Robust Matrix Estimation with Side Information

2026-03-25

AI Summary

This work addresses key limitations in high-dimensional matrix estimation—namely, insufficient integration of row/column side information, restricted nonlinear modeling capabilities, and inadequate noise handling—by proposing a four-component decomposition framework. The target matrix is decoupled into a row-column nonlinear interaction term, row-specific and column-specific main effects, and a low-rank residual structure. Each component is estimated via sieve basis projection and nuclear norm regularization, then aggregated to yield a flexible estimator that accommodates nonlinear interactions, leverages unilateral features, and explicitly models noise. The method is applicable under both missing-at-random (MAR) and missing-not-at-random (MNAR) mechanisms, including block-wise missingness in causal panel settings. Theoretical analysis establishes convergence rates under varying strengths of auxiliary information, while simulations and an empirical study on tobacco sales demonstrate substantial improvements over conventional low-rank and spectral methods in matrix completion and treatment effect estimation.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

We introduce a flexible framework for high-dimensional matrix estimation to incorporate side information for both rows and columns. Existing approaches, such as inductive matrix completion, often impose restrictive structure-for example, an exact low-rank covariate interaction term, linear covariate effects, and limited ability to exploit components explained only by one side (row or column) or by neither-and frequently omit an explicit noise component. To address these limitations, we propose to decompose the underlying matrix as the sum of four complementary components: (possibly nonlinear) interaction between row and column characteristics; row characteristic-driven component, column characteristic-driven component, and residual low-rank structure unexplained by observed characteristics. By combining sieve-based projection with nuclear-norm penalization, each component can be estimated separately and these estimated components can then be aggregated to yield a final estimate. We derive convergence rates that highlight robustness across a range of model configurations depending on the informativeness of the side information. We further extend the method to partially observed matrices under both missing-at-random and missing-not-at-random mechanisms, including block-missing patterns motivated by causal panel data. Simulations and a real-data application to tobacco sales show that leveraging side information improves imputation accuracy and can enhance treatment-effect estimation relative to standard low-rank and spectral-based alternatives.

Problem

• matrix estimation
• side information
• high-dimensional
• low-rank
• missing data

Innovation

• matrix estimation
• side information
• nonlinear interaction
• nuclear-norm penalization
• missing data