Marcel Duchamp

3 Standard Stoppages

Network of Stoppages

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)

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.

Persistent Homologytime

H0

H1

Barcode

Persistence Diagram

Stability

Data set Persistence diagram

close Data set

close Persistence diagram

Stability

Data set Persistence diagramMetric on

sMetric on

sLipschitz

Computing Persistent HomologyInput: Boundary Matrix D Find V, R such that

D = RVV 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.

Nested DissectionA method for solving symmetric positive definite linear systems.

Ax = bIf 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.

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?

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?