Robust Matrix Estimation with Side Information

TL;DR

This paper proposes a flexible, robust matrix estimation framework that incorporates side information for rows and columns, improving accuracy over traditional low-rank methods, especially in complex, partially observed data scenarios.

Contribution

It introduces a novel decomposition of the matrix into four components and combines sieve projection with nuclear-norm penalization for improved estimation.

Findings

Enhanced imputation accuracy with side information

Robust convergence rates across various model configurations

Effective handling of missing data mechanisms, including block-missing patterns

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.