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Abstract Representing data in lower dimensional spaces has been used extensively in many disciplines such as natural language and image processing, data mining, and information retrieval. Recommender systems deal with challenging issues such as scalability, noise, and sparsity and thus, matrix and tensor factorization techniques appear as an interesting tool to be exploited. That is, we can deal with all aforementioned challenges by applying matrix and tensor decomposition methods (also known as factorization methods). In this chapter, we provide some basic definitions and preliminary concepts on dimensionality reduction methods of matrices and tensors. Gradient descent and alternating least squares methods are also discussed. Finally, we present the book outline and the goals of each chapter. Keywords: Matrix decomposition · Tensor decomposition

Recommender Systemsfor Social Tagging SystemsSocial tagging systems are Web 2.0 applications that promote user participation through facilitated content sharing and annotation of that content with freely chosen keywords, called tags. Despite the potential of social tagging to improve organization and sharing of content, without efficient tools for content filtering and search, users are prone to suffer from information overload as more and more users, content, and tags become available on-line. Recommender systems are among the best known techniques for helping users to filter out and discover relevant information in large datasets. However, social tagging systems put forward new challenges for recommender systems since – differently from the standard recommender setting where users are mainlyinterested in content – in social tagging systems users may additionally beinterested in finding tags and even other users.