5. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
6. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
Compute Persistence
7. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
Compute Persistence
Geometry
8. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
Compute Persistence
Geometry Topology
9. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
Compute Persistence
Geometry Topology
What to compute?
10. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
Compute Persistence
Geometry Topology
What to compute? How to compute it?
11. A Topological Data Analysis Pipeline
Data Points a Function
(Lipschitz)
A (filtered) Simplicial Complex
(to approx. the function)
Compute Persistence
Geometry Topology
What to compute? How to compute it?
Q:Can we use the geometry to speed
up the persistence computation?
12. We want to build a filtered simplicial complex.
Associate a birth time with each simplex in complex K.
At timeα, we have a complex Kα consisting of all simplices
born at or before time α.
time
13. We want to build a filtered simplicial complex.
Associate a birth time with each simplex in complex K.
At timeα, we have a complex Kα consisting of all simplices
born at or before time α.
time
This can be represented by its boundary matrix D, with a
fixed row order.
17. Computing Persistent Homology
Input: Boundary Matrix D
Find V, R such that
D = RV
V is upper-triangular
R is “reduced” (i.e. no two columns
have lowest nonzeros in the same row)
18. Computing Persistent Homology
Input: Boundary Matrix D
Find V, R such that
D = RV
V is upper-triangular
R is “reduced” (i.e. no two columns
have lowest nonzeros in the same row)
Output is a collection of pairs corresponding
to the lowest nonzeros in R.
19. Computing Persistent Homology
Input: Boundary Matrix D
Find V, R such that
D = RV
V is upper-triangular
R is “reduced” (i.e. no two columns
have lowest nonzeros in the same row)
It’s just Gaussian elimination!
Output is a collection of pairs corresponding
to the lowest nonzeros in R.
21. The State of the Art
Asymptotics
“The Persistence Algorithm” [ELZ05] O(n3
)
22. The State of the Art
Asymptotics
“The Persistence Algorithm” [ELZ05]
via Matrix Multiplication [MMS11]
O(n3
)
O(n!
)
23. The State of the Art
Asymptotics
“The Persistence Algorithm” [ELZ05]
via Matrix Multiplication [MMS11]
Output-Sensitive [CK12] O(|D(1 ) |n2
)
O(n3
)
O(n!
)
24. The State of the Art
Asymptotics
Heuristics
“The Persistence Algorithm” [ELZ05]
via Matrix Multiplication [MMS11]
Output-Sensitive [CK12] O(|D(1 ) |n2
)
O(n3
)
O(n!
)
25. The State of the Art
Asymptotics
Heuristics
“The Persistence Algorithm” [ELZ05]
via Matrix Multiplication [MMS11]
Output-Sensitive [CK12]
Persistent Cohomology [MdSV-J11]
O(|D(1 ) |n2
)
O(n3
)
O(n!
)
26. The State of the Art
Asymptotics
Heuristics
“The Persistence Algorithm” [ELZ05]
via Matrix Multiplication [MMS11]
Output-Sensitive [CK12]
Persistent Cohomology [MdSV-J11]
Discrete Morse Reduction [MN13]
O(|D(1 ) |n2
)
O(n3
)
O(n!
)
27. The State of the Art
Asymptotics
Heuristics
“The Persistence Algorithm” [ELZ05]
via Matrix Multiplication [MMS11]
Output-Sensitive [CK12]
Persistent Cohomology [MdSV-J11]
Discrete Morse Reduction [MN13]
Clear and Compress [BKR13]
O(|D(1 ) |n2
)
O(n3
)
O(n!
)
31. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
32. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
33. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
Find a vertex separator S such that
34. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
Find a vertex separator S such that
- |S| = O(nβ
)
35. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
Find a vertex separator S such that
- |S| = O(nβ
)
- each connected piece has O(n) vertices.
36. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
Find a vertex separator S such that
- |S| = O(nβ
)
- each connected piece has O(n) vertices.
The Punchline:
37. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
Find a vertex separator S such that
- |S| = O(nβ
)
- each connected piece has O(n) vertices.
The Punchline:
Inverting A can be done in O(nβω
) time.
38. Nested Dissection
A method for solving symmetric positive definite linear systems.
Ax = b
If A is n x n, consider the n vertex graph with an edge (i,j) for each
nonzero entry A(i, j) of A.
Find a vertex separator S such that
- |S| = O(nβ
)
- each connected piece has O(n) vertices.
The Punchline:
Inverting A can be done in O(nβω
) time.
Also works for computing ranks of singular,
nonsymmetric matrices over finite fields.
43. Sphere Separators
Sparse: Intersect only a
sublinear number of disks.
Balanced: Contains the
centers of at least and at
most a constant fraction of
the disks.
44. Sphere Separators
Sparse: Intersect only a
sublinear number of disks.
Balanced: Contains the
centers of at least and at
most a constant fraction of
the disks.
O(n1 1
d )
45. Sphere Separators
Sparse: Intersect only a
sublinear number of disks.
Balanced: Contains the
centers of at least and at
most a constant fraction of
the disks.
O(n1 1
d )
n
d + 2
48. Persistent Homology and Nested Dissection
A Conundrum:
Nested dissection gives an improvement by
choosing a good pivot order.
49. Persistent Homology and Nested Dissection
A Conundrum:
Nested dissection gives an improvement by
choosing a good pivot order.
Persistent homology restricts the pivot order.
50. Persistent Homology and Nested Dissection
A Conundrum:
Nested dissection gives an improvement by
choosing a good pivot order.
Persistent homology restricts the pivot order.
51. Persistent Homology and Nested Dissection
A Conundrum:
Nested dissection gives an improvement by
choosing a good pivot order.
Persistent homology restricts the pivot order.
The first trick:
52. Persistent Homology and Nested Dissection
A Conundrum:
Nested dissection gives an improvement by
choosing a good pivot order.
Persistent homology restricts the pivot order.
The first trick:
In the output-sensitive persistence algorithm of
Chen and Kerber, only ranks need to be computed.
53. Persistent Homology and Nested Dissection
A Conundrum:
Nested dissection gives an improvement by
choosing a good pivot order.
Persistent homology restricts the pivot order.
The first trick:
In the output-sensitive persistence algorithm of
Chen and Kerber, only ranks need to be computed.
58. Mesh Filtrations
Geometric
Approximation
Topologically
Equivalent
1. Compute the function on the vertices.
!
2. Approximate the sublevel with a union of Voronoi cells.
!
3. Filter the Delaunay triangulation appropriately.
Gives a guaranteed approximation for
a range of interesting functions.
Bonus: The theory of geometric
separators was invented for graphs
coming from meshes like this.
60. Putting it all together
Mesh Filtrations Geometric Separators
Nested Dissection Output-Sensitive
Persistence Algorithm
61. Putting it all together
Mesh Filtrations Geometric Separators
Nested Dissection Output-Sensitive
Persistence Algorithm
62. Theorem.
Let F be a filtration on the Delaunay triangulation of a set of ⌧-well-spaced
points. For a constant 0, the subset D of the persistence diagram of
F consisting of those pairs with persistence at least can be computed in
O(|D(1 ) |n!(1 1
d )
) time, where D(1 ) is the set of pairs in the persistence
diagram with persistence at least (1 ) for any constant > 0.
Putting it all together
Mesh Filtrations Geometric Separators
Nested Dissection Output-Sensitive
Persistence Algorithm
64. Summary
We can beat the matrix multiplication bound
for computing the persistent homology of a wide
class of functions defined on quality meshes.
65. Summary
We can beat the matrix multiplication bound
for computing the persistent homology of a wide
class of functions defined on quality meshes.
66. Summary
We can beat the matrix multiplication bound
for computing the persistent homology of a wide
class of functions defined on quality meshes.
The main technique is a combination of recent
advances in nested dissection, persistence
computation, and geometric separators.
67. Summary
We can beat the matrix multiplication bound
for computing the persistent homology of a wide
class of functions defined on quality meshes.
The main technique is a combination of recent
advances in nested dissection, persistence
computation, and geometric separators.
Thanks.
