 {0}, y D where y = (b, a) is y = (a, b) reflected at the diagonal. 19 / 28

85 Extending the domain Extend domain from Ω to R 2 Remove boundary condition, change initial condition: t u = x u in Ω [0, ), u = δ y δ y on R 2 {0}, y D where y = (b, a) is y = (a, b) reflected at the diagonal. Restricting to Ω yields a solution for the original equation. 19 / 28

86 Smoothing persistence diagrams 20 / 28

87 Smoothing persistence diagrams 20 / 28

88 Smoothing persistence diagrams 20 / 28

89 Smoothing persistence diagrams 20 / 28

90 A closed-form solution Closed form solution (convolve with a Gaussian): u(x, t) = 1 4πt x y 2 x y 2 exp exp. y D 4t 4t 21 / 28

91 A closed-form solution Closed form solution (convolve with a Gaussian): u(x, t) = 1 4πt x y 2 x y 2 exp exp. y D 4t 4t Closed form expression for PSS kernel: k σ (D f, D g ) = 1 8πσ y z exp 2 y D f 8σ z D g exp y z 2. 8σ 21 / 28

92 Stability Let Φ σ (D f ) be the smoothing of a persistence diagram D f. Theorem For two persistence diagrams D f and D g and σ > 0 we have Φ σ (D f ) Φ σ (D g ) L 2 C σ d W 1 (D f, D g ). 22 / 28

93 Stability Let Φ σ (D f ) be the smoothing of a persistence diagram D f. Theorem For two persistence diagrams D f and D g and σ > 0 we have Φ σ (D f ) Φ σ (D g ) L 2 C σ d W 1 (D f, D g ). Here: stability with respect to d W1 (D f, D g ), where d Wp (D f, D g ) = ( inf µ D f D g x µ(x) p ) 1 p x D f 22 / 28

94 Stability Let Φ σ (D f ) be the smoothing of a persistence diagram D f. Theorem For two persistence diagrams D f and D g and σ > 0 we have Φ σ (D f ) Φ σ (D g ) L 2 C σ d W 1 (D f, D g ). Here: stability with respect to d W1 (D f, D g ), where d Wp (D f, D g ) = ( inf µ D f D g x µ(x) p ) 1 p x D f Note: bottleneck distance d B (D f, D g ) = lim p d Wp (D f, D g ) 22 / 28

95 Persistence landscapes Introduced by Bubenik et at. (2013). death birth 23 / 28

96 Persistence landscapes Introduced by Bubenik et at. (2013). death birth 23 / 28

97 Persistence landscapes Introduced by Bubenik et at. (2013). λ k (x) death (a, b) x a b birth 23 / 28

98 Persistence landscapes Introduced by Bubenik et at. (2013). death (a, b) λ k (x) a b x 23 / 28

99 Persistence landscapes Introduced by Bubenik et at. (2013). death (a, b) λ k (x) λ 1 a b x 23 / 28

100 Persistence landscapes Introduced by Bubenik et at. (2013). death (a, b) λ k (x) λ 2 λ 1 a b x 23 / 28

101 Persistence landscapes Introduced by Bubenik et at. (2013). death (a, b) λ k (x) λ 2 λ 1 a λ 3 b x 23 / 28

102 Persistence landscapes The persistence landscape of a persistence diagram D f can be interpreted as a function Φ L (D f ) in L p (R 2 ). λ(k, x) 0 x 1 2 k / 28

103 Persistence landscapes The persistence landscape of a persistence diagram D f can be interpreted as a function Φ L (D f ) in L p (R 2 ). λ(k, x) 0 x 1 2 k... Stablility: Φ L (D f ) Φ L (D g ) d B (D f, D g ) 24 / 28

104 Persistence landscapes The persistence landscape of a persistence diagram D f can be interpreted as a function Φ L (D f ) in L p (R 2 ). λ(k, x) 0 x 1 2 k... Stablility: Φ L (D f ) Φ L (D g ) d B (D f, D g ) But: no stability for Φ σ (D f ) Φ σ (D g ) L 2 24 / 28

105 Smoothed persistence diagrams vs. landscapes SHREC 2014 dataset of shapes: 300 surfaces, 20 classes (poses). Sublevel set filtration of heat kernel signature (HKS) on each surface Additional scale parameter t 2 Tasks: Retrieval: Get the nearest neighbour in L 2. Classification: Train a SVM with the kernel. 25 / 28

106 Experiments: Retrieval HKS t i d k L d kσ d k L d kσ t t t t t t t t t t Top 3 (2014) Table: Nearest-neighbor retrieval performance. Left: SHREC 2014 (synthetic); Right: SHREC 2014 (real). 26 / 28

107 Experiments: Classification HKS t i k L k σ t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± Table: Classification performance on SHREC 2014 (synthetic). 27 / 28

108 Experiments: Classification HKS t i k L k σ t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± t ± ± Table: Classification performance on SHREC 2014 (real). 27 / 28