Matthew L. Wright St. Olaf College June 15, 217
Persistent homology detects topological features of data. For this data, the persistence barcode reveals one significant hole in the point cloud.
Problem: Persistent homology is sensitive to outliers.
Problem: Persistent homology is sensitive to outliers. Red points in dense regions Purple points in sparse regions Can we avoid thresholds?
distance Two-dimensional (2-D) persistence: Allows us to work with data indexed by two parameters, such as distance and density. We obtain a bifiltration: a set of simplicial complexes indexed by two parameters, with inclusion maps in two directions. codensity
distance Problem: There is no barcode for 2D persistence. Concept: Visualize a barcode along any onedimensional slice of a 2D persistence module. Example: Along any onedimensional slice, a persistence barcode exists. codensity
Fix a field k. A 2-D persistence module M is a collection of k-vector spaces M u u Z2 and linear maps M u M v u v such that the following diagram commutes for all u v w in Z 2 : M u M v M w (That is, u 1 v 1 w 1 and u 2 v 2 w 2.) 2-D persistence module diagram: M 1,3 M 1,2 M 2,2 M 1,1 M 2,1 M 2,3 M 3,3 M 3,2 M 3,1 The homology of a bifiltration is a 2-D persistence module.
2-D persistence module diagram: A 2-D persistence module is a bigraded module over k[x, y]. M 1,3 M 1,2 M 2,2 M 2,3 M 3,3 M 3,2 Problem: The structure of bigraded modules is much more complicated than that of graded modules. M 1,1 M 2,1 M 3,1 There is no analog of a barcode for 2-D persistence modules. [Carlsson and Zomorodian, 27] The homology of a bifiltration is a 2-D persistence module. Question: How can we visualize 2-D persistence?
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. M,3 M 1,3 M 2,3 M 3,3 M,2 M 1,2 M 2,2 M 3,2 M,1 M 1,1 M 2,1 M 3,1 M, M 1, M 2, M 3, x-grades
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. dim M,3 dim M,2 dim M 1,3 dim M 1,2 dim M 2,2 dim M 2,3 dim M 3,3 dim M 3,2 dim M,1 dim M 1,1 dim M 2,1 dim M 3,1 dim M, dim M 1, dim M 2, dim M 3, x-grades
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. We can create a color code for dim M a,b and visualize dimension as an image. dim M,3 dim M,2 dim M,1 dim M 1,3 dim M 1,2 dim M 2,2 dim M 1,1 dim M 2,1 dim M 2,3 dim M 3,3 dim M 3,2 dim M 3,1 dim M, dim M 1, dim M 2, dim M 3, zero dimension high dimension x-grades
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. We can create a color code for dim M a,b and visualize dimension as an image. dim M,3 dim M,2 dim M,1 dim M 1,3 dim M 1,2 dim M 2,2 dim M 1,1 dim M 2,1 dim M 2,3 dim M 3,3 dim M 3,2 dim M 3,1 dim M, dim M 1, dim M 2, dim M 3, zero dimension high dimension x-grades
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. We can create a color code for dim M a,b and visualize dimension as an image. zero dimension high dimension x-grades
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. 2. The rank invariant: For u v, rank(u, v) is the dimension of homology at u that also exists at v. Let L be the line through u and v. The restriction of M to L is a 1-D persistence module M L. Thus, M L has a barcode, B M L, which is a set of intervals in L. Then rank(u, v) is the number of bars in this barcode that stretch from u to v. B M L v u x-grades In this example, rank u, v = 2. L
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. 2. The rank invariant. 3. The bigraded Betti numbers. x-grades
y-grades The bigraded Betti numbers are functions, ξ, ξ 1, ξ 2 Z 2 Z For a finitely-presented 2-D persistence module M, with minimal free resolution F 2 F 1 F M, ξ i a, b is the number of elements in a basis for F i at grade a, b. For more details, see, for example: Eisenbud, The Geometry of Syzygies. x-grades
The bigraded Betti numbers are functions, ξ, ξ 1, ξ 2 Z 2 Z ξ (a, b) is the dimension of homology that appears at (a, b). ξ corresponds to left endpoints of bars. Example: A single homology generator appears at (a, b). b a Then ξ a, b = 1. If m homology generators appear at a, b, then ξ a, b = m.
The bigraded Betti numbers are functions, ξ, ξ 1, ξ 2 Z 2 Z ξ (a, b) is the dimension of homology that appears at (a, b). ξ corresponds to left endpoints of bars. ξ 1 (a, b) is the dimension of homology that disappears at (a, b). ξ 1 corresponds to right endpoints of bars. Example: A homology class disappears at (c, d). d b a Then ξ 1 c, d = 1. If m homology classes disappear at c, d, then ξ 1 c, d = m. c
y-grades Visualizing 2-D Persistence We can visualize: 1. The dimension of each homology vector space M i,j. 2. The rank invariant. 3. The bigraded Betti numbers. Our software program, RIVET, allows us to visualize these three invariants. x-grades
Rank Invariant Visualization and Exploration Tool with Mike Lesnick http://rivet.online
For More Information RIVET is available at: http://rivet.online Michael Lesnick and Matthew Wright. Interactive Visualization of 2-D Persistence Modules. Dec. 215, arxiv:1512.18. Michael Lesnick and Matthew Wright. Computing Bigraded Betti Numbers in Cubic Time. In preparation. Development of RIVET is supported by NSF DMS-166967.