使用 AJIVE 估计共享子空间：多个数据矩阵的力量与局限
Estimating shared subspace with AJIVE: the power and limitation of multiple data matrices
AI总结：
本文系统分析 AJIVE 在多矩阵 JIVE 共享子空间估计中的表现，证明其在高 SNR 下随矩阵数增加达到最优误差，而低 SNR 下存在不可消除的根本局限。
摘要：
整合数据分析通常需要厘清多个数据集之间的联合变异与个体变异，而 Joint and Individual Variation Explained（JIVE）模型常被用于应对这一挑战。尽管已有许多方法用于在 JIVE 下估计共享子空间，但对其性能的理论理解仍然有限，尤其是在涉及多个矩阵以及子空间错位程度不同的情形下。本文通过系统分析多矩阵设置中的共享子空间估计来弥补这一空白。我们聚焦于 Angle-based Joint and Individual Variation Explained（AJIVE）这一两阶段谱方法，并建立新的性能保证，以揭示其优势与局限。具体而言，我们证明在高信噪比（SNR）区间，AJIVE 的估计误差随矩阵数量增加而下降，展示了多矩阵整合的力量。相反，在低 SNR 设置下，AJIVE 表现出不会衰减的误差，凸显了根本性局限。为补充这些结果，我们推导了 minimax 下界，表明 AJIVE 在高 SNR 区间达到最优速率。此外，我们分析了一种借助 oracle 的谱估计量，以证明低 SNR 情形下的非衰减误差是一个根本性障碍。大量数值实验证实了我们的理论发现，并为 SNR、矩阵数量与子空间错位之间的相互作用提供了洞见。
英文摘要：
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.
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