Persistent Homology

Miguel Lopez

February 1, 2019

Miguel Lopez Persistent Homology February 1, 2019 1 / 12 Filtrations Homology Persistence Applications: Singh, Gurjeet et al. Topological analysis of population activity in visual cortex Journal of vision vol. 8,8 11.1-18. 30 Jun. 2008 Lockwood, Svetlana and Krishnamoorthy, B. (2015). Topological Features in Cancer Gene Expression Data. Biocomputing 2015. November 2014, 108-119 Silva, Vin and Ghrist, Robert. (2007). Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology. 7. 10.2140/agt.2007.7.339

Overview Topics: Simplicies

Miguel Lopez Persistent Homology February 1, 2019 2 / 12 Homology Persistence Applications: Singh, Gurjeet et al. Topological analysis of population activity in visual cortex Journal of vision vol. 8,8 11.1-18. 30 Jun. 2008 Lockwood, Svetlana and Krishnamoorthy, B. (2015). Topological Features in Cancer Gene Expression Data. Biocomputing 2015. November 2014, 108-119 Silva, Vin and Ghrist, Robert. (2007). Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology. 7. 10.2140/agt.2007.7.339

Overview Topics: Simplicies Filtrations

Miguel Lopez Persistent Homology February 1, 2019 2 / 12 Persistence Applications: Singh, Gurjeet et al. Topological analysis of population activity in visual cortex Journal of vision vol. 8,8 11.1-18. 30 Jun. 2008 Lockwood, Svetlana and Krishnamoorthy, B. (2015). Topological Features in Cancer Gene Expression Data. Biocomputing 2015. November 2014, 108-119 Silva, Vin and Ghrist, Robert. (2007). Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology. 7. 10.2140/agt.2007.7.339

Overview Topics: Simplicies Filtrations Homology

Miguel Lopez Persistent Homology February 1, 2019 2 / 12 Applications: Singh, Gurjeet et al. Topological analysis of population activity in visual cortex Journal of vision vol. 8,8 11.1-18. 30 Jun. 2008 Lockwood, Svetlana and Krishnamoorthy, B. (2015). Topological Features in Cancer Gene Expression Data. Biocomputing 2015. November 2014, 108-119 Silva, Vin and Ghrist, Robert. (2007). Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology. 7. 10.2140/agt.2007.7.339

Overview Topics: Simplicies Filtrations Homology Persistence

Miguel Lopez Persistent Homology February 1, 2019 2 / 12 Overview Topics: Simplicies Filtrations Homology Persistence Applications: Singh, Gurjeet et al. Topological analysis of population activity in visual cortex Journal of vision vol. 8,8 11.1-18. 30 Jun. 2008 Lockwood, Svetlana and Krishnamoorthy, B. (2015). Topological Features in Cancer Gene Expression Data. Biocomputing 2015. November 2014, 108-119 Silva, Vin and Ghrist, Robert. (2007). Coverage in sensor networks via persistent homology. Algebraic and Geometric Topology. 7. 10.2140/agt.2007.7.339

Miguel Lopez Persistent Homology February 1, 2019 2 / 12 Simplices

Definition d A k-simplex is the convex hull of k + 1 affinely independent points in R .

A simplicial complex is a finite collection of simplices K such that σ ∈ K and τ ≤ σ implies τ ∈ K, and σ, σ0 ∈ K implies σ ∩ σ0 is either empty or a face of both.

Miguel Lopez Persistent Homology February 1, 2019 3 / 12 Vietoris-Rips Complex

Definition

A filtration is a sequence of simplicial complexes {Ki } such that σ ∈ Ki only if each face of σ is in some Kj for j ≤ i.

For each point in a finite set S the Vietoris-Rips Complex for a fixed r is the simplicial complex {σ ∈ S | diam(σ) ≤ 2r}.

Miguel Lopez Persistent Homology February 1, 2019 4 / 12 Vietoris-Rips Complex

Definition

A filtration is a sequence of simplicial complexes {Ki } such that σ ∈ Ki only if each face of σ is in some Kj for j ≤ i.

For each point in a finite set S the Vietoris-Rips Complex for a fixed r is the simplicial complex {σ ∈ S | diam(σ) ≤ 2r}.

Miguel Lopez Persistent Homology February 1, 2019 4 / 12 Simplicial Homology Chains

Definition Given a simpicial complex K, a p-chain over Z/2Z is a formal sum of p-dimensional simplices

n X ai σi i=1 where ai ∈ {0, 1} and σi is a p-dimensional simplex in K.

Definition The boundary operator ∂ is a linear operator that sends a p-dimensional simplex to the formal sum of its (p − 1)-dimensional faces.

Miguel Lopez Persistent Homology February 1, 2019 5 / 12 The boundary group Bp consists of all p-chains σ such that there exists some (p + 1)-chain τ with ∂(τ) = σ.

We define the p-th homology group as Hp(K) = Zp(K)/Bp(K).

Simplicial Homology

∂(τ) = σ1 + σ2 + σ3 ∂(σ1 + σ2 + σ3) = ∂(σ1) + ∂(σ2) + ∂(σ3) = (v1 + v3) + (v2 + v3) + (v1 + v2) = 0

A p-cycle is a p-chain σ such that ∂(σ) = 0. We define Zp as the group of all p-cycles.

Miguel Lopez Persistent Homology February 1, 2019 6 / 12 We define the p-th homology group as Hp(K) = Zp(K)/Bp(K).

Simplicial Homology

∂(τ) = σ1 + σ2 + σ3 ∂(σ1 + σ2 + σ3) = ∂(σ1) + ∂(σ2) + ∂(σ3) = (v1 + v3) + (v2 + v3) + (v1 + v2) = 0

A p-cycle is a p-chain σ such that ∂(σ) = 0. We define Zp as the group of all p-cycles.

The boundary group Bp consists of all p-chains σ such that there exists some (p + 1)-chain τ with ∂(τ) = σ.

Miguel Lopez Persistent Homology February 1, 2019 6 / 12 Simplicial Homology

∂(τ) = σ1 + σ2 + σ3 ∂(σ1 + σ2 + σ3) = ∂(σ1) + ∂(σ2) + ∂(σ3) = (v1 + v3) + (v2 + v3) + (v1 + v2) = 0

A p-cycle is a p-chain σ such that ∂(σ) = 0. We define Zp as the group of all p-cycles.

The boundary group Bp consists of all p-chains σ such that there exists some (p + 1)-chain τ with ∂(τ) = σ.

We define the p-th homology group as Hp(K) = Zp(K)/Bp(K).

Miguel Lopez Persistent Homology February 1, 2019 6 / 12 Persistence

i,j For a filtration {Fi }, the inclusion maps f : Fi → Fj are simplicial maps so i,j they induce homomorphisms fp :Hp(Fi ) → Hp(Fj ) for each dimension p. This produces a sequence

Hp(F1) → Hp(F2) → · · · → Hp(Fn) = Hp(F ).

i,j The p-th persistent homology groups are the images of the fp i.e. i,j i,j Hp = imfp . i−1,i A class of p-cycles c ∈ Hp(Fi ) is born at Fi if c ∈/ Hp . If a class c is born entering Fi then it dies entering Fj if f it merges with an older class as we go from Fj−1 to Fj .

If c is born at Fi and dies entering Fj then we define the persistence of c as pers(c) = j − i. i,j i,j i,j We define the Betti Number of Hp as βp = rank Hp .

Miguel Lopez Persistent Homology February 1, 2019 7 / 12 Persistence Example

A cycle is born at F3 and at F4 and one of them dies entering F5. 5 H0(F1) = (Z/2Z) H1(F3) = H1(F5) = Z/2Z 2 H1(F4) = (Z/2Z) 3,4 4,5 H1 = H1 = Z/2Z 3,5 2,i H1 = H1 = 0 1,2 4 H0 = (Z/2Z) . 3,4 4,5 β1 = β1 = 1 3,5 β1 = 0

Miguel Lopez Persistent Homology February 1, 2019 8 / 12 Sampling Points From A Circle

Below are set of data points uniformly sampled points from a wedge sum of circles with uniform noise added. The persistence diagrams are computed using the ‘Statistical Tools for Topological Data Analysis’ R package.

Miguel Lopez Persistent Homology February 1, 2019 9 / 12 Sampling Points From A Circle

Below are set of data points uniformly sampled points from a wedge sum of circles with uniform noise added. The persistence diagrams are computed using the ‘Statistical Tools for Topological Data Analysis’ R package.

Miguel Lopez Persistent Homology February 1, 2019 9 / 12 Sampling From A Torus Rips Complex

Rips complex from points sampled from torus with added noise.

Miguel Lopez Persistent Homology February 1, 2019 10 / 12 Sampling From A Torus Rips Complex

Rips complex from points sampled from torus with added noise.

Miguel Lopez Persistent Homology February 1, 2019 10 / 12 Sampling From A Torus Persistence Diagram

Miguel Lopez Persistent Homology February 1, 2019 11 / 12 Barcode diagrams display the lifetime of each homology generator. Here 1-cycles are shown in red and 2-cycles are shown in blue.

Miguel Lopez Persistent Homology February 1, 2019 12 / 12