发表机构
University of São Paulo; Columbia University（圣保罗大学; 哥伦比亚大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对含秩1项的矩形随机矩阵，确定估计秩1方向时最大误差收敛的精确尾条件，指出不满足条件时收敛失败，正则变化尾介于2-4时误差趋于无穷。
AI 中文摘要
我们考虑一个矩形随机矩阵，该矩阵由一个具有独立元素的矩阵加上一个秩1项构成。我们通过主导左奇异向量估计该项左侧的方向，误差会乘以该添加项大小的平方。对于均值为0、方差为1的元素，我们确定了以下陈述对每个确定性方向序列成立的精确尾条件：在添加项的所有大小上的最大误差收敛到由行与列的极限比值确定的同一固定值。该条件（弱于四阶矩的存在性）要求尾概率乘以阈值的四次方趋于0。若该条件不成立，即使在矩阵抽取前固定大小，当两个方向均为坐标向量时，收敛到该值也已失败。对于阶数介于2和4之间的正则变化尾，最大误差趋于无穷大。
英文摘要
We consider a rectangular random matrix formed by adding a rank-one term to a matrix with independent entries. The direction on the left side of that term is estimated by the leading left singular vector, and the error is multiplied by the square of the size of the added term. For entries with mean zero and variance one, we identify the exact tail condition for the following statement to hold for every deterministic sequence of directions: the largest error over all sizes of the added term converges to the same fixed value determined by the limiting ratio of rows to columns. This condition (weaker than the existence of fourth moments) requires that the tail probability, multiplied by the fourth power of the threshold, to tend to zero. If it fails, convergence to this value already fails when both directions are coordinate vectors, even at a size fixed before the matrix is drawn. For regularly varying tails of order between two and four, the largest error tends to infinity.
