Geometric Data. Goal: describe the structure of the geometry underlying the data, for interpretation or summary

Transcription

1 Geometric Data - ma recherche s inscrit dans le contexte de l analyse exploratoire des donnees, dont l obj Input: point cloud equipped with a metric or (dis-)similarity measure data point image/patch, geometric shape, protein conformation, patient, LinkedIn user...

2 Geometric Data - ma recherche s inscrit dans le contexte de l analyse exploratoire des donnees, dont l obj Input: point cloud equipped with a metric or (dis-)similarity measure data point image/patch, geometric shape, protein conformation, patient, LinkedIn user... Goal: describe the structure of the geometry underlying the data, for interpretation or summary

3 Geometric Data - ma recherche s inscrit dans le contexte de l analyse exploratoire des donnees, dont l obj Input: point cloud equipped with a metric or (dis-)similarity measure data point image/patch, geometric shape, protein conformation, patient, LinkedIn user... Goal: describe the structure of the geometry underlying the data, for interpretation or summary

4 Mathematical framework geometric data set / underlying space compact metric space ambient / extrinsic distance intrinsic distance 2

5 Mathematical framework geometric data set / underlying space compact metric space distance between compact metric spaces Gromov-Hausdorff (GH) distance Z γx X γy Y 2

6 Mathematical framework geometric data set / underlying space compact metric space Euclidean distance distance between compact metric spaces Gromov-Hausdorff (GH) distance dgh = 2

7 Mathematical framework geometric data set / underlying space compact metric space Euclidean distance distance between compact metric spaces Gromov-Hausdorff (GH) distance dgh > 2

8 Mathematical framework geometric data set / underlying space compact metric space geodesic distance distance between compact metric spaces Gromov-Hausdorff (GH) distance dgh = 2

9 Mathematical framework geometric data set / underlying space compact metric space distance between compact metric spaces Gromov-Hausdorff (GH) distance descriptor persistence diagram (choose the filtration) - multi-scale reflects the structure of the shape across scales - global/local attached to the whole shape / to a base point(s) - stable variations with GH-distance and base point location are controlled 2

10 Why use descriptors isometries GH distance hard to compute [Bronstein 2, Kimmel 26] [Mémoli 27] shape space descriptors space [Agarwal [Agarwal et al. et 25] al. 25] show that it is NP-hard to approximate the GH distance within a factor equality distance easy to compute 3

11 Why use descriptors isometries GH distance hard to compute [Bronstein 2, Kimmel 26] [Mémoli 27] shape space descriptors space [Agarwal [Agarwal et al. et 25] al. 25] show that it is NP-hard to approximate the GH distance within a factor Ideally, descriptors distance = GH distance equality distance easy to compute 3

12 Why use descriptors isometries GH distance hard to compute [Bronstein 2, Kimmel 26] [Mémoli 27] shape space descriptors space [Agarwal [Agarwal et al. et 25] al. 25] show that it is NP-hard to approximate the GH distance within a factor Ideally, descriptors distance = GH distance In reality, equality distance easy to compute 3

13 Why use descriptors Some descriptors for images / 3d shapes / metric spaces: diameter curvature (mean, Gaussian, sectional) shape context (distribution of distances) [Agarwal et al. 25] show that it is NP-hard to approximate the GH distance within a factor heat kernel signature (heat diffusion) wave kernel signature (Maxwell s equations) spin image (local neighborhood parametrization) SIFT features (local distribution of gradient orientations) etc. 3

14 Why use descriptors Some descriptors for images / 3d shapes / metric spaces: diameter curvature (mean, Gaussian, sectional) shape context (distribution of distances) [Agarwal et al. 25] show that it is NP-hard to approximate the GH distance within a factor heat kernel signature (heat diffusion) wave kernel signature (Maxwell s equations) spin image (local neighborhood parametrization) SIFT features (local distribution of gradient orientations) etc. geometry statistics 3

15 Menu. Global topological descriptors 2. Local topological descriptors 45

16 Global topological descriptors Input: a compact metric space (X, d X ) ThisDescriptor: signature reveals the structure of the metric space across scales dgm F(X, d X ), where F(X, d X ) is some simplicial filtration over X derived from d X (proxy for union of balls) dimensional homology generators -dimensional homology generators 6

17 Global topological descriptors Input: a compact metric space (X, d X ) This signature reveals the structure of the metric space across scales Descriptor: dgm F(X, d X ), where F(X, d X ) is some simplicial filtration over X derived from d X (proxy for union of balls) t C t (X, d X ) popular choices: - Čech/Nerve filtration C(X, d X ) R 2t (X, d X ) - (Vietoris)-Rips filtration R(X, d X ) 6

18 Global topological descriptors Input: a compact metric space (X, d X ) This signature reveals the structure of the metric space across scales Descriptor: dgm F(X, d X ), where F(X, d X ) is some simplicial filtration over X derived from d X (proxy for union of balls) t C t (X, d X ) = R 2t (X, d X ) popular choices: - Čech/Nerve filtration C(X, d X ) - (Vietoris)-Rips filtration R(X, d X ) 6

19 Some examples Descriptors of some elementary shapes (approximated from finite samples): Euclidean geodesic 7

20 This equality is fine, since both curves are isometric when equipped with the geodesic distance (their total lengths are the same). The Euclidean distance Descriptors of some elementary shapes (approximated from finite samples): geodesic Euclidean Some examples 7

21 This equality is not fine, since the two spaces are not isometric. Note that the e Descriptors of some elementary shapes (approximated from finite samples): geodesic Euclidean Some examples 7

22 Stability Theorem: [Chazal, de Silva, O. 23] For any compact metric spaces (X, d X ) and (Y, d Y ), d B (dgm R(X, d X ), dgm R(Y, d Y )) 2d GH (X, Y ). 8

23 Stability Theorem: [Chazal, de Silva, O. 23] For any compact metric spaces (X, d X ) and (Y, d Y ), d B (dgm R(X, d X ), dgm R(Y, d Y )) 2d GH (X, Y ). The bound is worst-case tight... X = d GH (X, Y ) = ε dgm R(X, d X ) = {(, ), (, )} Y = + 2ε dgm R(Y, d Y ) = {(, ), (, + 2ε)} d B (dgm R(X, d X), dgm R(Y, d Y )) = 2ε 8

24 Stability Theorem: [Chazal, de Silva, O. 23] For any compact metric spaces (X, d X ) and (Y, d Y ), d B (dgm R(X, d X ), dgm R(Y, d Y )) 2d GH (X, Y ). The bound is worst-case tight... but it is still only an upper bound this means that the distance between our signatures is still only a lower bound on the G X = d GH (X, Y ) = 2 - any one-to-one mapping X Y has a metric distortion of - any correspondence has a metric d dgm R(X, d X ) = {(, ), (, ), (.)} Y = 2 dgm R(Y, d Y ) = {(, ), (, ), (, )} d B (dgm R(X, d X), dgm R(Y, d Y )) = 8

25 Stability Theorem: [Chazal, de Silva, O. 23] For any compact metric spaces (X, d X ) and (Y, d Y ), d B (dgm R(X, d X ), dgm R(Y, d Y )) 2d GH (X, Y ). Variants and extensions: - other filtrations: v Cech / Nerve, witness complex, etc. - larger classes of metric spaces: precompact, totally bounded, etc. - (dis-)similarity measures 8

26 Stability finite the issue with infinite spaces is that they give rise to infinite Rips complexes, whose filtrations may or may not be tame and th Theorem: [Chazal, de Silva, O. 23] For any compact metric spaces (X, d X ) and (Y, d Y ), d B (dgm R(X, d X ), dgm R(Y, d Y )) 2d GH (X, Y ). Proof outline: (X, d X ) (Z, d Z ) (R X + Y, l ) γ X id (γ X (X) γ Y (Y ), d Z ) γ γ Y id (Y, d Y ) 8

27 Toy application (unsupervised shape classification) 9

28 Toy application (unsupervised shape classification) camels cats elephants faces heads horses camels cats elephants faces heads horses 9

29 Toy application (unsupervised shape classification) camel cat elephant face head horse

30 Local topological descriptors Goal: associate PD to every point choice of filtration(s)? stability guarantees?

31 Local topological descriptors Input: a compact intrinsic metric space (X, d X ), a basepoint x X Construction: filtration of the sublevel sets of d x ( ) = d X (x, ) Descriptor: persistence diagram of the filtration, denoted dgm d x In practice: compute descriptor from point cloud using a pair of Rips complexes [Chazal et al. 29]

32 Stability Theorem: [Carrière, O., Ovsjanikov 25] Let (X, d X ) and (Y, d Y ) be compact intrinsic metric spaces with positive convexity radius (ϱ(x), ϱ(y ) > ). Let x X and y Y. If d GH ((X, x ), (Y, y )) min{ϱ(x), ϱ(y )}, then 2 d B (dgm d x, dgm d y ) 2 d GH ((X, x ), (Y, y )).

33 Stability Theorem: [Carrière, O., Ovsjanikov 25] Let (X, d X ) and (Y, d Y ) be compact intrinsic metric spaces with positive convexity radius (ϱ(x), ϱ(y ) > ). Let x X and y Y. If d GH ((X, x ), (Y, y )) min{ϱ(x), ϱ(y )}, then 2 d B (dgm d x, dgm d y ) 2 d GH ((X, x ), (Y, y )). Definitions: convexity radius Gromov-Hausdorff distance between pointed spaces

34 Stability Theorem: [Carrière, O., Ovsjanikov 25] Let (X, d X ) and (Y, d Y ) be compact intrinsic metric spaces with positive convexity radius (ϱ(x), ϱ(y ) > ). Let x X and y Y. If d GH ((X, x ), (Y, y )) min{ϱ(x), ϱ(y )}, then 2 d B (dgm d x, dgm d y ) 2 d GH ((X, x ), (Y, y )). Prerequisite: d GH (X, Y ) < 2 min{ϱ(x), ϱ(y )} X = Y = d GH (X, Y ) < = ϱ(y ) d B (dgm f, dgm g) =

35 Toy application (unsupervised shape segmentation) Experimental results: - input: shapes from the TOSCA database, in mesh form (triangulated) - select a few base points by hand on each shape - approximate geodesic distances to base points using the -skeleton graph - use the PDs of the PL interpolations over the meshes as descriptors 2

36 Toy application (unsupervised shape segmentation) Experimental results: 2

37 Toy application (unsupervised shape segmentation) Experimental results: 2

38 Toy application (unsupervised shape segmentation) Experimental results: mapping to R 3 via MDS k-means in R 3 note that blue and green are not contiguous after MDS 2

39 Toy application (unsupervised shape segmentation) Experimental results: mapping to R 3 via MDS k-means in R 3 2

40 Toy application (supervised shape segmentation) Goal: segment 3d shapes based on examples Approach: - train a (multiclass) classifier on PDs extracted from the training shapes - apply classifier to PDs extracted from each test shape Head Torso Label =? Foot Hand Training Test 3

41 Toy application (supervised shape segmentation) Strategy: use k-nn classifier in diagram space (equipped with d B ) 3

42 Recap finite metric space / basepoint filtration persistence diagram Pros: topological descriptors carry information of a different nature -dimensional homology generators -dimensional homology generators they enjoy stability properties, e.g. d B (R(X), R(Y )) 2d GH(X, Y ) 4

43 Recap finite metric space / basepoint filtration persistence diagram Pros: topological descriptors carry information of a different nature -dimensional homology generators -dimensional homology generators they enjoy stability properties, e.g. d B (R(X), R(Y )) 2d GH(X, Y ) Cons: this implies the space that oflinear persistence classifiers diagrams cannotisbe not used a vector/hilbert PDs have space been mostly used in unsuper bad for supervised learning and statistics having descriptors long construction can be slow times to is compute fine as long and (more as comparisons importantly) canto becompare made fast, but here t bad for applications 4