The Persistence Lattice Jo ao Pita Costa z (in a joint work with - - PowerPoint PPT Presentation
The Persistence Lattice Jo ao Pita Costa z (in a joint work with Primo Skraba) Jo zef Stefan Institute Ljubljana, Slovenia Novi Sad Algebra Conference, June 8, 2013 Motivation & Background Order Structure Algebraic
Jo˜ ao Pita Costa (in a joint work with Primoˇ z ˇ Skraba)
Joˇ zef Stefan Institute Ljubljana, Slovenia
Novi Sad Algebra Conference, June 8, 2013
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Applied Topology
Persistence Lattice Algebraic Topology Computational Geometry Data Analysis
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
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 :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Persistence of ❍✵ of sublevel-sets by the height function with six critical points on a topological sphere.
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Setting: X space and ❢ ✿ X ✦ ❘. ❢✶✭✭✶❀ ☛❪✮
☛ ✒ ☞
❀ ❂
✵ ✒ ✶ ✒ ✷ ✒ ✿ ✿ ✿ ✒ ◆✶ ✒ ◆ ❂
❍✭
✵✮
❍✭
✶✮
❍✭
✷✮
❍✭
✸✮
❍✭
✹✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Setting: X space and ❢ ✿ X ✦ ❘. Filtration: sequence of sub-level sets ❢✶✭✭✶❀ ☛❪✮
☛ ✒ ☞
❀ ❂
✵ ✒ ✶ ✒ ✷ ✒ ✿ ✿ ✿ ✒ ◆✶ ✒ ◆ ❂
❍✭
✵✮
❍✭
✶✮
❍✭
✷✮
❍✭
✸✮
❍✭
✹✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Setting: X space and ❢ ✿ X ✦ ❘. Filtration: sequence of sub-level sets ❢✶✭✭✶❀ ☛❪✮ X☛ ✒ X☞ ❀ ❂
✵ ✒ ✶ ✒ ✷ ✒ ✿ ✿ ✿ ✒ ◆✶ ✒ ◆ ❂
❍✭
✵✮
❍✭
✶✮
❍✭
✷✮
❍✭
✸✮
❍✭
✹✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Setting: X space and ❢ ✿ X ✦ ❘. Filtration: sequence of sub-level sets ❢✶✭✭✶❀ ☛❪✮
☛ ✒ ☞
❀ ❂ X✵ ✒ X✶ ✒ X✷ ✒ ✿ ✿ ✿ ✒ X◆✶ ✒ X◆ ❂ X ❍✭
✵✮
❍✭
✶✮
❍✭
✷✮
❍✭
✸✮
❍✭
✹✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Setting: X space and ❢ ✿ X ✦ ❘. Filtration: sequence of sub-level sets ❢✶✭✭✶❀ ☛❪✮
☛ ✒ ☞
❀ ❂ X✵ ✒ X✶ ✒ X✷ ✒ ✿ ✿ ✿ ✒ X◆✶ ✒ X◆ ❂ X We get: a diagram of vector spaces and linear maps. ❍✭X✵✮ ❍✭X✶✮ ❍✭X✷✮ ❍✭X✸✮ ❍✭X✹✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Setting: X space and ❢ ✿ X ✦ ❘. Filtration: sequence of sub-level sets ❢✶✭✭✶❀ ☛❪✮
☛ ✒ ☞
❀ ❂ X✵ ✒ X✶ ✒ X✷ ✒ ✿ ✿ ✿ ✒ X◆✶ ✒ X◆ ❂ X We get: a diagram of vector spaces and linear maps. ❍✭X✵✮ ❍✭X✶✮ ❍✭X✷✮ ❍✭X✸✮ ❍✭X✹✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability of the Persistence Diagram.
D Cohen-Steiner, H Edelsbrunner, and J Harer, Stability of persistence diagrams. Discrete Comput Geom (2005) JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
What if we have more than one parameter?
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
A bifiltration parametrized along curvature ❦ and radious ✎
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ X✹✵ X✹✶ X✹✷ X✹✸
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ X✹✵ X✹✶ X✹✷ X✹✸
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ X✹✵ X✹✶ X✹✷ X✹✸
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ X✹✵ X✹✶ X✹✷ X✹✸
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ X✹✵ X✹✶ X✹✷ X✹✸
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
The Missing Data Problem
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
The Missing Data Problem
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
The Missing Data Problem
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
The Missing Data Problem
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
What can the order tell us?
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Boolean algebras Persistence lattices Heyting algebras vector lattices totally ordered sets distributive lattices subspace lattices modular lattices semi-modular lattices partition lattices lattices skew lattices partially ordered sets metric lattices geometric lattices projective lattices
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
A Morse-filtration is a partial order on the parameter ❳☛ ✒ ❳☞ ✮ ☛ ❁ ☞ ❍✐❀❥
✄ ✭ ✮ ❂ ✐♠✭❍✄✭ ✐✮ ✦ ❍✄✭ ❥✮✮
❳◆ ❳◆✶ . . . ❳✷ ❳✶
■ ❍✄✭ ✐✮ ❴ ❍✄✭ ❥✮ ❂ ❍✄✭❳♠❛①✭✐❀❥✮✮ ■ ❍✄✭ ✐✮ ❫ ❍✄✭ ❥✮ ❂ ❍✄✭❳♠✐♥✭✐❀❥✮✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
A Morse-filtration is a partial order on the parameter ❳☛ ✒ ❳☞ ✮ ☛ ❁ ☞ ❍✐❀❥
✄ ✭ ✮ ❂ ✐♠✭❍✄✭ ✐✮ ✦ ❍✄✭ ❥✮✮
❳◆ ❳◆✶ . . . ❳✷ ❳✶
■ ❍✄✭ ✐✮ ❴ ❍✄✭ ❥✮ ❂ ❍✄✭❳♠❛①✭✐❀❥✮✮ ■ ❍✄✭ ✐✮ ❫ ❍✄✭ ❥✮ ❂ ❍✄✭❳♠✐♥✭✐❀❥✮✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
A Morse-filtration is a partial order on the parameter ❳☛ ✒ ❳☞ ✮ ☛ ❁ ☞ Persistent homology classes ❍✐❀❥
✄ ✭X✮ ❂ ✐♠✭❍✄✭X✐✮ ✦ ❍✄✭X❥✮✮
❳◆ ❳◆✶ . . . ❳✷ ❳✶
■ ❍✄✭ ✐✮ ❴ ❍✄✭ ❥✮ ❂ ❍✄✭❳♠❛①✭✐❀❥✮✮ ■ ❍✄✭ ✐✮ ❫ ❍✄✭ ❥✮ ❂ ❍✄✭❳♠✐♥✭✐❀❥✮✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
A Morse-filtration is a partial order on the parameter ❳☛ ✒ ❳☞ ✮ ☛ ❁ ☞ Persistent homology classes ❍✐❀❥
✄ ✭X✮ ❂ ✐♠✭❍✄✭X✐✮ ✦ ❍✄✭X❥✮✮
❳◆ ❳◆✶ . . . ❳✷ ❳✶
■ ❍✄✭X✐✮ ❴ ❍✄✭X❥✮ ❂ ❍✄✭❳♠❛①✭✐❀❥✮✮ ■ ❍✄✭X✐✮ ❫ ❍✄✭X❥✮ ❂ ❍✄✭❳♠✐♥✭✐❀❥✮✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
A Morse-filtration is a partial order on the parameter ❳☛ ✒ ❳☞ ✮ ☛ ❁ ☞ Persistent homology classes ❍✐❀❥
✄ ✭X✮ ❂ ✐♠✭❍✄✭X✐✮ ✦ ❍✄✭X❥✮✮
❳◆ ❳◆✶ . . . ❳✷ ❳✶
Definition
For any two elements ❍✄✭X✐✮ and ❍✄✭X❥✮, the rank of the persistent homology classes is ✐♠✭❍✄✭X✐ ❫ X❥✮ ✦ ❍✄✭X✐ ❴ X❥✮✮.
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸
✐❥ ❫ ❦❵ ✮ ②③
② ❂ ♠✐♥✭✐❀ ❦✮❀ ③ ❂ ♠✐♥✭❥❀ ❵✮
✐❥ ❴ ❦❵ ✮ ②③
② ❂ ♠❛①✭✐❀ ❦✮❀ ③ ❂ ♠❛①✭❥❀ ❵✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ Set: X✐❥ ❫ X❦❵ ✮ X②③, with ② ❂ ♠✐♥✭✐❀ ❦✮❀ ③ ❂ ♠✐♥✭❥❀ ❵✮ X✐❥ ❴ X❦❵ ✮ X②③, with ② ❂ ♠❛①✭✐❀ ❦✮❀ ③ ❂ ♠❛①✭❥❀ ❵✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ Set: X✐❥ ❫ X❦❵ ✮ X②③, with ② ❂ ♠✐♥✭✐❀ ❦✮❀ ③ ❂ ♠✐♥✭❥❀ ❵✮ X✐❥ ❴ X❦❵ ✮ X②③, with ② ❂ ♠❛①✭✐❀ ❦✮❀ ③ ❂ ♠❛①✭❥❀ ❵✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ Set: X✐❥ ❫ X❦❵ ✮ X②③, with ② ❂ ♠✐♥✭✐❀ ❦✮❀ ③ ❂ ♠✐♥✭❥❀ ❵✮ X✐❥ ❴ X❦❵ ✮ X②③, with ② ❂ ♠❛①✭✐❀ ❦✮❀ ③ ❂ ♠❛①✭❥❀ ❵✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ Set: X✐❥ ❫ X❦❵ ✮ X②③, with ② ❂ ♠✐♥✭✐❀ ❦✮❀ ③ ❂ ♠✐♥✭❥❀ ❵✮ X✐❥ ❴ X❦❵ ✮ X②③, with ② ❂ ♠❛①✭✐❀ ❦✮❀ ③ ❂ ♠❛①✭❥❀ ❵✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ Set: X✐❥ ❫ X❦❵ ✮ X②③, with ② ❂ ♠✐♥✭✐❀ ❦✮❀ ③ ❂ ♠✐♥✭❥❀ ❵✮ X✐❥ ❴ X❦❵ ✮ X②③, with ② ❂ ♠❛①✭✐❀ ❦✮❀ ③ ❂ ♠❛①✭❥❀ ❵✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺ X✷ ❴ X✼
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Requirements
Diagram is commutative and connected. ❆ ❇ ❆ ✔ ❇ ❢ ✿ ❆ ✦ ❇✿ ❆ ❇ ❢ ❣ ✐❞❇ ✐❞❆
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Requirements
Diagram is commutative and connected. the reverse maps exist in the case of isomorphisms. ❆ ❇ ❆ ✔ ❇ ❢ ✿ ❆ ✦ ❇✿ ❆ ❇ ❢ ❣ ✐❞❇ ✐❞❆
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Requirements
Diagram is commutative and connected. the reverse maps exist in the case of isomorphisms. the composition will not commute with identity unless the map is an isomorphism. ❆ ❇ ❆ ✔ ❇ ❢ ✿ ❆ ✦ ❇✿ ❆ ❇ ❢ ❣ ✐❞❇ ✐❞❆
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Requirements
Diagram is commutative and connected. the reverse maps exist in the case of isomorphisms.
Partial order of vector spaces
For all vector spaces ❆ and ❇, ❆ ✔ ❇ if there exists a linear map ❢ ✿ ❆ ✦ ❇✿ ❆ ❇ ❢ ❣ ✐❞❇ ✐❞❆
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Requirements
Diagram is commutative and connected. the reverse maps exist in the case of isomorphisms.
Partial order of vector spaces
For all vector spaces ❆ and ❇, ❆ ✔ ❇ if there exists a linear map ❢ ✿ ❆ ✦ ❇✿ ❆ ❇ ❢ ❣ ✐❞❇ ✐❞❆
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Equalizers
❆ ❈ f g
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Equalizers
❊ ❆ ❈ e f g
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Equalizers
❊ ❆ ❈ ❊✵ e e’ f g
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Equalizers
❊ ❆ ❈ ❊✵ e e’ ✣ f g
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Equalizers
❊ ❆ ❈ ❊✵ e e’ ✣ f g The kernel set is ❊ ❂ ❢ ① ✷ ❳ ❥ ❢✭①✮ ❂ ❣✭①✮ ❣ ❂ ❦❡r✭❢ ❣✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Coequalizers
❍ ❉ ❈ ❍✵ h h’ ✣ ❢ ❣ ❍ is the quotient of ❨ by the equivalence ❤✭❢✭①✮❀ ❣✭①✮✮❥① ✷ ❳✐, i.e., ❍ ❂ ❈❂✐♠✭❢ ❣✮ ❂ ❝♦❦❡r✭❢ ❣✮
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❊ ❆ ✟ ❇ ❈ e ❢✐ ❢❥ ❆ ❇ ❆ ✟ ❇ ❈
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❍ ❉ ❆ ✟ ❇ h ❣✐ ❣❥ ❆ ❇ ❆ ✟ ❇ ❉
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❊ ❆ ✟ ❇ ❈✶❀ ❈✷ e ❢✐ ❢❥ ❆ ❇ ❈✶ ❈✷
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❊ ❆ ✟ ❇ ❈✶ ✟ ❈✷ e ❢✐ ❢❥ ❆ ❇ ❈✶ ✟ ❈✷
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❍ ❉✶❀ ❉✷ ❆ ✟ ❇ h f g ❆ ❇ ❉✶ ❉✷
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❍ ❉✶ ✟ ❉✷ ❆ ✟ ❇ h f g ❆ ❇ ❉✶ ✟ ❉✷
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Meet Operation
The join of two elements ❆ and ❇ is the equalizer of ❆ ❫ ❇ ✦ ❆ ✟ ❇ ✓ ❈❦ given by: ❆ ❫ ❇ ❂ ❢① ✷ ❆ ✟ ❇❥❢✐✭①✮ ❂ ❢❥✭①✮, for all ✐❀ ❥ ✷ ■❣
Join Operation
The meet of two elements ❆ and ❇ is the coequalizer of ❉❦ ✓ ❆ ✟ ❇ ✦ ❆ ❴ ❇ given by: ❆ ❴ ❇ ❂ ❆ ✟ ❇❂❤❣✐✭①✮ ✘ ❣❥✭①✮ ❥ ① ✷ ❉❦, for all ✐❀ ❥ ✷ ■✐
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❆ ❇
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❆ ❇ ❆ ❫ ❇
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❆ ❇ ❆ ❴ ❇
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❆ ❇ ❆ ❫ ❇ ❆ ❴ ❇
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Theorem (JPC & P ˇ S 2013)
The persistence lattice is a complete lattice with ❫ ❆❦ ❂ ❢ ① ✷ ✟❦❆❦ ✿ ❢❆✐✭①✮ ❂ ❢❆❥✭①✮ ❣❀ ❴
❦
❆❦ ❂ ✭✟❦❆❦✮❂❤ ❬ ✒❆✐❆❥✐✿ where ✒❆✐❆❥ ❂ ❤✭❢❆✐✭①✮❀ ❢❆❥✭①✮✮✐
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What lattice do we get?
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Theorem (JPC & P ˇ S 2013)
Let ❆ and ❇ be vector spaces. Then, ✵ ✦ ❆ ❫ ❇ ✦ ❆ ✟ ❇ ✦ ❆ ❴ ❇ ✦ ✵ is a short exact sequence. ❢ ✿ ❆ ❫ ❇ ✦ ❆ ✟ ❇ ❣ ✿ ❆ ✟ ❇ ✦ ❆ ❴ ❇ ✐♠❢ ❂ ❦❡r ❣ ❆ ❴ ❇ ✘ ❂ ❆ ✟ ❇❂❢✭❆ ❫ ❇✮✿
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Theorem (JPC & P ˇ S 2013)
Let ❆ and ❇ be vector spaces. Then, ✵ ✦ ❆ ❫ ❇ ✦ ❆ ✟ ❇ ✦ ❆ ❴ ❇ ✦ ✵ is a short exact sequence.
Sketch of the Proof.
The equalizer map ❢ ✿ ❆ ❫ ❇ ✦ ❆ ✟ ❇ is injective. The coequalizer map ❣ ✿ ❆ ✟ ❇ ✦ ❆ ❴ ❇ is surjective. Moreover ✐♠❢ ❂ ❦❡r ❣ so that ❆ ❴ ❇ ✘ ❂ ❆ ✟ ❇❂❢✭❆ ❫ ❇✮✿
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Theorem (JPC & P ˇ S 2013)
The persistence lattice of a given persistence diagram is distributive. ❆ ❇ ❳ ❳ ❴ ❆ ❂ ❳ ❴ ❇ ❳ ❫ ❆ ❂ ❳ ❫ ❇ ❆ ✘ ❂ ❇
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Theorem (JPC & P ˇ S 2013)
The persistence lattice of a given persistence diagram is distributive.
Proof.
Let ❆, ❇ and ❳ be vector spaces such that ❳ ❴ ❆ ❂ ❳ ❴ ❇ and ❳ ❫ ❆ ❂ ❳ ❫ ❇ in order to show that ❆ ✘ ❂ ❇.
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Definition
A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲, ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇, i.e., ❳ ❫ ❆ ❆ ❳ ❇ ✣❆ ✣❳ ✤
■
❳ ❭ ❬ ✜ ❳ ❯ ✮ ❱ ❂ ✐♥t✭✭❳ ❯✮ ❬ ❱ ✮
■
① ✮ ② ❂ ❲❢ ③ ✿ ① ❫ ③ ✔ ② ❣
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Definition
A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲, ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇, i.e., ❳ ❫ ❆ ❆ ❳ ❇ ✣❆ ✣❳ ✤
■
❳ ❭ ❬ ✜ ❳ ❯ ✮ ❱ ❂ ✐♥t✭✭❳ ❯✮ ❬ ❱ ✮
■
① ✮ ② ❂ ❲❢ ③ ✿ ① ❫ ③ ✔ ② ❣
JPC :: NSAC 2013 The Persistence Lattice
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Definition
A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲, ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇, i.e., ❳ ❫ ❆ ❆ ❳ ❇ ✣❆ ✣❳ ✤
■
❳ ❭ ❬ ✜ ❳ ❯ ✮ ❱ ❂ ✐♥t✭✭❳ ❯✮ ❬ ❱ ✮
■
① ✮ ② ❂ ❲❢ ③ ✿ ① ❫ ③ ✔ ② ❣
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Definition
A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲, ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇, i.e., ❳ ❫ ❆ ❆ ❳ ❇ ✣❆ ✣❳ ✤
Example
■ The open sets of any top space ❳ under ❭, ❬, ✜, ❳ and
❯ ✮ ❱ ❂ ✐♥t✭✭❳ ❯✮ ❬ ❱ ✮
■ Complete distributive lattices with ① ✮ ② ❂ ❲❢ ③ ✿ ① ❫ ③ ✔ ② ❣
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Theorem (JPC & P ˇ S 2013)
The persistence lattice of a given persistence diagram is distributive, complete and bounded. It is completely distributive thus constituting a complete Heyting algebra.
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Arrow Operation for standard persistence
❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✦ ❇ . . . ❳✐ ❇ ❆ ❳❥ . . . ❆ ✮ ❇ ❂ ✭ ❇❀ if ❇ ✔ ❆ ✶❀ if ❆ ✔ ❇
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Arrow operation for multidimensional persistence
❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✦ ❇ X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸
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Arrow operation for multidimensional persistence
❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✦ ❇ X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸
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Arrow operation for multidimensional persistence
❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✦ ❇ X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸
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Stability
❆ ❇ ❈ ❉
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Stability
❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ ❈ ❴ ❉ ❈ ❉ ❈ ❫ ❉
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Stability
✭❆ ❴ ❈✮ ❴ ✭❇ ❴ ❉✮ ❆ ❴ ❈ ❇ ❴ ❉ ✭❆ ❴ ❈✮ ❫ ✭❇ ❴ ❉✮ ❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ ❈ ❴ ❉ ❈ ❉ ❈ ❫ ❉ ✭❆ ❫ ❈✮ ❴ ✭❇ ❫ ❉✮ ❆ ❫ ❈ ❇ ❫ ❉ ✭❆ ❫ ❈✮ ❫ ✭❇ ❫ ❉✮
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Stability
✭❆ ❴ ❈✮ ❴ ✭❇ ❴ ❉✮ ❆ ❴ ❈ ❇ ❴ ❉ ✭❆ ❴ ❈✮ ❫ ✭❇ ❴ ❉✮ ❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ ❈ ❴ ❉ ❈ ❉ ❈ ❫ ❉ ✭❆ ❫ ❈✮ ❴ ✭❇ ❫ ❉✮ ❆ ❫ ❈ ❇ ❫ ❉ ✭❆ ❫ ❈✮ ❫ ✭❇ ❫ ❉✮
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■ Other views on stability ■ General decompositions and diagrams ■ New algorithms and analysis ■ Impact of the Heyting algebra structure ■ Study of the dual space
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JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
JPC :: NSAC 2013 The Persistence Lattice
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Implementing pullbacks and pushouts
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❆ ❇ ❈ ❢ ❣ ❦❡r✭❆ ✟ ❇
✭❢❀❣✮
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P ❆ ❇ ❈ ❢ ❣ ❦❡r✭❆ ✟ ❇
✭❢❀❣✮
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P ❆ ❇ ❈ ❢ ❣ Compute ❦❡r✭❆ ✟ ❇
✭❢❀❣✮
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We start out with two maps ❢❀ ❣ represented by matrices ❋❀ ●. To compute the pullback of f and g, we construct the matrix corresponding to ✭❢❀ ❣✮: ❋
The Persistence Lattice
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We start out with two maps ❢❀ ❣ represented by matrices ❋❀ ●. To compute the pullback of f and g, we construct the matrix corresponding to ✭❢❀ ❣✮: Compute kernel ❋
■
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We start out with two maps ❢❀ ❣ represented by matrices ❋❀ ●. To compute the pullback of f and g, we construct the matrix corresponding to ✭❢❀ ❣✮: Compute kernel
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Computing the pullback
P . ˇ Skraba and M. Vejdemo-Johansson, Persistence modules: algebra and algorithms. Mathematics of Computation (submitted, 2013) JPC :: NSAC 2013 The Persistence Lattice
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Computing the pullback
P . ˇ Skraba and M. Vejdemo-Johansson, Persistence modules: algebra and algorithms. Mathematics of Computation (submitted, 2013) JPC :: NSAC 2013 The Persistence Lattice
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❆ ❇ ❉ ❢ ❣ ❝♦❦❡r✭❉
✭❢❀❣✮
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◗ ❆ ❇ ❉ ❢ ❣ ❝♦❦❡r✭❉
✭❢❀❣✮
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◗ ❆ ❇ ❉ ❢ ❣ Compute ❝♦❦❡r✭❉
✭❢❀❣✮
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Using a duality for Heyting algebras
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HA := Heyting algebras ✘ ❂ Esa := Esakia Spaces and homomorphisms and homeomorphisms
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Esakia Spaces
An Esakia Space ✭❳❀ ✔ ✜✮ is a set ❳ equipped with a partial order ✔ and a topology ✜ such that:
■ ✭❳❀ ✜✮ is compact; ■ ① ✂ ② implies ✾ ❯ of ❳ st. ① ✷ ❯ and ② ❂
✷ ❯;
■ for each clopen ❈ of ✭❳❀ ✜✮, the ideal ★ ❈ is clopen.
Esakia spaces are Hausdorff and zero-dimensional, constituting Stone spaces.
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Esakia Duality for Standard Persistence
join-irreducibles: all nonzero elements basic opens: ◆❛ ❂ ❢ ■ prime ideal ❥ ❛ ✷ ■ ❣ ✜ ❂ ❤◆❛❀ ❳ ◆❛ ❥ ❛ ✷ ❳✐ ✶ . . . X✸ X✷ X✶ X✵
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Esakia Duality for Multidimensional Persistence
join-irreducibles: X✵✐ and X❥✵ with ✐ ✻❂ ❥ basic opens: ◆❛ ❂ ❢ ■ prime ideal ❥ ❛ ✷ ■ ❣ ✜ ❂ ❤◆❛❀ ❳ ◆❛ ❥ ❛ ✷ ❳✐ X✷✷ X✶✷ X✷✶ X✵✷ X✶✶ X✷✵ X✵✶ X✶✵ X✵✵
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Other applications in the framework
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X✶ X✷ X✸ X✹ X✺ X✻ ✐♠ ✭❍✄✭X✐✮ ❫ ❍✄✭X❥✮ ✦ ❍✄✭X✐✮ ❴ ❍✄✭X❥✮✮ ✽✐❀ ❥
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X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✾ X✶✵ ✐♠ ✭❍✄✭X✐✮ ❫ ❍✄✭X❥✮ ✦ ❍✄✭X✐✮ ❴ ❍✄✭X❥✮✮ ✽✐❀ ❥
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X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✾ X✶✵ ✐♠ ✥❫
✐
❍✄✭X❥✮ ✦ ❴
✐
❍✄✭X✐✮ ✦
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X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✾ X✶✵ ✐♠ ✵ ❅ ❫
✐✷s♦✉r❝❡s
❍✄✭X❥✮ ✦ ❴
❥✷s✐♥❦s
❍✄✭X❥✮ ✶ ❆
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Bifiltrations: associativity.
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸
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Bifiltrations: associativity.
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ ❫ ❴
JPC :: NSAC 2013 The Persistence Lattice
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Bifiltrations: associativity.
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ ❫ ❴
JPC :: NSAC 2013 The Persistence Lattice
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Bifiltrations: associativity.
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ ❫ ❴ ❫ ❴ ❫ ❴
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Bifiltrations: associativity.
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ ❫ ❴ ❫ ❴ ❫ ❴
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Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Bifiltrations: sections.
X✵✵ X✵✶ X✵✷ X✵✸ X✶✵ X✶✶ X✶✷ X✶✸ X✷✵ X✷✶ X✷✷ X✷✸ X✸✵ X✸✶ X✸✷ X✸✸ ❫ ❴ ❫ ❴ ❫ ❴
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: common features.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: common features.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: common features.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺ X✷ ❴ X✼
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: associativity.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺ X✷ ❴ X✼ X✵ ❫ X✺ X✷ ❴ X✼
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: associativity.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺ X✷ ❴ X✼ X✵ ❫ X✺ X✷ ❴ X✼ ❴
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: associativity.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺ X✷ ❴ X✼ X✵ ❫ X✺ X✷ ❴ X✼ ❴ ❴
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
General Diagrams: associativity.
X✵ X✶ X✷ X✸ X✹ X✺ X✻ X✼ X✽ X✵ ❫ X✺ X✷ ❴ X✼ X✵ ❫ X✺ X✷ ❴ X✼ ❴ ❴ ❴
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability Teorems
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability of the Persistence Diagram.
D Cohen-Steiner, H Edelsbrunner, and J Harer, Stability of persistence diagrams. Discrete Comput Geom (2005) JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❫ ❇ ❆ ❇ ❈
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❴ ❇ ❆ ❫ ❇ ❆ ❇ ❈
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❴ ❈ ❆ ❫ ❈ ❆ ❈ ❇ ❫ ❉ ❇ ❉ ❇ ❴ ❉
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❴ ❈ ❆ ❫ ❈ ❆ ❈ ❇ ❫ ❉ ❇ ❉ ❇ ❴ ❉
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❴ ❈ ❆ ❫ ❈ ❆ ❈ ❇ ❫ ❉ ❇ ❉ ❇ ❴ ❉
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❇ ❈ ❉
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ ❈ ❴ ❉ ❈ ❉ ❈ ❫ ❉
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
✭❆ ❴ ❈✮ ❴ ✭❇ ❴ ❉✮ ❆ ❴ ❈ ❇ ❴ ❉ ✭❆ ❴ ❈✮ ❫ ✭❇ ❴ ❉✮ ❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ ❈ ❴ ❉ ❈ ❉ ❈ ❫ ❉ ✭❆ ❫ ❈✮ ❴ ✭❇ ❫ ❉✮ ❆ ❫ ❈ ❇ ❫ ❉ ✭❆ ❫ ❈✮ ❫ ✭❇ ❫ ❉✮
JPC :: NSAC 2013 The Persistence Lattice
Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications
Stability
✭❆ ❴ ❈✮ ❴ ✭❇ ❴ ❉✮ ❆ ❴ ❈ ❇ ❴ ❉ ✭❆ ❴ ❈✮ ❫ ✭❇ ❴ ❉✮ ❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ ❈ ❴ ❉ ❈ ❉ ❈ ❫ ❉ ✭❆ ❫ ❈✮ ❴ ✭❇ ❫ ❉✮ ❆ ❫ ❈ ❇ ❫ ❉ ✭❆ ❫ ❈✮ ❫ ✭❇ ❫ ❉✮
JPC :: NSAC 2013 The Persistence Lattice