On Inverse Problems in TDA

Transcription

1 Abel Symposium Geiranger, June 2018 On Inverse Problems in TDA Steve Oudot joint work with E. Solomon (Brown Math.) arxiv:

2 Features Data Rn The preimage problem in the data Sciences feature design or learning bag of words, word2vec shape contexts, heat kernels node2vec, Laplacian fact., rand. walks dim. reduction, auto-encoders, etc. 1

3 Features Data Rn The preimage problem in the data Sciences feature design or learning? Can the feature map be inverted? bag of words, word2vec s an important trend in AI research, and one - Right inverse ( current preimage): interpretable AI that is particularly difficult on graphs, time shape contexts, heat kernels - Left inverse (! preimage): reliable interpretation node2vec, Laplacian fact., rand. walks Scenarios: dictionaries, deep layers, stats, etc. dim. reduction, auto-encoders, etc. 1

4 Features Data Rn The preimage problem in the data Sciences feature design or learning? Can the feature map be inverted? bag of words, word2vec s an important trend in AI research, and one - Right inverse ( current preimage): interpretable AI that is particularly difficult on graphs, time shape contexts, heat kernels - Left inverse (! preimage): reliable interpretation node2vec, Laplacian fact., rand. walks Scenarios: dictionaries, deep layers, stats, etc. dim. reduction, auto-encoders, etc. 1

5 Features Data Rn The preimage problem in the data Sciences feature design or learning? Can the feature map be inverted? bag of words, word2vec s an important trend in AI research, and one - Right inverse ( current preimage): interpretable AI that is particularly difficult on graphs, time shape contexts, heat kernels - Left inverse (! preimage): reliable interpretation node2vec, Laplacian fact., rand. walks Scenarios: dictionaries, deep layers, stats, etc. dim. reduction, auto-encoders, etc. 1

6 TDA and the preimage problem (in Model (sampling) fer en ce) (decoding) Data Descriptor (TDA) 2

7 TDA and the preimage problem Mathematical abstraction: (in Model (sampling) fer en - compact metric spaces modulo isometries ce) - Gromov-Hausdorff distance (decoding) Data Descriptor (TDA) Lipschitz operator right inverse: realize barcode as the PH of some isom. class subsequent questions, e.g. can we infer the model from the descriptor of a finite compact metric space left inverse: characterize isom. class uniquely 2

8 TDA and the preimage problem Mathematical abstraction: (in Model (sampling) fer en - compact metric spaces modulo isometries ce) - Gromov-Hausdorff distance (decoding) Data Descriptor (TDA) Lipschitz operator right inverse: realize barcode as the PH of some isom. class subsequent questions, e.g. can we infer the model from the descriptor of a finite compact metric space left inverse: characterize isom. class uniquely 2

9 Right inverses for TDA Fact: [Folklore] Any (graded) persistence module R vect k can be realized as the (graded) P H of a piecewise-constant function on a bouquet of spheres. decomposes as a direct sum of interval modules, each of which is realized as the PH of a consta 3

10 Right inverses for TDA Fact: [Folklore] Any (graded) persistence module R vect k can be realized as the (graded) P H of a piecewise-constant function on a bouquet of spheres. decomposes as a direct sum of interval modules, each of which is realized as the PH of a consta Thm: [Curry, Reiss] [Botnan, Fluhr] Any (graded) barcode can be realized as the level-set P H of some stratified map on some stratified space. 3

11 Right inverses (local) for TDA point cloud Čech / Rips filtration barcode u R nd note: v bars R 2n 1 are ordered Thm: [Gameiro, Hiraoka, Obayashi] (i) Generic point cloud Ω u in R nd over which the correspondence u v can be extended to a map f : Ω R 2n 1 computing persistence barcodes. (ii) For Ω small enough, f is of class C. Observation: pairing given by order of distances is constant in small enough O. 4

12 Right inverses (local) for TDA point cloud Čech / Rips filtration barcode u R nd note: v bars R 2n 1 are ordered Thm: [Gameiro, Hiraoka, Obayashi] (i) Generic point cloud Ω u in R nd over which the correspondence u v can be extended to a map f : Ω R 2n 1 computing persistence barcodes. (ii) For Ω small enough, f is of class C. adapt Newton-Raphson continuation method to build right inverse of f in f(ω) (Jacobian matrix of f can be singular use pseudo-inverse) 4

13 Left inverses? Unions of (open) balls Čech/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 5

14 Left inverses? Unions of (open) balls Čech/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 5

15 Left inverses? Unions of (open) balls Čech/Rips/Delaunay filtrations Reeb graphs Reeb graphs are indistinguishable from their diagrams 5

16 Left inverses? Unions of (open) balls Čech/Rips/Delaunay filtrations Reeb graphs Real-valued functions Prop: [Folklore] Given f : X R and h : Y X homeomorphism, dgm f h = dgm f Too large a group of transformations... 5

17 Left inverses? Unions of (open) balls Čech/Rips/Delaunay filtrations Reeb graphs Real-valued functions possible solutions: richer topological invariants (e.g. persistent homotopy) use multiple filter functions (aggregation vs multipersistence) 5

18 Persistent Homology Transform (PHT) (X, d X ) (compact) implicit: PHT(X)={dgm = PHT(X, f w w F) W } PHT(X) F = {f w } w W note: here, as dgm before, f w dgm f cont R (diagrams, d b ) 6

19 Persistent Homology Transform (PHT) (X, d X ) (compact) implicit: PHT(X)={dgm = PHT(X, f w w F) W } PHT(X) F = {f w } w W note: here, as dgm before, f w dgm f cont R (diagrams, d b ) Thm: [Boyer, Curry, Mukherjee, Turner 2014, 2018] [Ghrist, Levanger, Mai 2018] Let F = {, w } w S d 1, where d is fixed. Then, PHT is injective on the class of semialgebraic sets in R d. tes: - semialgebraic sets are finite unions of solution sets of systems of polynomial w Still true for a fixed finite set of directions (of size exponential in d). [Curry, Mukherjee, Turner] X S d 1 6

20 Persistent Homology Transform (PHT) (X, d X ) (compact) implicit: PHT(X)={dgm = PHT(X, f w w F) W } PHT(X) F = {f w } w W note: here, as dgm before, f w dgm f cont R (diagrams, d b ) Thm: [Boyer, Curry, Mukherjee, Turner 2014, 2018] [Ghrist, Levanger, Mai 2018] Let F = {, w } w S d 1, where d is fixed. Then, PHT is injective on the class of semialgebraic sets in R d. S d 1 tes: - semialgebraic sets are finite unions of solution sets of systems of polynomial w Still Corollary: true for PHT a fixed is a finite sufficient set of statistic directions for such sets (of size parametric exponential inference d). [Curry, Mukherjee, Turner] X 6

21 PHT for length spaces Given a compact length space (X, d X ), take F = {d X (, x)} x X 7

22 PHT for length spaces Given a compact length space (X, d X ), take F = {d X (, x)} x X this construction Thm (localholds stability): for general [Carrière, metric O., spaces, Ovsjanikov however 2015] it makes sense mostly for comp Let (X, d X ) and (Y, d Y ) be compact length spaces with positive convexity radius (ϱ(x), ϱ(y ) > 0). Let x X and y Y. If d GH ((X, x), (Y, y)) 1 min{ϱ(x), ϱ(y )}, then 20 d b (dgm d X (, x), dgm d Y (, y)) 20 d GH ((X, x), (Y, y)). 7

23 PHT for length spaces Given a compact length space (X, d X ), 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 (ϱ(x), ϱ(y ) > 0). If d GH (X, Y ) 1 min{ϱ(x), ϱ(y )}, then 20 d H (PHT(X), PHT(Y )) 20 d GH ((X, x), (Y, y)). this construction Thm (localholds stability): for general [Carrière, metric O., spaces, Ovsjanikov however 2015] it makes sense mostly for comp Let (X, d X ) and (Y, d Y ) be compact length spaces with positive convexity radius (ϱ(x), ϱ(y ) > 0). Let x X and y Y. If d GH ((X, x), (Y, y)) 1 min{ϱ(x), ϱ(y )}, then 20 d b (dgm d X (, x), dgm d Y (, y)) 20 d GH ((X, x), (Y, y)). 7

24 PHT for length spaces Given a compact length space (X, d X ), 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 (ϱ(x), ϱ(y ) > 0). If d GH (X, Y ) 1 min{ϱ(x), ϱ(y )}, then 20 d H (PHT(X), PHT(Y )) 20 d GH ((X, x), (Y, y)). a b b d GH (T, X) #X 0 d H (PHT 2 (T ), PHT 2 (X)) is bounded away from 0 a 7

25 PHT for metric graphs fromfocus: now oncompact we will focus metric ongraphs compact (1-dimensional metric graphs, stratified which are length length spaces) spaces that adm 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, ote: 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. 8

26 PHT for metric graphs fromfocus: now oncompact we will focus metric ongraphs compact (1-dimensional metric graphs, stratified which are length length spaces) spaces that adm 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, ote: 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? 8

27 PHT for metric graphs Bad news: PHT is not injective on all compact metric graphs X Y PHT(X) = PHT(Y ) while X Y 8

28 PHT for metric graphs Bad news: 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 8

29 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 p 10 q Note: Ψ X injective Thus, Inj Ψ is a strict subset of the graphs with trivial automorphism group Aut(X) trivial 6 dgm d X (, p) = dgm d X (, q)

30 PHT for metric graphs Let Inj Ψ = {X compact metric graph s.t. Ψ X is injective} + Gromov s density result Thm 1: PHT is injective on Inj Ψ. 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 p 10 q Note: Ψ X injective Thus, Inj Ψ is a strict subset of the graphs with trivial automorphism group Aut(X) trivial 6 dgm d X (, p) = dgm d X (, q)

31 PHT for metric graphs Let Inj Ψ = {X compact metric graph s.t. Ψ X is injective} + Gromov s density result Thm 1: PHT is injective on Inj Ψ. 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. 8

32 Generic injectivity Generative model: metric graph combinatorial graph (V, E) + edge weights E R + this is a countable mixture space (for proba. eachmass fixed function numbers of, proba. vertices measure and edges withthere density are only R E finitely ) ma

33 Generic injectivity Generative model: metric graph combinatorial graph (V, E) + edge weights E R + this is a countable mixture space (for proba. eachmass fixed function numbers of, proba. vertices measure and edges withthere density are only R E finitely ) ma + Thm 4: Under this model, there is a full-measure subset of the metric graphs on which PHT is injective. 9

34 Generic injectivity Generative model: metric graph combinatorial graph (V, E) + edge weights E R + this is a countable mixture space (for proba. eachmass fixed function numbers of, proba. vertices measure and edges withthere density are only R E finitely ) ma + Thm 4: Under this model, there is a full-measure subset of the metric graphs on which PHT is injective. Questions: is PHT a sufficient statistic for metric graphs? are finitely many basepoints enough? algorithm? what about higher-dimensional stratified spaces? 9

35 Proof outline for Thm 1 Let Inj Ψ = {X compact metric graph s.t. Ψ X is injective} + Gromov s density result Thm 1: PHT is injective on Inj Ψ. 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. 10

36 Proof outline for Thm 1 Prop: hen X is topologically a circle, every point gives rise to the same barcode and therefore the ima If X is not a circle, then Ψ X is a local isometry: x U x y U x d X (x, y) = d b (Ψ X (x), Ψ X (y)) U x x (X, d X ) Ψ X (diagrams, d b ) 10

37 Proof outline for Thm 1 Prop: hen X is topologically a circle, every point gives rise to the same barcode and therefore the ima If X is not a circle, then Ψ X is a local isometry: x U x y U x d X (x, y) = d b (Ψ X (x), Ψ X (y)) Corollary: If Ψ X is injective, then Ψ X is a (global) isometry from (X, d X ) to (PHT(X), ˆd b ). n both sides, distances are given by shortest path lengths, and we can subdivide the paths enoug (X, d X ) Ψ X (diagrams, d b ) 10

38 Proof outline for Thm 2 Let Inj Ψ = {X compact metric graph s.t. Ψ X is injective} + Gromov s density result Thm 1: PHT is injective on Inj Ψ. 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. 11

39 Proof outline for Thm 2 Given (X, d X ), for any ε > 0 build an ε-approximation (X ε, d Xε ) in d GH Break symmetries by cactification: subdivide edges add hanging branches (thorns) with distinct lengths X ε X 11

40 Proof outline for Thm 2 Given (X, d X ), for any ε > 0 build an ε-approximation (X ε, d Xε ) in d GH Break symmetries by cactification: subdivide edges add hanging branches (thorns) with distinct lengths (X ε, d Xε ) parametrized by distances to thorn bases and tips X ε X these distances appear in the persistence diagrams 11

41 Proof outline for Thm 3 Let Inj Ψ = {X compact metric graph s.t. Ψ X is injective} + Gromov s density result Thm 1: PHT is injective on Inj Ψ. 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. 12

42 Proof outline for Thm 3 Prop: The map (X, d X, x) R dx (,x) is injective. d X (, x) x (X, d X ) x 12

43 Proof outline for Thm 3 Prop: The map (X, d X, x) R dx (,x) is injective. Thm: [Carrière, O. 2017] The map R f dgm f is GH-locally injective. 12

44 Generic injectivity Generative model: metric graph combinatorial graph (V, E) + edge weights E R + this is a countable mixture space (for proba. eachmass fixed function numbers of, proba. vertices measure and edges withthere density are only R E finitely ) ma + Thm 4: Under this model, there is a full-measure subset of the metric graphs on which PHT is injective. Proof outline: - for (almost) any fixed combinatorial graph G, Ψ G indeed, is generically the lack injective. of injectivity in - deal with exceptions (e.g. linear graphs) explicitly for these, the PHT has an image with a specific shape 13

45 Features Data Rn The preimage problem in the data Sciences feature design or learning? bag of words, word2vec shape contexts, heat kernels node2vec, Laplacian fact., rand. walks dim. reduction, auto-encoders, etc. 14