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 Don Sheehy Theory Lunch Joint work with Benoit Hudson, Gary Miller, and Steve Oudot
  2. 2. Computer Scientists want to know the shape of data. Clustering Principal Component Analysis Convex Hull Mesh Generation Surface Reconstruction
  3. 3. Point sets have no shape... so we have to add it ourselves. We build a simplicial complex.
  4. 4. We assume that the geometry is meaningful.
  5. 5. Distance functions add shape to data. dP (x) = min |x − p| p∈P α P = dP [0, α] −1 = ball(p, α) p∈P x
  6. 6. If you know the “scale” of the data, the situation is often easier.
  7. 7. There may not be a right scale at which to look at the data for two reasons. 1 No single scale captures the shape. 2 Interesting features appear at several scales. The Persistence Approach: Look at every scale and then see what persists.
  8. 8. Homology gives a quantitative description of qualitative aspects of shape, i.e. components, holes, voids. Reduces to linear algebra for simplicial complexes. 0th Homology group is the nullspace of the Laplacian.
  9. 9. Persistent Homology tracks homology across changes in scale. The input to the persistence algorithms is a filtration.
  10. 10. A filtration is a growing sequence of topological spaces. {Fα }α≥0 ∀α ≤ β, Fα ⊆ Fβ We care about two different types of filtrations. 1 Sublevel sets of well-behaved functions. 2 Filtered simplicial complexes.
  11. 11. The distance between two diagrams is the bottleneck of a matching. p2 q2 ∞ dB = max |pi − qi |∞ i q1 p3 p1 q3
  12. 12. Approximate persistence diagrams have features that are born and die within a constant factor of the birth and death times of their corresponding features. This is just the bottleneck distance of the log-scale diagrams. log a − log b < ε a log b < ε a b <1+ε
  13. 13. There are two phases, one is geometric the other is topological. Geometry Topology (linear algebra) Build a filtration, i.e. Compute the a filtered complex. persistence diagram (Run the Persistence Algorithm). We’ll focus on this side. Running time is polynomial in the size of the complex.
  14. 14. Idea 1: Use the Delaunay Triangulation Good: It works, (alpha-complex filtration). Bad: It can have size nO(d).
  15. 15. 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.
  16. 16. Topology is not Topography Sublevel sets (But in our case, there are some similarities)
  17. 17. Nobel Peace Prize!
  18. 18. Our Idea: Build a quality mesh. 2 We can build meshes of size 2 O(d )n.
  19. 19. 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.
  20. 20. 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.
  21. 21. 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.
  22. 22. Geometric Topological Approximation Approximation
  23. 23. If it’s so easy, why didn’t anyone think of this before? 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).
  24. 24. 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 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.
  25. 25. Thank you.