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

SLIDE 1

Towards concept analysis in categories: limit inferior as algebra, limit superior as coalgebra

Toshiki Kataoka

The University of Tokyo, JSPS Research Fellow

Dusko Pavlovic

University of Hawaii at Manoa

&

B ⇑B (⇑B)

Φ

A ⇓A (⇓A)

Φ

∆ ∆Φ

Φ

Φ∗

Φ

Φ

Φ

Φ∗

Φ

SLIDE 2

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Overview

Concept analysis
Concept analysis in categories
Dedekind–MacNeille completion
Generalizations of Dedekind–MacNeille completion
Bicompletions of categories

2

new problem new answer

SLIDE 3

Concept Analysis

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

SLIDE 4

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Example: Text Analysis

4

Corpus

This is a pen. Is that a pen? …

Proximity among words

pen apple banana pencil this

Co-occurrence

pen banana a this eat pencil delicious write draw

SLIDE 5

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Concept Analysis

Given data in a matrix
Extract information as vectors

5

between two

P Q R S Alice ✔ ✔ ✔ Bob ✔ ✔ Carol ✔ Dan ✔ Eve ✔

either side A B C D E P Q R S

SLIDE 6

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Preordered

Formal Concept Analysis

[Wille, 1982]

6

<{A,B,C,D,E}, {}> <{}, {P,Q,R,S}> <{A,B}, {P,Q}> <{A}, {P,Q,R}> <{C,D,E}, {S}>

P Q R S Alice ✔ ✔ ✔ Bob ✔ ✔ Carol ✔ Dan ✔ Eve ✔

Concept lattice

SLIDE 7

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Linear algebraic

P Q R S Alice

★★ ★★★ ★★★★

Bob

★★ ★★ ★★★

Carol

★★ ★★★

Dan

★★★★

Eve

★

Latent Semantic Analysis

(Principal Component Analysis)

7

M = 1 5       2 3 4 2 5 5 4 1      

†

singular vectors M = UΣV † where U =     

1 √ 6 2 √ 6

− 1

√ 5 1 √ 6 2 √ 5

1      Σ =   

√ 48 5 √ 42 5 √ 10 5

   V =       

1 √ 2 1 √ 2 1 √ 2

− 1

√ 2 1 √ 42 4 √ 42 1 √ 42

       singular value

[ 0 : 0 : 0 : 1 : 1 ] [ 1 : 2 : 1 : 0 ]

SLIDE 8

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Towards Unification

Fixed points (definition)
Completeness (theorem)
Minimality (theorem)

8

SLIDE 9

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

9

A B C D E P Q R S

Factorizations of Relations

minimal complete lattice that factorize the relation

SLIDE 10

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Singular Value Decomposition

Compact SVD: M = UΣV†
M: (given) real m×n matrix, rank r
U: m×r matrix, U†U = Ir
V: n×r matrix, V†V = Ir
Σ: diagonal r×r matrix
with positive reals on the diagonal

10

SLIDE 11

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Factorizations of Real Matrices

11

{P, Q, R, S} {X, Y, Z} R4 R3 R5 R3 {A, B, C, D, E} {X, Y, Z}



     

1 √ 6 2 √ 6

− 1

√ 5 1 √ 6 2 √ 5

1        † ∼ = ∼ =



        

2 5 3 5 4 5 2 5 5 5 5 5 4 5 1 5

         



        

1 √ 2 1 √ 2 1 √ 2

− 1

√ 2 1 √ 42 4 √ 42 1 √ 42

         



  

√ 48 5 √ 42 5 √ 10 5

   

least dimension to factorize

SLIDE 12

Concept Analysis in Categories

SLIDE 13

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Components and Functionalities

13

Structural components Functional modules …

SLIDE 14

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Engineering

Functionalities: top-down
Components: bottom-up
Relation:

specification

14

image by Herman Bruyninckx https://commons.wikimedia.org/ wiki/File:V-model-en.png (under GFDL)

V-model

SLIDE 15

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Reverse-Engineering

Want to know

15

Can be observed

Components Functionalities Components Functionalities

SLIDE 16

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Reverse-Engineering

Want to know

16

Can be observed

Components Functionalities Components Functionalities

SLIDE 17

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Actions on Relations

From super-components
From sub-functionalities

17

Φ: Aop × B → Set … … … …

SLIDE 18

Dedekind–MacNeille Completion

SLIDE 19

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Dedekind Cuts [Dedekind, 1872]

19

L U R = {L, U | Q = L U, L, U = , l L. u U. l u}/ where {< q}, { q} { q}, {> q} (q Q)

SLIDE 20

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Dedekind–MacNeille Completion [MacNeille, 1937]

20

Generalizes Dedekind cuts
(P, ≤): poset

Q = {, Q} {{ r} Q, { r} Q | r R} {Q, }

= {} R {}

P = {L, U | L P, U P, L = {l P | u U. l u}, U = {u P | l L. l u}} L, U L, U L L U U

lower bounds upper bounds

SLIDE 21

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

P

An Example

21

P (Hasse diagrams) P = {L, U | L is lower bounds of U, U is upper bounds of L}

SLIDE 22

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Categorically

22

(P, ≤): poset,
(↓P, ⊆) (resp. (↑P, ⊇)):

the family of lower (resp. upper) sets

P: fixed point of

the Galois connection P P P P

∆

SLIDE 23

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Properties

23

sup (and inf) existing in P

P is inf- and sup-complete. (Complete lattice)
P → P is an inf- and sup-dense embedding.
Thus, sup- and inf-preserving
The minimal bicompletion

SLIDE 24

Generalizations of Dedekind–MacNeille Completion

SLIDE 25

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Completions of Relations

Φ ⊆ P × Q: relation s.t. x’ ≤ x, xΦy, y ≤ y’ ⇒ x’Φy’

25

Q Q Φ P P

∆ Φ∗

Φ
Φ∗
P

P P P

∆

SLIDE 26

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Notations

26

C: V-enriched category,
C = [Cop, V]: category of presheaves,
C = [C, V]op: category of postsheaves,
: C C, ∆: C C: Yoneda embeddings,
A B denotes Aop B V.

SLIDE 27

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Isbell Completion

27

C C C C

H∗ ∆

H∗
Isbell duality
V = {0,1} (V-category = poset)
C: Dedekind–MacNeille

completion

V = [0,∞] (V-category =

Lawvere metric space)

C: (directed) tight span

[Willerton, 2013]

SLIDE 28

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

28

Q Q Φ P P

∆ Φ∗

Φ
Φ∗
C

C C C

H∗ ∆

H∗
B

B Φ A A

∆ Φ∗

Φ
Φ∗
P

P P P

∆

If V = [0,∞]

SLIDE 29

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Quantitative Concept Analysis [Pavlovic, 2012]

Lawvere metrized

29

P Q R S Alice

★★ ★★★ ★★★★

Bob

★★ ★★ ★★★

Carol

★★ ★★★

Dan

★★★★

Eve

★

l5 l3 l5 l4 l3 l3 l5 l2 l3 l2 l4 l4 l5 l1 l4 l5

A B C D E P Q R S

A B A B C D E = = P Q R S P Q R

d(A, P) = l2, d(A, Q) = l3, . . . where 0 ≤ l5 ≤ · · · ≤ l1 < l0 = ∞

SLIDE 30

Bicompletions

f Categories

SLIDE 31

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Question by Lambek (1966)

Does there exist
an inf- and sup-dense embedding to
an inf- and sup-complete category?

31

SLIDE 32

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Answer by Isbell (1968)

Any inf- and sup-dense embedding of

the group Z4 is not inf-complete (nor sup-complete)

32

SLIDE 33

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

33

P P L

Two Universalities

Cannot be a single self-dual universality

SLIDE 34

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

34

Q Q Φ P P

∆ Φ∗

Φ
Φ∗
C

C C C

H∗ ∆

H∗
B

B Φ A A

∆ Φ∗

Φ
Φ∗
✘

✘

P P P P

∆

Trouble with V = Set

SLIDE 35

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

35

Q Q Φ P P

∆ Φ∗

Φ
Φ∗
P

P P P

∆

B

⇑B (⇑B)

Φ

A ⇓A (⇓A)

Φ

∆ ∆Φ

Φ

Φ∗

Φ

Φ

Φ

Φ∗

Φ

⇑C (⇑C)

H

C ⇓C (⇓C)

H
H

H∗ ∆

H

H∗

C

Our Proposal

SLIDE 36

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

lim

− → F = lim − → ← − F (F : J → C, ← − F : Cop → Set) C CJ C ⇓C CJ ← − F F

lim
lim
Supremum along

Discrete Fibration

36

diagonal functor Yoneda embedding

SLIDE 37

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Limit inferior

The supremum of lower bounds

37

− → lim F := lim − → H∗− → F C CJ C ⇓C ⇑C CJ H− → F − → F F

lim
lim
H∗

SLIDE 38

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

38

Theorems

(⇓C)

← − H is the free −

→ lim-completion of C.

C → (⇓C)

← − H is lim

← −- and lim − →-preserving.

SLIDE 39

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Example: Completing Groups

39

(G-Set)op {1} ∪ {I · G | I ∈ Set} G Gop-Set ({1} ∪ {G · I | I ∈ Set})op

H

H∗ ∆

H

H∗

SLIDE 40

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Example: Completing Posets

40

P P (P)

H

P P P P (P)

H
H

H∗

∆
H

H∗

SLIDE 41

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

Summary

Algebras, instead of fixed points
Factorization (future work)
Add limits inferior
Not always limits superior
Minimality (future work)

41

B ⇑B (⇑B)

Φ

A ⇓A (⇓A)

Φ

∆ ∆Φ

Φ

Φ∗

Φ

Φ

Φ

Φ∗

Φ

SLIDE 42

Toshiki Kataoka (UTokyo, JSPS) Limit superior and limit inferior as algebras

References

Isbell, John R. "Small subcategories and completeness."

Theory of Computing Systems 2.1 (1968): 27-50.

Lambek, Joachim. "Completions of Categories: Seminar

lectures given 1966 in Zürich." Springer Berlin Heidelberg, 1966.

Pavlovic, Dusko. "Quantitative concept analysis." Formal

Concept Analysis. Springer Berlin Heidelberg, 2012. 260-277.

Wille, Rudolf. "Restructuring Lattice Theory: An Approach

Based on Hierarchies of Concepts." Ordered Sets. Springer Netherlands, 1982. 445-470.

42