The Hidden Topology of a Noisy Point Cloud

Transcription

1 The Hidden Topology of a Noisy Point Cloud (Part III) A cri&cal reading of Geometric Inference for Probability Measures by Chazal, Steiner & Merigot, A. Pedrini, M. Piastra

2 Measures A measure is a func3on that maps every (Borel) subset to a non- nega3ve number and which is countably addi&ve It is finite if A probability measure is such that (we will consider probability measures only, from now on) The support of is the smallest closed subset such that The hidden topology of a noisy point cloud 2

3 A measure from a point cloud Let be a finite point cloud The uniform measure on is defined as for every (Borel) subset 3 The uniform measure is a special case: it could be different The hidden topology of a noisy point cloud 3

4 Distance to a compact Distance to a compact subset Alterna3ve formula3on where centered in is a closed ball of radius The hidden topology of a noisy point cloud 4

5 From a compact to a measure Let be a measure having the compact as its support The distance to is The hidden topology of a noisy point cloud 5

6 A pseudo-distance Let be a measure having the compact as its support As a generaliza3on, define (for any ) The hidden topology of a noisy point cloud 6

7 A pseudo-distance For a point cloud, with uniform : where is the - nearest point to in The hidden topology of a noisy point cloud 7

8 A pseudo-distance where is the - nearest point from in The hidden topology of a noisy point cloud 8

9 A pseudo-distance However, is not distance- like This entails that neither the isotopy lemma nor the reconstruc&on theorem apply to Furthermore, the map is not con3nuous in any reasonable sense A slight perturba3on that turns into can induce substan3al differences between and The hidden topology of a noisy point cloud 9

10 A pseudo-distance is not distance like Counter- example: i.e. is the distance to the second- nearest point in The hidden topology of a noisy point cloud 10

11 A pseudo-distance is not distance like Counter- example: for to be semi- concave must be concave. However: is not concave The hidden topology of a noisy point cloud 11

12 A pseudo-distance The map in any reasonable sense is not con3nuous For : i.e. the func&on is not con&nuous The hidden topology of a noisy point cloud 12

13 Distance to a measure Let be a measure having the compact as its support Define, for any This is distance- like (see ar3cle) The hidden topology of a noisy point cloud 13

14 Distance to a measure In the case of a point cloud assuming the distance becomes The hidden topology of a noisy point cloud 14

15 Distance to a measure The hidden topology of a noisy point cloud 15

16 Distance to a measure is distance- like Not a proof, just an example: The hidden topology of a noisy point cloud 16

17 Distance to a measure is distance- like Not a proof, just an example: for to be semi- concave must be concave. We have: The hidden topology of a noisy point cloud 17

18 Distance to a measure is distance- like Not a proof, just an example: for to be semi- concave must be concave. We have: is concave The hidden topology of a noisy point cloud 18

19 Being distance-like The Isotopy Lemma holds for (see ar3cle) Given two numbers such that there are no - cri3cal points in, all the sublevel sets are isotopic The hidden topology of a noisy point cloud 19

20 Distance to a measure The map Not a proof, just an example is con3nuous For : (A rigorous defini&on requires Wasserstein distance) The hidden topology of a noisy point cloud 20

21 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 21

22 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 22

23 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 23

24 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 24

25 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 25

26 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 26

27 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 238 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 27

28 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 952 point samples from Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 28

29 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 952 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 29

30 Distances and level sets Topology of the sublevel sets of In black: Grey dots: 952 point samples from In color: Which values of guarantee the correct homotopy type? A rigorous answer requires Wasserstein distance The hidden topology of a noisy point cloud 30

31 Wasserstein distance The hidden topology of a noisy point cloud 31

32 Distance btw. two measures The intui&ve idea: transport An example There are two producers producing quan33es And two consumers consuming quan33es, with What is the most effec&ve way to transport quan&&es from producers to consumers? The hidden topology of a noisy point cloud 32

33 Distance btw. two measures The intui&ve idea: transport An example There are two producers producing quan33es And two consumers consuming quan33es, with i.e. two measures The total cost of transpor3ng quan33es around is: Transported quan3ty The hidden topology of a noisy point cloud 33

34 Distance btw. two measures The intui&ve idea: transport An example The most effec&ve transport plan is: under the constraints: This generalizes to any two measures defined on two finite point clouds The hidden topology of a noisy point cloud 34

35 Wasserstein distance A.k.a. Earth Mover s Distance, EMD Given two measures a transport plan is a joint measure such that for every (Borel) subset The cost of a transport plan is The hidden topology of a noisy point cloud 35

36 Wasserstein distance The Wasserstein distance of two measures is (Fear not: we won t compute it) The hidden topology of a noisy point cloud 36

37 Wasserstein distance A remarkable example: two point clouds of the same size and uniformly distributed mass A transport plan (in this case) is a bijec3on between the blue dots and the red dots: The cost of a transport plan is: In this case is: This image comes from: h\p:// tours.com/matlab/op3maltransp_4_matching_sliced/ The hidden topology of a noisy point cloud 37

38 Wasserstein distance A remarkable example: two point clouds of the same size and uniform mass A transport plan (in this case) is a bijec3on between the blue dots and the red dots: The cost of a transport plan is: In this case is: This image comes from: h\p:// tours.com/matlab/op3maltransp_4_matching_sliced/ The hidden topology of a noisy point cloud 38

39 Wasserstein distance A remarkable example: two point clouds of the same size and uniform mass An itera3ve MATLAB algorithm finds This image comes from: h\p:// tours.com/matlab/op3maltransp_4_matching_sliced/ The hidden topology of a noisy point cloud 39

40 Wasserstein distance A remarkable example: two gaussians Special case: See also: h\p://djalil.chafai.net/blog/2010/04/30/wasserstein- distance- between- two- gaussians/ The hidden topology of a noisy point cloud 40

41 Distance function stability The map with respect to is not con&nuous Same example as before, considering that: The hidden topology of a noisy point cloud 41

42 Distance function stability The map with respect to The map with respect to is not con&nuous is con&nuous Same example as before, considering that: The hidden topology of a noisy point cloud 42

43 Distance Function Stability If and are two probability measures on and then (Theorem 3.5) Relevance: the reconstruc3on theorem requires the two distance- like func3ons to be close enough i.e.. This theorem says that this can be guaranteed for when is small The hidden topology of a noisy point cloud 43

44 Reconstruction revisited (I) Let be probability measures. If it exists a such that then, for all and all the sublevel sets have the same homotopy type The hidden topology of a noisy point cloud 44

45 Distance to support The hidden topology of a noisy point cloud 45

46 Distance and distance-like Consider a compact and the distance Consider also a probability measure having as its support How does relate to? The hidden topology of a noisy point cloud 46

47 Distance and distance-like When the support of a probability measure is a compact, then is the uniform limit of as converges to The hidden topology of a noisy point cloud 47

48 Dimension at most k A probability measure with support has dimension at most iff is the smallest integer for which such that there exists an such that Ok, what does it mean? The hidden topology of a noisy point cloud 48

49 Dimension at most k A probability measure with support has dimension at most iff is the smallest integer for which such that there exists an such that Why the smallest? Assume then since but since The hidden topology of a noisy point cloud 49

50 Dimension at most k The hidden topology of a noisy point cloud 50

51 Dimension at most k The hidden topology of a noisy point cloud 51

52 Dimension at most k The hidden topology of a noisy point cloud 52

53 Dimension at most k The hidden topology of a noisy point cloud 53

54 Dimension at most k The hidden topology of a noisy point cloud 54

55 Dimension at most k The hidden topology of a noisy point cloud 55

56 Dimension at most k The squared lollipop has dimension at most 2, with The hidden topology of a noisy point cloud 56

57 Distance to measure s support Let be a probability measure with support having dimension at most, with constant Then The hidden topology of a noisy point cloud 57

58 Pu\ng everything together The hidden topology of a noisy point cloud 58

59 Compacts & Measures Assume we have: A compact set to be reconstructed A measure having as its support A noisy measure (e.g., where models noise) An empirical, noisy measure from a finite point sample of obtained The hidden topology of a noisy point cloud 59

60 Compacts & Measures Assume we have (examples): A circle to be reconstructed Source of coherent light from one side (i.e not uniform on ) A noisy measure (e.g., where models noise) An empirical, noisy measure from a finite point sample obtained The hidden topology of a noisy point cloud 60 of

61 Reconstruction (final) Then: The hidden topology of a noisy point cloud 61

62 Reconstruction (final) Then: The hidden topology of a noisy point cloud 62

63 Reconstruction (final) Let have dimension at most and support such that for some,. Let be another measure such that Then and have the same homotopy type The hidden topology of a noisy point cloud 63

64 Reconstruction (final) Given that, in a specific problem, the values are given The problem is then finding an such that: (Unfortunately, there is no known closed- form solu3on for to this inequality) The hidden topology of a noisy point cloud 64