2012 -TDA-Intro-part_2

DEPARTAMENTO DE INFORMATICA
INVESTIGACION Y POSTGRADO
Topological Data Analysis (TDA)
Simplicial Homology
Rodrigo Rojas Moraleda
May 29, 2013
Rodrigo Rojas Moraleda — Topological Data Analysis (TDA) 1/15

Outline
1 Last session
2 Simplicial Homology

1 Last session

Topology: Geometry continuity.

Homeomorphism.

Topological properties or topological invariants

Classify in sense of:
P = {P0, · · · , Pn}

P = {P0, · · · , Pn}
property preserved by homeomorphisms.

P = {P0, · · · , Pn}
Distinguishable

P = {P0, · · · , Pn}
Distinguishable
Completeness

However general topological classiﬁcation is unaﬀordable

Polyhedral space

1 Last session

Simplicial Homology
Convex Sets

Simplicial Homology
Convex Sets
X,Y ∈ Rn
PXY = {(1 − λ)X + λY : λ ∈ [0, 1]}

Simplicial Homology
Convex Sets
C ∈ Rn
, ∀X, Y ∈ C
{(1 − λ)X + λY : λ ∈ [0, 1]} ∈ C

Simplicial Homology
Convex Hull
S a ﬁnite point set
xi ∈ S, i = 1, .. | S |
αi ∈ R
Conv(S) =



|S|
i=1
αi xi | (∀i : αi ≥ 0) ∧
|S|
i=1
αi = 1



xi ∈ Conv(S {xi }), is a vertex of Conv(S)

Simplicial Homology
Convex Hull
V1
V2
X1
X2
(1 − λ)V1 + λV2 V1, V2 ∈ R2
, 0 ≤ λ ≤ 1

Simplicial Homology
Convex Hull
V1
V2
X3
X2
V3
λ1V1 + λ2V2 + λ3V3 V1, V2, V3 ∈ R2
, 3
i=1 λi = 1

Simplicial Homology
Convex Hull
V1
V2
X1
X3
X3 V3
V4
λ1V1 + λ2V2 + λ3V3 + λ4V4 V1, V2, V3, V4 ∈ R3
, 4
i=1 λi = 1

Simplicial Homology
p-simplex σ
given {x0, x1, · · · , xp} ∈ Rn
, p + 1 aﬃnely independent points.
p-simplex σ = Conv({x0, · · · , xp})
dimension of σ is p

Simplicial Homology
face of σ
V1
V3
V4
V2
V1
V2
V3
V1
V4
V2
V4
V2
V3
V4
V1
V3
V3
V4
V2
V2
V3
V4
V3
V2
V4
An l-simplex τ is called a face of a p-simplexσ if the vertices of τ are a sub-set
of the vertices of σ.
A k-simplex has exactly 2k+1
− 1 diﬀerent faces.
A tetrahedron consists has one face in dimension 3 (itself),four triangular faces,
six edges, and four vertices which adds up to 15 = 24
− 1.

Simplicial Homology
Simplicial complexes
A simplicial complex K is a ﬁnite collection of simplices such that σ ∈ K and τ
being a face of σ implies τ ∈ K, and σ, σ ∈ K implies σ σ is empty or a
face of both σ and σ .
Simplicial
complex
No simplicial
complex
Left: A simplicial complex of dimension 3, right: Not a simplicial complex.

Topology
Simplicial complexes - chains
Deﬁnition 3. A p-chain is a subset of p-simplices in a simplicial complex K.
Despite the name “chain,” a p-chain does not have to be connected.
Left: a 3-simplex. Center: one of its 2-chain (triangles). Right one of its
1-chain (the blue edges)

Topology
Simplicial complexes - chains group
Deﬁnition 4. The set of p-chains of a simplicial complex K form a p-chain
group Cp with the symmetric diﬀerence +2.
+ =
example of 1-chain addition:

2012 -TDA-Intro-part_2

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