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
   

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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
   

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3. Why Studying Topology in
   Computer Science?
   Topology: Science based on connectivity
   

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4. Why Studying Topology in
   Computer Science?
   Topology: Science based on connectivity
   

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5. Why Studying Topology in
   Computer Science?
   Topology: Science based on connectivity
   Persistence of Homology. Afra Zomorodian (after Salvador Dali)
   

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6. Why Studying Topology in
   Computer Science?
   Restoring missing data (i.e. in point clouds)
   

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7. Why Studying Topology in
   Computer Science?
   Big Data analysis
   

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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)
   

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9. Why Studying Topology in
   Computer Science?
   Applications to CS:
   The Borsuk-Ulam theorem
   

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10. Why Studying Topology in
    Computer Science?
    Applications to CS:
    The Borsuk-Ulam theorem
    -  Chromatic number of Kneser graphs
    

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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
    

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12. Why Studying Topology in
    Computer Science?
    Applications to CS:
    

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13. Why Studying Topology in
    Computer Science?
    Applications to CS:
    

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14. Why Studying Topology in
    Computer Science?
    Applications to CS:
    Dividing into
    disjoint subsystems
    

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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
    

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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
    

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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
    

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18. Major Changes in General
    Approach
    Major Changes
    A B
    Hom(A,B)
    A B
    Hom(B,A)
    Building dualities
    

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19. Major Changes in General
    Approach
    

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20. Major Changes in General
    Approach
    Computing Homologiy classes
    

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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.
    

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22. Major Changes in General
    Approach
    Basic idea: Marston Morse
    Scanning an object
    

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23. Major Changes in General
    Approach
    Persistent Homology
    H. Edelsbrunner (ECM, 2008)
    !
    !
    

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24. Major Changes in General
    Approach
    Persistent Homology
    H. Edelsbrunner (ECM, 2008)
    

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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
    

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26. From Simplicial Complexes to
    Polytopal Complexes
    Simplicial Complex
    

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27. From Simplicial Complexes to
    Polytopal Complexes
    n-dimensional simplex
    

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28. From Simplicial Complexes to
    Polytopal Complexes
    n-dimensional polytope
    

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29. From Simplicial Complexes to
    Polytopal Complexes
    Simplicial and polytopal complexes
    

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30. From Simplicial Complexes to
    Polytopal Complexes
    Non-Euclidian polytopal complexes
    

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31. Categorical Study of Polytopal
    Complexes
    Structure of polytopal complexes
    Polytopal Complexes Kozlov Complexes
    Lovasz Complexes
    Simplicial
    Complexes
    

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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
    

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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?
    

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34. Thanks !
    

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