Bounded Matrix Low Rank Approximation

Abstract

Low rank approximation is the problem of finding two matrices $\mathbf {P} \in \mathbb {R}^{m \times k}$ and $\mathbf {Q} \in \mathbb {R}^ {k \times n}$ for input matrix $\mathbf {R} \in \mathbb {R}^{m \times n}$, such that $\mathbf {R} \approx \mathbf {PQ}$. It is common in recommender systems rating matrix, where the input matrix $\mathbf {R}$ is bounded in the closed interval $[r_{min},r_{max}]$ such as [1, 5]. In this chapter, we propose a new improved scalable low rank approximation algorithm for such bounded matrices called bounded matrix low rank approximation (BMA) that bounds every element of the approximation $\mathbf {PQ}$. We also present an alternate formulation to bound existing recommender systems algorithms called BALS and discuss its convergence. Our experiments on real-world datasets illustrate that the proposed method BMA outperforms the state-of-the-art algorithms for recommender system such as stochastic gradient descent, alternating least squares with regularization, SVD++ and bias-SVD on real-world datasets such as Jester, Movielens, Book crossing, Online dating, and Netflix.