Sample efficient inductive matrix completion with noise and inexact side information

2026-05-19 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics, Mathematics & Computational Sciences · Depth: Expert, extended

Summary

This paper introduces a novel approach to Inductive Matrix Completion (IMC) that addresses the limitations of prior methods, specifically the trade-off between sample efficiency and noise handling. The authors propose a nonconvex projected gradient descent algorithm with spectral initialization to recover a low-rank matrix $\bm{L}^{\star}$ from partial, noisy observations, leveraging row and column side information. Their key technical contribution is establishing a regularity condition for the IMC loss function that holds at a reduced sample complexity, scaling with the side information dimension $a$ rather than the ambient dimension $n$. This yields linear convergence and estimation error dependent only on the effective problem size. The analysis is extended to the inexact side information setting, demonstrating maintained reduced sample complexity and order-optimal estimation error with respect to inexactness. Extensive simulations on synthetic data and real-world experiments on the MovieLens dataset validate these theoretical findings, showing IMC's superior performance at low sample sizes compared to standard matrix completion.

Key takeaway

For Data Scientists working with large, sparse datasets where side information is available, this work demonstrates that Inductive Matrix Completion (IMC) offers significant advantages in sample efficiency and estimation accuracy, even with noise or inexact side information. You should consider IMC, especially in low-sample regimes, as it can provide robust estimates where traditional matrix completion fails. When side information is imperfect, explore the proposed interpolation scheme to balance the benefits of IMC with the robustness of ordinary matrix completion, using cross-validation to tune the $\lambda$ parameter.

Key insights

Noisy inductive matrix completion can achieve reduced sample complexity and sharp error guarantees simultaneously.

Principles

• Side information reduces effective problem size.
• Inexact side information degrades error gracefully.
• Interpolation balances IMC efficiency and MC robustness.

Method

A projected gradient descent algorithm with spectral initialization minimizes a nonconvex objective, incorporating a balancing term and projecting onto an incoherence-preserving constraint set.

In practice

• Use IMC for low-sample scenarios.
• Consider interpolation for inexact side information.
• Cross-validate $\lambda$ for interpolation.

Topics

• Inductive Matrix Completion
• Sample Complexity Reduction
• Noisy Matrix Completion
• Inexact Side Information
• Projected Gradient Descent

Best for: AI Scientist, Research Scientist, Data Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.