Detection and approximation of linear structures in metric spaces

Transcription

1 Stanford - July 12, 2012 MMDS 2012 Detection and approximation of linear structures in metric spaces Frédéric Chazal Geometrica group INRIA Saclay Joint work with M. Aanjaneya, D. Chen, M. Glisse, L. Guibas, D. Morozov and on-going work with Jian Sun

2 Introduction Branching filamentary structures (metric graphs) appear in a wide of real world data sets. Data possibly not embedded in Euclidean space and only coming with (local) pairwise distance information metric spaces Problem: Can we recover the underlying metric graph structure from approximating data?

3 Problem statement Input: a metric space (Y, d Y ) (e.g. a point cloud with distance matrix). Y 1

4 Problem statement Input: a metric space (Y, d Y ) (e.g. a point cloud with distance matrix). Promise: the input is close to some unknown metric graph X Basically, a metric graph is a continuous metric space that has the structure of a graph, with edges and vertices, such that the edge interiors also belong X Y 1

5 Problem statement Input: a metric space (Y, d Y ) (e.g. a point cloud with distance matrix). Promise: the input is close to some unknown metric graph X Basically, a metric graph is a continuous metric space that has the structure of a graph, with edges and vertices, such that the edge interiors also belong A metric graph is a path metric space (X, d X ) that is homeomorphic to a 1-dimensional stratified space. A vertex of X is a 0-dimensional stratum of X and an edge of X is a 1-dimensional stratum of X. the distance between any pair of points is equal to the infimum of the lengths of the continuous curves joining them X Y 1

6 Problem statement Input: a metric space (Y, d Y ) (e.g. a point cloud with distance matrix). Promise: the input is close to some unknown metric graph X Basically, a metric graph is a continuous metric space that has the structure of a graph, with edges and vertices, such that the edge interiors also belong Output: - a metric graph (ˆX, dˆx) that is close and if possible homeomorphic to X X ˆX Y 1

7 Problem statement Input: a metric space (Y, d Y ) (e.g. a point cloud with distance matrix). Promise: the input is close to some unknown metric graph X Basically, a metric graph is a continuous metric space that has the structure of a graph, with edges and vertices, such that the edge interiors also belong Output: - a metric graph (ˆX, dˆx) that is close and if possible homeomorphic to X - a map Y ˆX that roughly preserves distances (quasi-isometry) X ˆX Y 1

8 Problem statement Input: a metric space (Y, d Y ) (e.g. a point cloud with distance matrix). Promise: the input is close to some unknown metric graph X Basically, a metric graph is a continuous metric space that has the structure of a graph, with edges and vertices, such that the edge interiors also belong Output: - a metric graph (ˆX, dˆx) that is close and if possible homeomorphic to X - a map Y ˆX that roughly preserves distances (quasi-isometry) In this talk: - A few basic topological and geometric ideas to address this problem. - Topology guaranteed graph reconstruction based upon degree inference. - Linear structure detection and tree reconstruction through metric hyperbolic geometry. 1

9 Metric spaces approximation Let (X, d X ), (Y, d Y ) be two metric spaces. Y y y C x x X An ε-correspondence between (X, d X ) and (Y, d Y ) is a set C X Y s. t. 1. for any x X (resp. y Y), there exists y Y (resp. x X) s. t. (x, y) C. 2. For any (x, y), (x, y ) C, d X (x, x ) d Y (y, y ) ε. The Gromov-Hausdorff d GH (X, Y) = 1 distance: inf{ε > 0 : there exists an ε-correspondence betweenxandy } 2

10 A first approach using the local topology of data [Aanjaneya, C., Chen, Glisse, Guibas, Morozov, SoCG 11, IJCGA (2012)] X Y The degree of a point x on X is the number of connected components of a sufficiently small (intrinsic) sphere centered at x. Vertices of X are the points with degree 1 or larger than 2. The degree of (most) points can be inferred from Y by looking at (intrinsic) spherical shells: - chose a parameter r > 0, - deg r (y) = nb. of clusters of B Y (y, 5r/3) \ B Y (y, r) (single linkage).

11 The algorithm Input: (Y, d Y ) approximating a metric graph (X, d X ) and parameter r > 0. Output: A metric graph (ˆX, dˆx). 1. Labelling points as edge or branch for all y Y do if deg r (y) = 2 then label y as an edge point. else label y as a branch point. Label all points within distance 2r from a preliminary branch point as branch points.

12 The algorithm Input: (Y, d Y ) approximating a metric graph (X, d X ) and parameter r > 0. Output: A metric graph (ˆX, dˆx). Graph structure reconstruction E points of Y labeled as edge points; V points of Y labeled as branch points. Clusterize E and V (single linkage) Vertices of ˆX clusters of V. Put an edge between vertices of ˆX if their corresponding clusters are close to a same cluster of E.

13 The algorithm Input: (Y, d Y ) approximating a metric graph (X, d X ) and parameter r > 0. Output: A metric graph (ˆX, dˆx). Reconstructing the metric To each edge ê of ˆX assign a length equal to the diameter of the corresponding cluster of E plus 4r.

14 Theoretical guarantees Let (Y, d Y ) be an ε-approximation of a metric graph (X, d X ) for some ε > 0. Let b > 0 be the length of the shortest edge of X. If r > 0 is correctly chosen with respect to ε and b then : Topological Reconstruction theorem: ˆX is homeomorphic to X. Metric Reconstruction theorem: there is an almost distance preserving map (quasi-isometry) φ : X ˆX. Quasi-isometric embedding theorem: there is an almost distance preserving map (quasi-isometry) ψ : Y ˆX.

15 Experimental results Data: GPS traces (sampled curves along a road network). (Y, d Y ): a neighborhood graph (Rips) built on the data set with its intrinsic metric.

16 Experimental results Data: earthquakes epicenters (preprocessed to remove outliers/noise ). (Y, d Y ): a neighborhood graph (Rips) with its intrinsic metric.

17 Advantages and drawbacks of the previous algorithm Good points: A simple algorithm for metric graph reconstruction: - coming with topological and metric guarantees, - relying on intrinsic metric information (no need of coordinates).

18 Advantages and drawbacks of the previous algorithm Good points: A simple algorithm for metric graph reconstruction: - coming with topological and metric guarantees, - relying on intrinsic metric information (no need of coordinates). Weak points and open questions: Choice of the parameter r (that relies on ε and b): how to do it in practice? How to make r dependent of the local quality of the approximation? Interpretation of the output of the algorithm when the sampling conditions are not fullfilled: the algorithm might not output a graph (edges adjacent to more than 2 vertices).

19 Another approach based upon basic metric geometry [F. C., Jian Sun, work in progress] Locally, metric graphs are trees... In metric trees the geodesic triangles/tetrahedra have a specific shape : metric trees are 0-hyperbolic. z y z y w x x

20 Another approach based upon basic metric geometry [F. C., Jian Sun, work in progress] Locally, metric graphs are trees... In metric trees the geodesic triangles/tetrahedra have a specific shape : metric trees are 0-hyperbolic. z y z y w x x

21 Another approach based upon basic metric geometry [F. C., Jian Sun, work in progress] Locally, metric graphs are trees... In metric trees the geodesic triangles/tetrahedra have a specific shape : metric trees are 0-hyperbolic. z y z y w x x Goal: Use these remarks/properties to reconstruct metric trees and graphs from approximations.

22 Let (X, d X ) be a metric space. δ-hyperbolic spaces z w y d(x, y) + d(z, w) d(x, w) + d(z, y) d(x, z) + d(y, w) x Given δ > 0, X is δ-hyperbolic if for any x, y, z, w X the two largest of the distance sums d X (x, w)+d X (y, z), d X (x, y)+d X (z, w) and d X (x, z)+d X (y, w) differ by at most 2δ.

23 δ-hyperbolic spaces Real Given tree: δ > 1. 0, any X is two δ-hyperbolic points are the if for ends any of x, a y, unique z, w geodesic X if the segment; two largest 2. of the the unio segments distance intersecting sums d X (x, exactly w)+d X at (y, one z), d of X (x, their y)+d ends X (z, is a w) geodesic and d X (x, segment. z)+d X (y, A geode w) 0-hyperbolic differ by at iff most it is 2δ. a real tree. Basic properties: Metric trees are 0-hyperbolic. This is indeed a characterization of (real) trees. z y w x d(x, y) + d(z, w) = d(x, w) + d(z, y)

24 δ-hyperbolic spaces Real Given tree: δ > 1. 0, any X is two δ-hyperbolic points are the if for ends any of x, a y, unique z, w geodesic X if the segment; two largest 2. of the the unio segments distance intersecting sums d X (x, exactly w)+d X at (y, one z), d of X (x, their y)+d ends X (z, is a w) geodesic and d X (x, segment. z)+d X (y, A geode w) 0-hyperbolic differ by at iff most it is 2δ. a real tree. Basic properties: Metric trees are 0-hyperbolic. This is indeed a characterization of (real) trees. if d GH (X, Y) < ε then (X is δ-hyperbolic) (Y is (δ + 2ε)-hyperbolic). z y w x d(x, y) + d(z, w) = d(x, w) + d(z, y)

25 δ-hyperbolic spaces Real Given tree: δ > 1. 0, any X is two δ-hyperbolic points are the if for ends any of x, a y, unique z, w geodesic X if the segment; two largest 2. of the the unio segments distance intersecting sums d X (x, exactly w)+d X at (y, one z), d of X (x, their y)+d ends X (z, is a w) geodesic and d X (x, segment. z)+d X (y, A geode w) 0-hyperbolic differ by at iff most it is 2δ. a real tree. Basic properties: Metric trees are 0-hyperbolic. This is indeed a characterization of (real) trees. if d GH (X, Y) < ε then (X is δ-hyperbolic) (Y is (δ + 2ε)-hyperbolic). Question: Can we use δ-hyperbolicity to characterize (metric) data that are close (w.r.t. d GH ) to a metric tree?

26 δ-hyperbolic spaces Real Given tree: δ > 1. 0, any X is two δ-hyperbolic points are the if for ends any of x, a y, unique z, w geodesic X if the segment; two largest 2. of the the unio segments distance intersecting sums d X (x, exactly w)+d X at (y, one z), d of X (x, their y)+d ends X (z, is a w) geodesic and d X (x, segment. z)+d X (y, A geode w) 0-hyperbolic differ by at iff most it is 2δ. a real tree. Basic properties: Metric trees are 0-hyperbolic. This is indeed a characterization of (real) trees. if d GH (X, Y) < ε then (X is δ-hyperbolic) (Y is (δ + 2ε)-hyperbolic). Question: Can we use δ-hyperbolicity to characterize (metric) data that are close (w.r.t. d GH ) to a metric tree? Not so easy: the Poincaré disc is log(3)-hyperbolic!

27 Distance functions and persistence tree Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Equivalence relation: x y iff d(x) = d(y) and there exists a continuous path from x to y contained in d 1 ([d(x), + )). r X

28 Distance functions and persistence tree Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Equivalence relation: x y iff d(x) = d(y) and there exists a continuous path from x to y contained in d 1 ([d(x), + )). T Persistence (merge) tree: T = X/ r X

29 Distance functions and persistence tree Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Equivalence relation: x y iff d(x) = d(y) and there exists a continuous path from x to y contained in d 1 ([d(x), + )). T Persistence (merge) tree: T = X/ Let Π : X T be the quotient map. r X Lemma: (i) T is a tree; its leaves are in one-to-one correspondence with the local maxima of d. (ii) T inherits a metric structure from d T (Πr, Πx) := d X (r, x) for all x X. In practice: T can be efficiently constructed using an union find data structure (0-persistence algorithm) on the input data endowed with a neighboring graph.

30 Tree reconstruction Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Lemma: if X is a metric tree then T is isometric to X. T r X

31 Tree reconstruction Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Lemma: if X is a metric tree then T is isometric to X. T r X What can we say about d GH (X, T) when X is δ-hyperbolic? In general nothing... But...

32 Tree reconstruction Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Not unique in general m x For x X let m x X be in the c.c. of d 1 ([d(x), + )) containing x that maximizes d on this c.c. and let f : X R + be defined by f(x) = (r m x ) x := d X (x, r)+d X (x, m x ) d X (r, m x ). x T r X Gromov product In practice: f can be easily computed as T is constructed (using the 0- persistence algorithm).

33 Tree reconstruction Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Not unique in general m x For x X let m x X be in the c.c. of d 1 ([d(x), + )) containing x that maximizes d on this c.c. and let f : X R + be defined by f(x) = (r m x ) x := d X (x, r)+d X (x, m x ) d X (r, m x ). x T r X Gromov product Theorem: Let M = f = max x X f(x). If X is δ-hyperbolic then d GH (X, T) < M + 9δ and for any x, y X, d T (Πx, Πy) d X (x, y) 2(M + 9δ).

34 Tree reconstruction Let (X, d X ) be a compact path metric space. Given r X, let d(.) = d X (r,.) the distance function to r in X (assume d has a finite number of local maxima). Not unique in general m x For x X let m x X be in the c.c. of d 1 ([d(x), + )) containing x that maximizes d on this c.c. and let f : X R + be defined by f(x) = (r m x ) x := d X (x, r)+d X (x, m x ) d X (r, m x ). x T r X Gromov product Theorem: Let M = f = max x X f(x). If X is δ-hyperbolic then d GH (X, T) < M + 9δ and for any x, y X, d T (Πx, Πy) d X (x, y) 2(M + 9δ). Corollary: there exists a constant C < 100 s.t. if T 0 is a tree and if d GH (X, T 0 ) < ε then d GH (T, T 0 ) < Cε.

35 Some practical considerations When X is a neighboring graph (e.g. Rips-Vietoris) built on top of a finite metric data set (Y, d Y )... T, Π : Y T and M can be computed efficiently using the 0-persistence algorithm in almost linear time (in the number of edges of X)!

36 Some practical considerations When X is a neighboring graph (e.g. Rips-Vietoris) built on top of a finite metric data set (Y, d Y )... T, Π : Y T and M can be computed efficiently using the 0-persistence algorithm in almost linear time (in the number of edges of X)! δ can also be computed from the data but in O( Y 4 ) can be improved to O( Y 3 ); ε-nets (making the complexity dependent of the geometry) random samples of tetrahedra?

37 Some practical considerations When X is a neighboring graph (e.g. Rips-Vietoris) built on top of a finite metric data set (Y, d Y )... T, Π : Y T and M can be computed efficiently using the 0-persistence algorithm in almost linear time (in the number of edges of X)! δ can also be computed from the data but in O( Y 4 ) can be improved to O( Y 3 ); ε-nets (making the complexity dependent of the geometry) random samples of tetrahedra? An algorithm that always output a tree and an upperbound on the Gromov- Hausdorff distance between the reconstructed tree and the data (no parameter to choose except to build X from Y). Warning: even if d GH (X, T 0 ) << 1, T is in general not homeomorphic to T 0.

38 From trees to graphs Locally, metric graphs are trees: if G is a metric graph and l(g) is the length of the shortest non-null homotopic simple path then for any r G, (B(r, l(g)/4), d G ) is a metric tree. r

39 Data can be semi-locally approximated by trees. From trees to graphs Locally, metric graphs are trees: if G is a metric graph and l(g) is the length of the shortest non-null homotopic simple path then for any r G, (B(r, l(g)/4), d G ) is a metric tree.

40 Data can be semi-locally approximated by trees. From trees to graphs Locally, metric graphs are trees: if G is a metric graph and l(g) is the length of the shortest non-null homotopic simple path then for any r G, (B(r, l(g)/4), d G ) is a metric tree. Using topological persistence: topology (homology) of G [C., de Silva, Oudot 12], estimation of l(g) [Dey, Sun, Wang 10].

41 Advantages and drawbacks of the previous approach Good points: A simple and efficient algorithm for tree recontruction/approximation. No parameter (except for the construction of the neighboring graph) and upper bounds! Semi-local exploration of the structure of metric data (δ-hyperbolicity can be localized).

42 Advantages and drawbacks of the previous approach Good points: A simple and efficient algorithm for tree recontruction/approximation. No parameter (except for the construction of the neighboring graph) and upper bounds! Semi-local exploration of the structure of metric data (δ-hyperbolicity can be localized). Weak points: No guarantees on the topology (homeomorphism). No global graph reconstruction: - covering the data by trees and gluying them together might lead to choice of parameters issues and pretty bad bounds on the metric approximation.

43 Advantages and drawbacks of the previous approach Good points: A simple and efficient algorithm for tree recontruction/approximation. No parameter (except for the construction of the neighboring graph) and upper bounds! Semi-local exploration of the structure of metric data (δ-hyperbolicity can be localized). Weak points: No guarantees on the topology (homeomorphism). No global graph reconstruction: - covering the data by trees and gluying them together might lead to choice of parameters issues and pretty bad bounds on the metric approximation. However...

44 Perspectives The previous ideas and results can be used to prove some approximation results for the Reeb graph of the distance to a point in a path metric space. Equivalence relation: x y iff d(x) = d(y) and x and y are contained in the same c.c. of d 1 (d(x)). G Reeb graph: G = X/ r X

45 Perspectives The previous ideas and results can be used to prove some approximation results for the Reeb graph of the distance to a point in a path metric space.

46 Thank you for your attention!