Video of the lecture.

Explained how Leray's nerve theorem motivates the use of the clique complex K_ε in data analysis.

Next lecture we'll see how these ideas lead to the Mapper clustering procedure for representing a matrix of distances between data points as a simplicial complex. The procedure was illustrated in this lecture by taking 1000 points in the plane, sampled at random from the image of the starfish. The following code produced the following 1-dimensional simplicial complex as a representation of the data.

gap> HapExample("1.3.5");
gap> #The example uses GraphViz software.
gap> file:=HapFile("data135.txt");
gap> Read(file);
gap> dx:=EuclideanApproximatedMetric;;
gap> dz:=EuclideanApproximatedMetric;;
gap> P:=30*[0..100];; P:=List(P, i->[i]);; r:=29;;
gap> epsilon:=32;;
gap> cluster:=function(S)
local Y, P, C;
if Length(S)=0 then return S; fi;
Y:=VectorsToOneSkeleton(S,epsilon,dx);
P:=PiZero(Y);
C:=Classify([1..Length(S)],P[2]);
return List(C,x->S{x});
end;;
gap> L:=List(S,v->Maximum(List(S,y->dx(v,y))));;
gap> n:=Position(L,Minimum(L));;
gap> f:=function(x); return [dx(S[n],x)]; end;;
gap> M:=Mapper(S,dx,f,dz,P,r,cluster);
gap> Display(GraphOfSimplicialComplex(M));