Geometry and Topology of Data ICERM, December 2017 Inverse problems in TDA Focus on metric graphs Steve Oudot — joint work with Elchanan (Isaac) Solomon (Brown University)
Persistence diagrams as descriptors for data ) ( g Model i n n i f l e p r m e n a c s e ( ) Data Descriptor (TDA) - genericity - stability - invariance - · · · 1
Persistence diagrams as descriptors for data ) ( g Model i n n i f l e p r m e n a c s e ( ) Data Descriptor (TDA) - genericity - stability autoencoders! - invariance - · · · 1
Persistence diagrams as descriptors for data ) ( g Model i n n i f l e p r m e n a c s e ( ) noise Data Descriptor signal (TDA) Challenges: - signal vs noise discrimination - model inference i.e. infer a model explaining the data / descriptor - hypothesis testing - · · · 1
Persistence diagrams as descriptors for data ) ( g Model i n n i f l e p r m e n a c s e ( ) noise Data Descriptor signal (TDA) Challenges: - signal vs noise discrimination - model inference model uniqueness? i.e. infer a model explaining the data / descriptor - hypothesis testing - · · · 1
Persistence diagrams as descriptors for data ) ( g Model i n n i f l e p r m e n a c s e ( ) Data Descriptor (TDA) Lipschitz operator left inverse? 1
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations offsets filtration barcode / diagram point cloud simplicial filtration 2
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations t C t ( X, d X ) R 2 t ( X, d X ) 2
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations 1 1 α ≥ π/ 2 dgm C ( P, ℓ 2 ) = { (0 , + ∞ ) } ⊔ { (0 , 1 2 ) } ⊔ { (0 , 1 2 ) } dgm R ( P, ℓ 2 ) = { (0 , + ∞ ) } ⊔ { (0 , 1) } ⊔ { (0 , 1) } ⇒ diagrams for different values of α are indistinguishable 2
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations Prop: [Folklore] For any metric tree ( X, d X ) : dgm R ( X, d X ) = dgm C ( X, d X ) = { (0 , + ∞ ) } ⇒ no information on the metric X is 0 -hyperbolic ⇒ metric balls are convex ⇒ geodesic triangles are tripods 2
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations • Reeb graphs ⇒ Reeb graphs are indistinguishable from their diagrams 2
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations • Reeb graphs • Real-valued functions Prop: [Folklore] Given f : X → R and h : Y → X homeomorphism, dgm f ◦ h = dgm f this is too large a class of transformations, for our purposes we rather target isometries ⇒ Persistence is invariant under reparametrizations 2
Lack of injectivity in general • Unions of (open) balls — ˇ Cech/Rips/Delaunay filtrations • Reeb graphs • Real-valued functions possible solutions: • richer topological invariants (e.g. persistent homotopy) • use several filter functions ( concatenation vs multipersistence) 2
Persistent Homology Transform (PHT) implicit: PHT( X ) = PHT( X, F ) PHT( X )= { dgm f w | w ∈ W } ( X, d X ) (compact) PHT( X ) · · · note: here, as before, dgm f contains dgm f w F = { f w } w ∈ W (diagrams, d b ) R 3
Persistent Homology Transform (PHT) implicit: PHT( X ) = PHT( X, F ) PHT( X )= { dgm f w | w ∈ W } ( X, d X ) (compact) PHT( X ) · · · note: here, as before, dgm f contains dgm f w F = { f w } w ∈ W (diagrams, d b ) R Thm: [Turner, Mukherjee, Boyer 2014] X S d − 1 Let F = {�· , w �} w ∈ S d − 1 , where d = 2 , 3 is fixed. Then, PHT is injective on the set of linear embeddings w of compact simplicial complexes in R d . Extension: [Turner et al., in progress] True for arbitrary d and semialgebraic compact sets. 3
Persistent Homology Transform (PHT) implicit: PHT( X ) = PHT( X, F ) PHT( X )= { dgm f w | w ∈ W } ( X, d X ) (compact) PHT( X ) · · · note: here, as before, dgm f contains dgm f w F = { f w } w ∈ W Q: can we derive an intrinsic version? (diagrams, d b ) R Thm: [Turner, Mukherjee, Boyer 2014] X S d − 1 Let F = {�· , w �} w ∈ S d − 1 , where d = 2 , 3 is fixed. Then, PHT is injective on the set of linear embeddings w of compact simplicial complexes in R d . Extension: [Turner et al., in progress] True for arbitrary d and semialgebraic compact sets. 3
PHT for intrinsic metrics Given ( X, d X ) compact, take F = { d X ( · , x ) } x ∈ X 4
PHT for intrinsic metrics Given ( X, d X ) compact, take F = { d X ( · , x ) } x ∈ X Thm (local stability): [Carri` ere, O., Ovsjanikov 2015] this construction holds for general metric spaces, however it makes sense mostly for compact Let ( X, d X ) and ( Y, d Y ) be compact length spaces with positive convexity radius ( ̺ ( X ) , ̺ ( Y ) > 0 ). Let x ∈ X and y ∈ Y . 1 If d GH (( X, x ) , ( Y, y )) ≤ 20 min { ̺ ( X ) , ̺ ( Y ) } , then d b (dgm d X ( · , x ) , dgm d Y ( · , y )) ≤ 20 d GH (( X, x ) , ( Y, y )) . 4
PHT for intrinsic metrics Given ( X, d X ) compact, take F = { d X ( · , x ) } x ∈ X Corollary (local stability of PHT): Let ( X, d X ) and ( Y, d Y ) be compact length spaces with positive convexity radius ( ̺ ( X ) , ̺ ( Y ) > 0 ). 1 If d GH ( X, Y ) ≤ 20 min { ̺ ( X ) , ̺ ( Y ) } , then d H ( PHT ( X ) , PHT ( Y )) ≤ 20 d GH (( X, x ) , ( Y, y )) . Thm (local stability): [Carri` ere, O., Ovsjanikov 2015] this construction holds for general metric spaces, however it makes sense mostly for compact Let ( X, d X ) and ( Y, d Y ) be compact length spaces with positive convexity radius ( ̺ ( X ) , ̺ ( Y ) > 0 ). Let x ∈ X and y ∈ Y . 1 If d GH (( X, x ) , ( Y, y )) ≤ 20 min { ̺ ( X ) , ̺ ( Y ) } , then d b (dgm d X ( · , x ) , dgm d Y ( · , y )) ≤ 20 d GH (( X, x ) , ( Y, y )) . 4
PHT for intrinsic metrics Given ( X, d X ) compact, take F = { d X ( · , x ) } x ∈ X Corollary (local stability of PHT): Let ( X, d X ) and ( Y, d Y ) be compact length spaces with positive convexity radius ( ̺ ( X ) , ̺ ( Y ) > 0 ). 1 If d GH (( X ) , ( Y )) ≤ 20 min { ̺ ( X ) , ̺ ( Y ) } , then d H ( PHT ( X ) , PHT ( Y )) ≤ 20 d GH (( X, x ) , ( Y, y )) . a # X →∞ d GH ( T, X ) − → 0 b b d H ( PHT ( T ) , PHT ( X )) is bounded away from 0 a 4
PHT for metric graphs from now on we will focus on compact metric graphs, which are length spaces that admit Focus: compact metric graphs (1-dimensional stratified length spaces) PHT: F = { d X ( · , x ) } x ∈ X , dgm = extended persistence diagram Thm (global stability): [Dey, Shi, Wang 2015] For any compact metric graphs X, Y , Note: this result does not extend to the class of compact length spaces. Indeed, a graph d H ( PHT ( X ) , PHT ( Y )) ≤ 18 d GH ( X, Y ) . Thm (density): [Gromov] Compact metric graphs are GH-dense among the compact length spaces. 5
PHT for metric graphs from now on we will focus on compact metric graphs, which are length spaces that admit Focus: compact metric graphs (1-dimensional stratified length spaces) PHT: F = { d X ( · , x ) } x ∈ X , dgm = extended persistence diagram Thm (global stability): [Dey, Shi, Wang 2015] For any compact metric graphs X, Y , Note: this result does not extend to the class of compact length spaces. Indeed, a graph d H ( PHT ( X ) , PHT ( Y )) ≤ 18 d GH ( X, Y ) . Thm (density): [Gromov] Compact metric graphs are GH-dense among the compact length spaces. Q: injectivity of PHT on metric graphs? 5
PHT for metric graphs Negative result: PHT is not injective on all compact metric graphs X Y PHT ( X ) = PHT ( Y ) while X �≃ Y 5
PHT for metric graphs Negative result: PHT is not injective on all compact metric graphs X Y PHT ( X ) = PHT ( Y ) while X �≃ Y Note: Aut ( X ) is non-trivial, hence Ψ X : x �→ dgm d X ( · , x ) is not injective So, maybe there is a connection between Ψ X being injective and PHTitself being injective... this is precisely what our results show 5
PHT for metric graphs Let Inj Ψ = { X compact metric graph s.t. Ψ X is injective } Thm 1: PHT is injective on Inj Ψ . Thm 2: Inj Ψ is GH-dense among the compact metric graphs. 1 1 0.9 1.1 5 1.1 5 10 ⇒ Note: Ψ X injective Aut ( X ) trivial Thus, Inj Ψ is a strict subset of the graphs with trivial automorphism group p q �⇒ 0.9 5 6 1 dgm d X ( · , p ) = dgm d X ( · , q ) 5
PHT for metric graphs Let Inj Ψ = { X compact metric graph s.t. Ψ X is injective } Thm 1: PHT is injective on Inj Ψ . + Gromov’s density result Thm 2: Inj Ψ is GH-dense among the compact metric graphs. Corollary: There is a GH-dense subset of the compact length spaces on which PHT is injective. 1 1 0.9 1.1 5 1.1 5 10 ⇒ Note: Ψ X injective Aut ( X ) trivial Thus, Inj Ψ is a strict subset of the graphs with trivial automorphism group p q �⇒ 0.9 5 6 1 dgm d X ( · , p ) = dgm d X ( · , q ) 5
PHT for metric graphs Let Inj Ψ = { X compact metric graph s.t. Ψ X is injective } Thm 1: PHT is injective on Inj Ψ . + Gromov’s density result Thm 2: Inj Ψ is GH-dense among the compact metric graphs. Corollary: There is a GH-dense subset of the compact length spaces on which PHT is injective. Thm 3: PHT is GH- locally injective on compact metric graphs. 5
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