Non-Negativity Constrained Missing Data Estimation for High-Dimensional and Sparse Matrices from Industrial Applications

High-dimensional and sparse (HiDS) matrices are commonly seen in big-data-related industrial applications like recommender systems. Latent factor (LF) models have proven to be accurate and efficient in extracting hidden knowledge from them. However, they mostly fail to fulfill the non-negativity constraints that describe the non-negative nature of many industrial data. Moreover, existing models suffer from slow convergence rate. An alternating-direction-method of multipliers-based non-negative LF (AMNLF) model decomposes the task of non-negative LF analysis on an HiDS matrix into small subtasks, where each task is solved based on the latest solutions to the previously solved ones, thereby achieving fast convergence and high prediction accuracy for its missing data. This paper theoretically analyzes the characteristics of an AMNLF model, and presents detailed empirical studies regarding its performance on nine HiDS matrices from industrial applications currently in use. Therefore, its capability of addressing HiDS matrices is justified in both theory and practice.

I. Introduction

High-dimensional and sparse (HiDS) matrices are found in many big-data-related industrial applications like recommender systems [1]–[5]. They are frequently encountered because of: 1) the great need to describe the relationship among involved entities with matrices and 2) the impossibility to fully observe the relationship among an exploding number of entities, for example, millions of users and items in a recommender system [1]–[5]. It is thus highly important to explore the full relationship among entities for various purposes, for example, predicting potential user preferences in online stores for personalized recommendation [1]–[5] and estimating missing links among users in social networks for community detection [6], [7], [26]. Therefore, to predict missing data of an HiDS matrix from industrial applications based on its known ones is a vital issue.