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```					Structural machine learning with
Galois lattice and Graphs

M. Liquière J.Sallantin

Montpellier, France
Liquiere@lirmm.fr
Sallantin@lirmm.fr
Galois Latice and Formal concept analysis

Propositional description                       Galois Lattice
a1 a2 a3 a4 … ap
e1       1 O 1 1 ….
e2       1 1 1 0 ….
e3       1 1 0 1 ….
e4       1 1 1 1…
...
en

Concept      Extension   Intention

{e1,e2,e4} | {a1,a3}
Two Lattices

Instance Lattice                       Description Lattice




Galois Lattice
Propositional description

Instance Lattice ( )                      Description Lattice
E= E1  E2                               D= D1  D2

E1                  E2                    D1                 D2

E’= E1  E2                             D’= D1  D2

Eg Ag                  Eg=h(E1  E2)
h(E) =  ( (E))
Ag=h’(A1  A2)
h’(D)=  ((D))
E2 A2
E1 A1
Es=h(E1  E2)
As=h’(A1  A2)
E s As
Galois Lattice and Structural description

Structural description

For an example e, d(e) -> Graph
Order -> Morphism (pre-order)
=      ->  (quotient order)
      -> Product, Reduction
Graph description

Rectangle

on           on

Rectangle                    Circle

right

O
O
H       H                    H                    H
Morphism

Rectangle
Rectangle
on           on
on           on
Rectangle                 Circle
Rectangle           Rectangle
right

morphism

G ≥ G’ <=> there is a morphism from G into G’
Equivalence

Rectangle                        Rectangle

on                               on          on

Rectangle                        Rectangle           Rectangle

G  G’ <=> G ≥ G’ and G’ ≥ G
Reduction

R(G)  G and R(G) is minimal (size)
G

R(G)

Let g be a labelled graph, R(g) is a core labelled graph.
≥ is an order for the set of core labelled Graphs
If ≥ is Polynomial then R is polynomial
Intersection, Product

Properties
G1  G2 ≥ G1, G1  G2 ≥ G2
If G ≥ G1 and G ≥ G2 then G ≥ G1  G2

Definition  operation for labelled graphs
For two labelled graphs G1:(V1,E1,L1) and G2:(V2,E2,L2)
The product G(V,E,L) = G1  G2 is defined by:
L = L1  L2
V  V1  V2 ={ v | v=[v1,v2] with L(v1) = L(v2) and L(v)=L(v1)}
U= {(v=[v1,v2],v'=[v'1,v'2]) | (v1,v'1) V1 and (v2,v'2) V2}(edge
oriented)
Product example

G1                                          G2

1     rectangl e                              1
rectangl e
2                                         2
on                                          on
5                                     5
3
rectangl e           4         circle        circle            4
rectangl e
3              righ t                                      righ ts

G=G1x G2

1,1    rectangl e
rectangl e
2,2
on                      1,5

rectangl e                        circle
5,3
3,5                righ t                        rectangl e
4,4                                  3,1
Intersection

G1                                 G2
rectangl e                        rectangl e

on                             on
rectangl e            circle     circle           rectangl e
righ t                         righ ts

G1 G2                                        G1 G2= R(G1 G2)
rectangl e                                                  rectangl e
rectangl e
on                                                          on
rectangl e            circle                                rectangl e            circle
righ t                 rectangl e                           righ t

The restriction of ≥ to the set of core labelled Graphs is a lattice
Theorem

With:
 a set of examples
D a description lattice for core labelled graphs
d a mapping d:  -> D
I an instance lattice (())
≥ the order relation DxD -> {True, false}
 the generalisation operation DxD -> D
We can define ,
g  D, (g)={e   | g ≥ d(e)}
h  I, (h)= e  h d(e)

The correspondance , defines a Galois connection between I and D

A concept C is a pair [Ext x Int] with
Int= (Ext) and Ext= (Int)
[E,I] ≥c [E’,I’] <=> E  E’ and I ≥ I’
Examples

e0                        e1                               e2                             e3                               e4

rectangle                                  rectangle                            square                        circle                           rectangle
on                           s
{0}                on
{1}              on                  {2}               on
{3}                  {4}                  on
rectangle             circle             circle            rectangles         rectangle           square                                     circle               square
right                                  right                                  on                        rectangle                           right

rectangle                                                                          rectangle

right
{0,1}                    on                                                        {1,4}                  on

rectangle              circle                                                       circle        right                      {2,4}     rectangle       on       square

rectangle
rectangle
on
on
{0,1,2}         on             rectangle
{0,1,4}                  on           right                        {0,1,3}                        circle
rectangle
circle

on                                       rectangle                         rectangle
{0,1,2,3}                                {0,1,2,4}                                                 on            {0,1,3,4}
rectangle                                   on                              circle

rectangle
{0,1,2,3,4}                on
An Algorithm
T<-¯ /* co ncept set emp ty */
/* descript ion of t he ex amples */
T 1<-{[{i}  d(ei)]} an d i | |]
k<-1
W hil e |T k| > 1 do
Fo r ea ch i<j an d i,j  |T k|] (so we have: [ei x di] [ej x dj] ) /* we create a new co ncept from t wo
co ncept salready foun d in t he prev io us step*/
dij<- didj /* descript ion for t he new co ncept */
i f dij- ¯ then
/* t est if t here is a co ncept with t he same descript ion */
i f dij  T k+1 then
eij <- eijej
el se
T k+1<- T k+1  eiej  dij]
/* t est if t he descript ion is a generalisat io n */
i f dij  di then
T k <- T k - [ei  di ]
i f dij  dj then
T k <- T k - [ej  dj]
En d-i f
en d-for
T<- T  T k
k<-k+1
en d-wh il e
T<- T  T k
Complexity

||
Maximal size for a Galois Lattice : 2

Operations     ≥,R,           Size of the Description for n graphs
Path           P               P              ( Pac)
Tree           P               P              ( Pac)
GLI            P               ?
D.Automata     P               Exponential
Graph          NP              Exponential
Graal Implementation

The following Operations are Graal’s parameters
Intersection                DxD -> D
Inclusion      or ≤         DxD ->{0,1}
Equivalence                 DxD ->{0,1}

For String description
      longest prefix => unique
      string  => order
      string =

For Labelled Graph description
      R(G1  G2)
≥      morphism => order for core Graph
      G1 ≥ G2 and G2≥ G1
Formal model -> Tools

Examples
Algorithm
Galois Lattice

Description Language
And operations
, , 
Conclusion

Results
Enlarge the domain of formal concept analysis

Generalization operator for Graphs

Graal implementation

Improvements

Logical rules

Concept selection

Heuristics (≥,,)

Categorical model

```
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