Conley’s Connection Matrix in
Topological Data Analysis
Konstantin Mischaikow
Dept. of Mathematics, Rutgers [email protected]
SMAI-SIGMA, Nov 2017
Long Term Goal
Rayleigh-Benard Convection - Experiments
M. Schatz, GaTech
R. P. Behringer’s lab
Dense Granular Media
Photo elastic particles placed between two cross polarizers.
Our Current Tool
R
R Persistence Module
Persistence Diagram
Persistent Homology
dWp(PD, PD0) := inf X
z2PD
kz (z)kp1
!1p
Theorem (Cohen-Steiner, Edelsbrunner, Harer) PD is a continuous function.
continuous functions space of persistence diagrams
PD: C0(X, R) ! Per F : X ! R
Filtration of sublevel sets
⇥(F, ✓) := {x 2 X | F (x) ✓}
Use homology to quantify the geometry of sublevel sets
{H⇤(⇥(F, ✓)) | ✓ 2 R}
H⇤(⇥(F,·))
Persistent Homology
Data
Pipeline for Data Analysis:
Complex Filtration Persistence
Diagram Interpretation
large huge huge small
Information Reduction
Data
Reduction
PH: C0(⌦, R) ! Per
Problems with Persistent Homology
A: Loss of relative geometry
Relative locations of critical points are different, but persistence diagrams are the same.
B: Tracking local features through time or parameter space is not smooth.
A Classical Tool
Morse Theory
R
Morse-Smale Homology
F : X ! R
˙
x = rF (x)
Gradient Vector Field
R Identify Critical Points
a c b d
e
Identify Morse Index
Identify Connecting Orbits
2
41 0 0 1 1 1
3 5 :
a
b ! 2 4c
d e
3 5
Morse-Smale boundary operator
Morse Homology
(in the context of these problems) A: Relative geometry is accessible
ab c de
fg 2
66 4
0 0 1 1 1 0 1 0 0 0 1 1
3 77 5 :
2 4e
f g
3
5 ! 2 66 4
a b c d
3 77 5
2 66 4
0 0 1 1 0 1 1 1 0 0 1 0
3 77 5 :
2 4e
f g
3
5 ! 2 66 4
a b c d
3 77 5
B: Boundary operator “changes” smoothly
a b c
a b
c c
a b
1
1 : ⇥ c⇤
!
a b
Problems with Morse Homology
MH: C2(⌦, R) \ Morse \ Smale ! Chain Complex
topology?
transverse intersection of stable and unstable
manifolds nondegenerate
critical points
Example: Global breakdown of Morse homology
a
c b d
1 0 1 1 :
c
d !
a b
1 1 1 0 :
c
d !
a b
An Alternative Approach
Remark/Warning: Because of my background I think of Morse homology in terms of dynamics.
Data Complex Filtration Persistence
Diagram Interpretation
large huge huge small
Information Reduction
Data
Reduction
Lattice Reduced
Complex
Persistence
Less
Information Reduction
Less Data Reduction
Goal:
Alternative Pipeline for Data Analysis
Let (L, ^, _, 0, 1) denote a finite bounded distributive lattice.
x < y if x ^ y = x
The set of join irreducible elements of L is
J_(L) := {x 2 L | if x = a _ b, then a = x or b = x}
The lattice of down sets of (P, <) is
O(P) := {U ⇢ P | if x 2 U and y < x then y 2 U} . Let (P, <) denote a partially ordered set (poset).
Fact: O and J are contravariant functors.
poset
Theorem: (Birkhoff)
O(J_(L)) ⇠= L J_(O(P)) ⇠= P
Birkhoff’s Theorem
;
{a} {b}
{ac} {ab} {bf}
{abc} {abd} {abe} {abf}
{abcd} {abde} {abcf} {abef} {abcde} {abcdf} {abdeg} {abcef} {abdef} {abcdeg} {abcdef} {abdef g} {abdef h}
{abcdef g} {abcdef h}
{abcdef h}
a b
c d e f
g h
(P, <) ⇠= J_(O(P))
Approximating Topological Spaces
Let X be a compact metric space. In the setting of dynamics this is the phase space
Particular choice of (family of) approximation
Let L be a finite bounded sublattice of R(X).
Space of approximations Typically an uncountable
lattice Let R(X) denote the lattice of
regular closed subsets of X.
cl(int(U)) = U
Smallest scale of
measurements or applicability of model
Let G(L) denoted atoms of L
Lattice/Dynamics
In the setting of dynamics this is the lattice of attracting
regions (attracting blocks) Fix A ⇢ L, a bounded sublattice.
For each p 2 P define a Morse tile M(p) := cl(A \ pred(A)) The Morse graph MG of A is the Hasse diagram for the poset (P := J_(A), <) obtained from Birkhoff.
Example
Morse tiles M(p)
Let F 0(x) = f(x).
-4 0 4
Atoms of lattice: G(L) = {[n, n + 1] | n = 4, . . . , 3} Phase space: X = [ 4, 4] ⇢ R
A
Lattice of attractors: A = {[ 3, 1], [1, 3], [ 3, 1] [ [1, 3], [ 4, 4]}
P
1 2
3
Birkhoff
How does this relate to a differential equation dxdt = f(x)?
-4 0 4
F
F F
Remark: This order structure is robust with respect to perturbations Elements of A represent regions of
phase space that are forward invari- ant with time.
Morse Tiles a
d c e
f h
b g
a b
c d e f
g h
;
{a} {b}
{ac} {ab} {bf}
{abc} {abd} {abe} {abf}
{abcd} {abde} {abcf} {abef} {abcde} {abcdf} {abdeg} {abcef} {abdef} {abcdeg} {abcdef} {abdef g} {abdef h}
{abcdef g} {abcdef h}
{abcdef h}
Lattice of Attracting Neighborhoods A
Theorems extracted from this information are valid for any differential equation x˙ = f(x) for which f is transverse to indicated curves in directions indicated by red arrows.
Morse graph MG
Conley Index
Let P := J_(A). For each p 2 P, the Conley index of the Morse tile M (p) is
CH⇤(p) := H⇤(A, pred(A)) where M(p) := cl(A \ pred(A)).
-4 0 4
F
F
F
-4 0 4
(0, Z, 0, . . .)
(Z, 0, . . .) (Z, 0, . . .)
Recall Example:
Phase space: X = [ 4, 4] ⇢ R
A
A = {[ 3, 1], [1, 3], [ 3, 1] [ [1, 3], [ 4, 4]}
Lattice of Attractors
1 2
3
(P, <)
Birkhoff
Let A be a lattice of attractors on X and (P := J_(A), <).
is called a connection matrix.
Theorem: (Franzosa, Robbin-Salamon) There exists a strictly upper triangular (with respect to <) boundary operator
: M
p2P
CH⇤(p) ! M
p2P
CH⇤(p)
such that the induced homology is isomorphic to H⇤(X), and further- more, if Q is a convex subset of P, then the induced homology on the restricted complex
: M
p2Q
CH⇤(p) ! M
p2Q
CH⇤(p) is CH⇤([Q]).
-4 0 4
F
F
F
-4 0 4
(0, Z, 0, . . .)
(Z, 0, . . .) (Z, 0, . . .)
Recall Example:
Phase space: X = [ 4, 4] ⇢ R
A
A = {[ 3, 1], [1, 3], [ 3, 1] [ [1, 3], [ 4, 4]}
Lattice of Attractors
1 2
3
(P, <)
Birkhoff
The associated connection matrix : L
p=1,2,3 CH(p) ! L
p=1,2,3 CH(p) is
:=
2
40 0 1 0 0 1 0 0 0
3 5
a
d c e
f h
b g
0 BB BB BB BB BB BB
@
0 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 ↵
0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
1 CC CC CC CC CC CC A
Connection matrix with Z2 coefficients.
(↵, ) 2 {(1, 0), (0, 1)} Connection matrices need not be unique (I):
a
c b d
1 0 1 1 :
c
d !
a b
1 1 1 0 :
c
d !
a b
Connection matrices need not be unique (II):
Computing
Cubical complex associated with L. Morse tiles associated with lattice A.
Run (repeated) discrete algebraic Morse theory reduction restricted to each Morse tile. (V. Nanda, K.M., D&CG, 2012)
Theorem: (S. Harker, K. M., K. Spendlove) Over field coefficients the boundary operator for a minimal chain complex is a connection matrix.
Remark: A different sequence of algebraic Morse reductions can pro- duce a different connection matrix.
Theorem: The set of connection matrices for a given lattice is given by T T 1 | T is upper trianguler w.r.t. <
Identifying the lattice A
p1 p0 p2
p3
Vertices: States Edges: Dynamics
Don’t know exact current state, so don’t know exact next state
Simple decomposition of Dynamics:
Recurrent
Nonrecurrent (gradient-like)
Linear time Algorithm!
Morse Graph
of state transition graph State Transition Graph
F : X !! XPoset
An Application
F = greyscale.
F : X ◆ X
go down + self loop
Identify Morse nodes with nontrivial Conley
index
F = greyscale.
F : X ◆ X
go down + self loop
Identify Morse nodes with nontrivial Conley
index
H0 generators H1 generators H2 generators
Thank-you for your Attention
Homology + Database Software chomp.rutgers.edu
