7. The Stability Theorem

Transcription

1 Vin de Silva Pomona College CBMS Lecture Series, Macalester College, June 2017

2 The Persistence Stability Theorem Stability Theorem (Cohen-Steiner, Edelsbrunner, Harer 2007) The following transformation is Lipschitz: data / persistence diagram / / / / Small changes in the data lead to small changes in the diagram. Factorization data persistence module persistence diagram Geometric Stability data persistence module is Lipschitz (in various settings) Algebraic Stability (Chazal, Cohen-Steiner, Glisse, Guibas, Oudot 2009) persistence module persistence diagram is 1-Lipschitz

3 Data Simplex stream Let S be a finite simplicial complex. Let f : S! R be a function with the property that f 1 ( 1, t] is a subcomplex of S for every t 2 R. Processing Run the persistence algorithm on the sequence of simplices ordered by f -value. Metric Two such functions f, g : S! R are compared by the supremum norm: kf gk 1 =max f ( ) g( ) 2S

4 Data Sublevelset filtration Let M be a compact smooth manifold. Let f : M! R be a Morse function (critical points nondegenerate) Processing Compute the persistence diagram of H f 1 ( 1, a 0] /... / H f 1 ( 1, a n] where a 0 < < a n are the critical values. Metric Two such functions f, g : M! R are compared by the supremum norm: kf gk 1 =max f (x) g(x) x2m

5 Data Sublevelset filtration Let P be a compact polyhedron (, realization of a simplicial complex). Let f : P! R be a piecewise-linear map. Processing Compute the persistence diagram of H f 1 ( 1, a 0] /... / H f 1 ( 1, a n] where a 0 < < a n are the critical values. Metric Two such functions f, g : P! R are compared by the supremum norm: kf gk 1 =max f (x) g(x) x2p

6 Data Metric data (fixed set) Let X be a finite set. Let d : X X! R be a metric. Processing Compute the persistence diagram of H VR((X, d), r 0) /... / H VR((X, d), r n) where r 0 < < r n are the values taken by the metric. Metric Two such metrics d, e : X X! R are compared by the supremum norm: kd ek 1 =max d(x, y) x,y2p e(x, y)

7 Data Metric data (subsets of a fixed space) Let (Z, d) beametricspace. Let X Z be a finite subset. Processing Compute the persistence diagram of H VR((X, d), r 0) /... / H VR((X, d), r n) where r 0 < < r n are the values taken by the metric restricted to X. Metric Two such subsets X, Y Z are compared by the Hausdor distance: d H(X, Y )=maxapple max d(x, Y ), max d(y, X ) x2x y2y

8 Data Metric data (general) Let (X, d) beafinitemetricspace. Processing Compute the persistence diagram of H VR((X, d), r 0) /... / H VR((X, d), r n) where r 0 < < r n are the values taken by the metric. Metric Two metric spaces (X, d), (Y, e) are compared using the Gromov Hausdor distance: d GH((X, d), (Y, e)) = inf Z,f,g dh(f (X ), g(y )) The min is taken over all metric spaces Z and isometric embeddings f : X! Z, g : Y! Z.

9 Persistence modules Persistence modules indexed by R (Chazal et al. 2009) For the stability theorem, we need to work with real index values. A persistence module indexed by R is specified by the following data: AvectorspaceV t for every t 2 R. Alinearmapvt s : V s! V t whenever s apple t. We require vt t =1 Vt for all t, andvuv t t s = vu s whenever s apple t apple u. Henceforth, all persistence modules will be indexed by R unless stated otherwise.

10 Persistence modules Persistence modules indexed by R (Chazal et al. 2009) For the stability theorem, we need to work with real index values. A persistence module indexed by R is specified by the following data: AvectorspaceV t for every t 2 R. Alinearmapvt s : V s! V t whenever s apple t. We require vt t =1 Vt for all t, andvuv t t s = vu s whenever s apple t apple u. Henceforth, all persistence modules will be indexed by R unless stated otherwise. Tameness Apersistencemoduleisoffinite type if there is a finite set of critical values a 0 < a 1 < < a n such that V t = 0 for all t < a 0;and V t is constant, and finite dimensional, over each interval [a k, a k+1 ). (Here we take a n+1 =+1.)

11 Persistence modules Persistence modules indexed by R (Chazal et al. 2009) For the stability theorem, we need to work with real index values. A persistence module indexed by R is specified by the following data: AvectorspaceV t for every t 2 R. Alinearmapvt s : V s! V t whenever s apple t. We require vt t =1 Vt for all t, andvuv t t s = vu s whenever s apple t apple u. Henceforth, all persistence modules will be indexed by R unless stated otherwise. Tameness Apersistencemoduleisq-tame if vt s has finite rank whenever s < t. Fact q-tame persistence modules have well-behaved persistence diagrams.

12 ! " < / < = / Persistence modules The interleaving distance (Chazal et al. 2009) An r-interleaving between persistence modules V, W is a collection of maps t : V t! W t+r, t : W t! V t+r such that the diagrams v s v t t+2r V s / t V t / s W s+r t w s+r t+r / W t+r t+r V t+2r V s+r v s+r t+r W s w s t s W t t / V t+r w t t+2r t+r " W t+2r commute for all s < t. A0-interleavingisanisomorphismofpersistencemodules. The interleaving distance between V, W is d I(V, W )=inf r 0 r there exists an r-interleaving between V, W

13 " < < = " Data! Persistence modules Simplex streams Let S be a simplicial complex and let f, g : S! R define simplex streams. Suppose kf gk 1 apple r. Writing [f ] t = f 1 ( 1, t], [g] t = g 1 ( 1, t], we have inclusions [f ] t [g] t+r, [g] t [f ] t+r. More generally, we have diagrams [f ] s / [f ] t / [f ] t+2r [f ] s+r / [f ] t+r [g] s+r / [g] t+r [g] s / [g] t / [g] t+2r which automatically commute, since all arrows are inclusion maps. Applying homology H,weobtainthemaps t, t and required relations for an r-interleaving between the persistence modules for the two simplex streams.

14 " < < = " Data! Persistence modules Sublevelset filtrations Let M be a compact smooth manifold with Morse functions f, g : M! R. Suppose kf gk 1 apple r. Writing [f ] t = f 1 ( 1, t], [g] t = g 1 ( 1, t], we have inclusions [f ] t [g] t+r, [g] t [f ] t+r. More generally, we have diagrams [f ] s / [f ] t / [f ] t+2r [f ] s+r / [f ] t+r [g] s+r / [g] t+r [g] s / [g] t / [g] t+2r which automatically commute, since all arrows are inclusion maps. Applying homology H,weobtainthemaps t, t and required relations for an r- interleaving between the persistence modules for the two sublevelset filtrations.

15 " < < = " Data! Persistence modules Sublevelset filtrations Let P be a compact polyhedron with piecewise linear maps f, g : P! R. Suppose kf gk 1 apple r. Writing [f ] t = f 1 ( 1, t], [g] t = g 1 ( 1, t], we have inclusions [f ] t [g] t+r, [g] t [f ] t+r. More generally, we have diagrams [f ] s / [f ] t / [f ] t+2r [f ] s+r / [f ] t+r [g] s+r / [g] t+r [g] s / [g] t / [g] t+2r which automatically commute, since all arrows are inclusion maps. Applying homology H,weobtainthemaps t, t and required relations for an r- interleaving between the persistence modules for the two sublevelset filtrations.

16 ! < < < " Data! Persistence modules Metric data (fixed set) Let X be a finite set and let d, e : X X! R be metrics. Suppose kd ek 1 apple r. Writing [d] t =VR((X, d), t) [e] t =VR((X, e), t) we have inclusions [d] t [e] t+r, [e] t [d] t+r. More generally, we have diagrams [d] s / [d] t / [d] t+2r [d] s+r / [d] t+r [e] s+r / [e] t+r [e] s / [e] t / [e] t+2r which automatically commute, since all arrows are inclusion maps. Applying homology H,weobtainthemaps t, t and required relations for an r-interleaving between the two Vietoris Rips homology persistence modules.

17 Data! Persistence modules More subtle The other two examples are slightly more subtle: Metric data (subsets of a fixed space) Let X, Y Z be finite subsets of a fixed metric space (Z, d). Metric data (general) Let (X, d), (Y, e) befinitemetricspaces.

18 Detour: Multivalued maps Multivalued maps Let X, Y be sets. A multivalued map F : X Y is a function F : X!P(Y) that carries each x 2 X to a nonempty subset of Y. Selections A selection from F is a function f : X! Y such that f (x) 2 F (x) for all x. Subordinate maps Let F, G : X Y.WesaythatG is subordinate to F,andwriteG F,if G(x) F (x) for all x. Equivalently, every selection from G is a selection from F.

19 Detour: Multivalued maps Multivalued simplicial maps Let S, T be simplicial complexes with vertex sets X, Y. A multivalued simplicial map F : S T is a multivalued map F : X Y, such that the image of every simplex 2 S F = S x2 F (x) (has the property that every finite subset) is a simplex of T. Induced homology map Let F : S T be a multivalued simplicial map. Then the maps f : H (S)! H (T ) induced by selections f are all equal to each other. Call this map F. Proof Indeed, any two selections are contiguous. Remark It is immediate that G F implies G = F.

20 Data! Persistence modules Metric data (subsets of a fixed space) Let X, Y Z be finite subsets of a fixed metric space (Z, d). Suppose d H(X, Y ) apple r/2. Thenwecandefinemultivaluedmaps: F : X Y ; G : Y X ; F (x) ={y 2 Y d(x, y) apple r/2}. G(y) ={x 2 Y d(y, x) apple r/2}. Using the triangle inequality, these define multivalued simplicial maps F :VR(X, t) VR(Y, t + r), G :VR(Y, t) VR(X, t + r). Applying homology, we get maps t = F : H (VR(X, t)) H (VR(Y, t + r)) t = G : H (VR(Y, t)) H (VR(X, t + r)) It remains to verify the required relations for an r-interleaving.

21 " " # # $ $ Data! Persistence modules Metric data (subsets of a fixed space) Let X, Y Z be finite subsets of a fixed metric space (Z, d). Write [X ] t =VR(X, t) and[y ] t =VR(Y, t). Then we have: [X ] s / [X ] t / [X ] t+2r [X ] s+r / : [X ] t+r : : : ; ; [Y ] s+r / [Y ] t+r [Y ] s / [Y ] t / [Y ] t+2r The two parallelograms commute. The triangles don t commute. However, the horizontal map in each triangle is subordinate to the composite of the two diagonal maps. Applying homology, the resulting diagrams do commute. Thus the maps t, t define an r-interleaving. Conclusion subset of a metric space / VR persistent homology is 2-Lipschitz.

22 Persistence Diagrams Diagrams A diagram in H is a multiset of points in the region of the extended plane Point metric H = {(p, q) 1< p < q apple +1} We use the `1 metric for points in this region: 8 >< max( p 1 p 2, q 1 q 2 ) if q 1, q 2 are finite d 1 ((p 1, q 1), (p 2, q 2)) = p 1 p 2 if q 1 = q 2 =+1 >: 1 otherwise We also note the distance from the diagonal: d 1 ((p, q), ) = 1 (q p) 2

23 Persistence Diagrams The bottleneck distance Let A, B be diagrams in H. An r-matching is a partial bijection between the multisets A, B such that: If a 2 A and b 2 B are matched, then d 1 (a, b) apple r. If a 2 A is not matched, then d 1 (a, ) apple r. If b 2 B is not matched, then d 1 (b, ) apple r. The bottleneck distance between A, B is d B(A, B) =inf r 0 r there exists an r-matching between A, B

24 Persistence modules! Persistence diagrams Algebraic Stability Theorem (Chazal et al. 2009) If two q-tame persistence modules are r-interleaved, then their persistence diagrams are r-matched. persistence module / persistence diagram is 1-Lipschitz Ingredients Box Lemma. Glisse Interpolation Lemma.

25 Persistence modules! Persistence diagrams Algebraic Stability Theorem (Chazal et al. 2009) If two q-tame persistence modules are r-interleaved, then their persistence diagrams are r-matched. Ingredients Box Lemma. persistence module / persistence diagram is 1-Lipschitz Glisse Interpolation Lemma. Box Lemma Let V, W be r-interleaved persistence modules with persistence diagrams A, B. Then the number of points of A inside any rectangle R is no greater than the number of points of B inside the rectangle R r obtained by thickening R on both sides by r (provided that R r does not meet the diagonal).

26 Persistence modules! Persistence diagrams Algebraic Stability Theorem (Chazal et al. 2009) If two q-tame persistence modules are r-interleaved, then their persistence diagrams are r-matched. Ingredients Box Lemma. persistence module / persistence diagram is 1-Lipschitz Glisse Interpolation Lemma. Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t.

27 > > / Persistence modules! Persistence diagrams Box Lemma Let V, W be r-interleaved persistence modules with persistence diagrams A, B. Then the number of points of A inside any rectangle R is no greater than the number of points of B inside the rectangle R r obtained by thickening R on both sides by r (provided that R r does not meet the diagonal). Outline Proof Let R =[a, b] [c, d] sothatr r =[a r, b + r] [c r, d + r]. Consider the diagram: a r vb a vc V a / b vd V b / c V c / V d b c r d W a r w a r b+r / W b+r / W c r w b+r c r w c r d+r W d+r

28 > > Persistence modules! Persistence diagrams Box Lemma Let V, W be r-interleaved persistence modules with persistence diagrams A, B. Then the number of points of A inside any rectangle R is no greater than the number of points of B inside the rectangle R r obtained by thickening R on both sides by r (provided that R r does not meet the diagonal). Outline Proof Let R =[a, b] [c, d] sothatr r =[a r, b + r] [c r, d + r]. Consider the diagram: V a v a b / V b V c v c d / V d a r b c r d W a r W b+r / W c r w b+r c r W d+r

29 Persistence modules! Persistence diagrams Box Lemma Let V, W be r-interleaved persistence modules with persistence diagrams A, B. Then the number of points of A inside any rectangle R is no greater than the number of points of B inside the rectangle R r obtained by thickening R on both sides by r (provided that R r does not meet the diagonal). Outline Proof Let R =[a, b] [c, d] sothatr r =[a r, b + r] [c r, d + r]. W a r / V a / V b / W b+r / W c r / V c / V d / W d+r Rectangle point counts are obtained by counting interval summands: # B \ R r : # A \ R :

30 Persistence modules! Persistence diagrams Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t. Remarks The lemma can be proved by direct algebraic construction; or by applying the Kan Extension Theorem from category theory. The lemma can be circumvented if the initial data can be interpolated.

31 Persistence modules! Persistence diagrams Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t. Remarks The lemma can be proved by direct algebraic construction; or by applying the Kan Extension Theorem from category theory. The lemma can be circumvented if the initial data can be interpolated. Simplex streams Let S be a simplicial complex and let f, g : S! R define simplex streams. Set f t =(1 t)f + tg.

32 Persistence modules! Persistence diagrams Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t. Remarks The lemma can be proved by direct algebraic construction; or by applying the Kan Extension Theorem from category theory. The lemma can be circumvented if the initial data can be interpolated. Sublevelset filtrations Let M be a compact smooth manifold with Morse functions f, g : M! R. Set f t =(1 t)f + tg.

33 Persistence modules! Persistence diagrams Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t. Remarks The lemma can be proved by direct algebraic construction; or by applying the Kan Extension Theorem from category theory. The lemma can be circumvented if the initial data can be interpolated. Sublevelset filtrations Let P be a compact polyhedron with piecewise linear maps f, g : P! R. Set f t =(1 t)f + tg.

34 Persistence modules! Persistence diagrams Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t. Remarks The lemma can be proved by direct algebraic construction; or by applying the Kan Extension Theorem from category theory. The lemma can be circumvented if the initial data can be interpolated. Metric data (fixed set) Let X be a finite set and let d, e : X X! R be metrics. Set d t =(1 t)d + te.

35 Persistence modules! Persistence diagrams Glisse Interpolation Lemma Let V, W be r-interleaved persistence modules. Then there is a 1-parameter family of persistence modules (V t t 2 [0, r]) such that V 0 = V and V r = W,withtheadditionalpropertythatV s, V t are s t -interleaved for all s, t. Remarks The lemma can be proved by direct algebraic construction; or by applying the Kan Extension Theorem from category theory. The lemma can be circumvented if the initial data can be interpolated. Metric data (subsets of a fixed space) Let X, Y Z be finite subsets of a fixed metric space (Z, d). Set X t = {(1 t)x + ty x 2 X, y 2 Y, d(x, y) apple d H(X, Y )}.

36 Persistence modules! Persistence diagrams Outline proof of Algebraic Stability Theorem The Box Lemma can be used to show that persistence module / persistence diagram is locally 1-Lipschitz (near persistence modules with a finite diagram). Local 1-Lipschitz implies global 1-Lipschitz, by applying a Heine Borel argument along a path provided by the Glisse Interpolation Lemma.