Creating a simplicial complex

MATH:7450 (22M:305) Topics in Topology: Scientific and Engineering Applications of Algebraic Topology Sept 25, 2013: Applicable Triangulations. Fall 2013 course offered through the University of Iowa Division of Continuing Education Isabel K. Darcy, Department of Mathematics Applied Mathematical and Computational Sciences, University of Iowa http://www.math.uiowa.edu/~idarcy/AppliedTopology.html

Creating a simplicial complex 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close”

Vietoris Rips complex = flag complex = clique complex 2.) Add all possible simplices of dimensional > 1.

Creating the Čech simplicial complex 1.) B1 … Bk+1 ≠ ⁄ , create k-simplex {v1, ... , vk+1}. 0 U U

Consider X an arbitrary topological space. Let V= {Vi | i = 1, …, n } where Vi X , The nerve of V = N(V) where The k -simplices of N(V) = nonempty intersections of k +1 distinct elements of V. For example, Vertices = elements of V Edges = pairs in Vwhich intersect nontrivially. Triangle = triples in V which intersect nontrivially. U http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

Nerve Lemma:If Vis a finite collection of subsets of X with all non-empty intersections of subcollections of Vcontractible, then N(V) is homotopicto the union of elements of V. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf

Choose data point v. The Voronoicell associated with v is H(v,w) U w ≠ v The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

Voronoi diagram Suppose your data points live in Rn. Choose data point v. The Voronoicell associated with v is H(v,w) U w ≠ v The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

The delaunay triangulation is the dual to the voronoidiagram If Cv≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} ⁄ U w in s The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

voronoi diagram

The delaunay triangulation is the dual to the voronoidiagram If Cv≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} ⁄ U w in s The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

Delaunay triangulation Čech

Alpha complex Nerve of {CvBv(r): v in data set} U The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }

Alpha complex Čech

Čech

Alpha complex

Let D = set of vertices. v0,v1,...,vkspan a Delaunay k-simplex iff the Voronoi cell associated with vi meet iff there is a point w ∈ Rn, whose k+1 nearest neighbours in D are v0,v1,...,vkand which is equidistant from them.

comptop.stanford.edu/preprints/witness.pdf

Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

v0,v1,...,vkspan a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v0,v1,...,vkand which is equidistant from them ?????????????????????? Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

v0,v1,...,vkspan a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v0,v1,...,vk and all the faces of {v0,v1,...,vk} belong to the witness complex. w is called a “weak” witness. W∞(D) = Witness complex Let D = set of point cloud data points. Choose L D, L = set of landmark points. U

W∞(D) = Witness complex

W1(D) = Lazy witness complex Let L = set of landmark points. 1-skeletion of W1(D) = 1-skeletion of W∞ (D). Create the flag (or clique) complex: Add all possible simplices of dimensional > 1.

Choosing Landmark points: A.) Random B.) Maxmin 1.) choose point l1 randomly 2.) If {l1, …, lk-1} have been chosen, choose lksuch that {l1, …, lk-1} is in D - {l1, …, lk-1} and min {d(lk, l1), …, d(lk, lk-1)} ≥ min {d(v, l1), …, d(v, lk-1)}

Choosing Landmark points

comptop.stanford.edu/preprints/witness.pdf

Strong witness complex: Let D = set of point cloud data points. Choose L D, L = set of landmark points. Let mv = dist (v, L) = min{ d(v, l ) : l in L } U {l1, …, lk+1} is a k-simplex iff d(v, li) ≤ mv + ε for all i v is the witness

Weak witness complex: Let D = set of point cloud data points. Choose L D, L = set of landmark points. U s = {l1, …, lk+1} is a k-simplex iff d(v, li) ≤ d(v, x) for all i and all x not in s v is the weak witness

Weak witness complex: Let D = set of point cloud data points. Choose L D, L = set of landmark points. U s = {l1, …, lk+1} is a k-simplex iff d(v, li) ≤ d(v, x) + e for all i and all x not in s v is the e-weak witness

The Theory of Multidimensional Persistence, Gunnar Carlsson, AfraZomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbnerbases, Gunnar Carlsson http://www.ima.umn.edu/videos/?id=862

From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www.ima.umn.edu/2008-2009/ND6.15-26.09/activities/Carlsson-Gunnar/imafour-handout4up.pdf

Creating a simplicial complex

Creating a simplicial complex

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