Probabilistic Matrix Completion

Abstract

Collaborative filtering (CF) is typically a matrix completion (MC) problem where the unknown values of the rating matrix are predicted by finding similar rating patterns based on the given entries. The most common paradigm of MC is to factorize the rating matrix into two low-rank matrices. The basic matrix factorization (MF) and its extensions, i.e. conventional MF-based models, have achieved great success in the past and recently models based on deep learning have become popular. However, some recent works have pointed out that many newly proposed methods are outperformed by conventional MF-based models, which demonstrates the simplicity but effectiveness of the basic MF and its extensions. Finding the basic MF cannot be formulated by Probabilistic Matrix Factorization (PMF), this paper proposes a new model called Probabilistic Matrix Completion (PMC), which can interpret the basic MF from a probabilistic perspective. Unlike PMF, which samples each latent vector for each row in the rating matrix indiscriminately, PMC considers different sample frequency between rows (and columns) and computes the prior distribution based on the observed entries. To further demonstrate the difference between PMF and PMC, we incorporate geometric structure into PMC and finally get a model named GPMC that can outperform various state-of-the-art CF methods in terms of rating prediction. We validate our claims on six real-world datasets.