Continuing the tale of Abby and doughnuts from part 1, this presentation shifts terminology and notation into formal concept analysis, and introduces lattice ideals and filters that allow us to better describe what's going on with collaborative filtering.

1. Mathematics
of
Incidence
part 2: formal concepts and formal concept lattices
!
Benjamin J. Keller
bjkeller.github.io
!
v.1, 3 October 2014
Creative Commons Attribution-ShareAlike 4.0 International License
bananas
apples
Abby
cherries
Charles
Brian

2. Recall: Collaborative Filtering
Use likes of users to recommend foods to Abby
Abby
Brian
Charles
David
cherries
doughnuts
eggs
apples
bananas
Recommend these three foods to Abby because
she likes food in common with users who like them

3. Recall: Likes represented as bipartite graph
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
bipartite graph (U,V,E) has vertices in disjoint sets U and V with edges (u,v) in E
from vertex u in U to vertex v in V

4. Recall: Biclique
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
A biclique (U,V,E) of a bipartite graph G is a subgraph of G
such that each u in U has an edge (u,v) with each v in V

5. Recall: Recommendations via bicliques
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Construct same recommendation
by composing bicliques
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby Charles
David
cherries
doughnuts
eggs
apples
bananas
Brian

6. Shift perspectives: Formal Concept Analysis
• Starts with Formal Context ⟨ G,M,I ⟩
• Set of objects G
• Set of attributes M
• Incidence relation I ⊆GxM
• Derive formal concepts (basically bicliques) from
incidence relation
• Constructs Formal Concept Lattice

7. "Likes" Formal Context
G = { Abby, Brian, Charles, David }
M = { apples, bananas, cherries, doughnuts, eggs }
I = {
(Abby, apples), (Abby, bananas),
(Brian, apples),(Brian, bananas),
(Brian, cherries), (Brian, doughnuts),
(Charles, apples), (Charles, bananas),
(Charles, cherries), (Charles, doughnuts),
(Charles, eggs), (David, bananas),
(David, doughnuts), (David, eggs)
}

8. Representing incidence relation
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Can represent incidence
relation as a bipartite graph,
but commonly represented as
a cross table
apples bananas cherries doughnuts eggs
Abby X X
Brian X X X X
Charles X X X X X
David X X X

9. Definition: Partially ordered set
(P,)is a partially ordered set (poset) when the order
satisfies the properties
if p  q and q  p then p = q 8p, q 2 P
(reflexivity)
(antisymmetry)
if p  q and q  r then p  r 8p, q, r 2 P
(transitivity)
p  p 8p 2 P

10. Definition: Lattice
For a poset (L,) and S ✓ L
W
S = p
the least upper bound of is
p 2 L 8s 2 S, s  p
where and
S
V
S = q
the greatest lower bound of is
q 2 L 8s 2 S, q  s
where and
S
L
exist for any set S ✓ L
V
S
W
If S and
then is a (complete) lattice

11. (Boolean) Lattice of powerset
• The powerset of G is set of
subsets of G
• Ordered by inclusion is a
poset
• And, is a lattice:
• Set union as least upper
bound
• Set intersection as greatest
lower bound
Abby,
Brian,
Charles,
David P(G)
Abby Brian Charles David
∅
Abby,
Brian
Abby,
Charles
Abby,
David
Brian,
Charles
Brian,
David
Charles,
David
Abby,
Brian,
Charles
Abby,
Brian,
David
Brian,
Charles,
David
Abby,
Charles,
David

12. Incidence relates powersets
(P(G),✓) (P(M),◆)
Abby,
Brian,
Charles,
David
∅
∅
Abby Brian Charles David
Abby,
Brian
Abby,
Charles
Abby,
David
Brian,
Charles
Brian,
David
Charles,
David
Abby,
Brian,
Charles
Abby,
Brian,
David
Brian,
Charles,
David
Abby,
Charles,
David
apples bananas cherries doughnuts eggs
apples,
bananas
apples,
cherries
apples,
doughnuts
apples,
eggs
bananas,
cherries
bananas,
doughnuts
bananas,
eggs
cherries,
doughnuts
cherries,
eggs
doughnuts,
eggs
apples,
bananas,
cherries
apples,
bananas,
eggs
apples,
bananas,
doughnuts
apples,
cherries,
doughnuts
apples,
cherries,
eggs
apples,
doughnuts,
eggs
bananas,
cherries,
doughnuts
bananas,
cherries,
eggs
bananas,
doughnuts,
eggs
cherries,
doughnuts,
eggs
apples,
bananas,
cherries,
doughnuts
apples,
bananas,
cherries,
eggs
apples,
bananas,
doughnuts,
eggs
apples,
cherries,
doughnuts,
eggs
bananas,
cherries,
doughnuts,
eggs
apples,
bananas,
cherries,
doughnuts,
eggs
the dual lattice (order is flipped),
so “upside-down”

13. Deriving functions from incidence
: P(G)!P(M)
(A) = {b 2 M | (a, b) 2 I}
for A ✓ G
: P(M)!P(G)
(B) = {a 2 G| (a, b) 2 I}
for B ✓ M

14. Incidence relates powersets
Abby,
Brian,
Charles,
David
∅
∅
Abby Brian Charles David
Abby,
Brian
Abby,
Charles
Abby,
David
Brian,
Charles
Brian,
David
Charles,
David
Abby,
Brian,
Charles
Abby,
Brian,
David
Brian,
Charles,
David
Abby,
Charles,
David
apples bananas cherries doughnuts eggs
apples,
bananas
apples,
cherries
apples,
doughnuts
apples,
eggs
bananas,
cherries
bananas,
doughnuts
bananas,
eggs
cherries,
doughnuts
cherries,
eggs
doughnuts,
eggs
apples,
bananas,
cherries
apples,
bananas,
eggs
apples,
bananas,
doughnuts
apples,
cherries,
doughnuts
apples,
cherries,
eggs
apples,
doughnuts,
eggs
bananas,
cherries,
doughnuts
bananas,
cherries,
eggs
bananas,
doughnuts,
eggs
cherries,
doughnuts,
eggs
apples,
bananas,
cherries,
doughnuts
apples,
bananas,
cherries,
eggs
apples,
bananas,
doughnuts,
eggs
apples,
cherries,
doughnuts,
eggs
bananas,
cherries,
doughnuts,
eggs
apples,
bananas,
cherries,
doughnuts,
eggs

15. Functions relate powersets
Abby,
Brian,
Charles
Abby
Abby,
Brian
Abby,
Charles
apples
apples,
bananas
({Abby}) = {apples, bananas}
({Abby, Brian}) = {apples, bananas}
({Abby,Charles}) = {apples, bananas}
({Abby, Brian,Charles}) = {apples, bananas}
({apples}) = {Abby, Brian,Charles}
({apples, bananas}) = {Abby, Brian,Charles}
Functions converge on largest set in each class of sets

16. Incidence relates power sets through functions
Abby,
Brian,
Charles,
David
∅
∅
Abby Brian Charles David
Abby,
Brian
Abby,
Charles
Abby,
David
Brian,
Charles
Brian,
David
Charles,
David
Abby,
Brian,
Charles
Abby,
Brian,
David
Brian,
Charles,
David
Abby,
Charles,
David
apples bananas cherries doughnuts eggs
apples,
bananas
apples,
cherries
apples,
doughnuts
apples,
eggs
bananas,
cherries
bananas,
doughnuts
bananas,
eggs
cherries,
doughnuts
cherries,
eggs
doughnuts,
eggs
apples,
bananas,
cherries
apples,
bananas,
eggs
apples,
bananas,
doughnuts
apples,
cherries,
doughnuts
apples,
cherries,
eggs
apples,
doughnuts,
eggs
bananas,
cherries,
doughnuts
bananas,
cherries,
eggs
bananas,
doughnuts,
eggs
cherries,
doughnuts,
eggs
apples,
bananas,
cherries,
doughnuts
apples,
bananas,
cherries,
eggs
apples,
bananas,
doughnuts,
eggs
apples,
cherries,
doughnuts,
eggs
bananas,
cherries,
doughnuts,
eggs
apples,
bananas,
cherries,
doughnuts,
eggs

17. Functions give us closure operators
!( ({apples})) = {apples, bananas}
!( ({apples, bananas})) = {apples, bananas}
(({Abby})) = {Abby, Brian,Charles}
(({Abby, Brian,Charles})) = {Abby, Brian,Charles}
Functions converge on largest set in each class of sets

18. Formal concept
(A,B)
where A ✓ G and B ✓ M
satisfying
!(A) = B
(B) = A
For formal concept (A,B)
A = ↵(A,B)
B = (A,B)
is the extent of the concept
is the intent of the concept

19. Formal concept construction
Like for biclique construction use closure
For arbitrary A ✓ G and B ✓ M define
!A = ( (#(A)),#(A))
μB = ( (B), (#(B)))
(I will just drop set notation for singleton sets)

20. Subconcepts
Subconcept order defined as for bicliques:
(A1,B1) (A2,B2)
whenever
A1 ✓ A2
or, equivalently,
B1 ◆ B2

21. Concept Lattice
({Abby,Brian,Charles,David},{bananas})
({Abby,Brian,Charles},{apples,bananas}) ({Brian, Charles, David},{bananas,doughnuts})
({Brian,Charles},{apples,bananas,cherries,doughnuts}) ({Charles, David},{bananas,doughnuts,eggs})
({Charles},{apples,bananas,cherries,doughnuts,eggs})
concept lattice is complete

22. Lattice of bicliques/Concept lattice
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
({Abby,Brian,Charles,David},{bananas})
({Abby,Brian,Charles},{apples,bananas}) ({Brian, Charles, David},{bananas,doughnuts})
({Brian,Charles},{apples,bananas,cherries,doughnuts}) ({Charles, David},{bananas,doughnuts,eggs})
({Charles},{apples,bananas,cherries,doughnuts,eggs})

23. Simplifying concept lattice
μm
g
m
({Abby,Brian,Charles,David},{bananas})
({Abby,Brian,Charles},{apples,bananas}) ({Brian, Charles, David},{bananas,doughnuts})
({Brian,Charles},{apples,bananas,cherries,doughnuts}) ({Charles, David},{bananas,doughnuts,eggs})
({Charles},{apples,bananas,cherries,doughnuts,eggs})
label elements with m 2 M μm
if is concept
if is concept
g 2 G g
everything above
everything below
includes
g
includes

24. Simplifying concept lattice
label elements with m 2 M μm
if is concept
if is concept
g 2 G g
({Abby,Brian,Charles,David},{bananas})
({Abby,Brian,Charles},{apples,bananas}) ({Brian, Charles, David},{bananas,doughnuts})
({Brian,Charles},{apples,bananas,cherries,doughnuts}) ({Charles, David},{bananas,doughnuts,eggs})
({Charles},{apples,bananas,cherries,doughnuts,eggs})

25. Simplified concept lattice
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby

26. Definition: Lattice filters
A filter is an upward closed set for a set of elements
S = {p 2 L| s # p, 8s 2 S}
A principal filter is filter for a single element
s = {p 2 L| s # p}
Corresponds directly to simplified lattice labeling
g = {p 2 L| g # p}

27. Principal Filters
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby
Abby Brian Charles

28. Defintion: Lattice ideals
A filter is an downward closed set for a set of elements
#S = {p 2 L| p # s, 8s 2 S}
A principal ideal is ideal for a single element
#s = {p 2 L| p # s}
Corresponds directly to simplified lattice labeling
#m = {p 2 L| p # μm}

29. Principal Ideals
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby
bananas
apples doughnuts
cherries eggs
Brian David
Charles
Abby
#eggs #doughnuts #bananas

30. Recall: recommendations
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
Abby
Brian
Charles
David
apples
bananas
cherries
doughnuts
eggs
every
biclique
above has
Abby
every
biclique
below has
doughnuts

31. Recommendations, filters and ideals
• Can recommend to Abby by
composing concepts from
principal filter of Abby with
concepts from principal ideal
of doughnuts
• Principal ideal for doughnuts
is maximal for foods (e.g.,
attributes) outside of Abby's
filter
bananas
Abby
apples doughnuts
cherries eggs
Brian David
Charles
Abby
#doughnuts

32. Questions linger:
• What is a good recommendation?
• Serendipity, who?
• And, what does the principal ideal for doughnuts
have to do with the principal ideal for Abby? (Or,
what is the deal with Abby and doughnuts?)

33. Reading
B.A. Davey and H.A. Priestley, Introduction to Lattices
and Order, Cambridge University Press, 2002
Covers posets and lattices in first two chapters, Formal Concept
Analysis in the third chapter
B.Ganter and R.Wille, Formal Concept Analysis:
Mathematical Foundations, Springer-Verlag, 1999
Harder to find mathematical reference on Formal Concept
Analysis. Related material in Chapters 0 and 1
I skipped Galois connections, relates to the formal
concept construction, both books cover those details

34. About me and these slides
I am Ben(jamin) Keller. I learn and, sometimes, create through
explaining. I had been involved in a big (US) federally funded
project that had the goal of helping biomedical scientists tell stories
about their experimental observations. The project is long gone, but
I’m still trying to grok how such a thing would work. Much of
biological data comes in the form of observations that are distilled
to something that looks like an incidence relation, which brings us
to this series of presentations.
My goal for the slides is to deal with the mathematics and analysis
of incidence in an approachable way, but the intuitive beginnings
will eventually allow us to embrace the more complex later.

35. This work is licensed under a
Creative Commons Attribution-ShareAlike 4.0
International License.