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 = max l =1 ... k A il · B lj . 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 operators. Theorem directly follows from properties of the concept-forming operators. 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     1 1 0 0 0 1 0 0   1 1 0 0 0 1 1 0 0 0 1 0 0     I =  =  ◦ 0 0 1 1 0       1 1 1 1 0 1 1 0   1 0 0 0 1 1 0 0 0 1 0 0 1 decompose I to A and B where I ij = ( A ◦ B ) ij = max l =1 ... k A il · B lj . 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 A F and B F we have I = A F ◦ B F . where for F = {� E 1 , G 1 � , � E 2 , G 2 � , . . . , � E |F| , G |F| �} A F and B F are defined as follows: ( A F ) il = E l ( i ) ( B F ) lj = G l ( 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¨ odel, 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” L U 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 A il ⊗ B lj ( A ⊳ B ) ij = � k l =1 A il → B lj ( A ⊲ B ) ij = � k l =1 B lj → A il 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 ) = � G ↓ ( x ) = � x ∈ X E ( x ) → I ( x , y ) y ∈ Y G ( y ) → I ( x , y ) Isotone: E ∩ ( y ) = � G ∪ ( x ) = � x ∈ X E ( x ) ⊗ I ( x , y ) y ∈ Y I ( x , y ) → G ( y ) E ∧ ( y ) = � G ∨ ( x ) = � x ∈ X I ( x , y ) → E ( x ) y ∈ Y G ( y ) ⊗ I ( x , y ) ∧ I . in binary case, they they are mutually definable: E ↑ I = E ∩ I = E ↑ 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 operators of A and B : Theorem Let A and B be m × k, and k × n matrices, let E ∈ L X , G ∈ L Y . 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 L F . Theorem � E , G � - formal concept from B ( X ∩ A ◦ B , Y ∪ A ◦ B , A ◦ B ) denote pre ( E , G ) = { H ∈ L F | H ∪ A = E , H ∩ B = G } . Then: pre ( E , G ) – interval in L F E ∩ A . . . least element, G ∪ B . . . greatest element L X L F L Y G ∪ B E 1 G 1 1 E ∩ A G ∪ B 1 E 2 G 2 2 E ∩ A 2 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 L F . L X L F L Y E ∧ A E 1 G 1 1 G ∨ B 1 E ∧ A E 2 G 2 2 G ∨ B 2 L X L F L Y L X L F L Y G ↓ B E ↑ A E 1 G 2 E 1 G 2 1 2 E ∩ A G ∨ B G ↓ B 1 2 E ↑ A E 2 G 1 E 2 G 1 2 1 E ∩ A G ∨ B 2 1 R. Belohlavek, J. Konecny (UP Olomouc) Operators and Spaces. . . CLA 2010 12 / 17

Results Definition V ⊆ L n is called an i-subspace if – V is closed under ⊗ -multiplication – V is closed under � -union V ⊆ L n is called a c-subspace if – V is closed under → -shift – V is closed under � -intersection Theorem C i ( I ) (i-subspace generated by columns of I) = Ext ( X ∧ , Y ∨ , I ) C c ( I ) (c-subspace generated by columns of I) = Ext ( X ↑ , Y ↓ , I ) R i ( I ) (i-subspace generated by rows of I) = Int ( X ∩ , Y ∪ , I ) R c ( 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: – I L J iff { M ◦ I | M ∈ L n × n } = { M ◦ J | M ∈ L n × n } , i.e. I and J generate the same principal left ideal (we say that I and J are L -equivalent). – I R J iff { I ◦ M | M ∈ L n × n } = { J ◦ M | M ∈ L n × n } , i.e. I and J generate the same principal right ideal (we say that I and J are R -equivalent). – I D J iff there exists matrix K ∈ L n × n such that I L K and K R J ( D is supremum of L and R ). Theorem (1) I L J iff R i ( I ) = R i ( J ) (iff Int ( X ∩ , Y ∪ , I ) = Int ( X ∩ , Y ∪ , J ) ) (2) I R J iff C i ( I ) = C i ( J ) (iff Ext ( X ∧ , Y ∨ , I ) = Ext ( X ∧ , Y ∨ , J ) ) R. Belohlavek, J. Konecny (UP Olomouc) Operators and Spaces. . . CLA 2010 14 / 17