Some Topics in Computational Topology

1. Some Topics in Computational Topology YusuWang Ohio State University AMS Short Course 2014

2. Introduction • Much recent developments in computational topology • Both in theory and in their applications • E.g, the theory of persistence homology • [Edelsbrunner, Letscher, Zomorodian, DCG 2002], [Zomorodian and Carlsson, DCG 2005], [Carlsson and de Silva, FoCM 2010], … • This short course: • A computational perspective: • Estimation and inference of topological information / structure from point clouds data

3. Develop discrete analog for the continuous case • Approximation from discrete samples with theoretical guarantees • Algorithmic issues ? Topological summary of hidden space Unorganized PCD

4. Main Topics • From PCDs to simplicial complexes • Sampling conditions • Topological inferences

5. Outline • From PCDs to simplicial complexes • Delaunay, Cech, Vietoris-Rips, witness complexes • Graph induced complex • Sampling conditions • Local feature size, and homological feature size • Topology inference • Homology inference • Handling noise • Approximating cycles of shortest basis of the first homology group • Approximating Reeb graph

6. From PCD to Simplicial Complexes Choice of SimplicialComplexes to build on top of point cloud data

7. Delaunay Complex • Given a set of points • Delaunay complex • A simplex is in if and only if • There exists a ball whose boundary contains vertices of , and that the interior of contains no other point from .

8. Delaunay Complex • Many beautiful properties • Connection to Voronoi diagram • Foundation for surface reconstruction and meshing in 3D • [Dey, Curve and Surface Reconstruction, 2006], • [Cheng, Dey and Shewchuk, Delaunay Mesh Generation, 2012] • However, • Computationally very expensive in high dimensions

9. Čech Complex • Given a set of points • Given a real value , the Čech complex is the nerve of the set • where • I.e, a simplex is in if • The definition can be extended to a finite sample of a metric space.

10. Rips Complex • Given a set of points • Given a real value , the Vietoris-Rips (Rips) complex is: • .

11. Rips and ČechComplexes • Relation in general metric spaces • Bounds better in Euclidean space • Simple to compute • Able to capture geometry and topology • We will make it precise shortly • One of the most popular choices for topology inference in recent years • However: • Huge sizes • Computation also costly

12. Witness Complexes • A simplex is weakly witnessed by a point x if for any and. • is strongly witnessed if in addition • Given a set of points and a subset • The witness complex is the collection of simplices with vertices from whose all subsimplices are weakly witnessed by a point in . • [de Silva and Carlsson, 2004] [de Silva 2003] • Can be defined for a general metric space • does not have to be a finite subset of points

13. Intuition • : landmarks from , a way to subsample.

14. Witness Complexes • Greatly reduce size of complex • Similar to Delaunay triangulation, remove redundancy • Relation to Delaunay complex • if • if is a smooth 1- or 2-manifold • [Attali et al, 2007] • However, • Does not capture full topology easily for high-dimensional manifolds

15. Remark • Rips complex • Capture homology when input points are sampled dense enough • But too large in size • Witness complex • Use a subsampling idea • Reduce size tremendously • May not be easy to capture topology in high-dimensions • Combine the two ? • Graph induced complex • [Dey, Fan, Wang, SoCG 2013]

16. Subsampling

17. Subsampling -cont

18. Subsampling -cont

19. Subsampling - cont

20. Graph Induced Complex • [Dey, Fan, Wang, SoCG 2013] • : finite set of points • : metric space • : a graph • a subset • the closest point of in

21. Graph Induced Complex • Graph induced complex • if and only if there is a (k+1)-clique in with vertices such that for any • Similar to geodesic Delaunay [Oudot, Guibas, Gao, Wang, 2010] • Graph induced complexdepends on the metric : • Euclidean metric • Graph based distance

22. Graph Induced Complex • Small size, but with homology inference guarantees • In particular: • inference from a lean sample

23. Graph Induced Complex • Small size, but with homology inference guarantees • In particular: • inference from a lean sample • Surface reconstruction in • Topological inference for compact sets in using persistence

24. Outline • From PCDs to simplicial complexes • Delaunay, Cech, Vietoris-Rips, witness complexes • Graph induced complex • Sampling conditions • Local feature size, and homological feature size • Topology inference • Homology inference • Handling noise • Approximating cycles of shortest basis of the first homology group • Approximating Reeb graph

25. Sampling Conditions

26. Motivation • Theoretical guarantees are usually obtained when input points sampling the hidden domain “well enough”. • Need to quantify the “wellness”. • Two common ones based on: • Local feature size • Weak feature size

27. Distance Function • a compact subset of • Distance function • is a -Lipschitz function • -offset of • Given any point

28. Medial Axis • The medial axis of is the closure of the set of points such that • means that there is a medial ball touching at more than point and whose interior is empty of points from . Courtesy of [Dey, 2006]

29. Local Feature Size • The local feature size at a point is the distance of to the medial axis of • That is, • This concept is adaptive • Large in a place without “features” • Intuitively: • We should sample more densely if local feature size is small. • The reach Courtesy of [Dey, 2006]

30. Gradient of Distance Function • Distance function not differentiable on the medial axis • Still can define a generalized concept of gradient • [Lieutier, 2004] • For • Let and be the center and radius of the smallest enclosing ball of point(s) in • The generalized gradient of distance function • Flow lines induced by the generalized gradient • Examples:

31. Critical Points • A critical point of the distance function is a point whose generalized gradient vanishes • A critical point is either in or in its medial axis

32. Weak Feature Size • Given a compact , let denote • the set of critical points of the distance function that are not in • Given a compact the weak feature size is • Equivalently, • is the infimum of the positive critical value of

33. Why Distance Field? • Theorem [Offset Homotopy] [Grove’93] If are such that there is no critical value of in the closed interval then deformation retracts onto . In particular, . • Remarks: • For the case of compact set , note that it is possible that , for sufficiently small may not be homotopy equivalent to . • Intuitively, by above theorem, we can approximate for any small positive from a thickened version (offset) of . • The sampling condition makes sure that the discrete sample is sufficient to recover the offset homology.

34. Typical Sampling Conditions • Hausdorff distance between two sets and • infinumumvalue such that and • No noise version: • A set of points is an -sample of if and • With noise version: • A set of points is an -sample of if • Theoretical guarantees will be achieved when is small with respect to local feature size or weak feature size

35. Outline • From PCDs to simplicial complexes • Delaunay, Cech, Vietoris-Rips, witness complexes • Graph induced complex • Sampling conditions • Local feature size, and homological feature size • Topology inference • Homology inference • Handling noise • Approximating cycles of shortest basis of the first homology group • Approximating Reeb graph

36. Homology Inference from PCD

37. Problem Setup • A hidden compact (or a manifold ) • An -sample of • Recover homology of from some complex built on • will focus on Čech complex and Rips complex

38. Union of Balls • Intuitively, approximates offset • The Čechcomplex is the Nerve of • By Nerve Lemma, is homotopy equivalent to

39. Smooth Manifold Case • Let be a smooth manifold embedded in Theorem [Niyogi, Smale, Weinberger] Let be such that . If , there is a deformation retraction from to Corollary A Under the conditions above, we have

40. How about using Rips complex instead of Čechcomplex? • Recall that inducing • Idea [Chazal and Oudot 2008]: • Forming interleaving sequence of homomorphism to connect them with the homology of the input manifold and its offsets

41. Convert to Cech Complexes • Lemma A [Chazal and Oudot, 2008]: • The following diagram commutes: Corollary B Let be s.t. . If , where the second isomorphism is induced by inclusion.

42. From Rips Complex • Lemma B: Given a sequence of homomorphisms between finite dimensional vector spaces, if then • Rips and Čechcomplexes: • Applying Lemma B

43. The Case of Compact • In contrast to Corollary A, now we have the following (using Lemma B). • Lemma C [Chazal and Oudot 2008]: Let be a finite set such that for some . Then for all such that and for all , we have , where is the homomorphism between homology groups induced by the canonical inclusion .

44. The Case of Compacts • One more level of interleaving. • Use the following extension of Lemma B: Given a sequence of homomorphisms between finite dimensional vector spaces, if then • Theorem [Homology Inference] [Chazal and Oudot 2008]: Let be a finite set such that for some . Then for all all , we have where is the homomorphism between homology groups induced by canonical inclusion .

45. Theorem [Homology Inference] [Chazal and Oudot 2008]: Let be a finite set such that for some . Then for all all , we have where is the homomorphism between homology groups induced by canonical inclusion .

46. Summary of Homology Inference • homotopy equivalent to • Critical points of distance field • approximates (may be interleaving) • E.g, [Niyogi, Smale, Weinberger, 2006], [Chazal and Oudot 2008] • homotopy equivalent to • Nerve Lemma • interleaves at homology level • [Chazal and Oudot 2008] • and interleave • Derive homology inference from the interleaving sequence of homomorphisms

47. Outline • From PCDs to simplicial complexes • Delaunay, Cech, Vietoris-Rips, witness complexes • Graph induced complex • Sampling conditions • Local feature size, and homological feature size • Topology inference • Homology inference • Handling noise • Approximating cycles of shortest basis of the first homology group • Approximating Reeb graph

48. Handling of Noise

49. Noise • Previous approaches can handle Hausdorff type noise • Where noise is within a tubular neighborhood of • How about more general noise? • E.g, Gaussian noise, background noise • Some approaches • Bound the probability that input samples fall in Hausdorff model • Denoise so that resulting points fall in Hausdorff model • E.g, [Niyogi, Smale, Weinberger 2006, 2008] • Distance to measure framework • [Chazal, Cohen-Steiner, Mérigot, 2011]

50. Overview • Input: • A set of points sampled from a probabilistic measure on potentially concentrated on a hidden compact (e.g, manifold) . • Goal: • Approximate topological features of Courtesy of Chazal et al 2011