Post on 10-Jul-2020
Braids Of Partitions
B Ravi Kiran*, Jean Serra**
*ENS Paris, DATA Lab
**Université Paris-Est, A3SI-ESIEE LIGM
Kiran-Serra Braids of Partitions 1
Problem Formulation
Kiran-Serra Braids of Partitions 2
Problem Formulation
c
d
a b
e h i
j k
f
g
a b
h i
a b c d e f g h i j k
Kiran-Serra Braids of Partitions 3
Constrained Optimization on Trees
Kiran-Serra Braids of Partitions 4
Kiran-Serra Braids of Partitions 5
Partitions & Partial Partitions
Kiran-Serra Braids of Partitions 6
Hierarchy of Partitions (HOP)
Kiran-Serra Braids of Partitions 7
c
d
a b
e h i
j k
f
g
Cuts in a Hierarchy
a b
h i
a b c d e f g h i j k
Energies on P.P.
Kiran-Serra Braids of Partitions 8
Dynamic Program
a b c d e f g h i j k
Kiran-Serra Braids of Partitions 9
Dynamic Program
c d h i j k
Kiran-Serra Braids of Partitions 10
Dynamic Program
c d h i j k
Kiran-Serra Braids of Partitions 11
Dynamic Program
c d
Kiran-Serra Braids of Partitions 12
Dynamic Program
c d h i j k
Kiran-Serra Braids of Partitions 13
Composition Laws
Kiran-Serra Braids of Partitions 14
h-increasingness on HOP
Kiran-Serra Braids of Partitions 15
Generalized Minkowski composition
Kiran-Serra Braids of Partitions 16
Uniqueness and Singular Energy
Kiran-Serra Braids of Partitions 17
Energetic Lattice
Kiran-Serra Braids of Partitions 18
Optimal λ-Cut Example
Kiran-Serra Braids of Partitions 19
Braids of Partitions
Kiran-Serra Braids of Partitions 20
Braids Examples
Kiran-Serra Braids of Partitions 21
Braids Examples
Kiran-Serra Braids of Partitions 22
Kiran-Serra Braids of Partitions 23
Why Braids
Kiran-Serra Braids of Partitions 24
h-increasingness on BOP
Kiran-Serra Braids of Partitions 25
Kiran-Serra Braids of Partitions 26
Searching better optimum Random re-compositions on hierarchies
Kiran-Serra Braids of Partitions 27
Kiran-Serra Braids of Partitions 28
Kiran-Serra Braids of Partitions 29
Observations
0 100 200 300 400 500 600 700 8001.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2x 10
7 Optimal vs Perturbed Cuts
(
)
Hierarchy
Perturbed
Kiran-Serra Braids of Partitions 30
Refinement order and global Infimum
1 1 2 1
3 2
5
1 1 2
5
5
1 1 2
5
5
Kiran-Serra Braids of Partitions 31
Decision Trees
Kiran-Serra Braids of Partitions 32
Kiran-Serra Braids of Partitions 33
Types of refinement relations
Kiran-Serra Braids of Partitions 34
Questions
Kiran-Serra Braids of Partitions 35
Conclusion
Kiran-Serra Braids of Partitions 36