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					Structural machine learning with
    Galois lattice and Graphs



     M. Liquière J.Sallantin

       LIRMM, 161 Rue ADA
         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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