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

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Braids Examples

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Kiran-Serra Braids of Partitions 23

Why Braids

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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

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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

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

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