Blind regression : understanding collaborative filtering from matrix completion to tensor completion

Abstract

Neighborhood-based Collaborative filtering (CF) methods have proven to be successful in practice and are widely applied in commercial recommendation systems. Yet theoretical understanding of their performance is lacking. In this work, we introduce a new framework of Blind Regression which assumes that there are latent features associated with input variables, and we observe outputs of some Lipschitz continuous function over those unobserved features. We apply our framework to the problem of matrix completion and give a nonparametric method which, similar to CF, combines the local estimates according to the distance between the neighbors. We use the sample variance of the difference in ratings between neighbors as the proximity of the distance. Through error analysis, we show that the minimum sample variance is a good proxy of the prediction error in the estimates. Experiments on real-world datasets suggests that our matrix completion algorithm outperforms classic user-user and item-item CF approaches. Finally, our framework easily extends to the setting of higher-order tensors and we present our algorithm for tensor completion. The result from real-world application of image inpainting demonstrates that our method is competitive with the state-of-the-art tensor factorization approaches in terms of predictive performance.