EigenRank: A ranking oriented approach to collaborative filtering By Nathan N. Liu and Qiang Yang Presented by Zachary 1

Outline Recommender system Motivation Memory-based CF – Similarity measure – User based – Item based EigenRank – Model – Prediction – Experiments and results 2

Recommender System Recommender systems try to recommend items to user based on – Your existing rating for some items 3

Motivation Ultimate goal of recommender system is to produce a list of items that a specific user would prefer. Rating V.S. Ranking 4 ItemTrue ratingPredicted rating MAE: 0.4MAE: 0.9

Problem definition Given – m users – n items – Users’ rating on items R (partial data) Produce – A list of items that active user might like m x n matrix, represents user u’s rating on item i. if u has not rated i. 5

Memory based CF We have no information regarding the content of an item Ratings are the only data we have For an active user (user based) – Find users similar to active user For rated items, they tend to give similar ratings – Based on their ratings for an item – Predict the rating that active user would assign 6

Memory based CF 7 Average rating that u gives Set of similar users (Neighborhood) Set of users that rated i Similarity between u and v

Similarity Measure For the memory based model to work, we need a similarity measure – Pearson Correlation Coefficient – Vector similarity – Adjusted cosine similarity 8

Pearson Correlation Coefficient 9

Vector similarity Also known as cosine similarity Measures the cosine value of two vectors in high dimension 10 Dot product Product of length

Adjusted cosine similarity Used to measure item similarity 11

User based V.S. Item based User based For active user u – Consider all similar user V – Combine their rating for i to predict – PCC or VS is often used Item based For active item i – Consider all similar item – Combine user u’s rating for all of them to predict – Adjusted cosine similarity is often used 12

EigenRank Ranking oriented model – No rating prediction – Output a ranking instead 13

EigenRank Define preference function 14 Indicate item i is prefered to j Magnitude denote the strength of preference Additional requirements

EigenRank The following definition is a valid preference function 15 The set of neighboring users who have rated both i and j

Kendall Rank Correlation Coefficient A similarity measure between two rankings of the same set of objects 16 Indicator function

Kendall Rank Correlation Coefficient KRCC is good at capturing the preference relationship between items rather than the actual rating 17 ItemUser1's ratingUser2's rating

EigenRank For a ranking, we define a value function V as: Then, to predict a ranking, we want to find maximizing V 18

EigenRank It is proved that solving the following problem is NP-Complete Resort to approximate solutions – Greedy algorithm – Random walk algorithm 19

Greedy algorithm Time complexity 20 Can be seen as utility of i Eliminate the effect of i

Random walk algorithm Model the problem as a first-order Markov Chain – States  items – Transition probability  preference function – Stationary distribution  a ranking 21

Random walk algorithm Transition probability from item i to j is: 22 Probability distribution after t steps Probability of being at item 2 after t steps Principle eigen vector of P

Experiments and Results Data set – EachMovie(6.1% non-zero entries) – Netflix (6.6% non-zero entries) – Random pick users who have rated more than 40 movies user for training 100 parameter tuning 500 active user – 50% training – 50% testing 23

Experiments and evaluation Metric used for evaluation – NDCG 24 Set of users included in test data Rate assigned by u to item at position p Normalization term

Experiments and evaluation Impact of neighborhood size 25

Experiments and evaluation 26

Q&A Any questions? 27

Appendix Personalization Vector 28

Appendix Impact of 29