Convex Factorization Machine for RegressionIntroductionWe propose the convex factorization machine (CFM), which is a convex variant of the widely used Factorization Machines (FMs). Specifically, we employ a linear+quadratic model and regularize the linear term with the regularizer and the quadratic term with the trace norm regularizer. Then, we formulate the CFM optimization as a semidefinite programming problem and propose an efficient optimization procedure with Hazan's algorithm. A key advantage of CFM over existing FMs is that it can find a globally optimal solution, while FMs may get a poor locally optimal solution since the objective function of FMs is nonconvex. In addition, the proposed algorithm is simple yet effective and can be implemented easily. Finally, CFM is a general factorization method and can also be used for other factorization problems including multiview matrix factorization problems. Through synthetic and movielens datasets, we first show that the proposed CFM achieves results competitive to FMs. Furthermore, in a toxicogenomics prediction task, we show that CFM outperforms a stateoftheart tensor factorization method. Main IdeaLet us denote the useritem matrix . In factorization machine framework (libFM), they formulate recommendation problems as regression problems, where the input is a feature vector that indicates the th user and the th item, and output is the rating of the useritem pair:
Note that, we can easily add user and article meta information such as user gender and article category by concatenating those information to . The optimization problem of the convex factorization machine is given as where is the trace norm, and
is the model. Features
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