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Some Topics in Computational Topology

Some Topics in Computational Topology. Yusu Wang Ohio State University AMS Short Course 2014. Introduction. Much recent developments in computational topology Both in theory and in their applications E.g , the theory of persistence homology

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Some Topics in Computational Topology

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  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

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