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EigenRank: A ranking oriented approach to collaborative filtering

By Nathan N. Liu and Qiang YangPresented by Zachary

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Outline

• Recommender system• Motivation• Memory-based CF– Similarity measure– User based– Item based

• EigenRank– Model– Prediction– Experiments and results

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Recommender System

• Recommender systems try to recommend items to user based on – Your existing rating for

some items

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Motivation

• Ultimate goal of recommender system is to produce a list of items that a specific user would prefer.

• Rating V.S. RankingItem True rating Predicted

ratingPredicted rating

1 4 4.5 2.8

2 5 4.3 4.1

3 2 2.1 2.6

MAE: 0.4 MAE: 0.9

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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.

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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

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Memory based CF

Average rating that u gives

Set of similar users(Neighborhood)

Set of users that rated i

Similarity between u and v

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Similarity Measure

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

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Pearson Correlation Coefficient

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Vector similarity

• Also known as cosine similarity• Measures the cosine value of two vectors in

high dimensionDot product

Product of length

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Adjusted cosine similarity

• Used to measure item similarity

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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

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EigenRank

• Ranking oriented model– No rating prediction– Output a ranking instead

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EigenRank

• Define preference function

Indicate item i is prefered to j

Magnitude denote the strength of preference

Additional requirements

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EigenRank

• The following definition is a valid preference function

The set of neighboring users who have rated both i and j

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Kendall Rank Correlation Coefficient

• A similarity measure between two rankings of the same set of objects

Indicator function

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Kendall Rank Correlation Coefficient

• KRCC is good at capturing the preference relationship between items rather than the actual rating

Item User1's rating User2's rating1 2 32 3 43 4 5

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EigenRank

• For a ranking , we define a value function V as:

• Then, to predict a ranking, we want to find maximizing V

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EigenRank

• It is proved that solving the following problem is NP-Complete

• Resort to approximate solutions– Greedy algorithm– Random walk algorithm

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Greedy algorithm• Time complexity

•

Can be seen as utility of i

Eliminate the effect of i

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Random walk algorithm

• Model the problem as a first-order Markov Chain– States items– Transition probability preference function

– Stationary distribution a ranking

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Random walk algorithm

• Transition probability from item i to j is:

Probability distribution after t steps Probability of being at item 2 after t steps

Principle eigen vector of P

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Experiments and Results

• Data set– EachMovie(6.1% non-zero entries)– Netflix (6.6% non-zero entries)– Random pick 10600 users who have rated more

than 40 movies• 10000 user for training• 100 parameter tuning• 500 active user

– 50% training– 50% testing

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Experiments and evaluation

• Metric used for evaluation– NDCG

Set of users included in test data

Rate assigned by u to item at position p

Normalization term

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Experiments and evaluation

• Impact of neighborhood size

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Experiments and evaluation

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Q&A

• Any questions?

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Appendix

• Personalization Vector

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Appendix

• Impact of