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Persistent Homology in Topological Data Analysis

Ben Fraser May 27, 2015. Persistent Homology in Topological Data Analysis. Data Analysis. Suppose we start with some point cloud data, and want to extract meaningful information from it. Data Analysis.

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Persistent Homology in Topological Data Analysis

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  1. Ben Fraser May 27, 2015 Persistent Homology in Topological Data Analysis

  2. Data Analysis • Suppose we start with some point cloud data, and want to extract meaningful information from it

  3. Data Analysis • Suppose we start with some point cloud data, and want to extract meaningful information from it • We may want to visualize the data to do so, by plotting it on a graph

  4. Data Analysis • Suppose we start with some point cloud data, and want to extract meaningful information from it • We may want to visualize the data to do so, by plotting it on a graph • However, in higher dimensions, visualization becomes difficult

  5. Data Analysis • Suppose we start with some point cloud data, and want to extract meaningful information from it • We may want to visualize the data to do so, by plotting it on a graph • However, in higher dimensions, visualization becomes difficult • A possible solution: dimensionality reduction

  6. Principal Component Analysis • Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component

  7. Principal Component Analysis • Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component • The smaller axes are those along which the data has less variance

  8. Principal Component Analysis • Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component • The smaller axes are those along which the data has less variance • We could discard these less important principal components to reduce the dimensionality of the data while retaining as much of the variance as possible

  9. Principal Component Analysis • Essentially, fits an ellipsoid to the data, where each of its axes corresponds to a principal component • The smaller axes are those along which the data has less variance • We could discard these less important principal components to reduce the dimensionality of the data while retaining as much of the variance as possible • Then may be easier to graph: identify clusters

  10. Principal Component Analysis • Done by computing the singular value decomposition of X (each row is a point, each column a dimension):

  11. Principal Component Analysis • Done by computing the singular value decomposition of X (each row is a point, each column a dimension): • Then a truncated score matrix, where L is the number of principal components we retain:

  12. Principal Component Analysis • 8-dim data → 2-dim to locate clusters:

  13. Principal Component Analysis • 3-dim → 2-dim collapses cylinder to circle:

  14. Principal Component Analysis • Scale sensitive! Same transformation produces poor result on same shape/different scale data

  15. Data Analysis • One weakness of PCA is its sensitivity to the scale of the data

  16. Data Analysis • One weakness of PCA is its sensitivity to the scale of the data • Also, it provides no information about the shape of our data

  17. Data Analysis • One weakness of PCA is its sensitivity to the scale of the data • Also, it provides no information about the shape of our data • We want something insensitive to scale which can identify shape (why?)

  18. Data Analysis • One weakness of PCA is its sensitivity to the scale of the data • Also, it provides no information about the shape of our data • We want something insensitive to scale which can identify shape (why?) • Because “data has shape, and shape has meaning” - Ayasdi (Gunnar Carlsson)

  19. Topological Data Analysis • Constructs higher-dimensional structure on our point cloud via simplicial complexes

  20. Topological Data Analysis • Constructs higher-dimensional structure on our point cloud via simplicial complexes • Then analyze this family of nested complexes with persistent homology

  21. Topological Data Analysis • Constructs higher-dimensional structure on our point cloud via simplicial complexes • Then analyze this family of nested complexes with persistent homology • Display Betti numbers in graph form

  22. Topological Data Analysis • Constructs higher-dimensional structure on our point cloud via simplicial complexes • Then analyze this family of nested complexes with persistent homology • Display Betti numbers in graph form • Essentially, we approximate the shape of the data by building a graph on it and considering cliques as higher dimensional objects, and counting the cycles of such objects.

  23. Algorithm • Since scale doesn't matter in this analysis, we can normalize the data.

  24. Algorithm • Since scale doesn't matter in this analysis, we can normalize the data. • Also, since we don't want to work with the entire data set (especially if it is very large), we want to choose a subset of the data to work with

  25. Algorithm • Since scale doesn't matter in this analysis, we can normalize the data. • Also, since we don't want to work with the entire data set (especially if it is very large), we want to choose a subset of the data to work with • We would ideally like this subset to be representative of the original data (but how?)

  26. Algorithm • Since scale doesn't matter in this analysis, we can normalize the data. • Also, since we don't want to work with the entire data set (especially if it is very large), we want to choose a subset of the data to work with • We would ideally like this subset to be representative of the original data (but how?) • This process is called landmarking

  27. Landmarking • The method used here is minMax

  28. Landmarking • The method used here is minMax • Start by computing a distance matrix D

  29. Landmarking • The method used here is minMax • Start by computing a distance matrix D • Then choose a random point l1 to add to the subset of landmarks L

  30. Landmarking • The method used here is minMax • Start by computing a distance matrix D • Then choose a random point l1 to add to the subset of landmarks L • Then choose each subsequent i-th point to add as that which has maximum distance from the landmark it is closest to:

  31. Landmarking • The method used here is minMax • Start by computing a distance matrix D • Then choose a random point l1 to add to the subset of landmarks L • Then choose each subsequent i-th point to add as that which has maximum distance from the landmark it is closest to: li = p such that dist(p,L) = max{dist(x,L) ∀ x ϵ X} dist(x,L) = min{dist(x,l) ∀ l ϵ L}

  32. Landmarking • Landmarking is not an exact science however: on certain types of data the method just used may result in a subset very unrepresentative of the original data. For example:

  33. Algorithm • As long as outliers are ignored, however, the method used works well to pick points as spread out as possible among the data

  34. Algorithm • As long as outliers are ignored, however, the method used works well to pick points as spread out as possible among the data • Next we keep only the distance matrix between the landmark points, and normalize it

  35. Algorithm • As long as outliers are ignored, however, the method used works well to pick points as spread out as possible among the data • Next we keep only the distance matrix between the landmark points, and normalize it • This is all the information we need from the data: the actual position of the points is irrelevant, all we need are the distances between the landmarks, on which we will construct a neighbourhood graph

  36. Neighbourhood Graph • Our goal is to create a nested sequence of graphs. To be precise, by adding a single edge at a time, between points x,y ϵ L, where dist(x,y) is the smallest value in D. Then replace the distance in D with 1.

  37. Neighbourhood Graph • Our goal is to create a nested sequence of graphs. To be precise, by adding a single edge at a time, between points x,y ϵ L, where dist(x,y) is the smallest value in D. Then replace the distance in D with 1. • At each iteration of adding an edge, we keep track of r = dist(x,y), r ϵ [0,1]: this is our proximity parameter, and will be important when we graph the Betti numbers later.

  38. Witness Complex Def: A point x is a weak witness to a p-simplex (a0,a1,...ap) in A if |x-a| < |x-b| ∀ a ϵ (a0,a1,...ap), and b ϵ A \ (a0,a1,...ap)

  39. Witness Complex Def: A point x is a weak witness to a p-simplex (a0,a1,...ap) in A if |x-a| < |x-b| ∀ a ϵ (a0,a1,...ap), and b ϵ A \ (a0,a1,...ap) Def: A point x is a strong witness to a p-simplex (a0,a1,...ap) in A if x is a weak witness and additionally, |x-a0| = |x-a1| = … = |x-ap|.

  40. Witness Complex Def: A point x is a weak witness to a p-simplex (a0,a1,...ap) in A if |x-a| < |x-b| ∀ a ϵ (a0,a1,...ap), and b ϵ A \ (a0,a1,...ap) Def: A point x is a strong witness to a p-simplex (a0,a1,...ap) in A if x is a weak witness and additionally, |x-a0| = |x-a1| = … = |x-ap| The requirement may be added that an edge is only added between two points if there exists a weak witness to that edge.

  41. Simplicial Complexes • Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex

  42. Simplicial Complexes • Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex • A simplex is a point, edge, triangle, tetrahedron, etc... (a k-simplex is a k+1-clique in the graph)

  43. Simplicial Complexes • Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex • A simplex is a point, edge, triangle, tetrahedron, etc... (a k-simplex is a k+1-clique in the graph) • A face of a simplex is a sub-simplex of it

  44. Simplicial Complexes • Next we want to construct higher dimensional structure on the neighbourhood graph: called a simplicial complex • A simplex is a point, edge, triangle, tetrahedron, etc... (a k-simplex is a k+1-clique in the graph) • A face of a simplex is a sub-simplex of it • A simplicial k-complex is a set S of simplices, each of dimension ≤ k, such that a face of any simplex in S is also in S, and the intersection of any two simplices is a face of both of them

  45. Simplicial Complexes • At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices

  46. Simplicial Complexes • At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices • The edge itself adds a single 1-simplex to the complex

  47. Simplicial Complexes • At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices • The edge itself adds a single 1-simplex to the complex • A k-simplex is formed if the intersection of neighbourhoods of a k-2 simplex contains the two points in the added edge

  48. Simplicial Complexes • At each iteration, we add an edge: all we need to do is see if that creates any new k-simplices • The edge itself adds a single 1-simplex to the complex • A k-simplex is formed if the intersection of neighbourhoods of a k-2 simplex contains the two points in the added edge • In other words, if every point in a k-2 simplex is joined to the two points in the edge, then together they form a k-simplex

  49. Boundary Matricies • Next we compute boundary matricies. Essentially, these store the information that k-1 simplices are faces of certain k simplices

  50. Boundary Matricies • Next we compute boundary matricies. Essentially, these store the information that k-1 simplices are faces of certain k simplices • For instance, in a simplicial complex with 100 triangles and 50 tetrahedra, the 4th boundary matrix has 100 rows and 50 columns, with zeros everywhere except where the given triangle is a face of the given tetrahedron, where it is 1.

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