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Persistent Homology and Entropy
N. Atienza, R. Gonzalez-Diaz, M. Soriano-Trigueros.
June 2018
N. Atienza et al. Persistent Homology & Entropy June 2018 1 / 36
Topological Data Analysis
N. Atienza et al. Persistent Homology & Entropy June 2018 2 / 36
Data Structure
directed multigraphs
simplicial sets
undirected graphs
abstract simplicial complexes
Abstract simplicial complexes are more suitable for computations.
N. Atienza et al. Persistent Homology & Entropy June 2018 3 / 36
Simplicial Complexes
Simplicial Complex2-simplexes {a, b, c}1-simplexes {a, b}, {b, c}, {a, c}
{b, d}, {c, d}, {d, e}0-simplexes {a}, {b}, {c}, {d}, {e}
N. Atienza et al. Persistent Homology & Entropy June 2018 4 / 36
Filtration
Fix a simplicial complex SC.Define the filter function f : SC → R s.t.σ C τ ⇒ f(σ) ≤ f(τ).A filtration is the assignment a 7→ f−1(−∞, a].
N. Atienza et al. Persistent Homology & Entropy June 2018 5 / 36
N. Atienza et al. Persistent Homology & Entropy June 2018 6 / 36
Filtration
Each simplicial complex is related with the following ones by inclusion
1 2 3
f−1(−∞, 1] f−1(−∞, 2] f−1(−∞, 3]
≤ ≤
i10 i21
N. Atienza et al. Persistent Homology & Entropy June 2018 7 / 36
Filtration (Abstract Concept)
K are all subcomplexes of a simplicial complex seen as acategory with morphism the inclusion.P is a poset seen as a category with morphism ≤.
FiltrationA filtration is a functor F : P→ K
In practice, P will be a subset of R.
N. Atienza et al. Persistent Homology & Entropy June 2018 8 / 36
Filtration (Abstract Concept)
K are all subcomplexes of a simplicial complex seen as acategory with morphism the inclusion.P is a poset seen as a category with morphism ≤.
FiltrationA filtration is a functor F : P→ K
In practice, P will be a subset of R.
N. Atienza et al. Persistent Homology & Entropy June 2018 8 / 36
Persistent Homology
Let Vec be the category of vector space over a field k.
Persistence ModuleA persistence module is a functor M : P→ Vec
We call n-persistent homology to the functor
M = Hn(−, k) ◦ F
Where Hn is the n-homology and usually k = Z/2Z.
N. Atienza et al. Persistent Homology & Entropy June 2018 9 / 36
Persistent Homology
Let Vec be the category of vector space over a field k.
Persistence ModuleA persistence module is a functor M : P→ Vec
We call n-persistent homology to the functor
M = Hn(−, k) ◦ F
Where Hn is the n-homology and usually k = Z/2Z.
N. Atienza et al. Persistent Homology & Entropy June 2018 9 / 36
Persistent Homology
Let Vec be the category of vector space over a field k.
Persistence ModuleA persistence module is a functor M : P→ Vec
We call n-persistent homology to the functor
M = Hn(−, k) ◦ F
Where Hn is the n-homology and usually k = Z/2Z.
N. Atienza et al. Persistent Homology & Entropy June 2018 9 / 36
Persistent Homology
Let Vec be the category of vector space over a field k.
Persistence ModuleA persistence module is a functor M : P→ Vec
We call n-persistent homology to the functor
M = Hn(−, k) ◦ F
Where Hn is the n-homology and usually k = Z/2Z.
N. Atienza et al. Persistent Homology & Entropy June 2018 9 / 36
Persistent Homology
1 2 3
⊕7Z/2Z ⊕2Z/2Z ⊕2Z/2Z
≤ ≤
i10
H0
i21
H0 H0
v21 v32
N. Atienza et al. Persistent Homology & Entropy June 2018 10 / 36
Persistent Homology
1 2 3
0 Z/2Z 0
≤ ≤
i10
H1
i21
H1 H1
v21 v32
N. Atienza et al. Persistent Homology & Entropy June 2018 11 / 36
Barcodes & Persistent Diagrams
If J is an interval in R, a interval module is kJ is:
vt =
{Z/2Z if t ∈ J0 otherwise.
vlt =
{Id if [t, l] ⊂ J0 otherwise.
Theorem (simplified)If M is a nice persistent homology module in R then
M ∼= ⊕ni=1kJi
where Ji = (xi, yi)
N. Atienza et al. Persistent Homology & Entropy June 2018 12 / 36
Barcodes & Persistent Diagrams
If J is an interval in R, a interval module is kJ is:
vt =
{Z/2Z if t ∈ J0 otherwise.
vlt =
{Id if [t, l] ⊂ J0 otherwise.
Theorem (simplified)If M is a nice persistent homology module in R then
M ∼= ⊕ni=1kJi
where Ji = (xi, yi)
N. Atienza et al. Persistent Homology & Entropy June 2018 12 / 36
Barcodes & Persistent Diagrams
We can represent the multiset {(xi, yi)}ni=1 in several ways
Short bars and points near the diagonal are usually considered noise
N. Atienza et al. Persistent Homology & Entropy June 2018 13 / 36
Persistent Homology
1 2 3
⊕7Z/2Z ⊕2Z/2Z ⊕2Z/2Z
≤ ≤
i10
H0
i21
H0 H0
v21 v32
N. Atienza et al. Persistent Homology & Entropy June 2018 14 / 36
Persistent Homology
1 2 3
0 Z/2Z 0
≤ ≤
i10
H1
i21
H1 H1
v21 v32
N. Atienza et al. Persistent Homology & Entropy June 2018 15 / 36
Barcodes & Persistent Diagrams
Figure: Computational Topology: An Introduction. H. Edelsbrunner, J. Harer.
N. Atienza et al. Persistent Homology & Entropy June 2018 16 / 36
Barcodes & Persistent Diagrams
A,B persistent diagrams,M ⊂ A×B is a partial matching if satisfies
∀ in A there is at most one β with (α, β) ∈M∀β there is at most one α with (α, β) ∈M
There exist a δ-matching if:for all pairs (α, β), ||α− β||∞ ≤ δfor all α unmatched, |α2 − α1| ≤ δ. Idem for β.
N. Atienza et al. Persistent Homology & Entropy June 2018 17 / 36
Stability Theorem
Bottleneck DistanceLet A,B be two persistent diagrams, then
d(A,B) = inf{δ| exists a δ-matching}
is their bottleneck distance
Stability theorem (simplified version)Let K be a simplicial complex and f, g : K → R filter functions. If A,Bare the corresponding persistent diagrams,
d(A,B) ≤ ||f − g||∞
N. Atienza et al. Persistent Homology & Entropy June 2018 18 / 36
Statistical disadvantage
Problems of barcodes:There are not obvious algebraic operation defined on them,
They do not have unique Fréchet mean,They are uncomfortable for applying the null hypothesissignificance test and confidence intervals.
N. Atienza et al. Persistent Homology & Entropy June 2018 19 / 36
Statistical disadvantage
Problems of barcodes:There are not obvious algebraic operation defined on them,They do not have unique Fréchet mean,
They are uncomfortable for applying the null hypothesissignificance test and confidence intervals.
N. Atienza et al. Persistent Homology & Entropy June 2018 19 / 36
Statistical disadvantage
Problems of barcodes:There are not obvious algebraic operation defined on them,They do not have unique Fréchet mean,They are uncomfortable for applying the null hypothesissignificance test and confidence intervals.
N. Atienza et al. Persistent Homology & Entropy June 2018 19 / 36
We can associate to each barcode a real number and use it to obtainstatistical information.
In some applications, counting the number of bars/points seems towork.
8
N. Atienza et al. Persistent Homology & Entropy June 2018 20 / 36
We can associate to each barcode a real number and use it to obtainstatistical information.
In some applications, counting the number of bars/points seems towork.
8
N. Atienza et al. Persistent Homology & Entropy June 2018 20 / 36
This is not stable respect to noise
N. Atienza et al. Persistent Homology & Entropy June 2018 21 / 36
Our research
Definitiongiven a finite random distribution A = {pi :
∑ni=1 pi = 1} we define its
shannon entropy as
E(A) = −n∑
i=1
pi log(pi).
N. Atienza et al. Persistent Homology & Entropy June 2018 22 / 36
Our Research
Persistent Entropy
Let {`i} be the length of the bars and L = `1 + . . .+ `n. The persistententropy of a barcode is
E = −n∑
i=1
`iLlog(`iL
).
N. Atienza et al. Persistent Homology & Entropy June 2018 23 / 36
Our Research
0.0 0.5 1.0 1.5 2.0 2.5 3.0
BF Dimension 1
0.0 0.5 1.0 1.5 2.0 2.5 3.0
BN Dimension 1
0′5343
N. Atienza et al. Persistent Homology & Entropy June 2018 24 / 36
Results
Stability TheoremLet K be a simplicial complex and let f, g : K → R be two monotonicfunctions. Let A,B be their barcodes.
||f − g||∞ ≤ δ ⇒ |E(A)− E(B)| ≤4δ
`max
[log(nmax)− log
(4δ
`max
)].
nmax is the maximum number of bars of the barcodes.`max is the maximum normalized length of the bars.
N. Atienza et al. Stability of persistent entropy and new summary functions for TDA.
N. Atienza et al. Persistent Homology & Entropy June 2018 25 / 36
Applications
Figure: R. Gonzalez-Diaz et al. A new topological entropy-based approach formeasuring similarities among piecewise linear functions.
N. Atienza et al. Persistent Homology & Entropy June 2018 26 / 36
Applications
1 Vertex {pi = (xi, yi)}.Edges {(pi, pi+1)}.
2 Define the filter functionf(pi) = yif(pi, pi+1) = max{yi, yi+1}
N. Atienza et al. Persistent Homology & Entropy June 2018 27 / 36
Applications
f−1(−∞, 1′5)
N. Atienza et al. Persistent Homology & Entropy June 2018 28 / 36
Applications
f−1(−∞, 2′7)
N. Atienza et al. Persistent Homology & Entropy June 2018 29 / 36
Applications
f−1(−∞, 4)
N. Atienza et al. Persistent Homology & Entropy June 2018 30 / 36
Applications
f−1(−∞, 5′7)
N. Atienza et al. Persistent Homology & Entropy June 2018 31 / 36
Applications
f−1(−∞, 8)
N. Atienza et al. Persistent Homology & Entropy June 2018 32 / 36
Our Research
N. Atienza et al. Persistent Homology & Entropy June 2018 33 / 36
Delving into TDA
Developing representation and stability results for new posets,specially Rn.Finding computational efficient functors, apart from homology,suitable for applications. (E.g. A∞-coalgebras).Understanding the homotopy types of most common filtrations.(E.g. What are the homotopy types of the Vietoris-Rips complex ofSn?)
N. Atienza et al. Persistent Homology & Entropy June 2018 34 / 36
Bibliography
Email: [email protected] Topology.
H. Edelsbrunner, J.L. Harer. Computational Topology: anintroduction.
Persistent modules.S.Y. Oudot. Persistence Theory: From Quiver Representations toData Analysis.F. Chazal, V. de Silva, M. Glisse, S.Y. Oudot. The Structure andStability of Persistence Modules.
Topological data analysis.G. Carlsson. Topology and DataR. Ghrist. Barcodes: the persistent topology of data.
N. Atienza et al. Persistent Homology & Entropy June 2018 35 / 36
N. Atienza et al. Persistent Homology & Entropy June 2018 36 / 36