Applications of Computational
Topology to AI
Alexander Gamkrelidze
I. Javakhishvili Tbilisi State University
Tbilisi, 7. 11. 2019
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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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Why Studying Topology in
Computer Science?
Topology: Science based on connectivity
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Why Studying Topology in
Computer Science?
Topology: Science based on connectivity
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Why Studying Topology in
Computer Science?
Topology: Science based on connectivity
Persistence of Homology. Afra Zomorodian (after Salvador Dali)
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Why Studying Topology in
Computer Science?
Restoring missing data (i.e. in point clouds)
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Why Studying Topology in
Computer Science?
Big Data analysis
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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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Why Studying Topology in
Computer Science?
Applications to CS:
The Borsuk-Ulam theorem
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Why Studying Topology in
Computer Science?
Applications to CS:
The Borsuk-Ulam theorem
- Chromatic number of Kneser graphs
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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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Why Studying Topology in
Computer Science?
Applications to CS:
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Why Studying Topology in
Computer Science?
Applications to CS:
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Why Studying Topology in
Computer Science?
Applications to CS:
Dividing into
disjoint subsystems
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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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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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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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Major Changes in General
Approach
Major Changes
A B
Hom(A,B)
A B
Hom(B,A)
Building dualities
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Major Changes in General
Approach
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Major Changes in General
Approach
Computing Homologiy classes
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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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Major Changes in General
Approach
Basic idea: Marston Morse
Scanning an object
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Major Changes in General
Approach
Persistent Homology
H. Edelsbrunner (ECM, 2008)
!
!
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Major Changes in General
Approach
Persistent Homology
H. Edelsbrunner (ECM, 2008)
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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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From Simplicial Complexes to
Polytopal Complexes
Simplicial Complex
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From Simplicial Complexes to
Polytopal Complexes
n-dimensional simplex
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From Simplicial Complexes to
Polytopal Complexes
n-dimensional polytope
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From Simplicial Complexes to
Polytopal Complexes
Simplicial and polytopal complexes
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From Simplicial Complexes to
Polytopal Complexes
Non-Euclidian polytopal complexes
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Categorical Study of Polytopal
Complexes
Structure of polytopal complexes
Polytopal Complexes Kozlov Complexes
Lovasz Complexes
Simplicial
Complexes
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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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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?