Applications of Computational Topology to Artificial Intelligence

Alexander Gamkrelidze
Professor, Ivane Javakhishvili Tbilisi State University

International Conference on Software Testing, Machine Learning and Complex Process Analysis (TMPA-2019)
7-9 November 2019, Tbilisi

Video: https://youtu.be/2jJoY-b4pDM

TMPA Conference website https://tmpaconf.org/
TMPA Conference on Facebook https://www.facebook.com/groups/tmpaconf/

Transcript

1. Applications of Computational Topology to AI Alexander Gamkrelidze I. Javakhishvili Tbilisi State University Tbilisi, 7. 11. 2019
2. Contents •  Why Studying Topology in Computer Science? •  Major Changes in General Approach •  Some Applications to AI •  From Simplicial Complexes to Polytopal Complexes •  Categorical Study of Polytopal Complexes •  Conclusions
3. Why Studying Topology in Computer Science? Topology: Science based on connectivity
4. Why Studying Topology in Computer Science? Topology: Science based on connectivity
5. Why Studying Topology in Computer Science? Topology: Science based on connectivity Persistence of Homology. Afra Zomorodian (after Salvador Dali)
6. Why Studying Topology in Computer Science? Restoring missing data (i.e. in point clouds)
7. Why Studying Topology in Computer Science? Big Data analysis
8. Why Studying Topology in Computer Science? Applications to CS: The Borsuk-Ulam theorem For any continous mapping f : Sn à Rn, there exists x so that f(x) = f(-x)
9. Why Studying Topology in Computer Science? Applications to CS: The Borsuk-Ulam theorem
10. Why Studying Topology in Computer Science? Applications to CS: The Borsuk-Ulam theorem -  Chromatic number of Kneser graphs
11. Why Studying Topology in Computer Science? Applications to CS: The Borsuk-Ulam theorem -  A plane with coloured points can be divided into disjoint convex hulls with points of all colours -  Dividing a system into equivalent disjoint subsystems
12. Why Studying Topology in Computer Science? Applications to CS:
13. Why Studying Topology in Computer Science? Applications to CS:
14. Why Studying Topology in Computer Science? Applications to CS: Dividing into disjoint subsystems
15. Major Changes in General Approach Since ancient times: Studying a system with structure (addition, multiplication, Lie bracket etc.) = Experimenting with elements Deeper look inside gives the information
16. Major Changes in General Approach New observation: Studying a system A = Studying Homomorphisms from A to a known system B Studying Homomorphisms out of A gives a deep insight
17. Major Changes in General Approach New observation: Studying a system A = Studying Homomorphisms from a known system B to A Studying Homomorphisms into A gives a deep insight
18. Major Changes in General Approach Major Changes A B Hom(A,B) A B Hom(B,A) Building dualities
19. Major Changes in General Approach
20. Major Changes in General Approach Computing Homologiy classes
21. Major Changes in General Approach Computing the generators of homologiy groups R. V. Gamkrelidze, Computation of the Chern cycles of algebraic manifolds Doklady Akad. Nauk SSSR (N.S.) 90 (1953), 719–722.
22. Major Changes in General Approach Basic idea: Marston Morse Scanning an object
23. Major Changes in General Approach Persistent Homology H. Edelsbrunner (ECM, 2008) ! !
24. Major Changes in General Approach Persistent Homology H. Edelsbrunner (ECM, 2008)
25. Applications to AI -  Denoising -  Expert analysis (i.e. divergency) -  Face recognition -  (Neural) network analysis -  Big data Connecting data points in the space: Simplicial complexes
26. From Simplicial Complexes to Polytopal Complexes Simplicial Complex
27. From Simplicial Complexes to Polytopal Complexes n-dimensional simplex
28. From Simplicial Complexes to Polytopal Complexes n-dimensional polytope
29. From Simplicial Complexes to Polytopal Complexes Simplicial and polytopal complexes
30. From Simplicial Complexes to Polytopal Complexes Non-Euclidian polytopal complexes
31. Categorical Study of Polytopal Complexes Structure of polytopal complexes Polytopal Complexes Kozlov Complexes Lovasz Complexes Simplicial Complexes
32. Categorical Study of Polytopal Complexes A B Hom(A,B) Hom(A,B) is a polytopal complex M. Bakuradze, A. Gamkrelidze, and J. Gubeladze Affine hom-complexes 2016
33. Conclusions -  Topology plays an important role in CS (AI) -  Actual trend: Describe objects with SCs and apply the persistent homology ideas; -  Problem: Some objects can not be described effectively by SCs; -  Question: Can we efficiently process objects described by PCs?
34. Thanks !