子空间局部律下的介观矩形尖峰：异常值与奇异子空间
Mesoscopic Rectangular Spikes under Subspace Local Laws: Outlier Values and Singular Subspaces
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中文总结 AI 辅助
本文研究子空间局部律下的介观矩形尖峰，通过该假设定位计数异常值、确定奇异子空间，在四类场景验证假设并计算Marchenko–Pastur噪声的常数，为高维数据相关问题提供理论支撑。
中文摘要 AI 辅助
矩形数据矩阵常被建模为噪声加上低秩信号。当秩固定时，经典结论为：信号方向需超过临界强度，才会产生超出噪声主体的奇异值；超过阈值后，观测的奇异向量会保留已确定、可计算的植入方向比例。本文研究当信号方向数随维度增长时的情形，核心依赖子空间局部律假设：从信号子空间内部看，噪声的预解式应类似标量。仅该假设即可定位并计数异常值，还能以全纯形式确定异常奇异子空间；结论通过谱投影算子表述，故在尖峰强度碰撞或比近似误差更接近时仍有意义（随秩增长必然出现此类情况）。随后在四种不同场景下验证该假设，涵盖随机定向信号方向的确定性噪声、含固定确定性元素的独立元素等，对Marchenko–Pastur噪声明确计算了所有常数。
英文摘要
A rectangular data matrix is often modelled as noise plus a signal of low rank. When that rank is fixed the picture is classical: a signal direction must exceed a critical strength before it produces a singular value outside the noise bulk, and above the threshold the singular vectors of the observation retain a definite, computable fraction of the planted direction. We ask what survives when the number of signal directions grows with the dimension. Everything here rests on one hypothesis, which we call a subspace local law: seen from inside the signal subspace, the resolvent of the noise should look like a scalar. We show that this hypothesis alone locates the outliers and counts them, and, in a holomorphic form, determines the outlier singular subspaces as well. The conclusions are stated through spectral projectors rather than individual singular vectors, so they remain meaningful when spike strengths collide or come closer together than the error of the approximation, which at growing rank they must. We then verify the hypothesis in four genuinely different settings, from deterministic noise viewed through randomly oriented signal directions to independent entries with fixed deterministic ones, and for Marchenko--Pastur noise we compute every constant explicitly.
