@inproceedings{rendle2010pairwise,
  abstract = {Tagging plays an important role in many recent websites. Recommender systems can help to suggest a user the tags he might want to use for tagging a specific item. Factorization models based on the Tucker Decomposition (TD) model have been shown to provide high quality tag recommendations outperforming other approaches like PageRank, FolkRank, collaborative filtering, etc. The problem with TD models is the cubic core tensor resulting in a cubic runtime in the factorization dimension for prediction and learning.</p> <p>In this paper, we present the factorization model PITF (Pairwise Interaction Tensor Factorization) which is a special case of the TD model with linear runtime both for learning and prediction. PITF explicitly models the pairwise interactions between users, items and tags. The model is learned with an adaption of the Bayesian personalized ranking (BPR) criterion which originally has been introduced for item recommendation. Empirically, we show on real world datasets that this model outperforms TD largely in runtime and even can achieve better prediction quality. Besides our lab experiments, PITF has also won the ECML/PKDD Discovery Challenge 2009 for graph-based tag recommendation.},
  acmid = {1718498},
  address = {New York, NY, USA},
  author = {Rendle, Steffen and Schmidt-Thieme, Lars},
  booktitle = {Proceedings of the third ACM international conference on Web search and data mining},
  doi = {10.1145/1718487.1718498},
  interhash = {ce8fbdf2afb954579cdb58104fb683a7},
  intrahash = {10fe730b391b08031f3103f9cdbb6e1a},
  isbn = {978-1-60558-889-6},
  location = {New York, New York, USA},
  numpages = {10},
  pages = {81--90},
  publisher = {ACM},
  title = {Pairwise interaction tensor factorization for personalized tag recommendation},
  url = {http://doi.acm.org/10.1145/1718487.1718498},
  year = 2010
}

@article{kolda2009tensor,
  abstract = {This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.},
  author = {Kolda, Tamara G. and Bader, Brett W.},
  doi = {10.1137/07070111X},
  interhash = {b30bb2d42e1a05fc41370c50844822ad},
  intrahash = {e52e5c7bff59fd01fb6497d3bb620077},
  issn = {00361445},
  journal = {SIAM Review},
  number = 3,
  pages = {455--500},
  publisher = {SIAM},
  title = {Tensor Decompositions and Applications},
  url = {http://dx.doi.org/10.1137/07070111X},
  volume = 51,
  year = 2009
}

@inproceedings{rendle2009learning,
  abstract = {Tag recommendation is the task of predicting a personalized list of tags for a user given an item. This is important for many websites with tagging capabilities like last.fm or delicious. In this paper, we propose a method for tag recommendation based on tensor factorization (TF). In contrast to other TF methods like higher order singular value decomposition (HOSVD), our method RTF ('ranking with tensor factorization') directly optimizes the factorization model for the best personalized ranking. RTF handles missing values and learns from pairwise ranking constraints. Our optimization criterion for TF is motivated by a detailed analysis of the problem and of interpretation schemes for the observed data in tagging systems. In all, RTF directly optimizes for the actual problem using a correct interpretation of the data. We provide a gradient descent algorithm to solve our optimization problem. We also provide an improved learning and prediction method with runtime complexity analysis for RTF. The prediction runtime of RTF is independent of the number of observations and only depends on the factorization dimensions. Besides the theoretical analysis, we empirically show that our method outperforms other state-of-the-art tag recommendation methods like FolkRank, PageRank and HOSVD both in quality and prediction runtime.},
  address = {New York, NY, USA},
  author = {Rendle, Steffen and Balby Marinho, Leandro and Nanopoulos, Alexandros and Schmidt-Thieme, Lars},
  booktitle = {KDD '09: Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining},
  doi = {10.1145/1557019.1557100},
  interhash = {1cc85ca2ec82db2a3caf40fd1795a58a},
  intrahash = {1bd672ffb8d6ba5589bb0c7deca09412},
  isbn = {978-1-60558-495-9},
  location = {Paris, France},
  pages = {727--736},
  publisher = {ACM},
  title = {Learning optimal ranking with tensor factorization for tag recommendation},
  url = {http://portal.acm.org/citation.cfm?doid=1557019.1557100},
  year = 2009
}