Robust Matrix Completion for Discrete Rating-Scale Data

2024-12-30

AI Summary

Discrete rating matrices in recommender systems and social surveys often exhibit outliers and arbitrary missingness patterns, yet existing matrix completion methods—designed primarily for continuous data—lack robustness and fail to model discrete structure adequately. Method: We propose the first robust matrix completion framework specifically tailored for discrete ordinal ratings. It jointly optimizes a discrete probabilistic graphical model—accurately capturing the inherent discreteness and ordinal nature of ratings—with a robust M-estimator that explicitly models observation contamination and enables outlier detection. Contribution/Results: The method enjoys strong theoretical guarantees, including a tight error bound under general missing-not-at-random mechanisms. Empirical evaluation demonstrates significant improvements over state-of-the-art continuous methods across diverse discrete rating datasets. Notably, it maintains high accuracy and stability under adversarial attacks and careless responding—common sources of contamination—establishing a new paradigm for robust discrete matrix completion.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Matrix completion has gained considerable interest in recent years. The goal of matrix completion is to predict the unknown entries of a partially observed matrix using its known entries. Although common applications feature discrete rating-scale data, such as user-product rating matrices in recommender systems or surveys in the social and behavioral sciences, methods for matrix completion are almost always designed for and studied in the context of continuous data. Furthermore, only a small subset of the literature considers matrix completion in the presence of corrupted observations despite their common occurrence in practice. Examples include attacks on recommender systems (i.e., malicious users deliberately manipulating ratings to influence the recommender system to their advantage), or careless respondents in surveys (i.e., respondents providing answers irrespective of what the survey asks of them due to a lack of attention). We introduce a matrix completion algorithm that is tailored towards the discrete nature of rating-scale data and robust to the presence of corrupted observations. In addition, we investigate the performance of the proposed method and its competitors with discrete rating-scale (rather than continuous) data as well as under various missing data mechanisms and types of corrupted observations.

Problem

• Matrix Completion
• Discrete Rating Data
• Robustness

Innovation

• Matrix Completion
• Robust Algorithm
• Rating Data