This webpage explains how to use GraphLab collaborative filtering library. In this library, multiple matrix decomposition algorithms are implemented.
See description in the following papers:
Probablistic matrix/tensor factorization:
A) Liang Xiong, Xi Chen, Tzu-Kuo Huang, Jeff Schneider, Jaime G. Carbonell, Temporal Collaborative Filtering with Bayesian Probabilistic Tensor Factorization. In Proceedings of SIAM Data Mining, 2010. html (source code is also available).
B) Salakhutdinov and Mnih, Bayesian Probabilistic Matrix Factorization using Markov Chain Monte Carlo. in International Conference on Machine Learning, 2008. pdf project website, since our code implements matrix factorization as a sepcial case of a tensor as well.
C) Alternating least squares: Yunhong Zhou, Dennis Wilkinson, Robert Schreiber and Rong Pan. Large-Scale Parallel Collaborative Filtering for the Netflix Prize. Proceedings of the 4th international conference on Algorithmic Aspects in Information and Management. Shanghai, China pp. 337-348, 2008. pdf
D) SVD++ algorithm: Koren, Yehuda. "Factorization meets the neighborhood: a multifaceted collaborative filtering model." In Proceeding of the 14th ACM SIGKDD international conference on Knowledge discovery and data mining, 426434. ACM, 2008. http://portal.acm.org/citation.cfm?id=1401890.1401944
E) SGD (sotchastic gradient descent) algorithm: Matrix Factorization Techniques for Recommender Systems Yehuda Koren, Robert Bell, Chris Volinsky In IEEE Computer, Vol. 42, No. 8. (07 August 2009), pp. 30-37.
F) Tikk, D. (2009). Scalable Collaborative Filtering Approaches for Large Recommender Systems. Journal of Machine Learning Research, 10, 623-656.
G) For Lanczos algorithm (SVD) see: wikipedia.
H) For NMF (non-negative matrix factorization) see: Lee, D..D., and Seung, H.S., (2001), 'Algorithms for Non-negative Matrix Factorization', Adv. Neural Info. Proc. Syst. 13, 556-562.
I) For Weighted-Alternating least squares: Collaborative Filtering for Implicit Feedback Datasets Hu, Y.; Koren, Y.; Volinsky, C. IEEE International Conference on Data Mining (ICDM 2008), IEEE (2008).
J) Pan, Yunhong Zhou, Bin Cao, Nathan N. Liu, Rajan Lukose, Martin Scholz, and Qiang Yang. 2008. One-Class Collaborative Filtering. In Proceedings of the 2008 Eighth IEEE International Conference on Data Mining (ICDM '08). IEEE Computer Society, Washington, DC, USA, 502-511.
K) For sparse factor matrices see: Xi Chen, Yanjun Qi, Bing Bai, Qihang Lin and Jaime Carbonell. Sparse Latent Semantic Analysis. In SIAM International Conference on Data Mining (SDM), 2011.
D. Needell, J. A. Tropp CoSaMP: Iterative signal recovery from incomplete and inaccurate samples Applied and Computational Harmonic Analysis, Vol. 26, No. 3. (17 Apr 2008), pp. 301-321.
L) For SVD see Wikipedia
M) For time-SVD++, see Yehuda Koren. 2009. Collaborative filtering with temporal dynamics. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and data mining (KDD '09). ACM, New York, NY, USA, 447-456. DOI=10.1145/1557019.1557072
N) For bias-SVD
Y. Koren. Factorization Meets the Neighborhood: a Multifaceted Collaborative Filtering Model. Equation (5), pdf.
O) For RBM:
G. Hinton. A Practical Guide to Training Restricted Boltzmann Machines. University of Toronto Tech report UTML TR 2010-003 pdf.