Singular vector and singular subspace distribution for the matrix denoising model
研究了低秩信号矩阵加随机噪声的矩阵去噪模型，在信号超临界且矩阵维度可比时，推导了主奇异向量及其张成子空间与真实信号对应量的夹角和距离的极限分布，并讨论了统计推断应用。
In this paper, we study the matrix denoising model $Y=S+X$, where $S$ is a low rank deterministic signal matrix and $X$ is a random noise matrix, and both are $M\times n$. In the scenario that $M$ and $n$ are comparably large and the signals are supercritical, we study the fluctuation of the outlier singular vectors of $Y$, under fully general assumptions on the structure of $S$ and the distribution of $X$. More specifically, we derive the limiting distribution of angles between the principal singular vectors of $Y$ and their deterministic counterparts, the singular vectors of $S$. Further, we also derive the distribution of the distance between the subspace spanned by the principal singular vectors of $Y$ and that spanned by the singular vectors of $S$. It turns out that the limiting distributions depend on the structure of the singular vectors of $S$ and the distribution of $X$, and thus they are nonuniversal. Statistical applications of our results to singular vector and singular subspace inferences are also discussed.