On best rank one approximation of tensors

In this paper we suggest a new algorithm for the computation of a best rank one approximation of tensors, called alternating singular value decomposition. This method is based on the computation of maximal singular values and the corresponding singular vectors of matrices. We also introduce a modification for this method and the alternating least squares method, which ensures that alternating iterations will always converge to a semi-maximal point. (A critical point in several vector variables is semi-maximal if it is maximal with respect to each vector variable, while other vector variables are kept fixed.) We present several numerical examples that illustrate the computational performance of the new method in comparison to the alternating least square method.Comment: 17 pages and 6 figure

Fast truncation of mode ranks for bilinear tensor operations

We propose a fast algorithm for mode rank truncation of the result of a bilinear operation on 3-tensors given in the Tucker or canonical form. If the arguments and the result have mode sizes n and mode ranks r, the computation costs $O(nr^3 + r^4)$. The algorithm is based on the cross approximation of Gram matrices, and the accuracy of the resulted Tucker approximation is limited by square root of machine precision.Comment: 9 pages, 2 tables. Submitted to Numerical Linear Algebra and Applications, special edition for ICSMT conference, Hong Kong, January 201

Fast ALS-based tensor factorization for context-aware recommendation from implicit feedback

Albeit, the implicit feedback based recommendation problem - when only the user history is available but there are no ratings - is the most typical setting in real-world applications, it is much less researched than the explicit feedback case. State-of-the-art algorithms that are efficient on the explicit case cannot be straightforwardly transformed to the implicit case if scalability should be maintained. There are few if any implicit feedback benchmark datasets, therefore new ideas are usually experimented on explicit benchmarks. In this paper, we propose a generic context-aware implicit feedback recommender algorithm, coined iTALS. iTALS apply a fast, ALS-based tensor factorization learning method that scales linearly with the number of non-zero elements in the tensor. The method also allows us to incorporate diverse context information into the model while maintaining its computational efficiency. In particular, we present two such context-aware implementation variants of iTALS. The first incorporates seasonality and enables to distinguish user behavior in different time intervals. The other views the user history as sequential information and has the ability to recognize usage pattern typical to certain group of items, e.g. to automatically tell apart product types or categories that are typically purchased repetitively (collectibles, grocery goods) or once (household appliances). Experiments performed on three implicit datasets (two proprietary ones and an implicit variant of the Netflix dataset) show that by integrating context-aware information with our factorization framework into the state-of-the-art implicit recommender algorithm the recommendation quality improves significantly.Comment: Accepted for ECML/PKDD 2012, presented on 25th September 2012, Bristol, U