Topological Persistence: Introduction, Stability,
Algorithms
Tamal K. Dey
Department of Computer Science and Engineering
The Ohio State University
Dey (2016)
Persistence
WSoCM 16
1 / 43
Topological Persistence
Data Analysis by Persistent Homology
Persistent homology [Edelsbrunner-Letscher-Zomorodian 00],
[Zomorodian-Carlsson 02]
[Ghrist 08]
Dey (2016)
Persistence
WSoCM 16
2 / 43
Topological Persistence
Homology (Co-Homology) with Z2
Definition
The p-dimensional homology group is defined as
Hp (K) = Zp (K)/Bp (K)
Dey (2016)
Persistence
WSoCM 16
3 / 43
Topological Persistence
Homology (Co-Homology) with Z2
Definition
The p-dimensional homology group is defined as
Hp (K) = Zp (K)/Bp (K)
Definition
Two p-chains c and c 0 are homologous if c = c 0 + ∂p+1 d for some
chain d
Dey (2016)
Persistence
WSoCM 16
3 / 43
Topological Persistence
Homology (Co-Homology) with Z2
Definition
The p-dimensional homology group is defined as
Hp (K) = Zp (K)/Bp (K)
Definition
Two p-chains c and c 0 are homologous if c = c 0 + ∂p+1 d for some
chain d
(a) trivial (null-homologous) cycle; (b), (c) nontrivial homologous cycles
Dey (2016)
Persistence
WSoCM 16
3 / 43
Topological Persistence
Complexes
Let P ⊂ Rd be a point set
Dey (2016)
Persistence
WSoCM 16
4 / 43
Topological Persistence
Complexes
Let P ⊂ Rd be a point set
B(p, r ) denotes an open d-ball centered at p with radius r
Dey (2016)
Persistence
WSoCM 16
4 / 43
Topological Persistence
Complexes
Let P ⊂ Rd be a point set
B(p, r ) denotes an open d-ball centered at p with radius r
Definition
The Čech complex C r (P): simplex σ ∈ C r (P) iff Vert(σ) ⊆ P and
∩p∈Vert(σ) B(p, r /2) 6= 0
Dey (2016)
Persistence
WSoCM 16
4 / 43
Topological Persistence
Complexes
Let P ⊂ Rd be a point set
B(p, r ) denotes an open d-ball centered at p with radius r
Definition
The Čech complex C r (P): simplex σ ∈ C r (P) iff Vert(σ) ⊆ P and
∩p∈Vert(σ) B(p, r /2) 6= 0
Definition
The Vietoris-Rips complex Rr (P): simplex σ ∈ Rr (P) iff Vert(σ) are
within pairwise Euclidean distance of r
Dey (2016)
Persistence
WSoCM 16
4 / 43
Topological Persistence
Complexes
Let P ⊂ Rd be a point set
B(p, r ) denotes an open d-ball centered at p with radius r
Definition
The Čech complex C r (P): simplex σ ∈ C r (P) iff Vert(σ) ⊆ P and
∩p∈Vert(σ) B(p, r /2) 6= 0
Definition
The Vietoris-Rips complex Rr (P): simplex σ ∈ Rr (P) iff Vert(σ) are
within pairwise Euclidean distance of r
Proposition
For any finite set P ⊂ Rd and any r ≥ 0, C r (P) ⊆ Rr (P) ⊆ C 2r (P)
Dey (2016)
Persistence
WSoCM 16
4 / 43
Topological Persistence
Point set P
Dey (2016)
Persistence
WSoCM 16
5 / 43
Topological Persistence
Balls B(p, r /2) for p ∈ P
Dey (2016)
Persistence
WSoCM 16
6 / 43
Topological Persistence
Čech complex C r (P)
Dey (2016)
Persistence
WSoCM 16
7 / 43
Topological Persistence
Rips complex Rr (P)
Dey (2016)
Persistence
WSoCM 16
8 / 43
Topological Persistence
Point cloud and persistence
r (x) = d(x, P): distance
to point cloud P
Sublevel sets r −1 [0, a] are
union of balls
Evolution of the sublevel
sets with increasing a–left
hole persists longer
Persistent homology
quantizes this idea
Dey (2016)
Persistence
WSoCM 16
9 / 43
Topological Persistence
Persistent Homology
f : T → R; Ta = f −1 (−∞, a], the
sublevel set
Ta ⊆ Tb for a ≤ b provides inclusion
map ι : Ta → Tb
Induced map ι∗ : Hp (Ta ) → Hp (Tb )
giving the sequence
0 → Hp (Ta1 ) → Hp (Ta2 ) → · · · → Hp (Tan ) → Hp (T)
Persistent homology classes: Image of
fpij : Hp (Tai ) → Hp (Taj )
Dey (2016)
Persistence
WSoCM 16
10 / 43
Topological Persistence
Continuous to Discrete
Replace T with a simplicial complex
K := K (T)
Dey (2016)
Persistence
WSoCM 16
11 / 43
Topological Persistence
Continuous to Discrete
Replace T with a simplicial complex
K := K (T)
Union of balls with its nerve Čech
complex
Dey (2016)
Persistence
WSoCM 16
11 / 43
Topological Persistence
Continuous to Discrete
Replace T with a simplicial complex
K := K (T)
Union of balls with its nerve Čech
complex
Dey (2016)
Persistence
WSoCM 16
11 / 43
Topological Persistence
Continuous to Discrete
Replace T with a simplicial complex
K := K (T)
Union of balls with its nerve Čech
complex
(a)
(b)
(c)
Dey (2016)
Persistence
(d)
WSoCM 16
11 / 43
Topological Persistence
Continuous to Discrete
Replace T with a simplicial complex
K := K (T)
Union of balls with its nerve Čech
complex
(a)
(b)
Evolution of sublevel sets becomes
Filtration:
(c)
(d)
∅ = K0 ⊆ K1 ⊆ · · · ⊆ Kn = K
0 → Hp (K1 ) → · · · → Hp (Kn ) = Hp (K ).
Dey (2016)
Persistence
WSoCM 16
11 / 43
Topological Persistence
Continuous to Discrete
Replace T with a simplicial complex
K := K (T)
Union of balls with its nerve Čech
complex
(a)
(b)
Evolution of sublevel sets becomes
Filtration:
(c)
(d)
∅ = K0 ⊆ K1 ⊆ · · · ⊆ Kn = K
0 → Hp (K1 ) → · · · → Hp (Kn ) = Hp (K ).
Birth and Death of homology classes
Dey (2016)
Persistence
WSoCM 16
11 / 43
Topological Persistence
Bar Codes
birth-death and bar codes
Dey (2016)
Persistence
WSoCM 16
12 / 43
Topological Persistence
Persistence Diagram
a bar [a, b] is represented as a point in the plane
Dey (2016)
Persistence
WSoCM 16
13 / 43
Topological Persistence
Persistence Diagram
a bar [a, b] is represented as a point in the plane
incorporate the diagonal in the diagram Dgm p (f )
Dey (2016)
Persistence
WSoCM 16
13 / 43
Topological Persistence
Persistence Diagram
a bar [a, b] is represented as a point in the plane
incorporate the diagonal in the diagram Dgm p (f )
Dey (2016)
Persistence
WSoCM 16
13 / 43
Topological Persistence
Persistence Diagram
a bar [a, b] is represented as a point in the plane
incorporate the diagonal in the diagram Dgm p (f )
Death
Birth
Dey (2016)
Persistence
WSoCM 16
13 / 43
Topological Persistence
Stability of Persistence Diagram
Bottleneck distance (C all bijections)
dB (Dgm p (f ), Dgm p (g )) := inf
sup
c∈C x∈Dgm (f )
p
kx − c(x)k∞
Theorem (Cohen-Steiner,Edelsbrunner,Harer 06)
dB (Dgm p (f ), Dgm p (g )) ≤ kf − g k∞
Dey (2016)
Persistence
WSoCM 16
14 / 43
Topological Persistence
Back to Point Data
dT be the distance function from the space T.
Dey (2016)
Persistence
WSoCM 16
15 / 43
Topological Persistence
Back to Point Data
dT be the distance function from the space T.
dP be the distance function from sample P
Dey (2016)
Persistence
WSoCM 16
15 / 43
Topological Persistence
Back to Point Data
Death
Birth
dT be the distance function from the space T.
dP be the distance function from sample P
Dey (2016)
Persistence
WSoCM 16
15 / 43
Topological Persistence
Back to Point Data
Death
Birth
dT be the distance function from the space T.
dP be the distance function from sample P
kdT − dP k∞ ≤ d(T, P) = ε
Dey (2016)
Persistence
WSoCM 16
15 / 43
Topological Persistence
Back to Point Data
Death
Birth
dT be the distance function from the space T.
dP be the distance function from sample P
kdT − dP k∞ ≤ d(T, P) = ε
dB (Dgm p (dT ), Dgm (dP )) ≤ ε
Dey (2016)
Persistence
WSoCM 16
15 / 43
Topological Persistence
Back to Point Data
Death
Birth
dT be the distance function from the space T.
dP be the distance function from sample P
kdT − dP k∞ ≤ d(T, P) = ε
dB (Dgm p (dT ), Dgm (dP )) ≤ ε
Dey (2016)
Persistence
WSoCM 16
15 / 43
Interleaving
Interleaving
What about stabilty when domains aren’t same.
Dey (2016)
Persistence
WSoCM 16
16 / 43
Interleaving
Interleaving
What about stabilty when domains aren’t same.
Interleaving [Chazal-Oudot 08], [CCGGO08]
Dey (2016)
Persistence
WSoCM 16
16 / 43
Interleaving
Interleaving
What about stabilty when domains aren’t same.
Interleaving [Chazal-Oudot 08], [CCGGO08]
Persistence modules
fαα
0
00
fαα0
· · · Fα −→ Fα0 −→ Fα00 · · · ∀α, α0 , α00 ∈ A ⊆ R
00
00
0
fαα = fαα0 ◦ fαα
Dey (2016)
Persistence
WSoCM 16
16 / 43
Interleaving
Interleaving
What about stabilty when domains aren’t same.
Interleaving [Chazal-Oudot 08], [CCGGO08]
Persistence modules
fαα
0
00
fαα0
· · · Fα −→ Fα0 −→ Fα00 · · · ∀α, α0 , α00 ∈ A ⊆ R
00
00
0
fαα = fαα0 ◦ fαα
One has persistence diagram Dgm p ({Fα })
Dey (2016)
Persistence
WSoCM 16
16 / 43
Interleaving
Interleaving
What about stabilty when domains aren’t same.
Interleaving [Chazal-Oudot 08], [CCGGO08]
Persistence modules
fαα
0
00
fαα0
· · · Fα −→ Fα0 −→ Fα00 · · · ∀α, α0 , α00 ∈ A ⊆ R
00
00
0
fαα = fαα0 ◦ fαα
One has persistence diagram Dgm p ({Fα })
Two persistence modules {Fα } and {Gα } ε-interleave if:
fαα
00
0
fαα0
· · · Fα −→ Fα0 −→ Fα00 · · ·
gαα
00
0
gαα0
· · · Gα −→ Gα0 −→ Gα00 · · ·
Dey (2016)
Persistence
WSoCM 16
16 / 43
Interleaving
Interleaving
/ Fα+2ε
;
Fα
!
Gα+ε
Dey (2016)
F= α+ε
Gα
Persistence
/
(1)
#
Gα+2ε
WSoCM 16
17 / 43
Interleaving
Interleaving
/ Fα+2ε
;
Fα
!
F= α+ε
/
Gα
Gα+ε
#
Gα+2ε
/ Fα0
Fα
"
/
Gα+ε
/
Gα
Persistence
"
Gα0 +ε
/ Fα0 +ε
<
F< α+ε
Dey (2016)
(1)
Gα0
WSoCM 16
17 / 43
Interleaving
Interleaving
Strong ε-interleaving if diagrams hold for ∀α, α0 ∈ A ⊆ R
Dey (2016)
Persistence
WSoCM 16
18 / 43
Interleaving
Interleaving
Strong ε-interleaving if diagrams hold for ∀α, α0 ∈ A ⊆ R
Weak ε-interleaving if diagrams hold for ε-periodic sets of α.
Dey (2016)
Persistence
WSoCM 16
18 / 43
Interleaving
Interleaving
Strong ε-interleaving if diagrams hold for ∀α, α0 ∈ A ⊆ R
Weak ε-interleaving if diagrams hold for ε-periodic sets of α.
dB (Dgm p ({Fα }, {Gα }) ≤ ε if {Fα } and {Gα } are strongly
ε-interleaved.
Dey (2016)
Persistence
WSoCM 16
18 / 43
Interleaving
Interleaving
Strong ε-interleaving if diagrams hold for ∀α, α0 ∈ A ⊆ R
Weak ε-interleaving if diagrams hold for ε-periodic sets of α.
dB (Dgm p ({Fα }, {Gα }) ≤ ε if {Fα } and {Gα } are strongly
ε-interleaved.
dB (Dgm p ({Fα }, {Gα }) ≤ 3ε if {Fα } and {Gα } are weakly
ε-interleaved.
Dey (2016)
Persistence
WSoCM 16
18 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Dey (2016)
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Most literature considers Rips complexes, but they are Huge
Dey (2016)
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Most literature considers Rips complexes, but they are Huge
Sparsified Rips complex [Sheehy 12]
Dey (2016)
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Most literature considers Rips complexes, but they are Huge
Sparsified Rips complex [Sheehy 12]
Graph Induced Complex (GIC) [D.-Fang-Wang 13]
Rr (P )
Dey (2016)
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Most literature considers Rips complexes, but they are Huge
Sparsified Rips complex [Sheehy 12]
Graph Induced Complex (GIC) [D.-Fang-Wang 13]
Rr (P )
Dey (2016)
Q
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Most literature considers Rips complexes, but they are Huge
Sparsified Rips complex [Sheehy 12]
Graph Induced Complex (GIC) [D.-Fang-Wang 13]
Rr (P )
Dey (2016)
0
Rr (Q)
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Choice of Complexes
Čech complexes are difficult to compute
Most literature considers Rips complexes, but they are Huge
Sparsified Rips complex [Sheehy 12]
Graph Induced Complex (GIC) [D.-Fang-Wang 13]
Rr (P )
Dey (2016)
0
Rr (Q)
G r (P, Q)
Persistence
WSoCM 16
19 / 43
Interleaving
Issues: Filtration Maps
Zigzag inclusions[Carlsson-de Silva-Morozov 09]
K1 ⊆ K2 ⊆ K3 ⊆ . . . ⊆ Kn
Dey (2016)
Persistence
WSoCM 16
20 / 43
Interleaving
Issues: Filtration Maps
Zigzag inclusions[Carlsson-de Silva-Morozov 09]
K1 ⊆ K2 ⊇ K3 ⊇ . . . ⊆ Kn
Dey (2016)
Persistence
WSoCM 16
20 / 43
Interleaving
Issues: Filtration Maps
Zigzag inclusions[Carlsson-de Silva-Morozov 09]
K1 ⊆ K2 ⊇ K3 ⊇ . . . ⊆ Kn
Simplicial maps instead of inclusions [D.-Fan-Wang 14]
f
f
f
n
1
2
K −→
K1 −→
K2 · · · −→
Kn = K 0
Dey (2016)
Persistence
WSoCM 16
20 / 43
Interleaving
Issues: Filtration Maps
Zigzag inclusions[Carlsson-de Silva-Morozov 09]
K1 ⊆ K2 ⊇ K3 ⊇ . . . ⊆ Kn
Simplicial maps instead of inclusions [D.-Fan-Wang 14]
f
f
f
n
1
2
K −→
K1 −→
K2 · · · −→
Kn = K 0
Efficient algorithm for Zigzag simplicial maps?
Dey (2016)
Persistence
WSoCM 16
20 / 43
Interleaving
Simplicial Maps
Dey (2016)
Persistence
WSoCM 16
21 / 43
Interleaving
Basic Definitions
Definition
Simplicial map: A map f : K → L is simplicial if vertices for every simplex
σ ∈ K are mapped to the vertices of the simplex f (σ) in L.
w
w
f
v
u
K
Dey (2016)
y
u
x
L
Persistence
x
WSoCM 16
22 / 43
Interleaving
Basic definitions
Definition
Elementary simplicial map: f : K → K 0 is elementary if the vertex map fv is
identical except possibly on X ⊆ V (K ) for which fv (X ) is singleton.
If X = ∅ and K 0 \ K is a single simplex, f is an elementary inclusion;
If |X | = 2 and f surjective, elementary collapse;
Dey (2016)
Persistence
WSoCM 16
23 / 43
Interleaving
Basic definitions
Definition
Elementary simplicial map: f : K → K 0 is elementary if the vertex map fv is
identical except possibly on X ⊆ V (K ) for which fv (X ) is singleton.
If X = ∅ and K 0 \ K is a single simplex, f is an elementary inclusion;
If |X | = 2 and f surjective, elementary collapse;
w
w
w
w
y
{u, v} → u
v
u
u
x
K
K
0
x
x
K
u
v
K
x
0
Elementary inclusion
Elementary collapse
Dey (2016)
,→
v
u
Persistence
WSoCM 16
23 / 43
Interleaving
Basic definitions
Definition
Stars and Links of X ⊆ K :
StX
StX
LkX
:= {σ | σ is a coface of a simplex in X }
:= {all faces of simplices in StX }
:= StX − StX
w
v
u
K
x
Stv = {v , vw , vx, uv , uvw , u, w , x, uw }
Lkv = {u, w , x, uw }, Lkuv = {w }
Dey (2016)
Persistence
WSoCM 16
24 / 43
Interleaving
Decomposition of simplicial maps
Proposition
If f : K → K 0 is a simplicial map, then there are elementary inclusions and
collapses fi
f1
f2
fn
K −→
K1 −→
K2 · · · −→
Kn = K 0
so that f = fn ◦ · · · ◦ f2 ◦ f1 .
Dey (2016)
Persistence
WSoCM 16
25 / 43
Interleaving
Decomposition of simplicial maps
Proposition
If f : K → K 0 is a simplicial map, then there are elementary inclusions and
collapses fi
f1
f2
fn
K −→
K1 −→
K2 · · · −→
Kn = K 0
so that f = fn ◦ · · · ◦ f2 ◦ f1 .
f
w
w
w
{u, v} → u
v
u
Dey (2016)
u
x
K
y
,→
u
K1
Persistence
x
x
K2
WSoCM 16
25 / 43
Interleaving
Zigzag simplicial maps
f
f
f
f
fn−1
1
2
3
4
K1 −→
K2 ←−
K3 −→
K4 ←−
. . . −→ Kn
Dey (2016)
Persistence
WSoCM 16
26 / 43
Interleaving
Simulating elementary maps by inclusions
If f : K → K 0 elementary inclusion
no action;
If f : K → K 0 elementary collapse
f ({u, v }) −→ u ∈ K 0
b := K + u ∗ Stv ;
Augmenting K as K
∗ is the join (coning) operation;
w
x
w
y
v
y
v
u
u
K̂ := K + u ∗ Stv
K
Dey (2016)
x
Persistence
WSoCM 16
27 / 43
Interleaving
Simulating elementary maps by inclusions
b
Claim: K 0 ⊆ K
Dey (2016)
Persistence
WSoCM 16
28 / 43
Interleaving
Simulating elementary maps by inclusions
b
Claim: K 0 ⊆ K
x
x
w
{u, v} → u
y
v
w
y
u
u
K
w
x
K0
y
v
u
K̂
Dey (2016)
Persistence
WSoCM 16
28 / 43
Interleaving
Zigzag persistence of simplicial maps
Proposition
For a zigzag sequence of elementary simplicial maps,
f
f
f
fn−1
1
2
3
K1 →
K2 ←
K3 →
. . . → Kn
one can compute its zigzag persistence through zigzag filtration:
i10
i1
i20
i2
i3
0
in−1
c1 ←- K2 ,→ K
c3 ←- K3 ,→ . . . ←- Kn
K1 ,→ K
(ik0 )∗ ’s are isomorphisms;
(ik )∗ = (ik0 )∗ ◦ (fk )∗ ;
The following two zigzag modules have the same persistence diagram:
f1
f2∗
f3∗
H∗ (K1 ) →∗ H∗ (K2 ) ← H∗ (K3 ) → · · · → H∗ (Kn )
i∗
i∗
i∗
i∗
c1 ) ' H∗ (K2 ) ' H∗ (K
c3 ) ←
H∗ (K1 ) →
H∗ (K
H∗ (K3 ) →
··· ←
H∗ (Kn )
Dey (2016)
Persistence
WSoCM 16
29 / 43
Interleaving
Annotations
Dey (2016)
Persistence
WSoCM 16
30 / 43
Interleaving
Persistence using annotation
Persistence of non-zigzag sequence of simplicial maps by
annotations
Maintains a consistent cohomology basis;
Cohomology basis elements are time stamped for tracking
persistence;
Creates much less intermediate simplices;
Dey (2016)
Persistence
WSoCM 16
31 / 43
Interleaving
Annotation a for simplicial complex K
00
00
00
01
10
00
00
u
Mapping a : K p → Zg2
v
u
11
v
K p : the set of p-simplices of K ;
aσ = a(σ) : a binary vector of length g ;
K1 = {
}
Z22
a valid if
g = rankHp (K ) ;
az1 = az2 iff [z1 ] = [z2 ];
Dey (2016)
Persistence
WSoCM 16
32 / 43
Interleaving
Annotation and Cohomology
Proposition
The following two statements are equivalent:
1
An annotation a : K p → Zg2 is valid
2
The cochains {φi }i=1,··· ,g given by φi (σ) = aσ [i] for all σ ∈ K p
are cocycles whose cohomology classes constitute a basis of
H p (K ).
Dey (2016)
Persistence
WSoCM 16
33 / 43
Interleaving
Elementary inclusion
Elementary inclusion
Obtain a valid annotation of Ki+1 from Ki after inserting
p-simplex σ = Ki+1 \ Ki ;
Two cases : a∂σ = 0 or a∂σ 6= 0;
σ
σ
1
0
0
0
a∂σ = 0 ;
Dey (2016)
0
a∂σ 6= 0;
Persistence
WSoCM 16
34 / 43
Interleaving
Elementary inclusion case 1
Elementary inclusion case 1 : a∂σ = 0
σ creates a p-cycle in Ki+1 ;
Augment the annotation [b1 , b2 , . . . , bg ] for p-simplex τ to
[b1 , b2 , . . . , bg , bg +1 ]
0
if τ 6= σ
bg +1 =
1
if τ = σ
The new element time stamped as i + 1;
0
0
0
Dey (2016)
1
1
1
00
00
1
01
v
u
00
Persistence
1
10
WSoCM 16
35 / 43
Interleaving
Elementary inclusion case 2
Elementary inclusion case 2 : a∂σ 6= 0
σ kills a (p − 1)-cycle ∂σ;
bu the last nonzero element in a∂σ = [b1 , b2 , . . . , bu , . . . , bg ];
aτ = aτ + a∂σ if (aτ )u = 1;
Remove u-th element from all annotations;
00
00
00
Dey (2016)
v
01
u
10
00
00
v
00
u
00
10
Persistence
0
0
v
0
u
0
1
WSoCM 16
36 / 43
Interleaving
Elementary collapse
Definition
For an elementary collapse fi : Ki → Ki+1 , a simplex σ ∈ Ki is called
vanishing if the cardinality of fi (σ) is one less than that of σ. Two
simplices σ and σ 0 are called mirror of each other if one adjoins u and
the other v , and share rest of the vertices.
w
u
w
u
v
f
uv vanishing simplex;
wu and wv mirror to each other;
Dey (2016)
Persistence
WSoCM 16
37 / 43
Interleaving
Elementary collapse
Obtain a valid annotation of Ki+1 from Ki after collapsing (u, v )
to u;
Two cases : (u, v ) satisfies link condition or not
Lkuv = Lku ∩ Lkv .
u
v
u
uv does not satisfy link condition;
Dey (2016)
Persistence
v
uv satisfies link condition;
WSoCM 16
38 / 43
Interleaving
Elementary collapse case 1
Elementary collapse case 1 : (u, v ) satisfying link condition
τ any mirror simplex containing u;
σ unique coface of τ containing the edge uv ;
aω = aω + aσ for any coface ω of τ of codimension one
00
00
00
00
01
10
00
00
u
Dey (2016)
11
v
00
00
00
10
10
00
11
u
00 00
00
Persistence
v
11
10
00
u
WSoCM 16
39 / 43
Interleaving
Elementary collapse case 2
Elementary collapse case 2 : (u, v ) not satisfying link condition
Insert necessary simplices σ to meet the link condition ;
Apply the elementary collapse case 1;
Collapsing edge uv :
000
000
010
100
000
1
Inclusion :
u
00 00
000
001
10
00
u
000
000
000
010
100
u
11
v
000
000
2
00
000
000
001
000
000
000
010
100
u
v
00
000
110
00
00
01
10
u
v
00
00
11
v
Elementary collapse case 1:
00
00
00
00
01
10
00
00
u
Dey (2016)
11
v
00
00
00
10
10
00
11
u
00 00
00
Persistence
v
11
10
00
u
WSoCM 16
40 / 43
Interleaving
Software
GUDHI library: https://project.inria.fr/gudhi/software/
Dey (2016)
Persistence
WSoCM 16
41 / 43
Interleaving
Software
GUDHI library: https://project.inria.fr/gudhi/software/
Based on Compressed Annotation Matrix (CAM)
[Boissonnat-D.-Maria 13]
Dey (2016)
Persistence
WSoCM 16
41 / 43
Interleaving
Software
GUDHI library: https://project.inria.fr/gudhi/software/
Based on Compressed Annotation Matrix (CAM)
[Boissonnat-D.-Maria 13]
Simpers: ∼tamaldey/SimpPers/SimpPers-software/
Dey (2016)
Persistence
WSoCM 16
41 / 43
Interleaving
Software
GUDHI library: https://project.inria.fr/gudhi/software/
Based on Compressed Annotation Matrix (CAM)
[Boissonnat-D.-Maria 13]
Simpers: ∼tamaldey/SimpPers/SimpPers-software/
Implements simplicial maps (under work)
Dey (2016)
Persistence
WSoCM 16
41 / 43
Interleaving
Software
GUDHI library: https://project.inria.fr/gudhi/software/
Based on Compressed Annotation Matrix (CAM)
[Boissonnat-D.-Maria 13]
Simpers: ∼tamaldey/SimpPers/SimpPers-software/
Implements simplicial maps (under work)
Data
CY8
K1
L35
Dey (2016)
|P|
|K| Dionysus
6K 21M
420
10K 74M
569
0.7K 18M
109
Persistence
PHAT
5.3
1785
17.5
CAM
6.4
19.7
5.1
WSoCM 16
41 / 43
Interleaving
Conclusions
Topological data analysis :
More efficient algorithms
Scalability
Synergy between Topology/Algorithms
Need both
Application areas:
Data mining, Machine learning
Scientific, Engineering, Social Studies
Dey (2016)
Persistence
WSoCM 16
42 / 43
Thank
www.cse.ohio-state.edu/~tamaldey/course/CTDA/CTDA.html
Thank You
Dey (2016)
Persistence
WSoCM 16
43 / 43