Persistence stability for geometric complexes

Transcription

1 Lake Tahoe - December, NIPS workshop on Algebraic Topology and Machine Learning Persistence stability for geometric complexes Frédéric Chazal Joint work with Vin de Silva and Steve Oudot (+ on-going work with M. Glisse, C. Labruère and B. Michel)

2 Introduction Data often come as (sampling of) metric spaces or sets/spaces endowed with a similarity measure with possibly complex topological/geometric structure. TDA: infer relevant topological and geometric features of these spaces. Goals: Better understanding of the phenomena from which data have been observed. Taking advantage of the structure of data for more efficient analysis (dimensionality reduction, parametrization,...). clustering, comparison, classification, visualization, learning,...

3 Introduction Data often come as (sampling of) metric spaces or sets/spaces endowed with a similarity measure with possibly complex topological/geometric structure. TDA: infer relevant topological and geometric features of these spaces. Challenges: no direct access to topological/geometric informationfrom data: need of intermediate constructions (e.g. simplicial complexes); distinguish topological signal from noise: robustness, stability results; topological information may be multiscale; statistical analysis of topological information. Curve or torus?

4 Topological signatures for data Metric data set Build a geometric filtered simplicial complex on top of the data Filtered simplicial complex Compute persistent homology of the complex. Signature: persistence diagram dimensional homology generators 1-dimensional homology generators Approach: Build a geometric filtered simplicial complex on top of (X, d X ) (d X metric/similarity on X). being a

5 Topological signatures for data Metric data set Build a geometric filtered simplicial complex on top of the data Filtered simplicial complex Compute persistent homology of the complex. Signature: persistence diagram dimensional homology generators 1-dimensional homology generators Approach: Build a geometric filtered simplicial complex on top of (X, d X ) (d X metric/similarity on X). being a Compute the persistent homology of the complex persistence diagram: multiscale topological signature.

6 Topological signatures for data camel cat elephant face head horse [C., Cohen-Steiner, Guibas, Mémoli, Oudot 09] Use the metric on the space of persistence diagrams Approach: Build a geometric filtered simplicial complex on top of (X, d X ) (d X metric/similarity on X). being a Compute the persistent homology of the complex persistence diagram: multiscale topological signature. Compare the signatures of close data sets robustness and stability results.

7 Topological signatures for data camel cat elephant face head horse [C., Cohen-Steiner, Guibas, Mémoli, Oudot 09] Use the metric on the space of persistence diagrams Approach: Build a geometric filtered simplicial complex on top of (X, d X ) (d X metric/similarity on X). being a Compute the persistent homology of the complex persistence diagram: multiscale topological signature. Compare the signatures of close data sets robustness and stability results. Statistical properties of signatures.

8 Filtered complexes A simplicial complex S built on top of a set X is a family S of finite subsets of X s.t. if σ S and σ σ then σ S.

9 Filtered complexes A filtered simplicial complex S built on top of a set X is a family (S a a R) of subcomplexes of some fixed simplicial complex S with vertex set X s. t. S a S b for any a b.

10 Filtered complexes Rips Čech Examples: Let (X, d X ) be a metric space. The Vietoris-Rips and Čech complexes Rips(X) and Čech(X) are the filtered complexes defined by: for a R, [x 0, x 1,, x k ] Rips(X, a) d X (x i, x j ) a, for all i, j [x 0, x 1,..., x k ] Čech(X, a) k where B(x, a) = {x X : d X (x, x ) a}. i=0 B(x i, a),

11 Persistent homology of filtered complexes An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties

12 Persistent homology of filtered complexes An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties

13 Persistent homology of filtered complexes An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties

14 Persistent homology of filtered complexes An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties

15 Persistent homology of filtered complexes An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties

16 Persistent homology of filtered complexes An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties

17 Persistent homology of filtered complexes Persistence bar code death An efficient way to encode the evolution of the topology (homology) of a filtered complex. Multiscale topological information. Barcodes/persistence diagrams can be efficiently computed. Stability properties 0 Add the diagonal Multiplicity: 2 birth Persistence diagram

18 Distance between persistence diagrams death Add the diagonal D 1 D 2 Multiplicity: 2 0 The bottleneck distance between two diagrams D 1 and D 2 is d B (D 1, D 2 ) = inf γ Γ sup p D 1 p γ(p) where Γ is the set of all the bijections between D 1 and D 2 and p q = max( x p x q, y p y q ). birth Persistence diagrams provide easy to compare topological signatures. Need to prove that close metric spaces/data sets have close signatues.

19 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. Examples: Let S be a filtered simplicial complex, i.e. a family (S a a R) of subcomplexes of some fixed simplicial complex S, s. t. S a S b for any a b. If V a = H(S a ) and v b a : H(S a ) H(S b ) is the linear map induced by the inclusion S a S b then (H(S a ) a R) is a persistence module. Given a metric space (X, d X ), H(Rips(X)) is a persistence module. Given a metric space (X, d X ), H(Čech(X)) is a persistence module.

20 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. Definition: A persistence module V is q-tame if for any a < b, v b a has a finite rank. Theorem [CCGGO 09-CdSGO 12]: q-tame persistence modules have well-defined persistence diagrams.

21 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. Definition: A persistence module V is q-tame if for any a < b, v b a has a finite rank. Theorem [CCGGO 09-CdSGO 12]: q-tame persistence modules have well-defined persistence diagrams. Theorem[CdSO 12]: Let X be a precompact metric space. Then H(Rips(X)) and H(Čech(X)) are q-tame. As a consequence dgm(h(rips(x))) and dgm(h(čech(x))) are well-defined! Recall that a metric space (X, d X ) is precompact if for any ɛ > 0 there exists a finite subset F ɛ X such that d H (X, F ɛ ) < ɛ (i.e. x X, p F ɛ s.t. d X (x, p) < ɛ).

22 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. A homomorphism of degree ɛ between two persistence modules U and V is a collection Φ of linear maps (φ a : U a V a+ɛ a R) U a V a+ɛ U b V b+ɛ such that v b+ɛ a+ɛ φ a = φ b u b a for all a b. An ε-interleaving between U and V is specified by two homomorphisms of degree ɛ Φ : U V and Ψ : V U s.t. Φ Ψ and Ψ Φ are the shifts of degree 2ɛ between U and V. U a u a+2ɛ a U a+2ɛ φ a V a+ɛ ψ a+ɛ v a+3ɛ a+ɛ V a+3ɛ

23 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. Stability Theorem [CCGGO 09-CdSGO 12]: If U and V are q-tame and ɛ-interleaved for some ɛ 0 then d B (dgm(u), dgm(v)) ɛ

24 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. Stability Theorem [CCGGO 09-CdSGO 12]: If U and V are q-tame and ɛ-interleaved for some ɛ 0 then d B (dgm(u), dgm(v)) ɛ Strategy: build filtered complexes on top of metric spaces that induce q-tame homology persistence modules and that turns out to be ɛ-interleaved when the considered spaces are O(ɛ)-close.

25 Tame persistent modules Definition: A persistence module V is an indexed family of vector spaces (V a a R) and a doubly-indexed family of linear maps (v b a : V a V b a b) which satisfy the composition law v c b v b a = v c a whenever a b c, and where v a a is the identity map on V a. Stability Theorem [CCGGO 09-CdSGO 12]: If U and V are q-tame and ɛ-interleaved for some ɛ 0 then d B (dgm(u), dgm(v)) ɛ Strategy: build filtered complexes on top of metric spaces that induce q-tame homology persistence modules and that turns out to be ɛ-interleaved when the considered spaces are O(ɛ)-close. Need to be defined.

26 Multivalued maps and correspondences Y C X C T X A multivalued map C : X Y from a set X to a set Y is a subset of X Y, also denoted C, that projects surjectively onto X through the canonical projection π X : X Y X. The image C(σ) of a subset σ of X is the canonical projection onto Y of the preimage of σ through π X. Y

27 Multivalued maps and correspondences Y C X C T X A multivalued map C : X Y from a set X to a set Y is a subset of X Y, also denoted C, that projects surjectively onto X through the canonical projection π X : X Y X. The image C(σ) of a subset σ of X is the canonical projection onto Y of the preimage of σ through π X. The transpose of C, denoted C T, is the image of C through the symmetry map (x, y) (y, x). A multivalued map C : X Y is a correspondence if C T is also a multivalued map. Y

28 Multivalued maps and correspondences Y C X C T X A multivalued map C : X Y from a set X to a set Y is a subset of X Y, also denoted C, that projects surjectively onto X through the canonical projection π X : X Y X. The image C(σ) of a subset σ of X is the canonical projection onto Y of the preimage of σ through π X. Example: ɛ-correspondence and Gromov-Hausdorff distance. Let (X, d X ) and (Y, d Y ) be compact metric spaces. A correspondence C : X Y is an ɛ-correspondence if (x, y), (x, y ) C, d X (x, x ) d Y (y, y ) ε. Y y y Y C d GH (X, Y ) = 1 2 x x X inf{ε 0 : there exists an ε-correspondence between Xand Y }

29 Multivalued simplicial maps Y C X C T X Let S and T be two filtered simplicial complexes with vertex sets X and Y respectively. A multivalued map C : X Y is ε-simplicial from S to T if for any a R and any simplex σ S a, every finite subset of C(σ) is a simplex of T a+ε. Y

30 Multivalued simplicial maps Y C X C T X Let S and T be two filtered simplicial complexes with vertex sets X and Y respectively. A multivalued map C : X Y is ε-simplicial from S to T if for any a R and any simplex σ S a, every finite subset of C(σ) is a simplex of T a+ε. Y Proposition: Let S, T be filtered complexes with vertex sets X, Y respectively. If C : X Y is a correspondence such that C and C T are both ε-simplicial, then together they induce a canonical ε-interleaving between H(S) and H(T), the interleaving homomorphisms being H(C) and H(C T ).

31 The example of the Rips and Čech filtration Proposition: Let (X, d X ), (Y, d Y ) be metric spaces. For any ɛ > 2d GH (X, Y ) the persistence modules H(Rips(X)) and H(Rips(Y )) are ɛ-interleaved.

32 The example of the Rips and Čech filtration Proposition: Let (X, d X ), (Y, d Y ) be metric spaces. For any ɛ > 2d GH (X, Y ) the persistence modules H(Rips(X)) and H(Rips(Y )) are ɛ-interleaved. Proof: Let C : X Y be a correspondence with distortion at most ɛ. If σ Rips(X, a) then d X (x, x ) a for all x, x σ. Let τ C(σ) be any finite subset. For any y, y τ there exist x, x σ s. t. y C(x), y C(x ) so d Y (y, y ) d X (x, x ) a + ɛ and τ Rips(Y, a + ɛ) C is ɛ-simplicial from Rips(X) to Rips(Y ). Symetrically, C T is ɛ-simplicial from Rips(Y ) to Rips(X).

33 The example of the Rips and Čech filtration Proposition: Let (X, d X ), (Y, d Y ) be metric spaces. For any ɛ > 2d GH (X, Y ) the persistence modules H(Rips(X)) and H(Rips(Y )) are ɛ-interleaved. Proof: Let C : X Y be a correspondence with distortion at most ɛ. If σ Rips(X, a) then d X (x, x ) a for all x, x σ. Let τ C(σ) be any finite subset. For any y, y τ there exist x, x σ s. t. y C(x), y C(x ) so d Y (y, y ) d X (x, x ) a + ɛ and τ Rips(Y, a + ɛ) C is ɛ-simplicial from Rips(X) to Rips(Y ). Symetrically, C T is ɛ-simplicial from Rips(Y ) to Rips(X). Proposition: Let (X, d X ), (Y, d Y ) be metric spaces. For any ɛ > 2d GH (X, Y ) the persistence modules H(Čech(X)) and H(Čech(Y )) are ɛ-interleaved.

34 The example of the Rips and Čech filtration Proposition: Let (X, d X ), (Y, d Y ) be metric spaces. For any ɛ > 2d GH (X, Y ) the persistence modules H(Rips(X)) and H(Rips(Y )) are ɛ-interleaved. Proof: Let C : X Y be a correspondence with distortion at most ɛ. If σ Rips(X, a) then d X (x, x ) a for all x, x σ. Let τ C(σ) be any finite subset. For any y, y τ there exist x, x σ s. t. y C(x), y C(x ) so d Y (y, y ) d X (x, x ) a + ɛ and τ Rips(Y, a + ɛ) C is ɛ-simplicial from Rips(X) to Rips(Y ). Symetrically, C T is ɛ-simplicial from Rips(Y ) to Rips(X). Proposition: Let (X, d X ), (Y, d Y ) be metric spaces. For any ɛ > 2d GH (X, Y ) the persistence modules H(Čech(X)) and H(Čech(Y )) are ɛ-interleaved. Remark: Similar results for witness complexes (fixed landmarks)

35 Tameness of the Rips and Čech filtrations Theorem: Let X be a precompact metric space. Then H(Rips(X)) and H(Čech(X)) are q-tame. As a consequence dgm(h(rips(x))) and dgm(h(čech(x))) are well-defined! Theorem: Let X, Y be precompact metric spaces. Then d b (dgm(h(čech(x))), dgm(h(čech(y )))) 2d GH(X, Y ), d b (dgm(h(rips(x))), dgm(h(rips(y )))) 2d GH (X, Y ). Important remark: We never use the triangle inequality! The previous approch and results easily extend to other settings like, e.g. spaces endowed with a similarity measure.

36 Some more statistical remarks [on-going work with M. Glisse, C. Labruère and B. Michel] Given a, d > 0 let X = X (a, d) the set of precompact mm-spaces (X, d X, µ X ) s. t. that x X, r 0, µ X (B(x, r)) min(1, ar d ).

37 Some more statistical remarks Given a, d > 0 let X = X (a, d) the set of precompact mm-spaces (X, d X, µ X ) s. t. that x X, r 0, µ X (B(x, r)) min(1, ar d ). Let X = (X, d X, µ X ) X. Let x = {x 1,, x n } be n i.i.d samples drawn from µ X. Theorem: [on-going work with M. Glisse, C. Labruère and B. Michel] P (d b (dgm(rips(x)), dgm(rips(x))) > 4ɛ) 2d aɛ d exp( naɛd ).

38 Some more statistical remarks Given a, d > 0 let X = X (a, d) the set of precompact mm-spaces (X, d X, µ X ) s. t. that x X, r 0, µ X (B(x, r)) min(1, ar d ). Let X = (X, d X, µ X ) X. Let x = {x 1,, x n } be n i.i.d samples drawn from µ X. Theorem: [on-going work with M. Glisse, C. Labruère and B. Michel] P (d b (dgm(rips(x)), dgm(rips(x))) > 4ɛ) 2d aɛ d exp( naɛd ). Convergence results, confidence intervals, tests,... No need to go through a notion of mean. Similar results with other geometric complexes more resilient to noise and outliers.

39 Some more statistical remarks Given a, d > 0 let X = X (a, d) the set of precompact mm-spaces (X, d X, µ X ) s. t. that x X, r 0, µ X (B(x, r)) min(1, ar d ). Let X = (X, d X, µ X ) X. Let x = {x 1,, x n } be n i.i.d samples drawn from µ X. Theorem: [on-going work with M. Glisse, C. Labruère and B. Michel] P (d b (dgm(rips(x)), dgm(rips(x))) > 4ɛ) 2d aɛ d exp( naɛd ). Convergence results, confidence intervals, tests,... No need to go through a notion of mean. Similar results with other geometric complexes more resilient to noise and outliers. Practical question: Estimation of a and d?

40 Some more statistical remarks [on-going work with M. Glisse, C. Labruère and B. Michel] X E [ d b (dgm(rips(x)), dgm(rips(x))) ] n Example: Corollary: E µ n [d b (dgm(rips(x)), dgm(rips(x)))] C where C only depends on a and d. ( log n n ) 1 d Theorem: Let a > 0 and 1 d D. There exists a constant C > 0 which depends only on a and d such that for any estimator dgm n of dgm(rips(x)): [ lim inf n n1/d sup E µ n d b (dgm(rips(m µ )), dgm ] n ) C. (X,µ) X (a,d) Remark: better bounds if X is a submanifold of R D - see [Genovese, Perone-Pacifico,Verdinelli, Wasserman 2011, 2012]

41 Some more statistical remarks [on-going work with M. Glisse, C. Labruère and B. Michel] - The considered circle and square have the same diagram D (α-shape filtration). - 20, 000 samples of 300 points on each shape histograms of the distances to D. An open question: what are the relations between the distribution of PD or errors and geometry of the sampled spaces?

42 Take-home message Persistence diagrams of geometric complexes built on top of data provide a very general and flexible way to define multiscale topological signatures for data. Correspondences provide an efficient way to compare data endowed with some structure: here metrics ( Gromov-Hausdorff distance), but one can also consider data with scalar fields, measures,... Gromov-Who-ever youwant distance. Notion of ɛ-simplicial correspondence leads to robustness of signatures. Interesting statistical properties of topological signatures. References: F. Chazal, V. de Silva, S. Oudot, Persistence Stability for Geometric complexes, arxiv: , July F. Chazal, V. de Silva, M. Glisse, S. Oudot, The Structure and Stability of Persistence Modules, arxiv: , July 2012.