Matthew L. Wright. St. Olaf College. June 15, 2017

Transcription

1 Matthew L. Wright St. Olaf College June 15, 217

2 Persistent homology detects topological features of data. For this data, the persistence barcode reveals one significant hole in the point cloud.

3 Problem: Persistent homology is sensitive to outliers.

4 Problem: Persistent homology is sensitive to outliers. Red points in dense regions Purple points in sparse regions Can we avoid thresholds?

5 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

6 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

7 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.

8 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?

9 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

10 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

11 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

12 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

13 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

14 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

15 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

16 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

17 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.

18 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

19 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

20 Rank Invariant Visualization and Exploration Tool with Mike Lesnick

21 For More Information RIVET is available at: Michael Lesnick and Matthew Wright. Interactive Visualization of 2-D Persistence Modules. Dec. 215, arxiv: Michael Lesnick and Matthew Wright. Computing Bigraded Betti Numbers in Cubic Time. In preparation. Development of RIVET is supported by NSF DMS