1. Mapper

6. Example: Point cloud data representing a hand.
A) Data Set
Example: Point cloud data
representing a hand.
B) Function f : Data Set  R
Example: x-coordinate
f : (x, y, z)  x
Put data into overlapping bins.
Example: f-1(ai, bi)
Cluster each bin & create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters

7. Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition
Singh, Gurjeet; Memoli, Facundo; Carlsson, Gunnar

9. Mapper Data Overlapping bins Graph

10. Example: Point cloud data representing a hand.
A) Data Set
Example: Point cloud data
representing a hand.

11. Function f : Data Set  R Ex 1: x-coordinate f : (x, y, z)  x

12. Function f : Data Set  R Ex 1: x-coordinate f : (x, y, z)  x

13. Put data into overlapping bins. Example: f-1(ai, bi)
( ( ) ( ) ( ) ( ) ( ) )
Function f : Data Set  R
Ex 1: x-coordinate
f : (x, y, z)  x

14. Put data into overlapping bins. Example: f-1(ai, bi)
( ( ) ( ) ( ) ( ) ( ) )
Function f : Data Set  R
Ex 1: x-coordinate
f : (x, y, z)  x

15. D) Cluster each bin
Need covering Resolution multiscale

16. Vertex = a cluster of a bin.
D) Cluster each bin
Vertex = a cluster of a bin.
Need covering Resolution multiscale

17. Vertex = a cluster of a bin. Edge = nonempty intersection
D) Cluster each bin
& create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters
Need covering Resolution multiscale

18. Vertex = a cluster of a bin. Edge = nonempty intersection
D) Cluster each bin
& create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters
Need covering Resolution multiscale

19. Example: Point cloud data representing a hand.
A) Data Set
Example: Point cloud data
representing a hand.
B) Function f : Data Set  R
Example: x-coordinate
f : (x, y, z)  x
Put data into overlapping bins.
Example: f-1(ai, bi)
Cluster each bin & create network.
Vertex = a cluster of a bin.
Edge = nonempty intersection
between clusters

20. To understand software, it helps to apply it to a variety of examples.

21. What graph do you get when you apply mapper to the ideal trefoil knot if filter = projection to x-axis?
( ( ) ( ) ( ) ( ) ( ) )

22. 1.) Put data into overlapping bins.
( ( ) ( ) ( ) ( ) ( ) )
Example: f-1(ai, bi)

23. Put data into overlapping
bins.

24. Cluster each bin.

25. Cluster each bin.
Each cluster is represented by a vertex.

27. Create a graph representing the data set:
Cluster each bin.
Each cluster is represented by a vertex.
Draw an edge between 2 vertices if
their clusters have points in common.

28. Create a graph representing the data set:
Cluster each bin.
Each cluster is represented by a vertex.
Draw an edge between 2 vertices if
their clusters have points in common.

31. D C B A E

32. A E B C D

33. A B C D E 1 2 3 4

34. A B C D E 4 2 3 1

35. 4 1 3 2 E D A B C

36. A B C D E A B C D E 1 2 3 4 A B C D E 4 2 3 1 A B C D E 4 2 3 1

37. A B C D E 4 2 3 1

38. What graph do you get when you apply mapper to the above blob if filter = projection to x-axis?

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50. To understand software, it helps to apply it to a variety of examples.