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Homology and Topological Persistence

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Homology and Topological Persistence ENS Lyon January 2010 Homology and topological persistence F. Chazal Geometrica Group INRIA Saclay To download these slides: http://geometrica.saclay.inria.fr/team/Fred.Chazal/Teaching/persistence.pdf If you have any question: [email protected] Motivation: getting topological information without reconsctructing How to determine the number of \cycles" of the underlying shape from the point cloud approximation? Motivation: getting topological information without reconsctructing How to determine the number of \cycles" of the underlying shape from the point cloud approximation? Motivation: getting topological information without reconsctructing How to determine the number of \cycles" of the underlying shape from the point cloud approximation? Motivation: getting topological information without reconsctructing How to determine the number of \cycles" of the underlying shape from the point cloud approximation? Motivation: getting topological information without reconsctructing Homology How to determine the number of \cycles" of the underlying shape from the point cloud approximation? Persistent homology Simplices 0-simplex: 1-simplex: 2-simplex: 3-simplex: vertex edge triangle tetraedron d v0; v1; ··· ; vk 2 R are affinely independant if k k ! X X tivi = 0 and ti = 0 ) t0 = t1 = ··· = tk = 0 i=0 i=0 In this case σ = [v0; v1; ··· ; vk] is a simplex of dimension d. A simplex generated by a subset of the vertices v0; v1; ··· ; vk of σ is a face of σ. Simplicial complexes A (finite) simplicial complex C is a (finite) union of simplices s.t. i) for any σ 2 C, all the faces of σ are in C, ii) the intersection of any two simplices of C is either empty or a simplex which is their common face of highest dimension. Simplicial complexes A (finite) simplicial complex C is a (finite) union of simplices s.t. i) for any σ 2 C, all the faces of σ are in C, ii) the intersection of any two simplices of C is either empty or a simplex which is their common face of highest dimension. Simplicial complexes A (finite) simplicial complex C is a (finite) union of simplices s.t. i) for any σ 2 C, all the faces of σ are in C, ii) the intersection of any two simplices of C is either empty or a simplex which is their common face of highest dimension. Faces: the simplices of C. j-skeleton: the subcomplex made of the simplices of dimension at most j. Dimension of C: the maximum of the dimensions of the faces. C is homoge- nousof dimension n if any of its faces is a face of a n-dimensional simplex. Filtrations of simplicial complexes A filtration of a (finite) simplicial complex K is a sequence of subcomplexes such that i) ; = K0 ⊂ K1 ⊂ · · · ⊂ Km = K, ii) Ki+1 = Ki [ σi+1 where σi+1 is a simplex of K. Example: filtration associated to a function • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 2 0 3 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 2 2 0 1 3 3 3 3 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 2 2 0 0 1 3 3 3 3 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 2 2 0 0 1 3 3 3 3 1 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 2 2 0 0 1 3 3 3 3 1 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 0 2 2 2 0 1 3 3 3 3 1 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 0 2 2 2 0 1 3 3 3 3 1 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 0 2 2 2 0 1 3 3 3 3 1 3 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 0 2 2 2 0 1 3 3 3 3 1 3 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: filtration associated to a function 0 2 2 2 0 1 3 3 3 3 1 3 1 • f a real valued function defined on the vertices of K • For σ = [v0; ··· ; vk] 2 K, f(σ) = maxi=0;··· ;k f(vi) • The simplices of K are ordered according increasing f values (and di- mension in case of equal values on different simplices). ) The sublevel sets filtration. Exercise: show that this is a filtration. Example: The Cˇechcomplex • Let U = (Ui)i2I be a covering of a topological space X by open sets: X = [i2I Ui. • The Cˇechcomplex C(U) associated to the covering U is the simplicial complex defined by: - the vertex set of C(U) is the set of the open sets Ui k - [Ui0 ; ··· ;Uik ] is a k-simplex in C(U) iff \j=0Uij 6= ;. Example: The Cˇechcomplex Nerve theorem (Leray): If all the intersections between opens in U are either empty or contractible then C(U) and X = [i2I Ui are homotopy equivalent. ) The combinatorics of the covering (a simplicial complex) carries the topology of the space. Example: The Cˇechcomplex Nerve theorem (Leray): If all the intersections between opens in U are either empty or contractible then C(U) and X = [i2I Ui are homotopy equivalent. ) The combinatorics of the covering (a simplicial complex) carries the topology of the space. Warning: even when the open sets are euclidean balls, the computation of the Cˇechcomplex is a very difficult task! Example: the Rips complex Rips vs Cechˇ Let L = fp0; ··· png be a (finite) point cloud (in a metric space). α The Rips complex R (L): for p0; ··· pk 2 L, α σ = [p0p1 ··· pk] 2 R (L) iff 8i; j 2 f0; ··· kg; d(pi; pj) ≤ α • Easy to compute and fully determined by its 1-skeleton • Rips-Cechˇ interleaving: for any α > 0, α α α 2α C 2 (L) ⊆ R (L) ⊆ C (L) ⊆ R (L) ⊆ · · · Homology of simplicial complexes • 2 connected components • Intuitively: 2 cycles Topological invariants: - Number of connected components (Simplicial) homology and - Number of cycles: how to define a cycle? Betti numbers - Number of voids: how to define a void? - ... In the following: homology with coeeficient in Z=2 Refs: J.R. Munkres, Elements of Algebraic Topology, Addison-Wesley, 1984. A. Hatcher, Algebraic Topology, Cambridge University Press 2002. The space of k-chains Let K be a d-dimensional simplicial complex. Let k 2 f0; 1; ··· ; dg and fσ1; ··· ; σpg be the set of k-simplices of K. k-chain: p X c = "iσi with "i 2 Z=2Z = f0; 1g i=1 Sum of k-chains: p p 0 X 0 X 0 c + c = ("i + "i)σi and λ.c = (λεi)σi i=1 i=1 0 where the sums "i + "i and the products λεi are modulo 2.
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