Correlated Matrix Factorization for Recommendation with Implicit Feedback

Abstract

As a typical latent factor model, Matrix Factorization (MF) has demonstrated its great effectiveness in recommender systems. Users and items are represented in a shared low-dimensional space so that the user preference can be modeled by linearly combining the item factor vector <inline-formula><tex-math notation= LaTeX >$V$</tex-math><alternatives><inline-graphic xlink:href= he-ieq1-2840993.gif /></alternatives></inline-formula> using the user-specific coefficients <inline-formula><tex-math notation= LaTeX >$U$</tex-math><alternatives><inline-graphic xlink:href= he-ieq2-2840993.gif /></alternatives></inline-formula>. From a generative model perspective, <inline-formula><tex-math notation= LaTeX >$U$</tex-math><alternatives><inline-graphic xlink:href= he-ieq3-2840993.gif /></alternatives></inline-formula> and <inline-formula><tex-math notation= LaTeX >$V$</tex-math><alternatives><inline-graphic xlink:href= he-ieq4-2840993.gif /></alternatives></inline-formula> are drawn from two <italic>independent</italic> Gaussian distributions, which is not so faithful to the reality. Items are produced to maximally meet users’ requirements, which makes <inline-formula><tex-math notation= LaTeX >$U$</tex-math><alternatives><inline-graphic xlink:href= he-ieq5-2840993.gif /></alternatives></inline-formula> and <inline-formula><tex-math notation= LaTeX >$V$</tex-math><alternatives><inline-graphic xlink:href= he-ieq6-2840993.gif /></alternatives></inline-formula> strongly correlated. Meanwhile, the linear combination between <inline-formula><tex-math notation= LaTeX >$U$</tex-math><alternatives><inline-graphic xlink:href= he-ieq7-2840993.gif /></alternatives></inline-formula> and <inline-formula><tex-math notation= LaTeX >$V$</tex-math><alternatives><inline-graphic xlink:href= he-ieq8-2840993.gif /></alternatives></inline-formula> forces a bijection (one-to-one mapping), which thereby neglects the mutual correlation between the latent factors. In this paper, we address the upper drawbacks, and propose a new model, named Correlated Matrix Factorization (CMF). Technically, we apply Canonical Correlation Analysis (CCA) to map <inline-formula><tex-math notation= LaTeX >$U$</tex-math><alternatives><inline-graphic xlink:href= he-ieq9-2840993.gif /></alternatives></inline-formula> and <inline-formula><tex-math notation= LaTeX >$V$</tex-math><alternatives><inline-graphic xlink:href= he-ieq10-2840993.gif /></alternatives></inline-formula> into a new semantic space. Besides achieving the optimal fitting on the rating matrix, one component in each vector (<inline-formula><tex-math notation= LaTeX >$U$</tex-math><alternatives><inline-graphic xlink:href= he-ieq11-2840993.gif /></alternatives></inline-formula> or <inline-formula><tex-math notation= LaTeX >$V$</tex-math><alternatives><inline-graphic xlink:href= he-ieq12-2840993.gif /></alternatives></inline-formula>) is also tightly correlated with every single component in the other. We derive efficient inference and learning algorithms based on variational EM methods. The effectiveness of our proposed model is comprehensively verified on four public datasets. Experimental results show that our approach achieves competitive performance on both prediction accuracy and efficiency compared with the current state of the art.