Group-Aware Matrix Estimation and Latent Subspace Recovery

2026-05-19

AI Summary

This work addresses the challenge that in heterogeneous data, row samples may belong to multiple meta-categories simultaneously, which undermines the ability of standard low-rank matrix completion methods to preserve subgroup-specific structures. To tackle this, the paper introduces GAME, the first convex optimization estimator capable of handling overlapping subgroups. GAME applies nuclear-norm regularization to submatrices corresponding to overlapping subgroups, thereby balancing information sharing across groups and preservation of local latent structures within a unified coordinate system. Theoretical analysis provides finite-sample guarantees for both reconstruction error and subspace recovery, revealing explicit dependencies on sampling density, subgroup rank, and overlap structure. Experiments demonstrate that GAME significantly outperforms existing baselines on synthetic data and real-world applications—including recommender systems, ecology, and neuroscience—yielding improved reconstruction accuracy and fidelity of recovered latent subspaces.

Abstract

Modern matrix completion problems often involve heterogeneous data whose rows simultaneously belong to many meta-categories, such as demographic and age groups in recommendation systems, or region and recording session labels in neural electrophysiological experiments. Standard low-rank estimators impose a single global latent geometry, which can recover average structure but may smooth away subgroup-specific variation, especially when observations are unevenly distributed across groups. We introduce Group-Aware Matrix Estimation (GAME), a convex estimator for overlapping subgroup-wise low-rank matrix estimation. GAME regularizes category-specific submatrices through overlapping nuclear-norm penalties, allowing related groups to borrow information while preserving local latent structure in a shared coordinate system. We provide finite-sample guarantees for both reconstruction error and subgroup-specific subspace recovery, showing how performance depends on sampling density, subgroup rank, and overlap structure. Experiments on synthetic, recommendation, ecological, and neuroscience datasets show that GAME is most beneficial in structured missingness regimes, where subgroup-aware regularization improves both reconstruction accuracy and latent subspace fidelity. Across these benchmarks, GAME is competitive or best among global low-rank, side-information, and modern imputation baselines, with the largest gains when subgroups exhibit distinct low-rank structure.

Problem

• matrix completion
• heterogeneous data
• subgroup-specific variation
• overlapping groups
• latent subspace

Innovation

• Group-Aware Matrix Estimation
• overlapping nuclear-norm regularization
• latent subspace recovery
• heterogeneous matrix completion
• structured missingness