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TDA: Lecture 2 Complexes and Filtrations TDA: Lecture 2 Complexes and filtrations Wesley Hamilton University of North Carolina - Chapel Hill [email protected] 02/05/20 1 / 89 1 Complexes Simplicial complexes Cubical complexes 2 Filtrations 3 From data to filtrations Vietoris-Rips filtrations Sub/superlevel set filtrations Alpha shapes 2 / 89 Complexes Complexes Complexes are the combinatorial building blocks used in TDA. The two types of complexes we'll focus on are simplicial complexes and cubical complexes. Simplicial complexes are easier to \construct" with, and more common in algebraic topology. Cubical complexes are better adapted to image/pixelated/voxel data. 3 / 89 Complexes Simplicial complexes Simplices A geometric n-simplex σ is the set ( n ) X X σ = ai ei : ai = 1; ai ≥ 0 ; i=0 i n where the ei are standard basis vectors for R . 4 / 89 Complexes Simplicial complexes Examples of simplices, dim 0 and 1 5 / 89 Complexes Simplicial complexes Examples of simplices, dim 2 and 3 6 / 89 Complexes Simplicial complexes Simplices and barycentric coordinates Given an n-simplex σ, we can specify points within σ using barycentric coordinates: n X (a0; :::; a1) 7! ai ei 2 σ: i=0 7 / 89 The jth face of a simplex σ is the set @j σ. Complexes Simplicial complexes Faces and facets The face maps @j , defined on n-simplexes, are the restriction of a simplex σ's barycentric coordinates to aj = 0, i.e. n X X @j σ = f ai ei : ai = 1; ai ≥ 0; aj = 0g: i=0 i 8 / 89 Complexes Simplicial complexes Faces and facets The face maps @j , defined on n-simplexes, are the restriction of a simplex σ's barycentric coordinates to aj = 0, i.e. n X X @j σ = f ai ei : ai = 1; ai ≥ 0; aj = 0g: i=0 i The jth face of a simplex σ is the set @j σ. 9 / 89 Complexes Simplicial complexes Examples of faces 10 / 89 Complexes Simplicial complexes Simplicial complexes A simplicial complex K is a set of simplexes (any dimension) such that 1 Every face of a simplex from K is also in K, and 2 If two simplexes in K have a non-empty intersection, then said intersection is a face of each simplex. 11 / 89 Complexes Simplicial complexes Examples of simplicial complexes 12 / 89 Complexes Simplicial complexes Non-examples of simplicial complexes 13 / 89 Complexes Simplicial complexes Non-examples of simplicial complexes 14 / 89 The sets in ∆ are the faces of the simplicial complex. The \intersection" property for (geometric) simplicial complexes is automatically satisfied by abstract simplicial complexes. Given a geometric simplicial complex, we can recover an abstract simplicial complex. Also vice-versa. Complexes Simplicial complexes Abstract vs. geometric simplicial complexes An abstract simplicial complex on a set S (such as S = f1; :::; ng) is a collection ∆ of non-empty subsets of S, such that when Y ⊂ X 2 ∆, Y 2 ∆ as well. 15 / 89 Also vice-versa. Complexes Simplicial complexes Abstract vs. geometric simplicial complexes An abstract simplicial complex on a set S (such as S = f1; :::; ng) is a collection ∆ of non-empty subsets of S, such that when Y ⊂ X 2 ∆, Y 2 ∆ as well. The sets in ∆ are the faces of the simplicial complex. The \intersection" property for (geometric) simplicial complexes is automatically satisfied by abstract simplicial complexes. Given a geometric simplicial complex, we can recover an abstract simplicial complex. 16 / 89 Complexes Simplicial complexes Abstract vs. geometric simplicial complexes An abstract simplicial complex on a set S (such as S = f1; :::; ng) is a collection ∆ of non-empty subsets of S, such that when Y ⊂ X 2 ∆, Y 2 ∆ as well. The sets in ∆ are the faces of the simplicial complex. The \intersection" property for (geometric) simplicial complexes is automatically satisfied by abstract simplicial complexes. Given a geometric simplicial complex, we can recover an abstract simplicial complex. Also vice-versa. 17 / 89 The issue is that the faces of f0; 1; 2g aren't also in K. K = ff0g; f1g; f2g; f0; 1g; f0; 2g; f1; 2g; f0; 1; 2gg is a \quick" fix. Complexes Simplicial complexes Abstract simplicial complex non-example K = ff0; 1; 2gg 18 / 89 K = ff0g; f1g; f2g; f0; 1g; f0; 2g; f1; 2g; f0; 1; 2gg is a \quick" fix. Complexes Simplicial complexes Abstract simplicial complex non-example K = ff0; 1; 2gg The issue is that the faces of f0; 1; 2g aren't also in K. 19 / 89 Complexes Simplicial complexes Abstract simplicial complex non-example K = ff0; 1; 2gg The issue is that the faces of f0; 1; 2g aren't also in K. K = ff0g; f1g; f2g; f0; 1g; f0; 2g; f1; 2g; f0; 1; 2gg is a \quick" fix. 20 / 89 Complexes Simplicial complexes Abstract simplicial complex example K = ff0g; f1g; f2g; f3g; f0; 1g; f0; 2g; f1; 2g; f1; 3g; f0; 3g; f2; 3g; f0; 2; 3gg 21 / 89 Complexes Simplicial complexes Abstract simplicial complex example K = ff0g; f1g; f2g; f3g; f0; 1g; f0; 2g; f1; 2g; f1; 3g; f0; 3g; f2; 3g; f0; 2; 3gg 22 / 89 An n-interval is a product of n elementary intervals. An n-interval with n > 1 is degenerate if any of its factors is a singleton. Complexes Cubical complexes Cubes An (elementary) interval is an interval I ⊂ R of the form I = [l; l + 1]; or I = [l; l] where l 2 Z. 23 / 89 Complexes Cubical complexes Cubes An (elementary) interval is an interval I ⊂ R of the form I = [l; l + 1]; or I = [l; l] where l 2 Z. An n-interval is a product of n elementary intervals. An n-interval with n > 1 is degenerate if any of its factors is a singleton. 24 / 89 Complexes Cubical complexes Examples of cubes, dim 0 and 1 25 / 89 Complexes Cubical complexes Examples of cubes, dim 2 and 3 26 / 89 For a degenerate cube I = [l1; l1+1] × · · · [li ; li ] × · · · × [lk ; lk+1], we define the ith (upper and lower) face(s) to be empty. Complexes Cubical complexes Faces and facets For a non-degenerate cube I = [l1; l1+1] × · · · × [lk ; lk+1], the ith upper and lower faces are: + @i I = [l1; l1+1] × · · · × [li+1] × · · · × [lk ; lk+1], − @i I = [l1; l1+1] × · · · × [li ] × · · · × [lk ; lk+1]: 27 / 89 Complexes Cubical complexes Faces and facets For a non-degenerate cube I = [l1; l1+1] × · · · × [lk ; lk+1], the ith upper and lower faces are: + @i I = [l1; l1+1] × · · · × [li+1] × · · · × [lk ; lk+1], − @i I = [l1; l1+1] × · · · × [li ] × · · · × [lk ; lk+1]: For a degenerate cube I = [l1; l1+1] × · · · [li ; li ] × · · · × [lk ; lk+1], we define the ith (upper and lower) face(s) to be empty. 28 / 89 Complexes Cubical complexes Cubical face example 29 / 89 Complexes Cubical complexes Cubical complexes A cubical complex K is a set of cubes (any dimension) such that 1 Every face of a cube from K is also in K, and 2 If two cubes in K have a non-empty intersection, then said intersection is a face of each simplex. 30 / 89 Complexes Cubical complexes Examples of cubical complexes 31 / 89 Filtrations Filtrations Suppose our (simplicial or cubical) complex comes with more information: each simplex is added to the complex at some time/index (dynamic formulation), or we have a function defined on the simplices/cubes (function formulation). 32 / 89 A filtration F of a complex K is a sequence of subsets Kα ⊂ K, such that whenever α < β, Kα ⊂ Kβ. Note that we haven't specified what the indices α are; they could by integers, reals, or any other object from a poset. Filtrations Filtrations We can use this extra information to filter our complex, and analyze pieces at different times (dynamic) or level sets (function). 33 / 89 Note that we haven't specified what the indices α are; they could by integers, reals, or any other object from a poset. Filtrations Filtrations We can use this extra information to filter our complex, and analyze pieces at different times (dynamic) or level sets (function). A filtration F of a complex K is a sequence of subsets Kα ⊂ K, such that whenever α < β, Kα ⊂ Kβ. 34 / 89 Filtrations Filtrations We can use this extra information to filter our complex, and analyze pieces at different times (dynamic) or level sets (function). A filtration F of a complex K is a sequence of subsets Kα ⊂ K, such that whenever α < β, Kα ⊂ Kβ. Note that we haven't specified what the indices α are; they could by integers, reals, or any other object from a poset. 35 / 89 Filtrations Examples of filtrations (indexed by integers) 36 / 89 Filtrations Examples of filtrations (indexed by a function) 37 / 89 Filtrations Examples of filtrations (indexed by a function) 38 / 89 Example coming soon... Filtrations Examples of filtrations We can also have filtrations over lattices: the index set might be Z × Z = f(m; n): m; n 2 Zg, the partial order is (x1; y1) ≤ (x2; y2) if x1 ≤ x2 and y1 ≤ y2. 39 / 89 Filtrations Examples of filtrations We can also have filtrations over lattices: the index set might be Z × Z = f(m; n): m; n 2 Zg, the partial order is (x1; y1) ≤ (x2; y2) if x1 ≤ x2 and y1 ≤ y2. Example coming soon... 40 / 89 There are many ways to do this, depending on what extra information you have: Vietoris-Rips filtrations (need a metric), Sub/super level set filtrations (need a function), Cubical filtrations (need a voxelization and function), 2 3 Alpha-shape filtrations (need the points to be in R or R ), Graph-based filtrations, etc..
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