Recommender Systems, Matrices and Graphs

Recommender Systems, MaTRICES

and Graphs

Roelof Pieters roelof@vionlabs.com

14 May 2014 @ KTH

About meInterests in: • IR, RecSys, Big Data, ML, NLP, SNA,

Graphs, CV, Data Visualization, Discourse Analysis

History: • 2002-2006: almost-BA Computer Science @

Amsterdam Tech Uni (dropped out in 2006) • 2006-2010: BA Cultural Anthropology @

Leiden & Amsterdam Uni’s • 2010-2012: MA Social Anthropology @

Stockholm Uni • 2011-Current: Working @ Vionlabs

se.linkedin.com/in/roelofpieters/ roelof@vionlabs.com

Say Hello! St: Eriksgatan 63

112 33 Stockholm - Sweden Email: hello@vionlabs.com

Tech company here in Stockholm with Geeks and Movie lovers… Since 2009:

• Digital ecosystems for network operators, cable TV companies, and film distributor such as Tele2/Comviq, Cyberia, and Warner Bros

• Various software and hardware hacks for different companies: Webbstory, Excito, Spotify, Samsung

Focus since 2012: • Movie and TV recommendation

service FoorSee

WE LOVE MOVIES….

Outline•Recommender Systems

•Algorithms*

•Graphs

(* math magicians better pay attention here)

•Taxonomy

•History

•Evaluating Recommenders

•Algorithms*

•Graphs

Information Retrieval• Recommender

Systems as part of Information Retrieval

Document(s)Document(s)Document(s)Document(s)Document(s)

Retrieval

• Information Retrieval is the activity of obtaining information resources relevant to an information need from a collection of information resources.

IR: Measure Success

• Recall: success in retrieving all correct documents

• Precision: success in retrieving the most relevant documents

• Given a set of terms and a set of document terms select only the most relevant documents (precision), and preferably all the relevant ones (recall)

“generate meaningful recommendations to a (collection of) user(s) for items or products that

might interest them”

Recommender Systems

Where can RS be found?• Movie recommendation (Netflix) • Related product recommendation (Amazon) • Web page ranking (Google) • Social recommendation (Facebook) • News content recommendation (Yahoo) • Priority inbox & spam filtering (Google) • Online dating (OK Cupid) • Computational Advertising (Yahoo)

•Taxonomy

•History

•Algorithms*

•Graphs

Taxonomy of RS

• Collaborative Filtering (CF)

• Content Based Filtering (CBF)

• Knowledge Based Filtering (KBF)

• Hybrid

Taxonomy of RS

• Collaborative Filtering (CF)!

• Hybrid

Collaborative Filtering:• relies on past user behavior

• Implicit feedback

• Explicit feedback

• requires no gathering of external data

• sparse data

• domain free

• cold start problem

Collaborative

(Dietmar et. al. At ‘AI 2011)

User based Collaborative Filtering

Taxonomy of RS

• Content Based Filtering (CBF)!

• Hybrid

Content Filtering• creates profile for user/movie

• requires gathering external data

• dense data

• domain-bounded

• no cold start problem

Content based

Item based Collaborative Filtering

Taxonomy of RS

• Knowledge Based Filtering (KBF)!

• Hybrid

Knowledge based

Knowledge based Content Filtering

Taxonomy of RS

• Hybrid

Hybrid

•Taxonomy

•History

•Algorithms*

•Graphs

History

• 1992-1995: Manual Collaborative Filtering

• 1994-2000: Automatic Collaborative Filtering + Content

• 2000+: Commercialization…

Tapestry (1992)

(Golberg et. al 1992)

Grouplens (1994)

(Resnick et. al 1994)

2000+: Commercial CF’s• 2001: Amazon starts using item based collaborative

filtering (Patent filed at 1998)

• 2000: Pandora starts music genomeproject, where each song“is analyzed using up to 450 distinct musical characteristics by a trained music analyst.”

• 2006-2009: Netflix Contents: 2 of many algorithms put in use by Netflix replacing “Cinematch": Matrix Factorization (SVD) and Restricted Boltzmann Machines (RBM)

(http://www.pandora.com/about/mgp)

(http://www.netflixprize.com)

Annual Conferences

• RecSys (since 2007) http://recsys.acm.org

• SIGIR (since 1978) http://sigir.org/

• KDD (official since 1998) http://www.kdd.org/

• KDD Cup

Ongoing Discussion• Evaluation • Scalability • Similarity versus Diversity • Cold start (items + users) • Fraud • Imbalanced dataset or Sparsity • Personalization • Filter Bubbles • Privacy • Data Collection

•Taxonomy

•History

•Algorithms*

•Graphs

Evaluating Recommenders• Least mean squares prediction error

• RMSE

• Similarity measure enough ?

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

Evaluating Recommenders

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

rmse(S) =s|S|�1

(i,u)2S

(r̂ui � rui)2

•Algorithms*

•Graphs

•Algorithms*

•Content based Algorithms *

•Collaborative Algorithms *

•Classification

•Rating/Ranking *

•Graphs

• content is exploited (item to item filtering)

• content model:

• keywords (ie TF-IDF)

• similarity/distance measures:

• Euclidean distance:

• L1 and L2-norm

• Jaccard distance

Content-based Filtering

• (adjusted) Cosine distance

• Edit distance

• Hamming distance

• Euclidean distance

• Cosine distance

dot product x.y is 1 × 2 + 2 × 1 + (−1) × 1 = 3

x = [1,2, −1] and = [2,1,1].

L2-norm =√12 + 22 + (−1)2 = 6

• Euclidean distance

• Cosine distance

dot product x.y is 1 × 2 + 2 × 1 + (−1) × 1 = 3

x = [1,2, −1] and = [2,1,1].

cosine of angle: 3/(√6√6) =1/2cos distance of 1/2: 60 degrees,

L2-norm =√12 + 22 + (−1)2 = 6

Examples

• Item to Query

• Item to Item

• Item to User

Examples

• Item to Query!

• Item to Item

• Item to User

Example: Item to QueryTitle Price Genre Rating

The Avengers 5 Action 3,7

Spiderman II 10 Action 4,5

user query q : “price (6) AND genre(Adventure) AND rating (4)”

weights of features: 0.22 0.450.33

Sim(q,”The Avengers”) = 0.22 x (1 - 1/25) + 0.33 x 0 + 0.45 x (1 - 0.3/5) = 0.6342

1-25 price range no matchdiff of 1 diff of 0.3 0-5 rating range

Sim(q,”Spiderman II”) = 0.5898 (0.6348 if we count rating 4.5 > 4 as match)

Weighted Sum:

Examples

• Item to Query

• Item to Item!

• Item to User

Example: Item to Item Similarity

Title ReleaseTime Genres Actors Rating

TA 90s, start 90s, 1993 Action, Comedy, Romance X,Y,Z 3,7

S2 90s, start 90s, 1991 Action W,X,Z 4,5

numericArray of Booleans

Sim(X,Y) = 1 - d(X,Y) or

Sim(X,Y) = exp(- d(X,Y))

where 0 ≤ wi ≤ 1, and i=1..n (number of features).

Set of hierarchical related symbols

Title ReleaseTime Genres Actors Rating

TA 90s, start 90s, 1993 Action, Comedy, Romance X,Y,Z 3,7

S2 90s, start 90s, 1991 Action W,X,Z 4,5

numericArray of Booleans Set of hierarchical related symbols

X1 = (90s,S90s,1993)

X2 = (1,1,1)

X3 = (0,1,1,1)

X4 = 3.7

W 0.5 0.3 0.2

X1 = (90s,S90s,1991)

X2 = (1,0,0)

X3 = (1,1,0,1)

X4 = 4.5

weights of feature all the same

weights of categories within “Release time” different

X1 = (90s,S90s,1993)

X2 = (1,1,1)

X3 = (0,1,1,1)

X4 = 3.7

W 0.5 0.3 0.2

X1 = (90s,S90s,1991)

X2 = (1,0,0)

X3 = (1,1,0,1)

X4 = 4.5

exp(- (1/√4) √d1(X1,Y1)2 +…+d4(X4,Y4 )2 ) =

exp(- )

exp(-(1/√4) √(1-(0.3+0.5))2 + (1-1/3)2 +(1-2/4)2 + (1-0.8/5)2 ) =exp(-(1/√4) √(1.5745 ) = exp(-0.339) = 0.534

Sim( dest1,dest2 ) =

(content factors)

Examples

• Item to Query

• Item to Item

• Item to User

Example: Item to User

Title Roelof Klas Mo Max X(Action)

)The Avengers 5 1 2 5 0.8 0.1

Spiderman II ? 2 1 ? 0.9 0.2

American Pie 2 5 ? 1 0.05 0.9

X(1) =1

0.80.1

For each user u, learn a parameter ��∈ R(n+1).Predict user u as rating movie i with (��)Tx(i)

)The Avengers 5 1 2 5 0.8 0.1

Spiderman II ? 2 1 ? 0.9 0.2

American Pie 2 5 ? 1 0.05 0.9

Mo (�(3)) and Klas (�(2))

predict rating Mo (�(3)), American pie (X(3))

�(2) �(3)�(1) �(4)

X(3) =1

0.050.9

temp �(3) =

)The Avengers 5 1 2 5 0.8 0.1

Spiderman II ? 2 1 ? 0.9 0.2

American Pie 2 5 ? 1 0.05 0.9

Mo (�(3)) and Klas (�(2))

�(2) �(3)�(1) �(4)

10.050.9

dot product

≈ 4.5

)The Avengers 5 1 2 5 0.8 0.1

Spiderman II ? 2 1 ? 0.9 0.2

American Pie 2 5 4.5 1 0.05 0.9

Mo (�(3)) and Klas (�(2))

�(2) �(3)�(1) �(4)

10.050.9

dot product

≈ 4.5

)The Avengers 5 1 2 5 0.8 0.1

Spiderman II ≈4 2 1 ≈4 0.9 0.2

American Pie 2 5 4.5 1 0.05 0.9

How do we learn these user factor parameters?

�(2) �(3)�(1) �(4)

problem formulation:!

• r(i,u) = 1 if user u has rated movie i, otherwise 0

• y(i,u) = rating by user u on movie i (if defined)

• �(u) = parameter vector for user u

• x(i) = feature vector for movie i

• For user u, movie i, predicted rating: (��

)T(x(i))

• temp m(u) = # of movies rated by user u

min ∑ ( (�(u))T!(i) - "(i,u) )2 + ∑ (��)2

ƛ——2

k=1(u)

�(u)

12——

�� m(u)m(u)

Say what?• learning �(u) =

(A. Ng. 2013)

min ∑ ∑ ((�(u))T!(i) - "(i,u))2 + ∑ ∑ (��)2ƛ—2

problem formulation:!

• learning �(u):

• learning �(1), �(2) , … , ��#�� :

�� #��

min ∑ ( (�(u))T!(i) - "(i,u) )2 + ∑ (��)2

ƛ—2

k=1(u)

�(u)

12—��

��

k=1(u)

regularization term

squared error term

actualpredicted

learn for “all” users

Example: Item to Userremember:y = rating �� parameter vector for a userx = feature vector for a movie

•Algorithms*

•Classification

•Rating/Ranking *

•Graphs

Collaborative Filtering:• User-based approach!

• Find a set of users Si who rated item j, that are most similar to ui

• compute predicted Vij score as a function of ratings of item j given by Si (usually weighted linear combination)

• Item-based approach!

• Find a set of most similar items Sj to the item j which were rated by ui

• compute predicted Vij score as a function of ui's ratings for Sj

Collaborative Filtering:• Two primary models:

• Neighborhood models!

• focus on relationships between movies or users

• Latent Factor models

• focus on factors inferred from (rating) patterns

• computerized alternative to naive content creation

• predicts rating by dot product of user and movie locations on known dimensions

(Sarwar, B. et al. 2001)

Neighborhood (user oriented)

(pic from Koren et al. 2009)

Neighbourhood Methods• Problems:

• Ratings biased per user • Ratings biased towards certain items • Ratings change over time • Ratings can rapidly change through real time

events (Oscar nomination, etc) • Bias correction needed

Latent Factors

• latent factor models map users and items into a latent feature space

• user's feature vector denotes the user's affinity to each of the features

• item's feature vector represents how much the item itself is related to the features.

• rating is approximated by the dot product of the user feature vector and the item feature vector.

Latent Factors (users+movies)

(pic from Koren et al. 2009)

Latent Factors (x+y)

(http://xkcd.com/388/)

xkcd.com

Latent Factor models• Matrix Factorization:

• characterizes items + users by vectors of factors inferred from (ratings or other user-item related) patterns

• Given a list of users and items, and user-item interactions, predict user behavior

• can deal with sparse data (matrix)

• can incorporate additional information74

Matrix Factorization

• Dimensionality reduction

• Principal Components Analysis, PCA

• Singular Value Decomposition, SVD

• Non Negative Matrix Factorization, NNMF

Matrix Factorization: SVDSVD, Singular Value Decomposition

• transforms correlated variables into a set of uncorrelated ones that better expose the various relationships among the original data items.

• identifies and orders the dimensions along which data points exhibit the most variation.

• allowing us to find the best approximation of the original data points using fewer dimensions.

SVD: Matrix Decomposition

U: document-to-concept similarity matrix !V: term-to-concept similarity matrix !ƛ : its diagonal elements: ‘strength’ of each concept !

(pic by Xavier Amatriain 2013)

SVD for Collaborative Filtering

each item i associated with vector qi ∈ ℝf each user u associated with vector pu ∈ ℝf qi measures extent to which item possesses factors pu measures extent of interest for user in items which possess high on factors user-item interactions modeled as dot products within the factor space, measured by qiT pu user u rating on item i approximates: rui = qiT pu

SVD for Collaborative Filtering

• compute u,i mappings: qi,pu ∈ ℝf

• factor user, item matrix

• imputation (Sarwar et.al. 2000)

• model only observed ratings + regularization (Funk 2006; Koren 2008)

• learn factor vectors qi and pu by minimizing (regularized) squared error on set of known ratings: approximate user u rating of item i, denoted by rui, leading to Learning Algorithm:

SVD Visualized

regression line reducing two dimensional space into one dimensional one

reducing three dimensional (multidimensional) space into two dimensional plane

SVD Visualized

SVD: Code Away!

Stochastic Gradient Descent

• optimizable by Stochastic Gradient Descent (SGD) (Funk 2006)

• incremental learning

• loops trough ratings and computes prediction error for predicted rating on rui :

• modify parameters by magnitude proportional to y in opposite direction of the gradient, giving learning rule:

Gradient Descent

Alternating Least Squares

• optimizable by Alternating Least Squares (ALS) (2006)

• both qi and pu unknown: minimum function not convex—> can not be solved for a minimum.

• ALS rotates between fixing qi’s and pu’s

• Fix qi or pu makes optimization problem quadratic —> one not optimized can now be solved

• qi and pu independently computed of other item/user factors: parallelization

• Best for implicit data (dense matrix)85

Alternating Least Squares• rotates between fixing qi’s and pu’s

• when all pu’s fixed, recompute qi’s by solving a least squares problem:

• Fix matrix P as some matrix P, so that minimization problem:

• or fix Q similarly as:

• Learning Rule:

• Add Biases:

• Add Input Sources: Implicit Feedback:pu in rui becomes (pu + + (…) )Add Temporal Aspect / time-varying parameters

• Vary Confidence Levels of Inputs

Develop Further…

pic: Lei Guo 2012

(Salakhutdinov & Mnih 2008; Koren 2010)

Develop Further…

• Final Algoritm:

confidence bias terms

regularization

(Paterek,A. 2007)

• Final Algorithm with Temporal dimensions:

Develop Further…

• So what if we don’t have any content factors known?

• Probabilistic Matrix Factorization to the rescue!

• describe each user and each movie by a small set of attributes

Probabilistic Matrix Factorization

• Imagine we have the following rating data: we could say that Roelof and Klas like Action movies, but don’t like Comedy’s, while its the opposite for Mo and Max

Title Roelof Klas Mo Max

The Avengers 5 1 1 4

Spiderman II 4 2 1 5

American Pie 3 5 4 1

Shrek 1 4 5 2

• This could be represented by the PMF model by using three dimensional vectors to describe each user and each movie.

• example latent vectors: • AV: [0, 0.3]

• SPII: [1, 0.3]

• AP [1, 0.3]

• SH [1, 0.3]

• Roelof: [0, 3]

• Klas: [8, 3]

• Mo [10, 3]

• Max [10, 3]

• predict rating by dot product of user vector with the item vector

• So predicting Klas’ rating for Spiderman II = 8*1 + 3*0.3 =

• But descriptions of users and movies not known ahead of time.

• PGM discovers such latent characteristics

ratings

•Algorithms*

•Classification

•Rating/Ranking *

•Graphs

Classification• k-Nearest Neighbors (KNN)

• Decision Trees

• Rule-Based

• Bayesian

• Artificial Neural Networks

• Support Vector Machines

Classification• k-Nearest Neighbors (KNN)!

• Decision Trees

• Rule-Based

• Bayesian

• Artificial Neural Networks

• Support Vector Machines

k-Nearest Neighbor s

• non parametric lazy learning algorithm

• data as feature space

• simple and fast

• k-nn classification

• k-nn regression: density estimation

kNN: Classification

• Classify

• several Xi used to classify Y

• compare (X1p,X2p) and (X1q,Xq) by Squared Euclidean distance: d2pq = (X1p - x1q)2 + (X2p - X2q)2

• find k-Nearest Neighbors

kNN: Classification• input: content extracted emotional values of 561

movies. thanks: Johannes Östling :) ie:

dimensions of movie

“Hamlet”:

<CODE>

k-Nearest Neighborsemotional dimension “Anger” vs “Love”

k-Nearest Neighbors

Negative: afraid, confused, helpless', hurt, sad, angry, depressed

Positive: good, interested, love, positive, strong

aggregate of positive and negative emotions

•Algorithms*

•Classification

•Rating/Ranking *

•Graphs

Rating predictions:• Pos — Neg

• Average

• Bayesian (Weighted) Estimates

• Lower bound of Wilson score confidence interval for a Bernoulli parameter

Rating predictions:• Pos — Neg!

• Average

P — N• (Positive ratings) - (Negative ratings)

• Problematic:

(http://www.urbandictionary.com/define.php?term=movies)

• Average!

Average• (Positive ratings) / (Total ratings)

• Problematic:

(http://www.amazon.co.uk/gp/bestsellers/electronics/)

• Average

• Bayesian (Weighted) Estimates!

Ratings• Top Ranking at IMDB (gives Bayesian estimate):

• Weighted Rating (WR) = (v / (v+m)) × R + (m / (v+m)) × C!

• Where:

R = average for the movie (mean) = (Rating)v = number of votes for the movie = (votes)m = minimum votes required to be listed in the Top 250 (currently 25000)C = the mean vote across the whole report (currently 7.0)

Bayesian (Weighted) Estimates

• weighted average on a per-item basis:

(source(s): http://www.imdb.com/title/tt0368891/ratings)

Bayesian (Weighted) Estimates @ IMDB

Bayesian Weights for m = 1250

0" 250" 500" 750" 1000" 1250" 2000" 3000" 4000" 5000"

specific" global"

• specific part for individual items

• global part is constant over all items • can be

precalculated

m=1250

• Average

Wilson Score interval

• 1927 by Edwin B. Wilson

• Given the ratings I have, there is a 95% chance that the "real" fraction of positive ratings is at least what?

Wilson Score interval• used by Reddit for comments ranking

• “rank the best comments highest regardless of their submission time”

• algorithm introduced to Reddit by Randall Munroe (the author of xkcd).

• treats the vote count as a statistical sampling of a hypothetical full vote by everyone, much as in an opinion poll.

Wilson Score interval• Endpoints for Wilson Score interval:

• Reddit’s comment Ranking function (phat+z*z/(2*n) - z*sqrt((phat*(1-phat) + z*z/(4*n))/n))

/(1+z*z/n)

Bernoulli anyone?

*as the trial (N) = 2 (2 throws of dice) its actually

not a real Bernoulli distribution

What’s next?

GRAPHS

•Algorithms*

•Graphs

Graph Based Approaches• Whats a Graph?!

• Why Graphs?

• Who uses Graphs?

• Talking with Graphs

• Graph example: Recommendations

• Graph example: Data Analysis

What’s a Graph?

has_genre

features_actor

Director

directed_bylik

Userlikes_user

likes_userfriends

follows

ents_movie

Comment

_commen

likes_actor

etceteralocations!

time!moods!

keywords!…

Vertices (Nodes)

Edges (Relations)

Graph Based Approaches• Whats a Graph?

• Why Graphs?!

Why Graphs?

• more complex (social networks…)

• more connected (wikis, pingbacks, rdf, collaborative tagging)

• more semi-structured (wikis, rss)

• more decentralized: democratization of content production (blogs, twitter*, social media*)

and just: MORE

Its the nature of todays Data, which is getting:

Data Trend“Every 2 days we create as much information as we did up to 2003”— Eric Schmidt, Google

Why Graphs?

Graphs vs Relational

relational

(pic by Michael Hunger, neo4j)

Why Graphs?Its Fast!

Matrix based Calculations: Exponential run-time

(items x users x factori x …)

Graphs vs Relational

relational

Why Graphs?Its Fast!

Graph based Calculations: Linear/Constant run-time (item of interest x relations)

Its White-Board

Friendly !

Why Graphs?

Its White-Board

Friendly !

Why Graphs?

Its White-Board

Friendly !

Why Graphs?

• Why Graphs?

• Who uses Graphs?!

Who uses Graphs?• Facebook: Open Graph (https://

developers.facebook.com/docs/opengraph)

• Google: Knowledge Graph (http://www.google.com/insidesearch/features/search/knowledge.html)

• Twitter: FlockDB (https://github.com/twitter/flockdb)

• Mozilla: Pancake (https://wiki.mozilla.org/Pancake)

• (…)

135(pic by Michael Hunger, neo4j)

• Why Graphs?

• Talking with Graphs!

Talking with Graphs

• Graphs can be queried!

• no unions for comparison, but traversals!

• many different graph traversal patterns

(xkcd)

graph traversal patterns

• traversals can be seen as a diffusion proces over a graph!

• “Energy” moves over a graph and spreads out through the network!

• energy:

(Ghahramani 2012)

Energy Diffusion

(pic by Marko A. Rodriguez, 2011)

Energy Diffusion

energy = 4

Energy Diffusion

energy = 3

Energy Diffusion

energy = 2

Energy Diffusion

energy = 1

• Why Graphs?

• Graph example: Recommendations!

Diffusion Example: Recommendations

• Energy diffusion is an easy algorithms for making recommendations!

• different paths make different recommendations!

• different paths for different problems can be solved on same graph/domain!

• recommendation = “jumps” through the data

Friend Recommendation

• Who are my friends’ friends that are not me or my friends

Friend Recommendation

• Who are my friends’ friends

• Who are my friends’ friends that are not me or my friends

G.V(‘me’).outE[knows].inV.outE.inV

G.V(‘me’).outE[knows].inV.aggregate(x).outE. inV{!x.contains(it)}

Product Recommendation

• Who likes what I like —> of these things, what do they like which I dont’ already like

Product Recommendation

• Who likes what I like

• Who likes what I like —> of these things, what do they like which I dont’ already like

G.V(‘me’).outE[likes].inV.inE[likes].outV

G.V(‘me’).outE[likes].inV.aggregate(x).inE[likes]. outV.outE[like].inV{!x.contains(it)}

G.V(‘me’).outE[likes].inV.inE[likes].outV.outE[like].inV

Recommendations atwith FoorSee

• Why Graphs?

Pulp Fiction

Graphs: Conclusion

• Fast!

• Scalable!

• Diversification!

• No Cold Start!

• Sparsity/Density not applicable

Graphs: Conclusion

• Natural Visualizable!

• Feedback / Understandable!

• Connectable to the “web” / semantic web!

• Social Network Analysis!

• Real Time Updates / Recommendations !

WARNING

Graphs are

Addictive!

Les Miserables

Facebook Network

References• J. Dietmar, G. Friedrich and M. Zanker (2011) “Recommender Systems”,

International Joint Conference on Artificial Intelligence Barcelona

• Z. Ghahramani (2012) “Graph-based Semi-supervised Learning”, MLSS, La Palma

• D. Goldbergs, D. Nichols, B.M. Oki and D. Terry (1992) “Using collaborative filtering to weave an information tapestry”, Communications of the ACM 35 (12)

• M. Hunger (2013) “Data Modeling with Neo4j”, http://www.slideshare.net/neo4j/data-modeling-with-neo4j-25767444

• S. Funk (2006) “Netflix Update: Try This at Home”, sifter.org/~simon/journal/20061211.html

References• Y. Koren (2008) “Factorization meets the Neighborhood: A

Multifaceted Collaborative Filtering Model”, SIGKDD, http://public.research.att.com/~volinsky/netflix/kdd08koren.pdf

• Y. Koren & C. Bell, (2007) “Scalable Collaborative Filtering with Jointly Derived Neighborhood Interpolation Weights”

• Y, Koren (2010) “Collaborative filtering with temporal dynamics”

• A. Ng. (2013) Machine Learning, ml-004 @ Coursera

• A. Paterek (2007) “Improving Regularized Singular Value Decomposition for Collaborative Filtering”, KDD

References• P. Resnick, N. Iacovou, M. Suchak, P. Bergstrom and J. Riedl (1994),

“GroupLens: An Open Architecture for Collaborative Filtering of Netnews”, Proceedings of ACM

• B.R. Sarwar et al. (2000) “Application of Dimensionality Reduction in Recommender System—A case Study”, WebKDD

• B. Saewar, G. Karypis, J. Konstan, J, Riedl (2001) “Item-Based Collaborative Filtering Recommendation Algorithms”

• R. Salakhutdinov & A. Mnih (2008) “Probabilistic Matrix Factorization”

• xkcd.com

Take Away Points

• Focus on the best Question, not just the Answer…!

• Best Match (most similar) vs Most Popular!

• Personalized vs Global Factors!

• Less is More ?!

• What is relevant?

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• Some examples of data extraction which is fed into our BAG (Big Ass Grap)…

Computer Vision

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Presentations & Public Speaking

Matrices for graphs, designs and codesmembers.upc.nl/w.haemers/opatija.pdf · Matrices for graphs, designs and codes Willem H. HAEMERS Department of Econometrics and OR, Tilburg University,

Graphs associated with matrices over nite elds and their …cklixx.people.wm.edu/graph-matrices.pdf · 2013-12-24 · Graphs associated with matrices over nite elds and their endomorphisms

Skew-adjacency matrices of graphs · graphs by the spectra of their derived sets of skew-adjacency matrices, but, as we have just seen, addressing that question leads to interesting

and Matrices Recurrences, Graphs,bang/gems/gems.pdf · Recurrences, Graphs, and Matrices Professor Lucas Bang Harvey Mudd College Department of Computer Science

Graphs and Matrices in Geometry Fiedler.pdf

Sparsiﬁcation,of,Graphs,and,Matrices, · Sparsiﬁcation,of,Graphs,and,Matrices,, DanielA. Spielman, Yale,University,,,,, joint,work,with, JoshuaBatson,(MIT) NikhilSrivastava(MSR)

Matrices, Graphs, and PDEs: A journey of unexpected relations · Matrices, Graphs, and PDEs: A journey of unexpected relations, BMS, Berlin 2014 Mario Arioli Some friends We do not

Graphs - math.uic.edujan/mcs360f17/graphs.pdf · Graphs 1 Graphs directed and undirected graphs weighted graphs adjacency matrices 2 Graph Representations abstract data type adjacency

Codes from incidence matrices of graphs - Clemson Universitycecas.clemson.edu/~keyj/Key/JDK_3ICTMA.pdf · J. D. Key (keyj@clemson.edu) Codes from incidence matrices of graphs 11-15

Laplacian Matrices of Graphs - cs.yale. · PDF fileLaplacian Matrices of Graphs: Spectral and Electrical Theory Daniel A. Spielman ... Spectral graph theory A very fast survey Trailer

Graphs and Adjacency Matrices - Home | Department of

Research Article The Distance Matrices of Some Graphs ...downloads.hindawi.com/journals/jam/2013/707954.pdf · of graphs whose distance matrices have exactly one positive eigenvalue

Sparse Matrices and Graphs: There and Back Again€¦ · Sparse Matrices and Graphs: There and Back Again John R. Gilbert University of California, Santa Barbara Simons Institute

Spectra of SparseRandom Matrices · Spectra of Sparse Random Matrices 5 simplicity to adjacency matrices of Erd¨os-Renyi random graphs, with P({cij}) = Y i

Testing Hereditary Properties of Ordered Graphs and Matrices › ~nogaa › PDFS › ord.pdf · 2017-07-02 · Testing Hereditary Properties of Ordered Graphs and Matrices Noga Alon

4.4. Gaussian Elimination for Sparse Matrices€¦ · 3.3 QR-Decomposition with Householder matrices 4 Sparse Matrices 4.1 General Properties, Storage 4.2 Sparse Matrices and Graphs

Skew-adjacency matrices of graphs€¦ · adjacency matrices in Section 3. These relations and those for other matrices of graphs may be found in [8]. InSection4,theskewco-spectralcharacterizationofodd-cyclegraphsisproved(Theorem4.2).Eq

Permutation decoding for codes from designs, finite ...€¦ · incidence matrices of classes of designs or graphs, and adjacency matrices of classes of regular graphs. These codes

NonNegative Matrices and Related Topics 5908-1 · NonNegative Matrices and Related Topics 1. The Perron-Frobenius Theorem 2. Graphs and Matrices 3. Stability 4. Applications and Extensions

totally unimodular matrices - Devi Ahilya … unimodular...Basic examples of totally unimodular matrices are incidence matrices of bipartite graphs & Directed graphs and Network matrices