Estimating shared subspace with AJIVE: the power and limitation of multiple data matrices

TL;DR

This paper analyzes the AJIVE method for estimating shared subspaces across multiple datasets, revealing its strengths in high-SNR conditions and limitations in low-SNR scenarios through theoretical guarantees and numerical experiments.

Contribution

It provides the first systematic performance analysis of AJIVE in multi-matrix settings, establishing error bounds and fundamental limitations based on SNR levels.

Findings

AJIVE's error decreases with more matrices in high-SNR regimes.

In low-SNR settings, AJIVE's error remains non-diminishing, indicating a fundamental limitation.

Theoretical bounds show AJIVE is optimal in high-SNR conditions.

Abstract

Integrative data analysis often requires disentangling joint and individual variations across multiple datasets, a challenge commonly addressed by the Joint and Individual Variation Explained (JIVE) model. While numerous methods have been developed to estimate the shared subspace under JIVE, the theoretical understanding of their performance remains limited, particularly in the context of multiple matrices and varying degrees of subspace misalignment. This paper bridges this gap by providing a systematic analysis of shared subspace estimation in multi-matrix settings. We focus on the Angle-based Joint and Individual Variation Explained (AJIVE) method, a two-stage spectral approach, and establish new performance guarantees that uncover its strengths and limitations. Specifically, we show that in high signal-to-noise ratio (SNR) regimes, AJIVE's estimation error decreases with the number of matrices, demonstrating the power of multi-matrix integration. Conversely, in low-SNR settings, AJIVE exhibits a non-diminishing error, highlighting fundamental limitations. To complement these results, we derive minimax lower bounds, showing that AJIVE achieves optimal rates in high-SNR regimes. Furthermore, we analyze an oracle-aided spectral estimator to demonstrate that the non-diminishing error in low-SNR scenarios is a fundamental barrier. Extensive numerical experiments corroborate our theoretical findings, providing insights into the interplay between SNR, the number of matrices, and subspace misalignment.