Mathematics of Incidencepart 3: What’s with Abby and Doughnuts?

(more about concept lattices) !Benjamin J. Kellerbjkeller.github.io linkedin.com/in/bjkeller!v.1, 10 October 2014

Creative Commons Attribution-ShareAlike 4.0 International License

a u1 u2 uk

s1s2sr(a) tsl

un(a)

�

⟘

⟙

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Recall: Abby and the doughnuts

bananas

doughnutsapples

eggscherries

Charles

DavidBrian

Abby

"Abby

#doughnuts

apples bananas cherries doughnuts eggs

Abby X XBrian X X X XCharles X X X X XDavid X X X

Can recommend foods to Abby based on shared “likes” with other users by composing concepts from principal filter for Abby with those from principal ideal for doughnuts

CF rating distributionSTUDYING RECOMMENDATION ALGORITHMS 147

Figure 6. Hits-buffs structure of the (reordered) MovieLens dataset.

4.2. Experiments

The goal of our experiments is to investigate the effect of the hammock width w on the aver-age characteristic path lengths of the induced Gs social network graph and Gr recommendergraph for the above datasets. We use versions of the EachMovie and MovieLens datasetssanitized by removing the rating information. So, even though rating information can easilybe used by an appropriate jump function (an example is given in Aggarwal et al. (1999),we explore a purely connection-oriented jump in this study. Then, for various values ofthe hammock width w, we form the social network and recommender graphs and calculatethe degree distributions (for the largest connected component). This was used to obtain thelengths predicted by Eqs. (1) and (2) from Section 3. We also compute the average pathlength for the largest connected component of both the secondary graphs using parallelimplementations of Djikstra’s and Floyd’s algorithms. The experimental observations arecompared with the formula predictions.

4.2.1. MovieLens. Figure 7 describes the number of connected components in Gr as aresult of imposing increasingly strict hammock jump constraints. Up to about w = 17, thegraph remains in one piece and rapidly disintegrates after this threshold. The value of thistransition threshold is not surprising, since the designers of MovieLens insisted that everyparticipant rate at least κ = 20 movies. As observed from our experiment results, after thethreshold and up to w = 28, there is still only one giant component with isolated peoplenodes (figure 8, left). Specifically, the degree distributions of the MovieLens social networkgraphs for w > 17 show us that the people nodes that are not part of the giant componentdo not form any other connected components and are isolated. We say that a jump shattersa set of nodes if the vertices that are not part of the giant component do not have any edges.This aspect of the formation of a giant component is well known from random graph theory(Bollabas, 1985). Since our construction views the movies as a secondary mode, we canensure that only the strictest hammock jumps shatter the NM movie nodes. Figure 8 (right)

MovieLens incidence (Mirza et al., JIIS 2003)

CF rating distributionSTUDYING RECOMMENDATION ALGORITHMS 147

Figure 6. Hits-buffs structure of the (reordered) MovieLens dataset.

4.2. Experiments

The goal of our experiments is to investigate the effect of the hammock width w on the aver-age characteristic path lengths of the induced Gs social network graph and Gr recommendergraph for the above datasets. We use versions of the EachMovie and MovieLens datasetssanitized by removing the rating information. So, even though rating information can easilybe used by an appropriate jump function (an example is given in Aggarwal et al. (1999),we explore a purely connection-oriented jump in this study. Then, for various values ofthe hammock width w, we form the social network and recommender graphs and calculatethe degree distributions (for the largest connected component). This was used to obtain thelengths predicted by Eqs. (1) and (2) from Section 3. We also compute the average pathlength for the largest connected component of both the secondary graphs using parallelimplementations of Djikstra’s and Floyd’s algorithms. The experimental observations arecompared with the formula predictions.

4.2.1. MovieLens. Figure 7 describes the number of connected components in Gr as aresult of imposing increasingly strict hammock jump constraints. Up to about w = 17, thegraph remains in one piece and rapidly disintegrates after this threshold. The value of thistransition threshold is not surprising, since the designers of MovieLens insisted that everyparticipant rate at least κ = 20 movies. As observed from our experiment results, after thethreshold and up to w = 28, there is still only one giant component with isolated peoplenodes (figure 8, left). Specifically, the degree distributions of the MovieLens social networkgraphs for w > 17 show us that the people nodes that are not part of the giant componentdo not form any other connected components and are isolated. We say that a jump shattersa set of nodes if the vertices that are not part of the giant component do not have any edges.This aspect of the formation of a giant component is well known from random graph theory(Bollabas, 1985). Since our construction views the movies as a secondary mode, we canensure that only the strictest hammock jumps shatter the NM movie nodes. Figure 8 (right)

MovieLens incidence (Mirza et al., JIIS 2003)

people who have rated more movies

movies with more ratings

(more “popular”)

A question of place

• Location in the lattice tells us something about users and foods

• User higher in lattice has only experienced more popular foods (e.g., those with more “likes”)

• Food lower in lattice only experienced by the more adventurous

bananas

doughnutsapples

eggscherries

Charles

DavidBrian

Abby

More popular items

Users who have experienced more

items

What happens if Abby likes cherries?

apples bananas cherries doughnuts eggs

Abby X X XBrian X X X XCharles X X X X XDavid X X X

bananas

doughnutsapples,cherries

eggs

Charles

DavidBrian

Abby

Makes cherries more popular, they move up the lattice

What happens if Abby likes grapes?

apples bananas cherries doughnuts eggs grapes

Abby X X XBrian X X X XCharles X X X X X XDavid X X X

(Charles likes everything)

grapes

bananas

doughnutsapples

eggscherries

Charles

DavidBrianAbby

Abby moves down the lattice

Definitions: top, co-atoms, bottom, atoms

co-atom – element of lattice covered by topa � > a 2 L

atom – element of lattice that covers bottom? � b b 2 L

top of lattice is least upper bound of all elements >written

bottom of lattice is greatest lower bound of all elements?written

(squirm) A concept lattice technicality

• Concept lattice is complete: it has a top and bottom

• Formally, defined as closure of empty sets

• top is concept with no attributes

• bottom is concept with no objects

• Been using closure on G (attributes shared by all objects) and M (objects with all attributes)

bananas

doughnutsapples

eggscherries

Charles

DavidBrian

Abby

�;

µ;

µM

�G

Being practically lazy

�G � µ; �; � µMCan show that where they only differ when there is an attribute shared by all objects or an object with all attributes

and

More importantly: �A � �G A ✓ G

µB ⌫ µM B ✓ Mfor allfor all

So ignore formal top and bottom and use

�G as top µM as bottom a � �G as a co-atom b � µM as an atom

Why do cherries become a co-atom?

apples bananas cherries doughnuts eggs

Abby X X XBrian X X X XCharles X X X X XDavid X X X

bananas

doughnutsapples,cherries

eggs

Charles

DavidBrian

Abby

Adding Abby likes cherries differentiates cherries from doughnuts (other foods liked by Brian and Charles)

Why does Abby become an atom?

apples bananas cherries doughnuts eggs grapes

Abby X X XBrian X X X XCharles X X X X X XDavid X X X

(Charles likes everything)

grapes

bananas

doughnutsapples

eggscherries

Charles

DavidBrianAbby

Adding Abby likes grapes differentiates Abby from Brian (e.g., other users who like apples and bananas)

Subtleties of place (CF)

• An atom represents

• at least one user liking unique food(s)

• user(s) who like more foods

• A co-atom represents

• at least one food liked by unique user(s)

• food(s) liked by more users

• Correspond to the long-tail of the user/food rating distributions

• Concepts between atoms and co-atoms closer to the fat end

• Ponder: Can we figure out where the long-tail starts?

So, what is it with Abby and doughnuts?

• Just that doughnuts are the most popular food that Abby has not liked

• Because the other foods are liked by subsets of same users, they fall in the principal ideal for doughnuts

• If there were another group with different tastes that Abby shared likes with, then there would be another most popular food

bananas

doughnutsapples

eggscherries

Charles

DavidBrian

Abby

"Abby

#doughnuts

Let's try it…

apples bananas cherries doughnuts eggs flounder lychee

Abby X XBrian X X X XCharles X X X X X X XDavid X X XEvan X XFan X X X X X

So, what is it with Abby, doughnuts and lychee?

• Based on new incidence, the ideal for both doughnuts and lychee fruit contains all of the concepts that could be used to recommend to Abby

• So, they correspond to the maximal concepts that generate this filter of composable concepts

• Ponder: do they have to be co-atoms?

lychee

flounder

Fan

Evan

bananas

doughnutsapples

eggs

cherries

Charles

DavidBrian

Abby

OK, let's back up…

A recommendation of item t to user a

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({a, u1, u2,…, uk}, {s1, s2,…, sl })

({u1, u2,…, uk}, {s1, s2,…, sl , t})

recommend t to a by composing concept from principal filter of a with a subconcept in principal ideal of t

Wait, let's figure out this co-atom thing

µt µs1, µs2, . . . µsls1, s2, . . . sl

Not necessary that any of orare co-atoms as shown, where areitems that a has rated, and t is the target item

si

µsi = µSi

But, for any there is a set of items

and satisfying

Si ⇢ �(µsi)

consisting only of co-atoms m 2 Si µm � �G

So, can always substitute in a set for any item that doesn’t correspond to a co-atom – but need to revisit this

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A different perspective

µT

T = {s1, s2, . . . , sl, t}

U = {a, u1, u2, . . . , uk}

µU

A different perspective

• Each recommendation corresponds to a selection of items and users

• So, can recommend items determined by selection of items liked by a and a selection of users who like those items

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So, what is a good recommendation?

µT

T = {s1, s2, . . . , sl, t}

U = {a, u1, u2, . . . , uk}

µUA composition of with where

are sets that are “strongly interdependent”

and

Expect that it would be something like

But, we don’t really know what that means

Questions of ponderousness

• What is dependency? And, what does it have to do with good recommendations?

• What happens with recommendations when don't have atoms/co-atoms?

• Can we figure out where the long-tail starts?

• Serendipity who?

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.

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International License.