发表机构
University of Chicago; the Wharton School, University of Pennsylvania（芝加哥大学; 宾夕法尼亚大学沃顿商学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对多视图数据共享子空间估计，提出偏差校正子空间交集（BCSI）方法，消除AJIVE的二阶偏差，实现极小极大最优，并给出有限样本风险界与数值验证。
AI 中文摘要
在噪声数据矩阵间估计共享的低维子空间是多视图矩阵估计中的一个基本问题。我们在双视图JIVE模型下研究该问题，其中每个数据矩阵包含共享和视图特定的低秩成分。我们证明，标准的插入式子空间交集方法（包括AJIVE）会因经验奇异向量的方向相关泄漏而产生二阶偏差。我们提出偏差校正子空间交集（BCSI），该方法在估计共享子空间之前消除此偏差。我们为BCSI建立了有限样本风险界，该界适用于视图维度、信号强度和视图特定秩不相等的情况，且不需要对信号矩阵施加条件数假设。当共享秩和视图特定秩相当时，这些界与我们的极小极大下界在通用常数范围内匹配。由此得到的极小极大速率包含一个新的二阶项，该项源于当视图特定子空间近似对齐时，相对于缩小的谱间隙的二次泄漏扰动。此项在之前的JIVE极小极大下界中不存在。数值实验表明，当泄漏偏差显著时，BCSI优于AJIVE。在此过程中，我们为矩形尖峰矩阵的偏差校正泄漏Gram矩阵建立了一个非渐近集中结果，该结果可能具有独立的研究价值。
英文摘要
Estimating a low-dimensional subspace shared across noisy data matrices is a fundamental problem in multi-view matrix estimation. We study this problem under the two-view JIVE model, where each data matrix contains shared and view-specific low-rank components. We demonstrate that standard plug-in subspace intersection, including AJIVE, suffers from a second-order bias caused by direction-dependent leakage of the empirical singular vectors. We propose bias-corrected subspace intersection (BCSI), which removes this bias before estimating the shared subspace. We establish finite-sample risk bounds for BCSI that accommodate unequal view dimensions, signal strengths, and view-specific ranks and require no condition-number assumptions on the signal matrices. When the shared and view-specific ranks are comparable, these bounds match our minimax lower bounds up to universal constants. The resulting minimax rate contains a new second-order term, arising from quadratic leakage perturbations relative to the shrinking spectral gap when the view-specific subspaces are nearly aligned. This term is absent from previous JIVE minimax lower bounds. Numerical experiments demonstrate the advantage of BCSI over AJIVE when the leakage bias is pronounced. Along the way, we establish a nonasymptotic concentration result for the bias-corrected leakage Gram matrix of a rectangular spiked matrix, which may be of independent interest.
