Yang & Ma's new algorithm slashes matrix completion data needs

Low-rank matrix completion powers systems like Netflix recommendations by filling missing entries from sparse observations. Inductive matrix completion (IMC) uses row/column side information (e.g., user demographics, movie genres) to narrow the search space. Prior work fell into two camps: methods that leveraged side information for sample efficiency but assumed noiseless settings, and robust methods that handled noise but required sample complexity matching the ambient matrix dimension—defeating the purpose of side information. Yang and Ma bridge this gap by analyzing a nonconvex projected gradient descent algorithm with spectral initialization. Their main technical contribution is proving a regularity condition for the IMC loss that holds at sample complexity determined by the effective problem size (the side information dimension 'a') rather than the ambient dimension 'n'. This yields linear convergence and estimation error that depend only on 'a', even under noise.

The authors extend their analysis to the more realistic scenario of inexact side information, showing the reduced sample complexity persists and the estimation error remains order-optimal relative to the inexactness. Extensive simulations and real-world experiments on the MovieLens dataset confirm the theory. For practitioners, this means building recommender systems with imperfect user profile data and noisy ratings no longer requires massive amounts of observed interactions—a critical advance for privacy-constrained or cold-start settings. The algorithm achieves accuracy comparable to methods using 10x more data, opening the door to personalization with minimal user input.