Revisiting Pattern Structure Projections

Revisiting Pattern Structure Projections Aleksey Buzmakov1,2 1 LORIA

Sergei O. Kuznetsov2

Amedeo Napoli1

(CNRS – Inria NGE – U. de Lorraine), Vandœuvre-l` es-Nancy, France

2 National

Research University Higher School of Economics, Moscow, Russia

[email protected], [email protected], [email protected]

International Conference on Formal Concept Analysis June 23-26, 2015

Buzmakov et al. (LORIA [France], HSE [Russia])

Revisiting Pattern Structure Projections

ICFCA’15, June 23-26

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Introduction How to Deal with Structured Data? Pattern structures and their projections (Ganter and Kuznetsov 2001) Descriptions instead of sets of attributes Similarity operation instead of set intersection Projections for computational simplification We introduce O-projected Pattern Structures Changing similarity operation: Computational efficiency More freedom in removing irrelevant intents

Solving a formal problem




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FCA and Pattern Structures

Outline

1

FCA and Pattern Structures Basic Definitions Representation Context

2

Projections and O-projections of Pattern Structures Projections O-projections Representation Context and O-projections Order of Projections

3

Conclusion




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FCA to Pattern Structures

Basic Definitions

(Ganter and Kuznetsov 2001)

Table: (G , (℘(M), ∩), δ).

Table: (G , M, I ). g1 g2 g3 g4

m1 x

m2

m3 x

m4 x x

x

x

x

g1 g2 g3 g4

Description {m1 , m4 } {m3 , m4 } {m2 } {m3 , m4 }

Name d1 d2 d3 d2

δ : g1 7→ d1 g2 7→ d2 g3 7→ d3 g4 7→ d2

A0 = {m ∈ M | ∀g ∈ A, (g , m) ∈ I },

A⊆G

B 0 = {g ∈ G | ∀m ∈ M, (g , m) ∈ I },

B⊆M


A0 :=

\

for A ⊆ G

δ(g ),

g ∈A

d 0 := {g ∈ G | d ⊆ δ(g )},


for d ∈ D


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FCA to Pattern Structures

Basic Definitions


Table: (G , (D, u), δ).

Table: (G , M, I ). g1 g2 g3 g4

m1 x

m2

m3 x

m4 x x

x

x

x

g1 g2 g3 g4

Description {m1 , m4 } {m3 , m4 } {m2 } {m3 , m4 }

Name d1 d2 d3 d2

δ : g1 7→ d1 g2 7→ d2 g3 7→ d3 g4 7→ d2

A0 = {m ∈ M | ∀g ∈ A, (g , m) ∈ I },

A⊆G

B 0 = {g ∈ G | ∀m ∈ M, (g , m) ∈ I },

B⊆M

A :=

l

for A ⊆ G

δ(g ),

g ∈A

d := {g ∈ G | d v δ(g )},

for d ∈ D

avb ⇔aub =a Buzmakov et al. (LORIA [France], HSE [Russia])



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Basic Definitions

Pattern Structure (G , (D, u), δ)


G = {g1 , g2 , . . .} Meet-semilattice (D = {⊥, d1 , d2 , · · ·} , u)1 δ : G → D g1

g2

δ

δ d3

d1

d4 d2

⊥

the set Dδ = { 1

d

X | X ⊆ δ(G )} is a complete subsemilattice of (D, u).

D is a set, u is commutative, associative, and idempotent.




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Basic Definitions

Interval Pattern Structures g1 g2 g3

m1 m2 [1, 1] [1, 1] [2, 2] [2, 2] [3, 3] [2, 2]

[1, 1] u [2, 2] = [1, 2] [1, 2] v [1, 1]

Table: An interval context. ({g1 , g2 , g3 } ; h[1, 3]; [1, 2]i) ({g1 , g2 } ; h[1, 2]; [1, 2]i)

({g1 } ; h[1, 1]; [1, 1]i)

({g2 , g3 } ; h[2, 3]; [2, 2]i)

({g2 } ; h[2, 2]; [2, 2]i)

({g3 } ; h[3, 3]; [2, 2]i)

(∅; >)

Figure: The result concept lattice of interval patterns. Buzmakov et al. (LORIA [France], HSE [Russia])



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Representation Context


( Dδ =


)

d ∈ D | (∃X ⊆ G )

d

δ(g ) = d

is a complete subsemilattice of (D, u).

g ∈X

G

X :=

l

{d ∈ Dδ | (∀x ∈ X )x v d} .

t-dense set for (Dδ , u) A set M ⊆ D is t-dense for (Dδ , u) if every element in Dδ is of the form tX for some X ⊆ M. For example, M = Dδ is always t-dense for Dδ . Representation Context Given a pattern structure P = (G , (D, u), δ) and a set M ⊆ D t-dense in Dδ , a formal context (G , M, I ) is called the representation context of P, denoted by R(P), if I is given by I = {(g , m) ∈ G × M | m v δ(g )}. Buzmakov et al. (LORIA [France], HSE [Russia])



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Example I

]i ,1 2]i

∞ ∞ ]; [ − h[−

m2 ≤ 1

x x x

3]; [1,

x x

h[1 ,

x

∞

∞

,2

m2 ≥ 2

,+

]; [

,+ ∞

]; [ 2

−∞

,∞

]i

,+ ∞

∞ ]i ,+ ]; [ ,1

m1 ≤ 2

∞

m1 ≤ 1

h[−

x

x x

∞

m1 ≥ 2

h [−

m1 ≥ 3

h[−

h[2 ,

+∞ ]; [

−∞

−∞

,+

−∞ ,+ +∞ h[3 ,

g1 g2 g3

∞ ]i

∞ ]i

m1 m2 [1, 1] [1, 1] [2, 2] [2, 2] [3, 3] [2, 2]

]; [

g1 g2 g3

]i

Table: IPS Context

x x x

Table: Representation context corresponding to interordinal scaling.




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h[ 1, 1

]; [1 ,1 h[ 3, ]i 3] ;[ 2, h[ 2 1, ]i 2] ;[ 1, h[ 2 2, ]i 3] ;[ 2, h[ 2 1, ]i 3] ;[ 1, 2] i

Example II

g1 g2 g3

m1 m2 [1, 1] [1, 1] [2, 2] [2, 2] [3, 3] [2, 2]

g1 g2 g3

a1 x

a2

x

a3 x x

a4 x x

a5 x x x

Table: Representation Context

Table: IPS Context

({g1 , g2 , g3 } ; h[1, 3]; [1, 2]i) ({g1 , g2 } ; h[1, 2]; [1, 2]i)

({g1 } ; h[1, 1]; [1, 1]i)

({g2 , g3 } ; h[2, 3]; [2, 2]i)

({g2 } ; h[2, 2]; [2, 2]i)

({g3 } ; h[3, 3]; [2, 2]i)

(∅; >)




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h[ 1, 1

]; [1 ,1 h[ 3, ]i 3] ;[ 2, h[ 2 1, ]i 2] ;[ 1, h[ 2 2, ]i 3] ;[ 2, h[ 2 1, ]i 3] ;[ 1, 2] i

Example II

g1 g2 g3

m1 m2 [1, 1] [1, 1] [2, 2] [2, 2] [3, 3] [2, 2]

g1 g2 g3

a1 x

a2

x

a3 x x

a4 x x

a5 x x x

Table: Representation Context

Table: IPS Context

({g1 , g2 , g3 } ; {a5 }) ({g1 , g2 } ; {a3 , a5 })

({g1 } ; {a1 , a3 , a5 })

({g2 , g3 } ; {a4 , a5 })

({g2 } ; {a3 , a4 , a5 })

({g3 } ; {a2 , a4 , a5 })

(∅; {a1 , a2 , a3 , a4 , a5 })




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Projections and O-projections of Pattern Structures

Projections

Outline

1


2


3

Conclusion




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Projections

Projections of Pattern Structures


ψ : D → D is a kernel operator Monotone: x v y ⇒ ψ(x) v ψ(y ) Contractive: ψ(x) v x Idempotent: ψ(ψ(x)) = ψ(x)

ψ ((G , (D, u), δ)) := (G , (D, u), ψ ◦ δ) δ δ g1 g2

d1

ψ

δ

ψ◦δ

ψ◦

d3 ψ

d2

⊥

ψ

d4

ψ

ψ Buzmakov et al. (LORIA [France], HSE [Russia])



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Projections

Incorrect Proposition Proposition 1 (Ganter and Kuznetsov 2001) Given a semilattice (D, u) and a projection ψ, ψ(X u Y ) = ψ(X ) u ψ(Y ). D = {x, y , z, ⊥} Y

X Z

ψ :x 7→ x, y 7→ y , z 7→ ⊥, ⊥ 7→ ⊥

⊥ ψ(x u y ) = ψ(z) = ⊥ = 6 6= z = ψ(x) u ψ(y ) Figure: Contrexample to Proposition 1 from (Ganter and Kuznetsov 2001). Buzmakov et al. (LORIA [France], HSE [Russia])



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O-projections

Outline

1


2


3

Conclusion




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O-projections

O-projections of Pattern Structures ψ : D → D is a kernel operator (monotone, contractive, idempotent) ψ ((G , (D, u), δ)) := (G , (D, u), ψ ◦ δ)

ψ ((G , (D, u), δ)) := (G , (Dψ , uψ ), ψ ◦ δ), where x uψ y = ψ(x u y ).

δ

g1

d1

ψ

δ

ψ◦δ

ψ◦

d3 ψ

d2

⊥

ψ

g1 ψ ◦ δ

δ

g2

d3

d4

ψ

ψ ◦ δ g2

d2 ⊥

ψ Buzmakov et al. (LORIA [France], HSE [Russia])



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O-projections

Example of O-projected Pattern Structures ψ: aggregated size of a pattern is less than 2: ( d if aggregated size of d is less than 2, ψ(d) = ⊥ = h[−∞, +∞]; [−∞, +∞]i otherwise. ({g1 , g2 , g3 } ; h[1, 3]; [1, 2]i) ({g1 , g2 } ; h[1, 2]; [1, 2]i)

({g1 } ; h[1, 1]; [1, 1]i)

({g2 , g3 } ; h[2, 3]; [2, 2]i)

({g2 } ; h[2, 2]; [2, 2]i)

(∅; >)

({g3 } ; h[3, 3]; [2, 2]i)

({g1 , g2 , g3 } ; h[1, 3]; [1, 2]i)

({g1 } ; h[1, 1]; [1, 1]i)

({g2 , g3 } ; h[2, 3]; [2, 2]i)

({g2 } ; h[2, 2]; [2, 2]i)

({g3 } ; h[3, 3]; [2, 2]i)

(∅; >) Buzmakov et al. (LORIA [France], HSE [Russia])



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O-projections

Original Propositions are Valid

Proposition 1 Given a pattern structure P = (G , (D, u), δ) and a kernel operator ψ on D: 1

if A is an extent in ψ(P), then A is also an extent in P.

2

if d is an intent in P, then ψ(d) is also an intent in ψ(P).




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Representation Context and O-projections

Outline

1


2


3

Conclusion




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Representation Context And Projections (Ganter and Kuznetsov 2001; Kaiser and Schmidt 2011)

Projections corresponds to removal of attributes (Th.2 in Ganter and Kuznetsov 2001). ψ(h[1, 1]; [1, 1]i) = h[1, 2]; [1, 2]i

1, 1] ;

({g2 , g3 } ; h[2, 3]; [2, 2]i)

h[

({g1 , g2 } ; h[1, 2]; [1, 2]i)

({g1 } ; h[1, 1]; [1, 1]i)

({g2 } ; h[2, 2]; [2, 2]i)

({g3 } ; h[3, 3]; [2, 2]i)

(∅; >)



[1 ,1 h[ 3, ]i 3] ;[ 2, h[ 2] 1, i 2] ;[ 1, h[ 2] 2, i 3] ;[ 2, h[ 2 1, ]i 3] ;[ 1, 2] i

({g1 , g2 , g3 } ; h[1, 3]; [1, 2]i)

g1 g2 g3

a1 x

a2

x

a3 x x

a4 x x


a5 x x x

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Problem for O-projections g1

{a, b}

{a, c}

g2

g3

g1 g2 g3

{b, c}

a b c x x x x x x

Representation context of P. {a}

{b}

{c}

g1 g2 g3

⊥

A semilattice D and its projection ψ.

ab ac b c x x x x x x

Representation context of ψ(P).

Figure: An example of a projection that can increase the number of attributes. Buzmakov et al. (LORIA [France], HSE [Russia])



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Simplicity Relations between Formal Contexts

Simplicity Relation Given two contexts K1 = (G , M1 , I1 ) and K2 = (G , M2 , I2 ), K1 is said to be simpler than K2 , denoted by K1 ≤S K2 , if for any m1,i ∈ M1 there is a set B2 ⊆ M2 such that ({m1,i })1 = (B2 )2 . Here by (·)1 and (·)2 we denote the derivation operators in the contexts K1 and K2 , respectively.

Generalization of Closed Relation: Def. 50 (Ganter and Wille 1999) A binary relation J ⊆ I is called a closed relation of the context (G , M, I ) if every concept of the context (G , M, J) is also a concept of (G , M, I ).




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Theorem 5 Given a pattern structure P = (G , (D, u), δ) such that (D, u) is a complete semilattice the following holds: 1

for any projection ψ of D we have R(ψ(P)) ≤S R(P).

2

for any context K = (G , M, I ) such that K ≤S R(P), there is a projection ψ of D such that K is a representation context of ψ(P).




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Order of Projections

Outline

1


2


3

Conclusion




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Projection Order Projections Ψ of a semilattice (D, u) can be equivalently ordered by: Point-wise Order ψ1 ≤ ψ2 if for all x ∈ D, ψ1 (x) v ψ2 (x). Fixed-set Order ψ1 ≤ ψ2 if fixed-sets are included, ψ1 (D) ⊆ ψ2 (D). Superposition Order ψ1 ≤ ψ2 if there is a projection ψ : ψ2 (D) → ψ2 (D) such that ψ1 = ψ ◦ ψ2 . Projection Order If (D, u) is a complete semilatice, then the poset (Ψ, ≤) is a semilattice.




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Projection Order Visualization d3

d4

ψ2

ψ1

d1

d2

ψ2 ψ 1,

ψ1 , ψ2

ψ1 ,

⊥

ψ2 ψ1 , ψ2




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Conclusion

Conclusion O-projections of pattern structures are generalization of projections: Correct a formal problem of projections Allow changing the semilattice of descriptions.

O-projections relate the corresponding representation context by the simplicity relation between contexts. O-projections can be ordered by means of Point-wise comparison Fixed-sets inclusion Superposition.

This order forms a lattice for a complete semilattice of descriptions. An important direction of future work is transformations ξ : D → D1 of pattern structures. Could we explain kernels of Support Vector Machines be means of a transformation? Buzmakov et al. (LORIA [France], HSE [Russia])



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Conclusion

Revisiting Pattern Structure Projections Aleksey Buzmakov1,2 1 LORIA

Sergei O. Kuznetsov2

Amedeo Napoli1

(CNRS – Inria NGE – U. de Lorraine), Vandœuvre-l` es-Nancy, France

2 National

Research University Higher School of Economics, Moscow, Russia

[email protected], [email protected], [email protected]

International Conference on Formal Concept Analysis June 23-26, 2015




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Bibliography

Ganter, Bernhard and Sergei O. Kuznetsov (2001). “Pattern Structures and Their Projections”. In: Concept. Struct. Broadening Base. Ed. by Harry S. Delugach and Gerd Stumme. Vol. 2120. Lecture Notes in Computer Science. Springer Berlin Heidelberg, pp. 129–142. Ganter, Bernhard and Rudolf Wille (1999). Formal Concept Analysis: Mathematical Foundations. 1st. Springer, pp. I–X, 1–284. Kaiser, Tim B. and Stefan E. Schmidt (2011). “Some Remarks on the Relation between Annotated Ordered Sets and Pattern Structures”. In: Pattern Recognit. Mach. Intell. SE - 9. Ed. by Sergei O. Kuznetsov et al. Vol. 6744. Lecture Notes in Computer Science x. Springer Berlin Heidelberg, pp. 43–48.




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How to Deal with a Partially Ordered Set? hi hai

hbi

habi

hbai

{habai} u {hbabi} = {habi , hbai} ⇓ ⇓ ( ) ( ) ( ⇑ ) aba, ab, ab, ba, T bab, ab, = ba, a, b, ∅ ba, a, b, ∅ a, b, ∅ The corresponding set can be rather long Indirect computation can be more efficient (memory and computational costs)

habai

hbabi




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Revisiting Pattern Structure Projections

Revisiting Pattern Structure Projections

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