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The Persistence Lattice Jo ao Pita Costa z (in a joint work with - PowerPoint PPT Presentation

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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

  1. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence The Missing Data Problem G. Carlsson and A. Zomorodian, The theory of multidimensional persistence. Discrete Comput Geom (2007) JPC :: NSAC 2013 The Persistence Lattice

  2. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence The Missing Data Problem G. Carlsson and A. Zomorodian, The theory of multidimensional persistence. Discrete Comput Geom (2007) JPC :: NSAC 2013 The Persistence Lattice

  3. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Partially ordered sets What can the order tell us? JPC :: NSAC 2013 The Persistence Lattice

  4. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Varieties Of Lattices Boolean algebras Persistence lattices Heyting algebras vector lattices totally ordered sets metric lattices distributive lattices subspace lattices projective lattices modular lattices geometric lattices partition lattices semi-modular lattices lattices skew lattices partially ordered sets JPC :: NSAC 2013 The Persistence Lattice

  5. ❍ ✐❀❥ ✄ ✭ ✮ ❂ ✐♠✭❍ ✄ ✭ ✐ ✮ ✦ ❍ ✄ ✭ ❥ ✮✮ ■ ❍ ✄ ✭ ✐ ✮ ❴ ❍ ✄ ✭ ❥ ✮ ❂ ❍ ✄ ✭ ❳ ♠❛①✭ ✐❀❥ ✮ ✮ ■ ❍ ✄ ✭ ✐ ✮ ❫ ❍ ✄ ✭ ❥ ✮ ❂ ❍ ✄ ✭ ❳ ♠✐♥✭ ✐❀❥ ✮ ✮ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Standard Persistence ❳ ◆ A Morse-filtration is a partial order on the parameter ❳ ◆ � ✶ ❳ ☛ ✒ ❳ ☞ ✮ ☛ ❁ ☞ . . . ❳ ✷ ❳ ✶ JPC :: NSAC 2013 The Persistence Lattice

  6. ❍ ✐❀❥ ✄ ✭ ✮ ❂ ✐♠✭❍ ✄ ✭ ✐ ✮ ✦ ❍ ✄ ✭ ❥ ✮✮ ■ ❍ ✄ ✭ ✐ ✮ ❴ ❍ ✄ ✭ ❥ ✮ ❂ ❍ ✄ ✭ ❳ ♠❛①✭ ✐❀❥ ✮ ✮ ■ ❍ ✄ ✭ ✐ ✮ ❫ ❍ ✄ ✭ ❥ ✮ ❂ ❍ ✄ ✭ ❳ ♠✐♥✭ ✐❀❥ ✮ ✮ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Standard Persistence ❳ ◆ A Morse-filtration is a partial order on the parameter ❳ ◆ � ✶ ❳ ☛ ✒ ❳ ☞ ✮ ☛ ❁ ☞ . . . ❳ ✷ ❳ ✶ JPC :: NSAC 2013 The Persistence Lattice

  7. ■ ❍ ✄ ✭ ✐ ✮ ❴ ❍ ✄ ✭ ❥ ✮ ❂ ❍ ✄ ✭ ❳ ♠❛①✭ ✐❀❥ ✮ ✮ ■ ❍ ✄ ✭ ✐ ✮ ❫ ❍ ✄ ✭ ❥ ✮ ❂ ❍ ✄ ✭ ❳ ♠✐♥✭ ✐❀❥ ✮ ✮ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Standard Persistence ❳ ◆ A Morse-filtration is a partial order on the parameter ❳ ◆ � ✶ ❳ ☛ ✒ ❳ ☞ ✮ ☛ ❁ ☞ . . Persistent homology classes . ❍ ✐❀❥ ✄ ✭ X ✮ ❂ ✐♠✭❍ ✄ ✭ X ✐ ✮ ✦ ❍ ✄ ✭ X ❥ ✮✮ ❳ ✷ ❳ ✶ JPC :: NSAC 2013 The Persistence Lattice

  8. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Standard Persistence ❳ ◆ 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

  9. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Standard Persistence ❳ ◆ 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

  10. ✐❥ ❫ ❦❵ ✮ ② ❂ ♠✐♥✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠✐♥✭ ❥❀ ❵ ✮ ②③ ✐❥ ❴ ❦❵ ✮ ② ❂ ♠❛①✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠❛①✭ ❥❀ ❵ ✮ ②③ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence X ✵✸ X ✶✸ X ✷✸ X ✸✸ X ✵✷ X ✶✷ X ✷✷ X ✸✷ X ✵✶ X ✶✶ X ✷✶ X ✸✶ X ✵✵ X ✶✵ X ✷✵ X ✸✵ JPC :: NSAC 2013 The Persistence Lattice

  11. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence X ✵✸ X ✶✸ X ✷✸ X ✸✸ X ✵✷ X ✶✷ X ✷✷ X ✸✷ X ✵✶ X ✶✶ X ✷✶ X ✸✶ X ✵✵ X ✶✵ X ✷✵ X ✸✵ Set: X ✐❥ ❫ X ❦❵ ✮ X ②③ , with ② ❂ ♠✐♥✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠✐♥✭ ❥❀ ❵ ✮ X ✐❥ ❴ X ❦❵ ✮ X ②③ , with ② ❂ ♠❛①✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠❛①✭ ❥❀ ❵ ✮ JPC :: NSAC 2013 The Persistence Lattice

  12. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence X ✵✸ X ✶✸ X ✷✸ X ✸✸ X ✵✷ X ✶✷ X ✷✷ X ✸✷ X ✵✶ X ✶✶ X ✷✶ X ✸✶ X ✵✵ X ✶✵ X ✷✵ X ✸✵ Set: X ✐❥ ❫ X ❦❵ ✮ X ②③ , with ② ❂ ♠✐♥✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠✐♥✭ ❥❀ ❵ ✮ X ✐❥ ❴ X ❦❵ ✮ X ②③ , with ② ❂ ♠❛①✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠❛①✭ ❥❀ ❵ ✮ JPC :: NSAC 2013 The Persistence Lattice

  13. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence X ✵✸ X ✶✸ X ✷✸ X ✸✸ X ✵✷ X ✶✷ X ✷✷ X ✸✷ X ✵✶ X ✶✶ X ✷✶ X ✸✶ X ✵✵ X ✶✵ X ✷✵ X ✸✵ Set: X ✐❥ ❫ X ❦❵ ✮ X ②③ , with ② ❂ ♠✐♥✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠✐♥✭ ❥❀ ❵ ✮ X ✐❥ ❴ X ❦❵ ✮ X ②③ , with ② ❂ ♠❛①✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠❛①✭ ❥❀ ❵ ✮ JPC :: NSAC 2013 The Persistence Lattice

  14. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence X ✵✸ X ✶✸ X ✷✸ X ✸✸ X ✵✷ X ✶✷ X ✷✷ X ✸✷ X ✵✶ X ✶✶ X ✷✶ X ✸✶ X ✵✵ X ✶✵ X ✷✵ X ✸✵ Set: X ✐❥ ❫ X ❦❵ ✮ X ②③ , with ② ❂ ♠✐♥✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠✐♥✭ ❥❀ ❵ ✮ X ✐❥ ❴ X ❦❵ ✮ X ②③ , with ② ❂ ♠❛①✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠❛①✭ ❥❀ ❵ ✮ JPC :: NSAC 2013 The Persistence Lattice

  15. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Multidimensional Persistence X ✵✸ X ✶✸ X ✷✸ X ✸✸ X ✵✷ X ✶✷ X ✷✷ X ✸✷ X ✵✶ X ✶✶ X ✷✶ X ✸✶ X ✵✵ X ✶✵ X ✷✵ X ✸✵ Set: X ✐❥ ❫ X ❦❵ ✮ X ②③ , with ② ❂ ♠✐♥✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠✐♥✭ ❥❀ ❵ ✮ X ✐❥ ❴ X ❦❵ ✮ X ②③ , with ② ❂ ♠❛①✭ ✐❀ ❦ ✮ ❀ ③ ❂ ♠❛①✭ ❥❀ ❵ ✮ JPC :: NSAC 2013 The Persistence Lattice

  16. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications General Diagrams? X ✷ X ✼ X ✶ X ✹ X ✽ X ✵ X ✸ X ✻ X ✺ JPC :: NSAC 2013 The Persistence Lattice

  17. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications General Diagrams? X ✷ X ✼ X ✶ X ✹ X ✽ X ✵ X ✸ X ✻ X ✺ X ✵ ❫ X ✺ JPC :: NSAC 2013 The Persistence Lattice

  18. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications General Diagrams? X ✷ ❴ X ✼ X ✷ X ✼ X ✶ X ✹ X ✽ X ✵ X ✸ X ✻ X ✺ X ✵ ❫ X ✺ JPC :: NSAC 2013 The Persistence Lattice

  19. ❆ ❇ ❆ ✔ ❇ ❢ ✿ ❆ ✦ ❇✿ ✐❞ ❆ ✐❞ ❇ ❢ ❆ ❇ ❣ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Diagrams of Spaces Requirements Diagram is commutative and connected. JPC :: NSAC 2013 The Persistence Lattice

  20. ❆ ❇ ❆ ✔ ❇ ❢ ✿ ❆ ✦ ❇✿ ✐❞ ❆ ✐❞ ❇ ❢ ❆ ❇ ❣ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Diagrams of Spaces Requirements Diagram is commutative and connected. the reverse maps exist in the case of isomorphisms. JPC :: NSAC 2013 The Persistence Lattice

  21. ❆ ❇ ❆ ✔ ❇ ❢ ✿ ❆ ✦ ❇✿ ✐❞ ❆ ✐❞ ❇ ❢ ❆ ❇ ❣ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Diagrams of Spaces 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

  22. ✐❞ ❆ ✐❞ ❇ ❢ ❆ ❇ ❣ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Diagrams of Spaces 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

  23. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Diagrams of Spaces 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

  24. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers Equalizers f ❆ ❈ g JPC :: NSAC 2013 The Persistence Lattice

  25. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers Equalizers e f ❊ ❆ ❈ g JPC :: NSAC 2013 The Persistence Lattice

  26. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers Equalizers e f ❊ ❆ ❈ g e’ ❊ ✵ JPC :: NSAC 2013 The Persistence Lattice

  27. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers Equalizers e f ❊ ❆ ❈ g e’ ✣ ❊ ✵ JPC :: NSAC 2013 The Persistence Lattice

  28. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers Equalizers e f ❊ ❆ ❈ g e’ ✣ ❊ ✵ The kernel set is ❊ ❂ ❢ ① ✷ ❳ ❥ ❢ ✭ ① ✮ ❂ ❣ ✭ ① ✮ ❣ ❂ ❦❡r ✭ ❢ � ❣ ✮ JPC :: NSAC 2013 The Persistence Lattice

  29. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers Coequalizers ❢ h ❉ ❈ ❍ ❣ ✣ h’ ❍ ✵ ❍ is the quotient of ❨ by the equivalence ❤ ✭ ❢ ✭ ① ✮ ❀ ❣ ✭ ① ✮✮ ❥ ① ✷ ❳ ✐ , i.e., ❍ ❂ ❈❂✐♠ ✭ ❢ � ❣ ✮ ❂ ❝♦❦❡r ✭ ❢ � ❣ ✮ JPC :: NSAC 2013 The Persistence Lattice

  30. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers ❢ ✐ e ❊ ❆ ✟ ❇ ❈ ❢ ❥ ❈ ❆ ✟ ❇ ❆ ❇ JPC :: NSAC 2013 The Persistence Lattice

  31. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers ❣ ✐ h ❆ ✟ ❇ ❉ ❍ ❣ ❥ ❆ ✟ ❇ ❆ ❇ ❉ JPC :: NSAC 2013 The Persistence Lattice

  32. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers ❢ ✐ e ❆ ✟ ❇ ❈ ✶ ❀ ❈ ✷ ❊ ❢ ❥ ❈ ✷ ❈ ✶ ❆ ❇ JPC :: NSAC 2013 The Persistence Lattice

  33. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers ❢ ✐ e ❆ ✟ ❇ ❈ ✶ ✟ ❈ ✷ ❊ ❢ ❥ ❈ ✶ ✟ ❈ ✷ ❆ ❇ JPC :: NSAC 2013 The Persistence Lattice

  34. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers f h ❉ ✶ ❀ ❉ ✷ ❆ ✟ ❇ ❍ g ❆ ❇ ❉ ✶ ❉ ✷ JPC :: NSAC 2013 The Persistence Lattice

  35. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Equalizers and Coequalizers f h ❉ ✶ ✟ ❉ ✷ ❆ ✟ ❇ ❍ g ❆ ❇ ❉ ✶ ✟ ❉ ✷ JPC :: NSAC 2013 The Persistence Lattice

  36. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Formal Definition 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 ✐❀ ❥ ✷ ■ ✐ JPC :: NSAC 2013 The Persistence Lattice

  37. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Intuition ❆ ❇ JPC :: NSAC 2013 The Persistence Lattice

  38. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Intuition ❆ ❇ ❆ ❫ ❇ JPC :: NSAC 2013 The Persistence Lattice

  39. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Intuition ❆ ❴ ❇ ❆ ❇ JPC :: NSAC 2013 The Persistence Lattice

  40. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Intuition ❆ ❴ ❇ ❆ ❇ ❆ ❫ ❇ JPC :: NSAC 2013 The Persistence Lattice

  41. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Completness Theorem (JPC & P ˇ S 2013) The persistence lattice is a complete lattice with ❫ ❆ ❦ ❂ ❢ ① ✷ ✟ ❦ ❆ ❦ ✿ ❢ ❆ ✐ ✭ ① ✮ ❂ ❢ ❆ ❥ ✭ ① ✮ ❣ ❀ ❴ ❬ ❆ ❦ ❂ ✭ ✟ ❦ ❆ ❦ ✮ ❂ ❤ ✒ ❆ ✐ ❆ ❥ ✐ ✿ ❦ where ✒ ❆ ✐ ❆ ❥ ❂ ❤ ✭ ❢ ❆ ✐ ✭ ① ✮ ❀ ❢ ❆ ❥ ✭ ① ✮✮ ✐ JPC :: NSAC 2013 The Persistence Lattice

  42. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties What lattice do we get? JPC :: NSAC 2013 The Persistence Lattice

  43. ❢ ✿ ❆ ❫ ❇ ✦ ❆ ✟ ❇ ❣ ✿ ❆ ✟ ❇ ✦ ❆ ❴ ❇ ✐♠ ❢ ❂ ❦❡r ❣ ❆ ❴ ❇ ✘ ❂ ❆ ✟ ❇❂❢ ✭ ❆ ❫ ❇ ✮ ✿ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Theorem (JPC & P ˇ S 2013) Let ❆ and ❇ be vector spaces. Then, ✵ ✦ ❆ ❫ ❇ ✦ ❆ ✟ ❇ ✦ ❆ ❴ ❇ ✦ ✵ is a short exact sequence. JPC :: NSAC 2013 The Persistence Lattice

  44. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 ❆ ❴ ❇ ✘ ❂ ❆ ✟ ❇❂❢ ✭ ❆ ❫ ❇ ✮ ✿ JPC :: NSAC 2013 The Persistence Lattice

  45. ❆ ❇ ❳ ❳ ❴ ❆ ❂ ❳ ❴ ❇ ❆ ✘ ❳ ❫ ❆ ❂ ❳ ❫ ❇ ❂ ❇ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Theorem (JPC & P ˇ S 2013) The persistence lattice of a given persistence diagram is distributive. JPC :: NSAC 2013 The Persistence Lattice

  46. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 ❆ ✘ ❂ ❇ . JPC :: NSAC 2013 The Persistence Lattice

  47. ❆ ❇ ❳ ✤ ✣ ❆ ✣ ❳ ❳ ❫ ❆ ❳ ❭ ❬ ✜ ❳ ■ ❯ ✮ ❱ ❂ ✐♥t ✭✭ ❳ � ❯ ✮ ❬ ❱ ✮ ① ✮ ② ❂ ❲ ❢ ③ ✿ ① ❫ ③ ✔ ② ❣ ■ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Definition A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲ , ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇ , i.e., JPC :: NSAC 2013 The Persistence Lattice

  48. ❳ ❭ ❬ ✜ ❳ ■ ❯ ✮ ❱ ❂ ✐♥t ✭✭ ❳ � ❯ ✮ ❬ ❱ ✮ ① ✮ ② ❂ ❲ ❢ ③ ✿ ① ❫ ③ ✔ ② ❣ ■ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Definition A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲ , ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇ , i.e., ❆ ❇ ❳ ✤ ✣ ❆ ✣ ❳ ❳ ❫ ❆ JPC :: NSAC 2013 The Persistence Lattice

  49. ❳ ❭ ❬ ✜ ❳ ■ ❯ ✮ ❱ ❂ ✐♥t ✭✭ ❳ � ❯ ✮ ❬ ❱ ✮ ① ✮ ② ❂ ❲ ❢ ③ ✿ ① ❫ ③ ✔ ② ❣ ■ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Definition A bounded distributive lattice ▲ is a Heyting algebra if, for all ❆❀ ❇ ✷ ▲ , ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✔ ❇ , i.e., ❆ ❇ ❳ ✤ ✣ ❆ ✣ ❳ ❳ ❫ ❆ JPC :: NSAC 2013 The Persistence Lattice

  50. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 ❣ JPC :: NSAC 2013 The Persistence Lattice

  51. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 . JPC :: NSAC 2013 The Persistence Lattice

  52. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Arrow Operation for standard persistence ❆ ✮ ❇ is the biggest ❳ such that ❆ ❫ ❳ ✦ ❇ . . . ❳ ✐ ❇ ✭ ❇❀ if ❇ ✔ ❆ ❆ ✮ ❇ ❂ ❆ ✶ ❀ if ❆ ✔ ❇ ❳ ❥ . . . JPC :: NSAC 2013 The Persistence Lattice

  53. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 ✸✵ JPC :: NSAC 2013 The Persistence Lattice

  54. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 ✸✵ JPC :: NSAC 2013 The Persistence Lattice

  55. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties 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 ✸✵ JPC :: NSAC 2013 The Persistence Lattice

  56. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Stability ❆ ❇ ❈ ❉ JPC :: NSAC 2013 The Persistence Lattice

  57. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Stability ❆ ❴ ❇ ❈ ❴ ❉ ❆ ❇ ❈ ❉ ❆ ❫ ❇ ❈ ❫ ❉ JPC :: NSAC 2013 The Persistence Lattice

  58. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Stability ✭ ❆ ❴ ❈ ✮ ❴ ✭ ❇ ❴ ❉ ✮ ❆ ❴ ❈ ❇ ❴ ❉ ✭ ❆ ❴ ❈ ✮ ❫ ✭ ❇ ❴ ❉ ✮ ❆ ❴ ❇ ❈ ❴ ❉ ❆ ❇ ❈ ❉ ❆ ❫ ❇ ❈ ❫ ❉ ✭ ❆ ❫ ❈ ✮ ❴ ✭ ❇ ❫ ❉ ✮ ❆ ❫ ❈ ❇ ❫ ❉ ✭ ❆ ❫ ❈ ✮ ❫ ✭ ❇ ❫ ❉ ✮ JPC :: NSAC 2013 The Persistence Lattice

  59. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algebraic Properties Stability ✭ ❆ ❴ ❈ ✮ ❴ ✭ ❇ ❴ ❉ ✮ ❆ ❴ ❈ ❇ ❴ ❉ ✭ ❆ ❴ ❈ ✮ ❫ ✭ ❇ ❴ ❉ ✮ ❆ ❴ ❇ ❈ ❴ ❉ ❆ ❇ ❈ ❉ ❆ ❫ ❇ ❈ ❫ ❉ ✭ ❆ ❫ ❈ ✮ ❴ ✭ ❇ ❫ ❉ ✮ ❆ ❫ ❈ ❇ ❫ ❉ ✭ ❆ ❫ ❈ ✮ ❫ ✭ ❇ ❫ ❉ ✮ JPC :: NSAC 2013 The Persistence Lattice

  60. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications (Other) Open Problems ■ Other views on stability ■ General decompositions and diagrams ■ New algorithms and analysis ■ Impact of the Heyting algebra structure ■ Study of the dual space JPC :: NSAC 2013 The Persistence Lattice

  61. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications HVALA JPC :: NSAC 2013 The Persistence Lattice

  62. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications HVALA THANK YOU JPC :: NSAC 2013 The Persistence Lattice

  63. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications HVALA THANK YOU OBRIGADO JPC :: NSAC 2013 The Persistence Lattice

  64. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications THE B-SIDES JPC :: NSAC 2013 The Persistence Lattice

  65. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Implementation Implementing pullbacks and pushouts JPC :: NSAC 2013 The Persistence Lattice

  66. ✭ ❢❀❣ ✮ ❦❡r✭ ❆ ✟ ❇ � � � ✦ ❈ ✮ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Pullback ❆ ❢ ❇ ❈ ❣ JPC :: NSAC 2013 The Persistence Lattice

  67. ✭ ❢❀❣ ✮ ❦❡r✭ ❆ ✟ ❇ � � � ✦ ❈ ✮ Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Pullback P ❆ ❢ ❇ ❈ ❣ JPC :: NSAC 2013 The Persistence Lattice

  68. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Pullback P ❆ ❢ ❇ ❈ ❣ ✭ ❢❀❣ ✮ Compute ❦❡r✭ ❆ ✟ ❇ � � � ✦ ❈ ✮ JPC :: NSAC 2013 The Persistence Lattice

  69. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algorithm We start out with two maps ❢❀ ❣ represented by matrices ❋❀ ● . To compute the pullback of f and g, we construct the matrix corresponding to ✭ ❢❀ � ❣ ✮ : � ● ❋ JPC :: NSAC 2013 The Persistence Lattice

  70. Motivation & Background Order Structure Algebraic Constructions The Persistence Lattice Further Applications Algorithm 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 � ● ❋ JPC :: NSAC 2013 The Persistence Lattice

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