3. Persistence. CBMS Lecture Series, Macalester College, June Vin de Silva Pomona College

Transcription

1 Vin de Silva Pomona College CBMS Lecture Series, Macalester College, June 2017

2 o o The homology of a planar region Chain complex We have constructed a diagram of vector spaces and linear 0 o 1 C 0 C 1 C 2(U) for any plane region U R 2. From this we define Z k (U) =ker(@ k 1 ) the space of k-cycles B k (U) =im(@ k ) the space of k-boundaries k k =0,wehaveB k apple Z k.since@ 1 =0wehaveZ 0 = C 0. Homology = Cycles/Boundaries ker(@ 1) H 0(U) = im(@ 0) H 1(U) = ker(@0) im(@ 1) = Z0(U) B = C0(U) 0(U) B 0(U) = Z1(U) B 1(U)

3 The homology of a planar region Homology = Cycles/Boundaries ker(@ 1) H 0(U) = im(@ 0) H 1(U) = ker(@0) im(@ 1) = Z0(U) B = C0(U) 0(U) B 0(U) = Z1(U) B 1(U) The Betti numbers of U are defined as follows: b 0(U) =dim(h 0(U)) b 1(U) =dim(h 1(U))

4 The homology of a planar region Theorem Let U R 2 be open. Then b 0(U) =#connectedcomponentsofu.

5 The homology of a planar region Theorem Let U R 2 be open. Then b 0(U) =#connectedcomponentsofu. Proof For simplicity assume finitely many components U 1,...,U k. Select points p i 2 U i.then[p i ] 2 C 0(U). [p 1],...,[p k ]spanc 0(U) modulob 0(U). [p 1],...,[p k ]arelinearlyindependentmodulob 0(U). We must show that every 0-chain can be written in the form = 1[p 1]+ + k [p k ]+@. Indeed, every generator can be written in this form. Specifically, where [a] =[p i ]+@ =[p i, a 1]+[a 1, a 2]+ +[a n, a] for some polygonal path P(p i, a 1, a 2,...,a n, a) inu.

6 The homology of a planar region Theorem Let U R 2 be open. Then b 0(U) =#connectedcomponentsofu. Proof For simplicity assume finitely many components U 1,...,U k. Select points p i 2 U i.then[p i ] 2 C 0(U). [p 1],...,[p k ]spanc 0(U) modulob 0(U). [p 1],...,[p k ]arelinearlyindependentmodulob 0(U). Suppose 1[p 1]+ + k [p k ]+@ =0 Consider the linear maps µ i : C 0(U)! F defined on generators by: ( 1 if a 2 U i µ i ([a]) = 0 if a 62 U i Then µ =0sincetheendpointsofeachedgemustlieinthesamecomponent. Applying µ i to the equation yields i =0.

7 The homology of a planar region Theorem Let U = R 2 {a 1, a 2,...,a`} where the a i are distinct. Then b 1(U) =`.

8 The homology of a planar region Theorem Let U = R 2 {a 1, a 2,...,a`} where the a i are distinct. Then b 1(U) =`. Proof Let i 2 Z 1(U) denoteasmallcounterclockwisesquarearounda i. 1,..., ` span Z 1(U) modulob 1(U). 1,..., ` are linearly independent modulo B 1(U). Suppose ` ` =0 Consider the linear maps w i : Z 1(U)! F defined by: w i ( ) =w(,a i ). We have seen that w =0inR 2 {a 1,...,a`}. Applying w i to the equation yields i =0.

9 The homology of a planar region Theorem Let U = R 2 {a 1, a 2,...,a`} where the a i are distinct. Then b 1(U) =`. Proof Let i 2 Z 1(U) denoteasmallcounterclockwisesquarearounda i. 1,..., ` span Z 1(U) modulob 1(U). 1,..., ` are linearly independent modulo B 1(U). We must show that every 1-cycle can be written in the form = ` ` Write i =w(,a i )andreplace by ( ` `). This has winding number zero about every a i. Partition the plane into a fine rectangular grid. By subtracting boundaries, we can reduce to a cycle of grid edges. By subtracting boundaries of grid rectangles, we can reduce to the case that w(,a) = 0 for every non-grid point a. By considering rays, it follows that we are reduced to the case =0.

10 The homology of a planar region Dual bases We have dual bases in the two proofs. We have [p 1], [p 2],...,[p k ] and µ 1,µ 2,...,µ k such that µ =0 and µ i [p j ]= ( 1 if i = j 0 i 6= j which guarantees that b 0 k. We have 1, 2,..., ` and w 1, w 2,...,w` such that w =0 and w i j = ( 1 if i = j 0 i 6= j which guarantees that b 1 `. For the upper bounds, we used ad hoc geometric constructions.

11 The cohomology of a planar region Cochains A k-cochain is a linear map T : C k denoted by the symbol C k :! F. The collection of all k-cochains is Cocycles C k (U; F) =Hom(C k (U; F), F) A k-cochain T is a k-cocycle if =0. Ak-cocycle can be thought of as a linear map T :(C k /B k )! F. This restricts to a linear map T :(Z k /B k )! F. The collection of k-cocycles is denoted Z k. Coboundaries A k-cochain of the form T = S@ is called a k-coboundary. Such a cochain restricts to the zero map on Z k and hence on H k = Z k /B k. The collection of k-coboundaries is denoted B k.wehaveb k Z k C k.

12 The cohomology of a planar region Cohomology We define the k-cohomology to be H k = Z k /B k. Duality Theorem The vector space H k is the dual of H k : H k = Hom(Hk, F) Construction of the isomorphism We have seen that a cocycle defines a map on homology. We get a linear map Z k! Hom(Z k /B k, F). Since coboundaries define the zero map, we get (Z k /B k )! Hom(Z k /B k, F). It is not di cult to show that the map is surjective and injective. Main Lemma: A linear map defined on a subspace can be extended to the whole vector space. (Fails for modules over a commutative ring.)

13 The cohomology of a planar region Example: winding number cocycles Define cochains W, X : C 1(R 2 {0})! R by W [a, b] = 1 ([a, b], 0) 2 8 >< +1 if [a, b] crosses R counterclockwise X [a, b] = 1 if [a, b] crosses R clockwise >: 0 if [a, b] doesn tmeetr They are cocycles, so they define maps H 1! R. They di er by a coboundary thanks to the formula W =( 1 arg )@ + X 2 ([a, b], 0) = arg (b) arg (a)+2 j so they define the same map on H 1.

14 o o Homology and cohomology of a planar region Dual complexes We have defined a homology chain complex 0 o C 0 C 1 C 2(U) and its dual, a cohomology cochain complex 0 / C 0 (U) 0 / C 1 (U) 1 / C 2 (U) using algebraic versions of points, edges, and triangles. (Here k is the operation pre-compose k.) Our homology theory is the homology of linear singular simplices.

15

16 Simplicial complexes Abstract simplicial complexes AsimplicialcomplexisasetS of nonempty finite sets which is closed under taking subsets: 2 S and ;6= implies 2 S Example The set M = {{2, 4}, {3, 4}, {1, 2, 3}} is not a simplicial complex, but its closure under taking subsets is: S = {{1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}} We write S = h1, 2, 3, 4, 12, 13, 24, 34, 123i = h24, 34, 123i. 3

17 Simplicial complexes Vertices The vertex set of a simplicial complex S is In the example above, V = {1, 2, 3, 4}. Simplices V = V (S) = [ S An element of S is called a simplex (plural simplices). If 2 S has k +1elements,wecallitak-simplex, and write dim = k. Subcomplexes A subcomplex is a subset of a simplical complex that is a simplicial complex. The k-skeleton of S is the subcomplex of simplices of dimension at most k.

18 Simplicial complexes Geometric realisation Let S be a simplicial complex with vertex set V. Its geometric realization is the subspace of R V defined as follows: n S = f : V! R f 0, f 1 (R {0}) 2 S, P o v2v f (v) =1 Each vertex w corresponds to a delta function: w (v) = ( 1 if v = w 0 if v 6= w (Caution: If V is infinite, there are two reasonable topologies on S : theproduct topology and the Whitehead topology. They agree when S is locally finite.) Shadow map Any function : V! R n gives rise to a map : S!R n,definedbythe formula (f )= X f (v) (v). v2v It is a continuous map (with respect to the Whitehead topology).

19 Simplicial complexes Shadow map Any function : V! R n gives rise to a map : S!R n,definedbythe formula (f )= X f (v) (v). v2v It is a continuous map (with respect to the Whitehead topology) Philosophy 3 In topological data analysis, we work with abstract simplicial complexes. If the vertices are taken from data points in R n,thenwemayalsoconsiderthe shadow map. This is rarely injective.

20 Simplicial complexes Vietoris Rips complex Let X be a metric space, and let R 0. The Vietoris Rips complex with diameter R is the simplicial complex with vertex set X defined by the condition: = {x 0, x 1,...,x n}2vr(x, R), d(x i, x j ) apple R for all 0 apple i, j apple n It is the clique complex for the neighbourhood graph of diameter R. Example Let X be a finite collection of robotic sensors located in the plane. The Coverage Theorem uses the 2-skeleton: VR(X, R) (2) VR(X, R) The fence cycle is built from 1-simplices. Coverage is determined in terms of (the shadows of) the 2-simplices.

21 Simplicial complexes The nerve of a set system Let U 1,...,U n be sets, and write U =(U 1,...,U n). The nerve of U is a simplicial complex on the vertex set {1, 2,..., n} defined by the following condition: 2 Nerve(U), \ k2 U k 6= ; U 2 U U 1 U U 7 U U 6 5 The Nerve Theorem (Leray, Borsuk, etc) For a family of convex subsets of a vector space, the nerve is a good model for the union of the family. 6

22 Simplicial complexes Čech complex Let X R N,andletr 0. The Čech complex with radius r is the simplicial complex with vertex set X defined by the condition: = {x 0, x 1,...,x n}2čech(x, r), there exists p 2 R N with p x i appler for all i It is the nerve of the set of closed disks with centers x i and radius r. By the Nerve theorem, it is a good model for the union of those disks. Dowker complex Let X, Y be sets, let : X Y! R, andlett 2 R. TheDowker complex with parameter t is the simplicial complex with vertex set X defined by the condition: = {x 0, x 1,...,x n}2dowker(, t) Dowker duality, there exists y 2 Y with (x i, y) apple t for all i Dowker(, t) and Dowker( t, t) are good models for each other.

23 Simplicial Complexes Comparing the Vietoris Rips and Čech complexes Let X R N.Then Čech(X, r) VR(X, 2r) and VR(X, R) Čech(X, Rp N/(2N +2)) The second inclusion is Jung s Theorem. Philosophy In this way the two methods of simplicial approximation are interleaved. In Euclidean space of any dimension, we deduce VR(X, R) Čech(X, R/p 2). In dimension 2, we get the p 3-Lemma VR(X, R) Čech(X, R/ p 3) which relates the two forms of sensor network coverage discussed earlier.

24

25 o o o Simplicial Homology Goal From a simplicial complex S, we wish to define a diagram of vector spaces and linear maps 0 o C 0 C C 2(S) o 3 C 3(S) 2 =0. Thisisthesimplicial chain complex of S. We also have the dual diagram 0 / C 0 (S) 0 / C 1 (S) 1 / C 2 (S) 2 / C 3 (S) 3 / satisfying 2 =0. Thisisthesimplicial cochain complex of S. Homology Z k =ker@ k 1, B k =im@ k, H k = Z k /B k. Cohomology Z k =ker k, B k =im k 1, H k = Z k /B k.

26 Simplicial Homology Simplicial k-chains We define C k (S; F) intermsofgeneratorsandrelations: agenerator[a 0, a 1,...,a k ]whenever{a 0, a 1,...,a k }2S, the relation [a 0, a 1,...,a k ]=0ifthea i are not distinct; the relation [a 0, a 1,...,a k ]=( 1) [a (0), a (1),...,a (k) ] for any permutation. Alphabetical convention Given a total ordering of the vertices of S, theelements [a 0, a 1,...,a k ] such that a 0 < a 1 < < a k and {a 0, a 1,...,a k }2S constiute a basis for C k.thusdimc k is equal to the number of k-simplices.

27 Simplicial Homology Simplicial k-chains We define C k (S; F) intermsofgeneratorsandrelations: agenerator[a 0, a 1,...,a k ]whenever{a 0, a 1,...,a k }2S, the relation [a 0, a 1,...,a k ]=0ifthea i are not distinct; the relation [a 0, a 1,...,a k ]=( 1) [a (0), a (1),...,a (k) ] for any permutation. Boundary map The boundary k : C k+1! C k is defined on generators by k [a 0,...,a k+1 ]= ( 1) i [...,â i,...] where [...,â i,...]denotestheresultofdeletinga i from [a 0,...,a k+1 ]. The boundary maps are well-defined, 2 =0. i=0

28 Simplicial Cohomology Simplicial k-cochains We define C k (S; F) tobethespaceoff-valuedfunctions F (a 0, a 1,...,a k ) defined for (a 0, a 1,...,a k )suchthat{a 0, a 1,...,a k }2S, and satisfying F (a 0, a 1,...,a k )=0ifthea i are not distinct, and F (a 0, a 1,...,a k )=( 1) F (a (0), a (1),...,a (k) ) for any permutation. Coboundary map The coboundary map k : C k! C k+1 is defined by the formula k F Xk+1 (a 0,...,a k+1 )= ( 1) i F (...,â i,...) where (...,â i,...)denotestheresultofdeletinga i from (a 0,...,a k+1 ). The coboundary maps are well-defined, and 2 =0. i=0

29

30 Simplicial maps Simplicial maps Let S, T be simplicial complexes with respective vertex sets V, W. A simplicial map f : S! T is a function f : V! W such that f {a 0, a 1,...,a k }2T whenever {a 0, a 1,...,a k }2S. Write f 2 T whenever 2 S for conciseness. Induced chain map Asimplicialmapf : S! T induces a family of maps f : C k (S)! C k (T ) by the formula f [a 0, a 1,...,a k ]=[f (a 0), f (a 1),...,f (a k )]. Induced homology map = thisinducesmapsonhomology f : H k (S)! H k (T ).

31 Simplicial maps Simplicial maps Let S, T be simplicial complexes with respective vertex sets V, W. A simplicial map f : S! T is a function f : V! W such that f {a 0, a 1,...,a k }2T whenever {a 0, a 1,...,a k }2S. Write f 2 T whenever 2 S for conciseness. Induced cochain map Asimplicialmapf : S! T induces a family of maps f : C k (T )! C k (S) by the formula f F (a 0, a 1,...,a k )=F (f (a 0), f (a 1),...,f (a k )). Induced cohomology map Since f = f,thisinducesmapsoncohomology f : H k (T )! H k (S).

32 Simplicial maps Contiguous maps Two simplicial maps f, g : S! T are contiguous if f [ g 2 T whenever 2 S. Theorem Contiguous simplicial maps f, g : S! T give rise to equal maps in homology f = g : H k (S)! H k (T ) and cohomology f = g : H k (T )! H k (S).

33

34 Persistent homology and cohomology Philosophy of Topological Data Analysis AfinitedatasetX is given: metric space or similarity space; or subset of R N. Construct a 1-parameter family of simplicial complexes S(r) that approximate X at di erent scales: S(r 0)! S(r 1)!! S(rn) Compute their homology H k (S(r 0))! H k (S(r 1))!! H k (S(r n)) or cohomology H k (S(r 0)) H k (S(r 1)) H k (S(r n)) Invariants of the resulting diagrams are multiscale descriptors of the data.