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 Mikael Vejdemo-Johansson and Primo Skraba AAA88 Conference , Warsaw, June 20, 2014 www.joaopitacosta.com/aaa88
Jo˜ ao Pita Costa
in a joint work with
Mikael Vejdemo-Johansson and Primoˇ z ˇ 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
Topological Data Analysis
Lattice Theory Algebraic Topology Computational Geometry 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
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
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
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
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
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
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
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
The collection of all open subsets of a topological space ❳ forms a complete Heyting algebra. Heyting algebra ❍ ❯ ❫ ❱ ❯ ❴ ❱ ✵ ✶ ❯ ✮ ❱ ✿❯ Topological space ❳ ❯ ❭ ❱ ❯ ❬ ❱ ✜ ❳ ✐♥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
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
intervals ❆ ❂ ❇✭❛✶❀ ❛✷✮ and ❇ ❂ ❇✭❜✶❀ ❜✷✮ represented in a persistence diagram by the points ❆✭❛✶❀ ❛✷✮ and ❇✭❜✶❀ ❜✷✮ in ❍. Define: ❆ ❫ ❇ ❂ ✭❛✶ ❴ ❜✶❀ ❛✷ ❫ ❜✷✮ and ❆ ❴ ❇ ❂ ✭❛✶ ❫ ❜✶❀ ❛✷ ❴ ❜✷✮
B A
A B A ∨ B A ∨ B A ∧ B A ∧ B
B = A ∨ B B = A ∨ B A = 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
intervals ❆ ❂ ❇✭❛✶❀ ❛✷✮ and ❇ ❂ ❇✭❜✶❀ ❜✷✮ represented in a persistence diagram by the points ❆✭❛✶❀ ❛✷✮ and ❇✭❜✶❀ ❜✷✮ in ❍. Define: ❆ ❫ ❇ ❂ ✭❛✶ ❴ ❜✶❀ ❛✷ ❫ ❜✷✮ and ❆ ❴ ❇ ❂ ✭❛✶ ❫ ❜✶❀ ❛✷ ❴ ❜✷✮
B A
A B A ∨ B A ∨ B A ∧ B A ∧ B
B = A ∨ B B = A ∨ B A = 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
Ordering bars in ❍
❆ ✔ ❇ iff ❆ ❫ ❇ ❂ ❆ iff ❜✶ ✔ ❛✶ and ❛✷ ✔ ❜✷ iff ❇✭❆✮ ✒ ❇✭❇✮✿
A B A B A ∧ B A ∨ B ↑ (A ∨ B)
A ∧ B A ∨ B
↓ (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
If ❆ ✔ ❇,then ❆ ✮ ❇ ❂ ✭ ✶ ❂ ✭✵❀ ✧✷✮ , if ❜✶ ✔ ❛✶ and ❛✷ ✔ ❜✷ ❇ ❂ ✭❜✶❀ ❜✷✮ , if ❛✶ ✔ ❜✶ and ❜✷ ✔ ❛✷ ✿ Otherwise, ❆ ✮ ❇ ❂ ✭ ✭❜✶❀ ✧✷✮ , if ❛✶ ✔ ❜✶ and ❛✷ ✔ ❜✷ ✭✵❀ ❜✷✮ , if ❜✶ ✔ ❛✶ and ❜✷ ✔ ❛✷ ✿
B = B ⇒ A A A ⇒ B
A ⇒ B A B B ⇒ A
A B A ⇒ B B ⇒ A
B ⇒ A A B 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
Assuming that ❍ is bounded by ✭✵❀ ✵✮ and ✭✧✶❀ ✧✷✮:
A Filters gen. by A Ideals gen. by A
Filter gen. by a bar ❆: ✧ ❆ ❂ ❬✵❀ ❛✶❪ ✂ ❬❛✷❀ ✧✷❪✿ Ideal gen. by a bar ❆: ★ ❆ ❂ ❬❛✶❀ ✧✶❪ ✂ ❬✵❀ ❛✷❪✿
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
The poset ❍ together with the operations ❫
✐
❆✐ ❂ ✭ ❴ ❢ ❛✶✐ ❣❀ ❫ ❢ ❛✷✐ ❣✮ and ❴
✐
❆✐ ❂ ✭ ❫ ❢ ❛✶✐ ❣❀ ❴ ❢ ❛✷✐ ❣✮✿ is a complete Heyting algebra. In particular, ❍ is completely distributive, i.e., the following identity holds ❳ ❫ ❴
✐✷■
❨✐ ❂ ❴
✐✷■
✭❳ ❫ ❨✐✮✿
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Ideal gen. by bars ❆ and ❇: ❬♠✐♥❢ ❛✶❀ ❜✶ ❣❀ ✧✶❪ ✂ ❬✵❀ ♠❛①❢ ❛✷❀ ❜✷ ❣❪✿ Filter gen. by bars ❆ and ❇: ❬✵❀ ♠❛① ❛✶❀ ❜✶❪ ✂ ❬♠✐♥❢ ❛✷❀ ❜✷ ❣❀ ✧✷❪✿
A B A ∨ B ↓ {A, B}
Ideal gen. by a family ❢ ❆✐ ❣✐✷■: ★ ❢ ❆✐ ❣✐✷■ ❂★ ❲
✐✷■ ❆✐ ❂ ❬♠✐♥❢ ❛✐ ❣❀ ✧✶❪ ✂ ❬✵❀ ♠❛①❢ ❛✐ ❣❪
Filter gen. by a family ❢ ❆✐ ❣✐✷■: ✧ ❢ ❆✐ ❣✐✷■ ❂✧ ❱
✐✷■ ❆✐ ❂ ❬✵❀ ♠❛①❢ ❛✐ ❣❪ ✂ ❬♠✐♥❢ ❛✐ ❣❀ ✧✷❪.
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
① is join-irreducible if ① ❂ ② ❴ ③ implies ① ❂ ② OR ① ❂ ③ ① is meet-irreducible if ① ❂ ② ❫ ③ implies ① ❂ ② OR ① ❂ ③ Join-irreducibles of ❍: bars with coordinates ✭①✶❀ ✵✮ or ✭✧✶❀ ①✷✮. Meet-irreducibles of ❍: bars with coordinates ✭✵❀ ①✷✮ or ✭①✶❀ ✧✷✮.
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Join-irreducibles of ❍: bars with coordinates ✭①✶❀ ✵✮ or ✭✧✶❀ ①✷✮. Meet-irreducibles of ❍: bars with coordinates ✭✵❀ ①✷✮ or ✭①✶❀ ✧✷✮.
∧-irreds: X = (0, x2) ∧-irreds: X = (x1, 1) ∨-irreds: X = (1, x2) ∨-irreds: X = (x1, 0)
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
An ideal ■ of ▲ is a prime ideal if for all ①❀ ② ✷ ▲, ① ❫ ② ✷ ■ implies ① ✷ ■ or ② ✷ ■. Join-irreducibles of ❍: bars with coordinates ✭①✶❀ ✵✮ or ✭✶❀ ①✷✮. ★ ✭①✶❀ ✧✷✮ ❂ ❬①✶❀ ✧✶❪ ✂ ❬✵❀ ✧✷❪ and ★ ✭✵❀ ①✷✮ ❂ ❬✵❀ ✧✶❪ ✂ ❬①✷❀ ✧✷❪ are prime ideals of ❍.
x1
x2
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Spac := Spatial Locales ✘ ❂ Sob := Sober Spaces and homomorphisms and homeomorphisms An open of the dual space as the sum of two filters intersecting only in ❃ and the point in the lattice it corresponds to.
Px Py P = Px ∧ Py
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Ring R scheme Spec(R) Spec(H) Locale H skew distributive lats ?
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Sh(PSL) PSL DL0 SDL CBKMG 2012 P1972
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Consider the functor ✣ ✿ ❍ ✦ ❙❡t defined by the sections ✣✭①✮ ❂ ❢ ★ ② ❥ ② ✔ ① ❣❀ for all ① ✷ ❍, and the restriction map ✤①
② ✿ ✣✭①✮ ✦ ✣✭②✮ defined by
✤①
②✭★ ✛✮ ❂ ✭★ ③✮ ❭ ✭★ ②✮ for all ①❀ ②❀ ③ ✷ ❍ such that ② ✔ ①. x xi z zi
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
Consider the functor ✣ ✿ ❍ ✦ ❙❡t defined by the sections ✣✭①✮ ❂ ❢ ★ ② ❥ ② ✔ ① ❣❀ for all ① ✷ ❍, and the restriction map ✤①
② ✿ ✣✭①✮ ✦ ✣✭②✮ defined by
✤①
②✭★ ✛✮ ❂ ✭★ ③✮ ❭ ✭★ ②✮ for all ①❀ ②❀ ③ ✷ ❍ such that ② ✔ ①. Fixing ✐ ✷ ■,
★ ③❭ ★ ①✐ ❂ ✭
❴
❥✷❏
★ ③❥✮❭ ★ ①✐ ❂
❴
❥✷❏
✭★ ③❥❭ ★ ①✐✮ ❂
❴
❥✷❏
★ ③✐❭ ★ ①❥ ❂★ ③✐ ❭ ✭
❴
❥✷❏
★ ①❥✮ ❂★ ③✐❭ ★ ① ❂★ ③✐
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
The category of presheaves ❙❡t❍♦♣ has exponentials: the exponential
The topos of sheaves on ❍ is given by the subject classifier including
✡✭①✮ ❂ ❢ ★ ② ❥ ② ✔ ① ❣✿
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
❯ ✶ ❊ ✡ ✾✦✣ ✤❊ tr✉❡ ✡✭①✮ ❂ ❢ ★ ② ❥ ② ✔ ① ❣✿ x y1 y2 y3
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
∆ Set SetHop
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
■ Computation of semisimplicial homology on the topos over ❍; ■ Study of the dual space and respective spectral space; ■ Interpretation of the arrow operation in the framework; ■ Integration of classical persistence results under this perspective; ■ Implementation of new algorithms.
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014
Topological Data Analysis Heyting Algebras Algebra of Lifetimes The Dual Space A Persistence Topos
JPC & MVJ & P ˇ S :: Toposys 2014 Toposys 2014