In this talk, I give a gentle introduction to geometric and topological data analysis and then segue into some natural questions that arise when one combines the topological view with the perhaps more well-studied linear algebraic view.

1. Marcel Duchamp

2. 3 Standard Stoppages

3. Network of Stoppages

4. Geometric and Topological Data Analysis
Geometric Data
Shape
Analysis
Geometry
Processing
Regression
(space between data)
Clustering
(meaning of data)
Geometry of Data
Persistent Homology
(beyond linear structure)
(beyond simple connectivity)

5. Homology
Homology turns topological questions into algebraic questions.
Augment the data points with
edges, triangles, tetrahedra, etc.
!
The kth boundary matrix ∂k maps
k-simplices to the (k-1)-simplices
in their boundary.
!
The kth homology group is the
quotient ker ∂k / im ∂k+1.
!
Homology encodes connected
components, holes, and voids.

6. Persistent Homology
time
H0
H1
Barcode
Persistence
Diagram

7. Stability
Data set Persistence diagram
close
Data set
close
Persistence diagram

8. Stability
Data set Persistence diagram
Metric on
s
Metric on
s
Lipschitz

9. 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.

10. 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 at most cn vertices (for some c < 1).
!
Repeat. Order the pivots going up from the leaves of the recursion.
The Punchline:
Inverting A can be done in O(nβω
) time.
Also works for computing ranks of singular,
nonsymmetric matrices over finite fields.

11. Reasonable complexes have small separators.
The theory of geometric separators applies to graphs of nice meshes.
!
Separators on graphs can be “lifted” to separators on complexes.
!
Improves the asymptotic complexity of static homology.
!
Persistence?

12. Thanks.
Some open problems.
!
How do we reconcile the filtration
order and the nested dissection order?
!
Is there a quotient version of
nested dissection?
!
Is there a reasonable separator theory
for filtrations?