Mathematical methods of Tensor Factorization applied to Recommender Systems Giuseppe Ricci, PhD Student in Computer Science University of Study of Bari A. Moro Advances in DataBases and Information Systems PhD Consortium, Genoa, 01 Septembre 2013 Semantic Web Access and Personalization research group http://www.di.uniba.it/~swap Dipartimento di Informatica

Information Overload & Recommender Systems On internet today, an overabundance of information can be accessed, making it difficult for users to process and evaluate options and make appropriate choices. Recommender Systems (RS) are techniques for information filtering which play an important role in e- commerce, advertising, e-mail filtering, etc.

What do RS do exactly? Predict how much you may like a certain product/service Compose a list of N best items for you Compose a list of N best users for a certain product/service Explain why these items are recommended to you Adjust the prediction and recommendation based on your feedback (ratings) and other people I1 I2 I3 I4 I5 I6 I7 I8 I9 U1 1 5 4 U2 4 2 5 U3 4 5 U4 5 2 4 A 1 3 1 3 1 4 5 8 user-item matrix

Matrix Factorization Matrix Factorization (MF) techniques fall in the class of collaborative filtering (CF) methods latent factor models: similarity between users and items is induced by some factors hidden in the data Latent factor models build a matrix of users and items and each element is associated with a vector of characteristics MF techniques represent users and items by vectors of features derived from ratings given by users for the items seen or tried Yehuda Koren, Robert Bell, and Chris Volinsky. Matrix factorization techniques for recommender systems. IEEE Computer, 42(8):30-37, 2009.

Matrix Factorization U set of users, D set of items, R rating matrix. MF aims to factorize R into two matrices P and Q such that their product approximates R: P row: strength of the association between user and k latent features. Q column: strength of the association between an item and the latent features. Once these vectors are discovered, recommendations are calculated using the expression of A MF used in literature: Singular Value Decomposition (SVD): introduced by Simon Funk in the NetFlix Prize has the objective of reducing the dimensionality, i. e. the rank, of the user-item matrix capture latent relationships between users and items T T ij i jR P Q r p q ijr

SVD Different SVD algorithms were used in RS literature: in [15], the authors uses a small SVD obtained retaining only k