Towards concept analysis in categories: ( B ) B B limit - PowerPoint PPT Presentation

∆ Φ � � Φ Towards concept analysis � � ∆ in categories: Φ ( ⇑ B ) B ⇑ B limit inferior as algebra, � Φ Φ limit superior as coalgebra Φ ∗ � Φ ∗ � � � � Φ ( ⇓ A ) A ⇓ A Toshiki Kataoka Dusko Pavlovic & � The University of Tokyo, University of Hawaii at Manoa JSPS Research Fellow � � Φ � Φ

Overview • Concept analysis • Concept analysis in categories new problem • Dedekind–MacNeille completion • Generalizations of Dedekind–MacNeille completion • Bicompletions of categories new answer 2 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Concept Analysis (or, knowledge acquisition, semantic indexing, data mining)

Example: Text Analysis Corpus Co-occurrence This is a pen. Is that a pen? … write draw pen this pencil a banana Proximity among words eat delicious pen pencil this banana apple 4 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Concept Analysis • Given data in a matrix between two • Extract information as vectors either side P Q R S ✔ ✔ ✔ Alice ✔ ✔ Bob P Q R S ✔ Carol ✔ Dan A B C D E ✔ Eve 5 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Formal Concept Analysis [Wille, 1982] • Preordered Concept lattice P Q R S <{A,B,C,D,E}, {}> ✔ ✔ ✔ Alice ✔ ✔ Bob <{A,B}, {P,Q}> ✔ Carol <{C,D,E}, ✔ Dan {S}> ✔ Eve <{A}, {P,Q,R}> <{}, {P,Q,R,S}> 6 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

★★ ★ ★★★★ ★★★ ★★ ★★★ ★★★★ ★★★ Latent Semantic Analysis (Principal Component Analysis) • Linear algebraic M = U Σ V † where [ 1 : 2 : 1 : 0 ] P Q R S  1  0 0 √ 6 [ 0 : 0 : 0 : 1 : 1 ] 2 − 1 0   Alice √ √ 6 5 U =   1 2   0 √ √  6 5  ★★ Bob 0 1 0 singular ★★   √ vectors Carol 48 0 0 5 √ 42 Σ =   0 0 Dan 5   √ 10 0 0 5 Eve singular 1 1  0  √ √ 2 2 value 1 − 1 † 0     2 3 4 0 √ √ 2 2   1 0 0 V =   2 5 0 0 √   42 M = 1   4     0 0 0 0 0 5   √   42 5   1 0 0 0 4 0 0   √ 42 0 0 0 1 7 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Towards Unification • Fixed points (definition) • Completeness (theorem) • Minimality (theorem) 8 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Factorizations of Relations minimal P Q R S complete lattice that factorize the relation A B C D E 9 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Singular Value Decomposition • Compact SVD: M = U Σ V † M : (given) real m × n matrix, rank r • U : m × r matrix, U † U = I r • V : n × r matrix, V † V = I r • Σ : diagonal r × r matrix • with positive reals on the diagonal • 10 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Factorizations of Real Matrices †   1 0 0 √ 6   2 − 1 0   √ √ 6 5     least 1 2 0   √ √ 6 5   dimension 0 1 0 { P, Q, R, S } { X, Y, Z } � to factorize   2 3 4 R 4 R 3 0 5 5 5    √  48 2 5 0 0 0 0   5 5 5      √  42 5 0 0 0 ∼ = ∼ = 0 0     5 5     √   4 10 0 0 0 0 0   5 5   R 5 R 3 1 0 0 0 5 � � { A, B, C, D, E } { X, Y, Z } �   1 1 0 √ √ 2 2   1 − 1 0   √ √ 2 2     1 0 0   √ 42     4 0 0   √ 42   1 0 0 √ 42 11 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Concept Analysis in Categories

Components and Functionalities Functional modules Structural components … 13 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Engineering • Functionalities: top-down • Components: bottom-up V-model • Relation: specification image by Herman Bruyninckx https://commons.wikimedia.org/ wiki/File:V-model-en.png (under GFDL) 14 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Reverse-Engineering • Want to know • Can be observed Components Functionalities Components Functionalities 15 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Reverse-Engineering • Want to know • Can be observed Components Functionalities Components Functionalities 16 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Actions on Relations … … … … • From super-components Φ : A op × B → Set • From sub-functionalities 17 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Dedekind–MacNeille Completion

Dedekind Cuts [Dedekind, 1872] R = { � L, U � | Q = L � U, L, U � = � , � l � L. � u � U. l � u } / � where � { < q } , { � q } � � � { � q } , { > q } � ( q � Q ) L U 19 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Dedekind–MacNeille Completion [MacNeille, 1937] • ( P , ≤ ): poset lower � P = { � L, U � | L � P, U � P, bounds L = { l � P | � u � U. l � u } , U = { u � P | � l � L. l � u }} upper � L � L � � bounds � L, U � � � L � , U � � � � U � U � • Generalizes Dedekind cuts � Q = { �� , Q � } � { � { � r } � Q , { � r } � Q � | r � R } � { � Q , �� } � = { �� } � R � { � } 20 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

An Example � P = { � L, U � | L is lower bounds of U, U is upper bounds of L } � P P (Hasse diagrams) 21 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Categorically • ( P , ≤ ): poset, • ( ↓ P , ⊆ ) (resp. ( ↑ P , ⊇ )): the family of lower (resp. upper) sets • � P : fixed point of � P the Galois connection ∆ � P P � � � P 22 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Properties • � P is inf- and sup-complete. (Complete lattice) • P → � P is an inf- and sup-dense embedding. • Thus, sup- and inf-preserving sup (and inf) existing in P • The minimal bicompletion 23 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Generalizations of Dedekind–MacNeille Completion

Completions of Relations • Φ ⊆ P × Q : relation s.t. x’ ≤ x , x Φ y , y ≤ y’ ⇒ x’ Φ y’ ∆ � Q Q � Φ Φ ∗ Φ Φ ∗ � � � P P � P � ∆ � P P � � � P 25 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Notations • C : V -enriched category, • � C = [ C op , V ] : category of presheaves, • � C = [ C , V ] op : category of postsheaves, • � : C � � C , ∆ : C � � C : Yoneda embeddings, • A � B denotes A op � B � V . 26 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Isbell Completion • V = {0,1} ( V -category = poset) • � C : Dedekind–MacNeille � C completion ∆ • V = [0, ∞ ] ( V -category = � C C H ∗ H ∗ � Lawvere metric space) � � C • � C : (directed) tight span Isbell duality [Willerton, 2013] 27 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

If V = [0, ∞ ] ∆ � B B � C ∆ � Φ � C C Φ H ∗ Φ ∗ H ∗ � Φ ∗ � � � C � � A A � � P ∆ � Q Q ∆ � P � Φ P Φ ∗ Φ � Φ ∗ � � � � P P � P � 28 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

★★★★ ★★ ★★★ ★ ★★ ★★★★ ★★★ ★★★ Quantitative Concept Analysis [Pavlovic, 2012] • Lawvere metrized A � B � C � D � E = � P Q R S B l 3 � l 5 E A � B l 5 Q Alice ★★ l 3 Bob l 3 � l 5 l 1 � l 4 ★★ Carol l 2 � l 3 Dan P D l 4 � l 5 Eve C l 5 S l 2 � l 4 l 4 P � Q � R R A d ( A, P ) = l 2 , d ( A, Q ) = l 3 , . . . where 0 ≤ l 5 ≤ · · · ≤ l 1 < l 0 = ∞ � = P � Q � R � S 29 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Bicompletions of Categories

Question by Lambek (1966) • Does there exist • an inf- and sup-dense embedding to • an inf- and sup-complete category? 31 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Answer by Isbell (1968) • Any inf- and sup-dense embedding of the group Z 4 is not inf-complete (nor sup-complete) 32 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)

Two Universalities • Cannot be a single self-dual universality � P P L 33 Limit superior and limit inferior as algebras Toshiki Kataoka (UTokyo, JSPS)