1. Sept 25, 2013: Applicable Triangulations.
MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology
Sept 25, 2013: Applicable Triangulations.
Fall 2013 course offered through the
University of Iowa Division of Continuing Education
Isabel K. Darcy, Department of Mathematics
Applied Mathematical and Computational Sciences, University of Iowa

2. Creating a simplicial complex
a one dimensional simplicial complex. Note that we have clustered our data into five disjoint connected sets. So this is one way to cluster our data – that is grouping our data points into disjoint sets based on some definition of similarity. In this case, we have 5 clusters.
We can now add higher dimensional simplices.
1.) Adding 1-dimensional edges (1-simplices)
Add an edge between data points that are “close”

3. Vietoris Rips complex = flag complex = clique complex
Thus we now have the Vietoris Rips simplicial complex. Note we get the same simplex by adding one dimension at a time
2.) Add all possible simplices of dimensional > 1.

4. Creating the Čech simplicial complex
Thus we now have the Vietoris Rips simplicial complex. Note we get the same simplex by adding one dimension at a time
1.) B1 … Bk+1 ≠ ⁄ , create k-simplex {v1, ... , vk+1}. U

5. Consider X an arbitrary topological space.
Let V = {Vi | i = 1, …, n } where Vi X ,
The nerve of V = N(V) where
The k -simplices of N(V) =
nonempty intersections of
k +1 distinct elements of V .
For example,
Vertices = elements of V
Edges = pairs in V which intersect nontrivially.
Triangle = triples in V which intersect nontrivially. U

6. Nerve Lemma: If V is a finite collection of subsets of X with all non-empty intersections of subcollections of V contractible, then N(V) is homotopic to the union of elements of V.

7. The Voronoi cell associated with v is
Choose data point v.
The Voronoi cell associated with v is
H(v,w) U
w ≠ v
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

8. Voronoi diagram Suppose your data points live in Rn.
Choose data point v.
The Voronoi cell associated with v is
H(v,w) U
w ≠ v
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

9. The delaunay triangulation is the dual to the voronoi diagram
If Cv ≠ 0, then
s is a simplex in the
delaunay triangulation.
Nerve of {Cv : v in data set} U
w in s ⁄
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

10. The delaunay triangulation is the dual to the voronoi diagram
If Cv ≠ 0, then
s is a simplex in the
delaunay triangulation.
Nerve of {Cv : v in data set} U
w in s ⁄
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

11. voronoi diagram

12. voronoi diagram

13. The delaunay triangulation is the dual to the voronoi diagram
If Cv ≠ 0, then
s is a simplex in the
delaunay triangulation.
Nerve of {Cv : v in data set} U
w in s ⁄
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

14. The delaunay triangulation is the dual to the voronoi diagram
If Cv ≠ 0, then
s is a simplex in the
delaunay triangulation.
Nerve of {Cv : v in data set} U
w in s ⁄
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

15. Delaunay triangulation
Čech

16. Alpha complex
Nerve of
{Cv Bv(r): v in data set} U
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

17. Alpha complex
Nerve of
{Cv Bv(r): v in data set} U
The Voronoi cell associated with v is
Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

18. Alpha complex
Čech

19. Čech

20. Alpha complex

21. Let D = set of vertices.
v0,v1,...,vk span a
Delaunay k-simplex
iff
the Voronoi cell
associated with vi meet
there is a point w ∈ Rn, whose k+1 nearest
neighbours in D are v0,v1,...,vk and which is
equidistant from them.

22. comptop.stanford.edu/preprints/witness.pdf

23. Witness complex
Let D = set of point cloud data points.
Choose L D, L = set of landmark points. U

24. v0,v1,...,vk span a k-simplex
iff
there is a point w ∈ D, whose k+1 nearest
neighbours in L are v0,v1,...,vk and which is
equidistant from them
??????????????????????
Witness complex
Let D = set of point cloud data points.
Choose L D, L = set of landmark points. U

25. v0,v1,...,vk span a k-simplex
iff
there is a point w ∈ D, whose k+1 nearest
neighbours in L are v0,v1,...,vk
and all the faces of {v0,v1,...,vk} belong to the witness complex.
w is called a “weak” witness.
W∞(D) = Witness complex
Let D = set of point cloud data points.
Choose L D,
L = set of landmark points. U

26. v0,v1,...,vk span a k-simplex
iff
there is a point w ∈ D, whose k+1 nearest
neighbours in L are v0,v1,...,vk
and all the faces of {v0,v1,...,vk} belong to the witness complex.
w is called a “weak” witness.
W∞(D) = Witness complex
Let D = set of point cloud data points.
Choose L D,
L = set of landmark points. U

27. W∞(D) = Witness complex

28. W∞(D) = Witness complex

29. W1(D) = Lazy witness complex
Let L = set of landmark points.
1-skeletion of W1(D) = 1-skeletion of W∞ (D).
Create the flag (or clique) complex:
Add all possible simplices of dimensional > 1.

30. W1(D) = Lazy witness complex
Let L = set of landmark points.
1-skeletion of W1(D) = 1-skeletion of W∞ (D).
Create the flag (or clique) complex:
Add all possible simplices of dimensional > 1.

31. W1(D) = Lazy witness complex
Let L = set of landmark points.
1-skeletion of W1(D) = 1-skeletion of W∞ (D).
Create the flag (or clique) complex:
Add all possible simplices of dimensional > 1.

32. Choosing Landmark points:
A.) Random
B.) Maxmin
1.) choose point l1 randomly
2.) If {l1, …, lk-1} have been chosen, choose lk such that {l1, …, lk-1} is in D - {l1, …, lk-1} and
min {d(lk, l1), …, d(lk, lk-1)} ≥ min {d(v, l1), …, d(v, lk-1)}

33. Choosing Landmark points
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.

34. Choosing Landmark points
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.

35. Choosing Landmark points
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.

36. Choosing Landmark points
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.

37. Choosing Landmark points
data points.
In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional.
For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site.

38. comptop.stanford.edu/preprints/witness.pdf

39. Strong witness complex:
Let D = set of point cloud data points.
Choose L D, L = set of landmark points.
Let mv = dist (v, L) = min{ d(v, l ) : l in L } U
{l1, …, lk+1} is a k-simplex
iff
d(v, li) ≤ mv + ε for all i
v is the witness

40. Weak witness complex:
Let D = set of point cloud data points.
Choose L D, L = set of landmark points. U
s = {l1, …, lk+1} is a k-simplex
iff
d(v, li) ≤ d(v, x) for all i
and all x not in s
v is the weak witness

41. Weak witness complex:
Let D = set of point cloud data points.
Choose L D, L = set of landmark points. U
s = {l1, …, lk+1} is a k-simplex
iff
d(v, li) ≤ d(v, x) + e for all i
and all x not in s
v is the e-weak witness

44. The Theory of Multidimensional Persistence, Gunnar Carlsson, Afra Zomorodian
"Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams,
multidimensional persistence, Gröbner bases, Gunnar Carlsson 

46. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima

47. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima

48. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima

49. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima

50. From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima

52. Computing Multidimensional Persistence,
Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian