Contributions of Lattice Theory to the Study of Computational - PowerPoint PPT Presentation

Contributions of Lattice Theory to the Study of Computational Topology Jo˜ ao Pita Costa in a joint work with z ˇ Mikael Vejdemo-Johansson and Primoˇ Skraba AAA88 Conference , Warsaw, June 20, 2014 www.joaopitacosta.com/aaa88

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Motivations Lattice Data Analysis Theory Topological Data Analysis Computational Algebraic Geometry Topology JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Topological Data Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology Persistence of ❍ ✵ of sublevel-sets of a real function. Mikael Vejdemo-Johansson, Sketches of a platypus: persistence homology and its foundations. arXiv:1212.5398v1 (2013) JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology Application: Image Analysis JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology R. Ghrist, Barcodes: the persistent topology of data. Bulletin of the American Math. Soc. 45.1 (2008): 61-75. JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology R. Ghrist, Barcodes: the persistent topology of data. Bulletin of the American Math. Soc. 45.1 (2008): 61-75. JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Persistent Homology Application: Tumor Detection JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Heyting Algebras A Boolean algebra ✭ ▲ ❀ ❫ ❀ ❴ ❀ ✿ ❀ ✵ ❀ ✶✮ is a distributive lattice ✭ ▲ ❀ ❫ ❀ ❴ ✮ with bounds ✵ and ✶ such that all elements ① ✷ ▲ have complement ② (noted ✿ ① ) satisfying ① ❫ ② ❂ ✵ and ① ❴ ② ❂ ✶ . A Heyting algebra ✭ ▲ ❀ ❫ ❀ ❴ ❀ ✮ ✮ is a distributive lattice ✭ ▲ ❀ ❫ ❀ ❴ ✮ such that for each pair ❛❀ ❜ ✷ ▲ there is a greatest element ① ✷ ▲ (noted ❛ ✮ ❜ ) such that ❛ ❫ ① ✔ ❜ . The pseudo complement of ① ✷ ▲ is ① ✮ ✵ (often also noted by ✿ ① ). Example Every Boolean algebra is a Heyting algebra with ❛ ✮ ❜ ❂ ✿ ❛ ❴ ❜ and ❛ ✮ ✵ ❂ ✿ ❛ . The open sets of a topological space ❳ constitute a complete Heyting algebra with ❆ ✮ ❇ ❂ ✐♥t ✭✭ ❳ � ❆ ✮ ❬ ❇ ✮ . JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Heyting Algebras The collection of all open subsets of a topological space ❳ forms a complete Heyting algebra. Topological space ❳ Heyting algebra ❍ ❯ ❭ ❱ ❯ ❫ ❱ ❯ ❬ ❱ ❯ ❴ ❱ ✜ ✵ ❳ ✶ ✐♥t ✭✭ ❳ � ❯ ✮ ❬ ❱ ❯ ✮ ❱ ✐♥t ✭ ❳ � ❯ ✮ ✿ ❯ JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Algebra of Lifetimes Mikael Vejdemo-Johansson, Sketches of a platypus: persistence homology and its foundations. arXiv:1212.5398v1 (2013) JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

❍ Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Algebra of Lifetimes Definition. Consider the complete lattice ✭❘❀ ❫ ❀ ❴ ✮ . Let ❆ and ❇ be intervals ❆ ❂ ❇ ✭ ❛ ✶ ❀ ❛ ✷ ✮ and ❇ ❂ ❇ ✭ ❜ ✶ ❀ ❜ ✷ ✮ represented in a persistence diagram by the points ❆ ✭ ❛ ✶ ❀ ❛ ✷ ✮ and ❇ ✭ ❜ ✶ ❀ ❜ ✷ ✮ in ❍ . Define: ❆ ❫ ❇ ❂ ✭ ❛ ✶ ❴ ❜ ✶ ❀ ❛ ✷ ❫ ❜ ✷ ✮ and ❆ ❴ ❇ ❂ ✭ ❛ ✶ ❫ ❜ ✶ ❀ ❛ ✷ ❴ ❜ ✷ ✮ B = A ∨ B B A ∨ B A = A ∧ B A A ∧ B A = A ∧ B B A ∧ B B = A ∨ B A A ∨ B JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Algebra of Lifetimes Definition. Consider the complete lattice ✭❘❀ ❫ ❀ ❴ ✮ . Let ❆ and ❇ be intervals ❆ ❂ ❇ ✭ ❛ ✶ ❀ ❛ ✷ ✮ and ❇ ❂ ❇ ✭ ❜ ✶ ❀ ❜ ✷ ✮ represented in a persistence diagram by the points ❆ ✭ ❛ ✶ ❀ ❛ ✷ ✮ and ❇ ✭ ❜ ✶ ❀ ❜ ✷ ✮ in ❍ . Define: ❆ ❫ ❇ ❂ ✭ ❛ ✶ ❴ ❜ ✶ ❀ ❛ ✷ ❫ ❜ ✷ ✮ and ❆ ❴ ❇ ❂ ✭ ❛ ✶ ❫ ❜ ✶ ❀ ❛ ✷ ❴ ❜ ✷ ✮ B = A ∨ B B A ∨ B A = A ∧ B A A ∧ B A = A ∧ B B A ∧ B B = A ∨ B A A ∨ B ❍ is a Heyting algebra. JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Algebra of Lifetimes Ordering bars in ❍ ❆ ✔ ❇ iff ❆ ❫ ❇ ❂ ❆ iff ❜ ✶ ✔ ❛ ✶ and ❛ ✷ ✔ ❜ ✷ iff ❇ ✭ ❆ ✮ ✒ ❇ ✭ ❇ ✮ ✿ ↑ ( A ∨ B ) A ∨ B B A ∧ B A A ∧ B B ↓ ( A ∧ B ) A ∨ B A JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Algebra of Lifetimes ✭ ✶ ❂ ✭✵ ❀ ✧ ✷ ✮ , if ❜ ✶ ✔ ❛ ✶ and ❛ ✷ ✔ ❜ ✷ If ❆ ✔ ❇ ,then ❆ ✮ ❇ ❂ ✿ ❇ ❂ ✭ ❜ ✶ ❀ ❜ ✷ ✮ , if ❛ ✶ ✔ ❜ ✶ and ❜ ✷ ✔ ❛ ✷ ✭ ✭ ❜ ✶ ❀ ✧ ✷ ✮ , if ❛ ✶ ✔ ❜ ✶ and ❛ ✷ ✔ ❜ ✷ Otherwise, ❆ ✮ ❇ ❂ ✿ ✭✵ ❀ ❜ ✷ ✮ , if ❜ ✶ ✔ ❛ ✶ and ❜ ✷ ✔ ❛ ✷ A ⇒ B A ⇒ B B = B ⇒ A B A B ⇒ A A A ⇒ B B A A B B ⇒ A A ⇒ B B ⇒ A JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014

Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos Ideals and Filters Assuming that ❍ is bounded by ✭✵ ❀ ✵✮ and ✭ ✧ ✶ ❀ ✧ ✷ ✮ : Filters gen. by A A Ideals gen. by A Filter gen. by a bar ❆ : ✧ ❆ ❂ ❬✵ ❀ ❛ ✶ ❪ ✂ ❬ ❛ ✷ ❀ ✧ ✷ ❪ ✿ Ideal gen. by a bar ❆ : ★ ❆ ❂ ❬ ❛ ✶ ❀ ✧ ✶ ❪ ✂ ❬✵ ❀ ❛ ✷ ❪ ✿ JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014