Operators and Spaces Associated to Matrices with Grades and Their - - PowerPoint PPT Presentation

Operators and Spaces Associated to Matrices with Grades and Their Decompositions II

Radim Belohlavek, Jan Konecny

• Dept. Computer Science,

Palacky University, Olomouc, Czech Republic

CLA 2010

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 1 / 17

Topic

In the past, we encountered relationships btw. FCA and Boolean matrices:

• R. Belohlavek, V. Vychodil: Discovery of optimal factors in binary

data via a novel method of matrix decomposition. Journal of Computer and System Sciences 76(1)(2010), pp. 3–20.

• E. Bartl, R. Belohlavek, J. Konecny: Optimal decompositions of

matrices with grades into binary and graded matrices, (to appear in Annals of Mathematics and Artificial Intelligence)

• R. Belohlavek: Optimal decomposition of matrices . . . (in revision for
• J. Logic and Computation)

Goal: Look in detail and establish connections Results on matrices with degrees from residuated lattices (Boolean matrices are particular case)

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 2 / 17

Literature on Boolean (and other) matrices

K.H. Kim: Boolean matrix theory and applications, New York, Dekker, 1982. Z.-Q. Cao, K.H. Kim, F.W. Roush: Incline algebra and applications, Chichester, Ellis Horwood, 1984. J.S. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Longman Scientific and Technical, U.K., 1992.

• N. Tatti, T. Mielik¨

ainen, A. Gionis, H. Mannila. What is the dimension of your binary data? In: The 2006 IEEE Conference on Data Mining (ICDM 2006), IEEE Computer Society, 2006, pp. 603–612.

• F. Geerts, B. Goethals, T. Mielik¨
• ainen. Tiling Databases. DS 2004,

LNCS 3245, pp. 278–289.

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 3 / 17

Example of result from Boolean matrix theory

Row space R(I) – the least set which contains all rows of I and is closed under max. (A ◦ B)ij = maxl=1...kAil · Blj.

Theorem

For Boolean matrices A and B: R(A ◦ B) ⊆ R(B). In terms of FCA: R(I) = (characteristic vectors of) intents of particular concept-forming

• perators. Theorem directly follows from properties of the concept-forming
• perators.
• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 4 / 17

Problem in Boolean matrix theory where FCA helps

Factor analyis: Given an object × attribute Boolean matrix I like I =     1 1 1 1 1 1 1 1 1 1     =     1 1 1 1 1     ◦   1 1 1 1 1 1   decompose I to A and B where Iij = (A ◦ B)ij = maxl=1...kAil · Blj. A . . . objects × factors matrix, B . . . factors × attributes matrix To find Booleam matrices A (n × k) and B (k × m) s.t. I = A ◦ B; k as small as possible

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 5 / 17

Formal concepts are optimal factors:

Theorem (Optimality)

Let I = A ◦ B for n × k and k × m matrices A and B. Then there exists F ⊆ B(X, Y , I) of formal concepts of I, s.t. we have |F| ≤ k and for the n × |F| and |F| × m matrices AF and BF we have I = AF ◦ BF. where for F = {E1, G1, E2, G2, . . . , E|F|, G|F|} AF and BF are defined as follows: (AF)il = El(i) (BF)lj = Gl(j) Fast approximation algorithm designed using this theorem.

• R. Belohlavek, V. Vychodil: Discovery of optimal factors in binary data via

a novel method of matrix decomposition. Journal of Computer and System Sciences 76(1)(2010), pp. 3–20.

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 6 / 17

Matrices we use:

entries from complete lattice L with operation ⊗ distributive wrt ∨. thus L with ⊗ forms residuated lattice: L = L, ∧, ∨, ⊗, →, 0, 1 L, ∧, ∨, 0, 1 . . . complete lattice L, ⊗, 1 . . . commutative monoid ⊗, → . . . adjoint pair (a ⊗ b ≤ c iff a ≤ b → c ) EXAMPLE: Lukasiewicz, G¨

• del, product algebras on [0, 1], finite chains,

MV-algebras, BL-algebras, Boolean algebras. L-set A in universe U . . . mapping A: U → L Interpretation of A(u): “degree to which u belongs to A” Binary L-relation R between sets U, V . . . mapping R : U × V → L, Interpretation of R(u, v): “degree to which u and v are R-related” LU denotes all L-sets A in universe U.

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 7 / 17

Matrix products (Bandler, Kohout)

Three matrix products A ⋆ B: (A ◦ B)ij = k

l=1 Ail ⊗ Blj

(A ⊳ B)ij = k

l=1 Ail → Blj

(A ⊲ B)ij = k

l=1 Blj → Ail

in case of Boolean matrices, they are mutually definable: (A ◦ B)ij = ¬(A ⊳ (¬B))ij = ¬((¬A) ⊲ B)ij we studied decompositions to these products; A ⋆ B, ⋆ ∈ {◦, ⊳, ⊲}. in terms of FCA: Context X, Y , I (X – objects, Y – attributes); Find contexts X, F, A and F, Y , B (F – factors); with |F| as small as possible, and A ⋆ B = I.

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 8 / 17

Concept-forming operators

Formal fuzzy context X, Y , I induces concept-forming operators: Antitone: E ↑(y) =

x∈X E(x) → I(x, y)

G ↓(x) =

y∈Y G(y) → I(x, y)

Isotone: E ∩(y) =

x∈X E(x) ⊗ I(x, y)

G ∪(x) =

y∈Y I(x, y) → G(y)

E ∧(y) =

x∈X I(x, y) → E(x)

G ∨(x) =

y∈Y G(y) ⊗ I(x, y)

in binary case, they they are mutually definable: E ↑I = E ∩I = E

∧I . ↑I , ↓I – concept-forming operators induced by I.

B(X ↑, Y ↓, I) ={E, G | E ↑ = G and E = G ↓} Int(X ↑, Y ↓, I) ={G | E, G ∈ B(X ↑, Y ↓, I)} Ext(X ↑, Y ↓, I) ={E | E, G ∈ B(X ↑, Y ↓, I)}

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 9 / 17

Results

Concept-forming operators of A ⋆ B are compositions of concept-forming

• perators of A and B:

Theorem

Let A and B be m × k, and k × n matrices, let E ∈ LX, G ∈ LY . We have E ∩A◦B = E ∩A∩B G ∪A◦B = G ∪B∪A E ∧A◦B = E ∧A∧B G ∨A◦B = G ∨B∨A E ↑A⊳B = E ∩A↑B G ↓A⊳B = G ↓B∪A E ↑A⊲B = E ↑A∧B G ↓A⊲B = G ∨B↓A From this we immediately get: Int(X ∩, Y ∪, A ◦ B) ⊆ Int(X ∩, Y ∪, B), which proves R(A ◦ B) ⊆ R(B), because R(B) = Int(X ∩, Y ∪, B) and R(A ◦ B) = Int(X ∩, Y ∪, A ◦ B).

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 10 / 17

Results: Formal concepts as intervals

Having contexts X, F, A and F, Y , B, formal concepts of X, Y , A ◦ B can be seen as particular intervals in LF.

Theorem

E, G - formal concept from B(X ∩A◦B, Y ∪A◦B, A ◦ B) denote pre(E, G) = {H ∈ LF | H∪A = E, H∩B = G}. Then: pre(E, G) – interval in LF E ∩A . . . least element, G ∪B . . . greatest element

LX LF LY E1 E2 E∩A

1

E∩A

2

G∪B

1

G∪B

2

G1 G2

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 11 / 17

Results: Formal concepts as intervals

Similarly for the other products: Having contexts X, F, A and F, Y , B, formal concepts of X, Y , A ⋆ B can be seen as particular intervals in LF.

LX LF LY E1 E2 E∧A

1

E∧A

2

G∨B

1

G∨B

2

G1 G2 LX LF LY E1 E2 E∩A

1

E∩A

2

G↓B

1

G↓B

2

G2 G1 LX LF LY E1 E2 E↑A

2

E↑A

1

G∨B

2

G∨B

1

G2 G1

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 12 / 17

Results

Definition

V ⊆ Ln is called an i-subspace if – V is closed under ⊗-multiplication – V is closed under -union V ⊆ Ln is called a c-subspace if – V is closed under →-shift – V is closed under -intersection

Theorem

Ci(I) (i-subspace generated by columns of I) = Ext(X ∧, Y ∨, I) Cc(I) (c-subspace generated by columns of I) = Ext(X ↑, Y ↓, I) Ri(I) (i-subspace generated by rows of I) = Int(X ∩, Y ∪, I) Rc(I) (c-subspace generated by rows of I) = Int(X ↑, Y ↓, I)

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 13 / 17

Results: Green’s relations

Definition

For n × n matrices I and J, Green’s relations L, R, and D are defined as follows: – ILJ iff {M ◦ I | M ∈ Ln×n} = {M ◦ J | M ∈ Ln×n}, i.e. I and J generate the same principal left ideal (we say that I and J are L-equivalent). – IRJ iff {I ◦ M | M ∈ Ln×n} = {J ◦ M | M ∈ Ln×n}, i.e. I and J generate the same principal right ideal (we say that I and J are R-equivalent). – IDJ iff there exists matrix K ∈ Ln×n such that ILK and KRJ (D is supremum of L and R).

Theorem

(1) ILJ iff Ri(I) = Ri(J) (iff Int(X ∩, Y ∪, I) = Int(X ∩, Y ∪, J)) (2) IRJ iff Ci(I) = Ci(J) (iff Ext(X ∧, Y ∨, I) = Ext(X ∧, Y ∨, J))

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 14 / 17

Results: Green’s relations

Theorem

h : U → Ln is a homomorphism iff there exists a matrix A such that h(u) = u ◦ A, for every u ∈ U.

Theorem

IDJ iff there exists a bijection h : B(X ∩, Y ∪, I) → B(X ∩, Y ∪, J): (h has two components: hInt : Int(X ∩, Y ∪, I) → Int(X ∩, Y ∪, J), hExt : Int(X ∩, Y ∪, I) → Ext(X ∩, Y ∪, J)) s.t. hInt(a ⊗ B) = a ⊗ hInt(B) hInt(Bi) = hInt( Bi) (thus h is isomorphism).

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 15 / 17

Results: Green’s relations

Similar results for triangular products:

Definition

For n × n matrices I and J, Green’s relations L⊳, R⊲, and D⊳⊲ are defined as follows: – IL⊳J iff {M ⊳ I | M ∈ Ln×n} = {M ⊳ J | M ∈ Ln×n}. – IR⊲J iff {I ⊲ M | M ∈ Ln×n} = {J ⊲ M | M ∈ Ln×n}. – ID⊳⊲J iff there exists matrix K ∈ Ln×n such that IL⊳K and KR⊲J.

Theorem

(1) IL⊳J iff Rc(I) = Rc(J) (2) IR⊲J iff Cc(I) = Cc(J)

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 16 / 17

Conclusions & Further Research

Conclusions: insight to Boolean matrix theory via FCA notions results established for matrices over L in particular: Green’s relation D characterizes isomorphism of concept lattices Further Research unifying framework further areas of Boolean matrix theory THANK YOU

• R. Belohlavek, J. Konecny (UP Olomouc)

Operators and Spaces. . . CLA 2010 17 / 17