Adaptive Sampling with Topological Scores

1. Adaptive Sampling with Topological Scores Dan Maljovec1, Bei Wang1, Ana Kupresanin2, Gardar Johannesson2, Valerio Pascucci1, and Peer-Timo Bremer2 1SCI Institute, University of Utah 2Lawrence Livermore National Laboratory

2. 1 Framework: 4 2 3

3. Choosing the “Right” Points 4 Models fit by using 10 training points: True Response Model:

4. Space-filling Sampling No prior knowledge of the dataset Where should we sample the model?

5. Space-filling Sampling Fill the domain space as evenly as possible

6. Space-filling Sampling Obtain responses and fit model

7. Space-filling Sampling 1 round:

8. Space-filling Sampling 2 rounds:

9. Space-filling Sampling 3 rounds:

10. Space-filling Sampling 4 rounds:

11. Space-filling Sampling 5 rounds:

12. Space-filling Sampling Refit after 5 rounds of “filling voids”:

13. Space-filling Sampling What have we learned? Initial fit Refit after adding 5 points

14. Topologically-Inspired Adaptive Sampling Starting over

15. Topologically-Inspired Adaptive Sampling 1 round:

16. Topologically-Inspired Adaptive Sampling 2 rounds:

17. Topologically-Inspired Adaptive Sampling 3 rounds:

18. Topologically-Inspired Adaptive Sampling 4 rounds:

19. Topologically-Inspired Adaptive Sampling 5 rounds:

20. Topologically-Inspired Adaptive Sampling Refit after 5 rounds of choosing the most topologically significant point:

21. Topologically-Inspired Adaptive Sampling What have we learned? Initial fit Refit after adding 5 points

22. Comparison Initial Fit Space-filling Method Adaptive Method

23. Comparison True Response Model Space-filling Method Adaptive Method

24. Measuring Topological Significance These points were selected in topologically significant regions: How do we measure topological impact?

25. Morse-Smale Complex A partition of the domain into monotonic regions ∩ = Stable Manifolds Unstable Manifolds Morse-Smale Complex

26. Computing the Morse-Smale Complex Traditionally: • Compute steepest ascent/descent gradient from each point in dataset • Color points twice once by following ascending gradient to maximum & another by descending gradient to minimum • Intersect partitions to form Morse-Smale Complex ∩ = Stable Manifolds Unstable Manifolds Morse-Smale Complex

27. Computing the Morse-Smale Complex Traditionally: • a. Compute steepest ascent gradient from each point in dataset • a. Color points based on maximum where flow halts

28. Computing the Morse-Smale Complex Traditionally: • b. Compute steepest descent gradient from each point in dataset • b. Color points based on minimum where flow halts

29. Computing the Morse-Smale Complex Traditionally: • Intersect partitions to form Morse-Smale Complex ∩ = Stable Manifolds Unstable Manifolds Morse-Smale Complex

30. Computing the Morse-Smale Complex Traditionally: • Compute steepest ascent/descent gradient from each point in dataset Assumes points have a neighborhood • Color points twice once by following ascending gradient to maximum & another by descending gradient to minimum • Intersect partitions to form Morse-Smale Complex ∩ = Stable Manifolds Unstable Manifolds Morse-Smale Complex

31. Computing the Approximate Morse-Smale Complex Modified Algorithm: • Compute approximate KNN for a point • Compute steepest ascent/descent gradient from each point’s neighborhood in dataset • Color points by gradient flows • Intersect partitions to form Morse-Smale Complex ∩ = Stable Manifolds Unstable Manifolds Morse-Smale Complex

32. Tracking Persistence Count the number of connected components belonging to the sub-level sets of the function y c a b d x

33. Tracking Persistence Every connected component has a birth y c a b d x

34. Tracking Persistence Every connected component has a birth y c a b d x

35. Tracking Persistence And every connected component has a death y c a b d x

36. Tracking Persistence Births and deaths of connected components are tied to critical points of the function y c a b d x

37. Tracking Persistence Births and deaths of connected components are tied to critical points of the function y c a b d x

38. Tracking Persistence y c a b d x

39. Tracking Persistence y death A persistence diagram plots birth-death pairings The function value difference between the critical point pairings is what we define as the persistence of a feature c a b d x birth

40. Persistence Simplification in 2D In surfaces (and higher dimension corollaries), we have saddle points

41. Persistence Simplification in 2D We can smooth away noise features by thresholdingcritical point pairs above a certain persistence level

42. Persistence Simplification in 2D

43. Bottleneck Distance y death Comparing two similar functions persistence diagrams x birth

44. Bottleneck Distance y death Optimization: minimize the distance between pairs of points in the two plots including the line y=x in both pairs Bottleneck distance – the maximum distance resulting from this pairing x birth

45. 1 Framework: 4 2 3

46. Selecting Initial Training Data Tests use Latin Hypercube Sampling

47. Fitting a Statistical Model Experimented with: • Gaussian Process Model • Sparse Online Gaussian Process (C++) • 2 Implementations of MARS with bootstrapping • Earth (CRAN) • MDA (CRAN) • Neural Network with bootstrapping • NNet (CRAN)

48. Classical Scoring Active-Learning McKay (ALM) • Sample high-frequency or low-confidence regions Delta • Distribute samples in the range space or areas of steep gradient Expected Improvement (EI) • Select points with high uncertainty or large discrepancy with existing data Distance (*DP) • Scaling factor applied to above, creating 3 new scoring functions (ALMDP, DeltaDP, EIDP)

49. Topological Scoring TOPOP • Average change in persistence of each point computed by comparing the Morse-Smale before and after inserting a candidate TOPOB • Bottleneck distance between persistence diagram of Morse-Smale before and after inserting a candidate point TOPOHP • Highest persistence critical point in Morse-Smale complex constructed from combined training and predicted candidate data

50. Topological Scoring Average Change in Persistence (TOPOP) • Average change in persistence of each point computed by comparing the Morse-Smale before and after inserting a candidate