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# Topological Inference via Meshing

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In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales.
To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation.
This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.

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### Topological Inference via Meshing

1. 1. Topological Inference via Meshing Benoit Hudson, Gary Miller, Steve Oudot and Don Sheehy SoCG 2010
2. 2. Mesh Generation and Persistent Homology
3. 3. The Problem Input: Points in Euclidean space sampled from some unknown object. Output: Information about the topology of the unknown object.
4. 4. Points, offsets, homology, and persistence.
5. 5. Points, offsets, homology, and persistence. Input: P ⊂ Rd
6. 6. Points, offsets, homology, and persistence. Input: P ⊂ Rd α P = ball(p, α) p∈P
7. 7. Points, offsets, homology, and persistence. Input: P ⊂ Rd α P = ball(p, α) p∈P
8. 8. Points, offsets, homology, and persistence. Input: P ⊂ Rd α P = ball(p, α) p∈P
9. 9. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P
10. 10. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Compute the Homology
11. 11. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Compute the Homology
12. 12. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Compute the Homology
13. 13. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Compute the Homology
14. 14. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Compute the Homology
15. 15. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Compute the Homology
16. 16. Points, offsets, homology, and persistence. Offsets Input: P ⊂ Rd α P = ball(p, α) p∈P Persistent Compute the Homology
17. 17. Persistence Diagrams
18. 18. Persistence Diagrams
19. 19. Persistence Diagrams Bottleneck Distance d∞ = max |pi − qi |∞ B i
20. 20. Approximate Persistence Diagrams Bottleneck Distance d∞ = max |pi − qi |∞ B i
21. 21. Approximate Birth and Death times Persistence Diagrams differ by a constant factor. Bottleneck Distance d∞ = max |pi − qi |∞ B i
22. 22. Approximate Birth and Death times Persistence Diagrams differ by a constant factor. Bottleneck Distance d∞ = max |pi − qi |∞ B i This is just the bottleneck distance of the log-scale diagrams.
23. 23. Approximate Birth and Death times Persistence Diagrams differ by a constant factor. Bottleneck Distance d∞ = max |pi − qi |∞ B i This is just the bottleneck distance of the log-scale diagrams. log a − log b < ε a log b < ε a b <1+ε
24. 24. We need to build a filtered simplicial complex. Associate a birth time with each simplex in complex K. At time , we have a complex K consisting of all simplices born at or before time . time
25. 25. There are two phases, one is geometric the other is topological. Geometry Topology (linear algebra) Build a filtration, i.e. Compute the a filtered simplicial persistence diagram complex. (Run the Persistence Algorithm).
26. 26. There are two phases, one is geometric the other is topological. Geometry Topology (linear algebra) Build a filtration, i.e. Compute the a filtered simplicial persistence diagram complex. (Run the Persistence Algorithm). We’ll focus on this side.
27. 27. There are two phases, one is geometric the other is topological. Geometry Topology (linear algebra) Build a filtration, i.e. Compute the a filtered simplicial persistence diagram complex. (Run the Persistence Algorithm). We’ll focus on this side. Running time: O(N3). N is the size of the complex.
28. 28. Idea 1: Use the Delaunay Triangulation
29. 29. Idea 1: Use the Delaunay Triangulation Good: It works, (alpha-complex filtration).
30. 30. Idea 1: Use the Delaunay Triangulation Good: It works, (alpha-complex filtration). Bad: It can have size nO(d).
31. 31. Idea 2: Connect up everything close.
32. 32. Idea 2: Connect up everything close. Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at most .
33. 33. Idea 2: Connect up everything close. Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at most . Rips Filtration: Add a k-simplex for every k+1 points that have all pairwise distances at most .
34. 34. Idea 2: Connect up everything close. Čech Filtration: Add a k-simplex for every k+1 points that have a smallest enclosing ball of radius at most . Rips Filtration: Add a k-simplex for every k+1 points that have all pairwise distances at most . Still nd, but we can quit early.
35. 35. Our Idea: Build a quality mesh.
36. 36. Our Idea: Build a quality mesh.
37. 37. Our Idea: Build a quality mesh. 2 We can build meshes of size 2 O(d )n.
38. 38. Meshing Counter-intuition Delaunay Refinement can take less time and space than Delaunay Triangulation.
39. 39. Meshing Counter-intuition Delaunay Refinement can take less time and space than Delaunay Triangulation. Theorem [Hudson, Miller, Phillips, ’06]: A quality mesh of a point set can be constructed in O(n log ∆) time, where ∆ is the spread.
40. 40. Meshing Counter-intuition Delaunay Refinement can take less time and space than Delaunay Triangulation. Theorem [Hudson, Miller, Phillips, ’06]: A quality mesh of a point set can be constructed in O(n log ∆) time, where ∆ is the spread. Theorem [Miller, Phillips, Sheehy, ’08]: A quality mesh of a well-paced point set has size O(n).
41. 41. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
42. 42. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
43. 43. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
44. 44. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
45. 45. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
46. 46. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
47. 47. The α-mesh filtration 1. Build a mesh M. 2. Assign birth times to vertices based on distance to P (special case points very close to P). 3. For each simplex s of Del(M), let birth(s) be the min birth time of its vertices. 4. Feed this filtered complex to the persistence algorithm.
48. 48. Approximation via interleaving.
49. 49. Approximation via interleaving. Definition: Two ﬁltrations, {Pα } and {Qα } are ε-interleaved if Pα−ε ⊆ Qα ⊆ Pα+ε for all α.
50. 50. Approximation via interleaving. Definition: Two ﬁltrations, {Pα } and {Qα } are ε-interleaved if Pα−ε ⊆ Qα ⊆ Pα+ε for all α. Theorem [Chazal et al, ’09]: If {Pα } and {Qα } are ε-interleaved then their persistence diagrams are ε-close in the bottleneck distance.
51. 51. The Voronoi filtration interleaves with the offset filtration.
52. 52. The Voronoi filtration interleaves with the offset filtration.
53. 53. The Voronoi filtration interleaves with the offset filtration.
54. 54. The Voronoi filtration interleaves with the offset filtration.
55. 55. The Voronoi filtration interleaves with the offset filtration. Theorem: α/ρ αρ For all α > rP , VM ⊂ P α ⊂ VM , where rP is minimum distance between any pair of points in P .
56. 56. The Voronoi filtration interleaves with the offset filtration. Theorem: α/ρ αρ For all α > rP , VM ⊂ P α ⊂ VM , where rP is minimum distance between any pair of points in P . Finer refinement yields a tighter interleaving.
57. 57. The Voronoi filtration interleaves with the offset filtration. Theorem: α/ρ αρ For all α > rP , VM ⊂ P α ⊂ VM , where rP is minimum distance between any pair of points in P . Finer refinement yields a tighter interleaving. Special case for small scales.
58. 58. Geometric Topologically Approximation Equivalent
59. 59. Geometric Topologically Approximation Equivalent
60. 60. The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Work 1 nO(d) 2. Assign birth times to Simple vertices based on distance to P mesh filtration (special case points very close to P).** Over-refine 3. For each simplex s of Del(M), the mesh let birth(s) be the min birth time of its vertices. Linear-Size 4. Feed this filtered complex to Meshing the persistence algorithm.
61. 61. The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Work 1 nO(d) 2. Assign birth times to Simple vertices based on distance to P mesh ρ 2 O(d2 ) n log ∆ filtration (special case points very close to P).** Over-refine 3. For each simplex s of Del(M), the mesh let birth(s) be the min birth time of its vertices. Linear-Size 4. Feed this filtered complex to Meshing the persistence algorithm.
62. 62. The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Work 1 nO(d) 2. Assign birth times to Simple vertices based on distance to P mesh ρ =~3 2 O(d2 ) n log ∆ filtration (special case points very close to P).** Over-refine 3. For each simplex s of Del(M), the mesh let birth(s) be the min birth time of its vertices. Linear-Size 4. Feed this filtered complex to Meshing the persistence algorithm.
63. 63. The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Over-refine it. Work 1 nO(d) 2. Assign birth times to Simple vertices based on distance to P mesh ρ =~3 2 O(d2 ) n log ∆ filtration (special case points very close to P).** Over-refine −O(d2 ) 1+ε n log ∆ 3. For each simplex s of Del(M), ε the mesh let birth(s) be the min birth time of its vertices. Linear-Size 4. Feed this filtered complex to Meshing the persistence algorithm.
64. 64. The Results Approximation ratio Complex Size 1. Build a mesh M. Previous Over-refine it. Work 1 nO(d) Use linear-size meshing. 2. Assign birth times to Simple vertices based on distance to P mesh ρ =~3 2 O(d2 ) n log ∆ filtration (special case points very close to P).** Over-refine −O(d2 ) 1+ε n log ∆ 3. For each simplex s of Del(M), ε the mesh let birth(s) be the min birth time of its vertices. Linear-Size −O(d2 ) 1 + ε + 3θ (εθ) n 4. Feed this filtered complex to Meshing the persistence algorithm.
65. 65. Thank you.