Persistent Homology Lecture

Persistent Homology Overview Topics: Simplicies Overview Topics: Simplicies Filtrations Overview Topics: Simplicies Filtrations Homology Overview Topics: Simplicies Filtrations Homology Persistence 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 Simplices Definition A k-simplex is the convex hull of k + 1 affinely independent points in Rd . 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. 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 }. 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 }. 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. 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. 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. 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 ). Persistence For a filtration {Fi }, the inclusion maps f i,j : Fi → Fj are simplicial maps so they induce homomorphisms fpi,j : Hp (Fi ) → Hp (Fj ) for each dimension p. This produces a sequence Hp (F1 ) → Hp (F2 ) → · · · → Hp (Fn ) = Hp (F ). The p-th persistent homology groups are the images of the fpi,j i.e. i,j Hi,j p = imfp . A class of p-cycles c ∈ Hp (Fi ) is born at Fi if c ∈ / Hi−1,i . p 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 We define the Betti Number of Hi,j p as βp = rank Hp . Persistence Example A cycle is born at F3 and at F4 and one of them dies entering F5 . H0 (F1 ) = (Z/2Z)5 H1 (F3 ) = H1 (F5 ) = Z/2Z H1 (F4 ) = (Z/2Z)2 4,5 H3,4 1 = H1 = Z/2Z 2,i H3,5 1 = H1 = 0 4 H1,2 0 = (Z/2Z) . β13,4 = β14,5 = 1 β13,5 = 0 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. 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. Sampling From A Torus Rips Complex Rips complex from points sampled from torus with added noise. Sampling From A Torus Rips Complex Rips complex from points sampled from torus with added noise. Sampling From A Torus Persistence Diagram Barcode diagrams display the lifetime of each homology generator. Here 1-cycles are shown in red and 2-cycles are shown in blue.