Weighted Regularized Matrix Factorization Algorithm
A Weighted Regularized Matrix Factorization Algorithm is a weighted matrix factorization algorithm that is a regularized matrix factorization algorithm with optimization criterion of [math]\displaystyle{ \Sigma_{u \in U} \Sigma_{i \in I} \ C_{ui}( \langle w_u, h_i \rangle - 1)^2 + \lambda \|{W}\|^2_f + \lambda \|{H}|^2_f }[/math] where [math]\displaystyle{ c_{ui} }[/math] are not model parameters.
- AKA: WR-MF.
- …
- Counter-Example(s):
- See: Unweighted Regularized Matrix Factorization.
References
2009
- (Rendle et al., 2009) ⇒ Steffen Rendle, Christoph Freudenthaler, Zeno Gantner, and Lars Schmidt-Thieme. (2009). “BPR: Bayesian Personalized Ranking from Implicit Feedback.” In: Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence. ISBN:978-0-9749039-5-8
- QUOTE: Both Pan et al., 2008 and Hu et al., 2008 have presented a matrix factorization method for item prediction from implicit feedback. Thus the model class is the same as we described in Section 4.3.1, i.e. [math]\displaystyle{ \hat{X} := WH^t }[/math] with the matrices [math]\displaystyle{ W : |U| \times k }[/math] and [math]\displaystyle{ H : |U| \times k }[/math]. The optimization criterion and learning method differ substantially from our approach. Their method is an adaption of a SVD, which minimizes the square-loss. Their extensions are regularization to prevent overfitting and weights in the error function to increase the impact of positive feedback. In total their optimization criterion is: [math]\displaystyle{ \Sigma_{u \in U} \Sigma_{i \in I} \ C_{ui}( \langle w_u, h_i \rangle - 1)^2 + \lambda \|{W}\|^2_f + \lambda \|{H}|^2_f }[/math] where [math]\displaystyle{ c_{ui} }[/math] are not model parameters but apriori given weights for each tuple [math]\displaystyle{ (u,i) }[/math]. Hu et al. have additional data to estimate [math]\displaystyle{ c_{ui} }[/math] for positive feedback and they set [math]\displaystyle{ c_{ui} = 1 }[/math] for the rest. Pan et al. suggest to set [math]\displaystyle{ c_{ui} = 1 }[/math] for positive feedback and choose lower constants for the rest.
First of all, it is obvious that this optimization is on instance level (one item) instead of pair level (two items) as BPR. Apart from this, their optimization is a least-square which is known to correspond to the MLE for normally distributed random variables. However, the task of item prediction is actually not a regression (quantitative), but a classification (qualitative) one, so the logistic optimization is more appropriate.
A strong point of WR-MF is that it can be learned in O(iter (jSj k2+k3 (jIj+jUj))) provided that cui is constant for non-positive pairs. Our evaluation indicates that LearnBPR usually converges after a subsample of [math]\displaystyle{ m \cdot |S| }[/math] single update steps even though there are much more triples to learn from.
- QUOTE: Both Pan et al., 2008 and Hu et al., 2008 have presented a matrix factorization method for item prediction from implicit feedback. Thus the model class is the same as we described in Section 4.3.1, i.e. [math]\displaystyle{ \hat{X} := WH^t }[/math] with the matrices [math]\displaystyle{ W : |U| \times k }[/math] and [math]\displaystyle{ H : |U| \times k }[/math]. The optimization criterion and learning method differ substantially from our approach. Their method is an adaption of a SVD, which minimizes the square-loss. Their extensions are regularization to prevent overfitting and weights in the error function to increase the impact of positive feedback. In total their optimization criterion is: [math]\displaystyle{ \Sigma_{u \in U} \Sigma_{i \in I} \ C_{ui}( \langle w_u, h_i \rangle - 1)^2 + \lambda \|{W}\|^2_f + \lambda \|{H}|^2_f }[/math] where [math]\displaystyle{ c_{ui} }[/math] are not model parameters but apriori given weights for each tuple [math]\displaystyle{ (u,i) }[/math]. Hu et al. have additional data to estimate [math]\displaystyle{ c_{ui} }[/math] for positive feedback and they set [math]\displaystyle{ c_{ui} = 1 }[/math] for the rest. Pan et al. suggest to set [math]\displaystyle{ c_{ui} = 1 }[/math] for positive feedback and choose lower constants for the rest.
2008a
- (Hu et al., 2008) ⇒ Yifan Hu, Yehuda Koren, and Chris Volinsky. (2008). “Collaborative Filtering for Implicit Feedback Datasets.” In: Proceedings of the 2008 Eighth IEEE International Conference on Data Mining. ISBN:978-0-7695-3502-9 doi:10.1109/ICDM.2008.22
- ABSTRACT: A common task of recommender systems is to improve customer experience through personalized recommendations based on prior implicit feedback. These systems passively track different sorts of user behavior, such as purchase history, watching habits and browsing activity, in order to model user preferences. Unlike the much more extensively researched explicit feedback, we do not have any direct input from the users regarding their preferences. In particular, we lack substantial evidence on which products consumer dislike. In this work we identify unique properties of implicit feedback datasets. We propose treating the data as indication of positive and negative preference associated with vastly varying confidence levels. This leads to a factor model which is especially tailored for implicit feedback recommenders.
2008b
- (Pan et al., 2008) ⇒ Rong 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. ISBN:978-0-7695-3502-9 doi:10.1109/ICDM.2008.16