Understanding the Sparsity

Abstract

An important problem of matrix completion/approximation based on Matrix Factorization (MF) algorithms is the existence of multiple global optima; this problem is especially serious when the matrix is sparse, which is common in real-world applications such as personalized recommender systems. In this work, we clarify data sparsity by bounding the solution space of MF algorithms. We present the conditions that an MF algorithm should satisfy for reliable completion of the unobservables, and we further propose to augment current MF algorithms with extra constraints constructed by compressive sampling on the unobserved values, which is well-motivated by the theoretical analysis. Model learning and optimal solution searching is conducted in a properly reduced solution space to achieve more accurate and efficient rating prediction performances. We implemented the proposed algorithms in the Map-Reduce framework, and comprehensive experimental results on Yelp and Dianping datasets verified the effectiveness and efficiency of the augmented matrix factorization algorithms.