Sept 25, 2013: Applicable Triangulations. 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 a one dimensional simplicial complex. Note that we have clustered our data into five disjoint connected sets. So this is one way to cluster our data – that is grouping our data points into disjoint sets based on some definition of similarity. In this case, we have 5 clusters. We can now add higher dimensional simplices. 1.) Adding 1-dimensional edges (1-simplices) Add an edge between data points that are “close”
Vietoris Rips complex = flag complex = clique complex Thus we now have the Vietoris Rips simplicial complex. Note we get the same simplex by adding one dimension at a time 2.) Add all possible simplices of dimensional > 1.
Creating the Čech simplicial complex Thus we now have the Vietoris Rips simplicial complex. Note we get the same simplex by adding one dimension at a time 1.) B1 … Bk+1 ≠ ⁄ , create k-simplex {v1, ... , vk+1}. 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 V which intersect nontrivially. Triangle = triples in V which intersect nontrivially. U http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf
Nerve Lemma: If V is a finite collection of subsets of X with all non-empty intersections of subcollections of V contractible, then N(V) is homotopic to the union of elements of V. http://www.math.upenn.edu/~ghrist/EAT/EATchapter2.pdf
The Voronoi cell associated with v is Choose data point v. The Voronoi cell associated with v is H(v,w) U w ≠ v data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 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 Voronoi cell associated with v is H(v,w) U w ≠ v data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 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 voronoi diagram If Cv ≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} U w in s ⁄ data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html 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 voronoi diagram If Cv ≠ 0, then s is a simplex in the delaunay triangulation. Nerve of {Cv : v in data set} U w in s ⁄ data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html The Voronoi cell associated with v is Cv= { x in Rn : d(x, v) ≤ d(x, w) for all w ≠ v }
voronoi diagram
voronoi diagram
The delaunay triangulation is the dual to the voronoi diagram 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 }
The delaunay triangulation is the dual to the voronoi diagram 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 {Cv Bv(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 Nerve of {Cv Bv(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,...,vk span a Delaunay k-simplex iff the Voronoi cell associated with vi meet there is a point w ∈ Rn, whose k+1 nearest neighbours in D are v0,v1,...,vk and 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,...,vk span a k-simplex iff there is a point w ∈ D, whose k+1 nearest neighbours in L are v0,v1,...,vk and 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,...,vk span 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
v0,v1,...,vk span 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
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.
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.
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 lk such 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 data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html
Choosing Landmark points data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html
Choosing Landmark points data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html
Choosing Landmark points data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html
Choosing Landmark points data points. In this very simplified case my data points lie in a two-dimensional plane. Normally data points are high dimensional. For example, I may be comparing the expression or thousands of genes in tumor cells to healthy cells using microarray data. OR I might be comparing politicians voting records. Or I might be comparing the stats of basketball players. These three applications were all, by the way, published by Lum et al this past February in Nature’s Scientific Reports. I have included a link to their paper on my youtube site. http://www.nature.com/srep/2013/130207/srep01236/full/srep01236.html
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, Afra Zomorodian "Persistence and Point Clouds" Functoriality, diagrams, difficulties in classifying diagrams, multidimensional persistence, Gröbner bases, Gunnar Carlsson http://www.ima.umn.edu/videos/?id=862
From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima
From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima
From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima
From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima
From Gunnar Carlsson, Lecture 7: Persistent Homology, http://www. ima
Computing Multidimensional Persistence, Gunnar Carlsson, Gurjeet Singh, and Afra Zomorodian