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Braids Of Partitions
B Ravi Kiran*, Jean Serra**
*ENS Paris, DATA Lab
**Université Paris-Est, A3SI-ESIEE LIGM
Kiran-Serra Braids of Partitions 1
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Problem Formulation
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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
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Constrained Optimization on Trees
Kiran-Serra Braids of Partitions 4
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Kiran-Serra Braids of Partitions 5
Partitions & Partial Partitions
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Kiran-Serra Braids of Partitions 6
Hierarchy of Partitions (HOP)
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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
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Energies on P.P.
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Dynamic Program
a b c d e f g h i j k
Kiran-Serra Braids of Partitions 9
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Dynamic Program
c d h i j k
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Dynamic Program
c d h i j k
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Dynamic Program
c d
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Dynamic Program
c d h i j k
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Composition Laws
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h-increasingness on HOP
Kiran-Serra Braids of Partitions 15
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Generalized Minkowski composition
Kiran-Serra Braids of Partitions 16
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Uniqueness and Singular Energy
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Energetic Lattice
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Optimal λ-Cut Example
Kiran-Serra Braids of Partitions 19
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Braids of Partitions
Kiran-Serra Braids of Partitions 20
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Braids Examples
Kiran-Serra Braids of Partitions 21
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Braids Examples
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Why Braids
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h-increasingness on BOP
Kiran-Serra Braids of Partitions 25
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Searching better optimum Random re-compositions on hierarchies
Kiran-Serra Braids of Partitions 27
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Kiran-Serra Braids of Partitions 29
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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
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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
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Decision Trees
Kiran-Serra Braids of Partitions 32
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Kiran-Serra Braids of Partitions 33
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Types of refinement relations
Kiran-Serra Braids of Partitions 34
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Questions
Kiran-Serra Braids of Partitions 35
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Conclusion
Kiran-Serra Braids of Partitions 36