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Singular vector and singular subspace distribution for the matrix denoising model
Annals of Statistics ( IF 4.5 ) Pub Date : 2021-01-29 , DOI: 10.1214/20-aos1960
Zhigang Bao , Xiucai Ding , and Ke Wang

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.

中文翻译:

矩阵去噪模型的奇异矢量和奇异子空间分布

在本文中,我们研究了矩阵去噪模型$ Y = S + X $,其中$ S $是低秩确定性信号矩阵,$ X $是随机噪声矩阵,都是$ M \ n n $。在$ M $和$ n $相对较大且信号超临界的情况下,我们在完全笼统地假设$ S $的结构和$ s的分布的情况下研究$ Y $的异常奇异矢量的波动X $。更具体地说,我们得出$ Y $的主要奇异矢量和它们的确定性对应物$ S $的奇异矢量之间的角度限制分布。此外,我们还导出了主空间奇异向量$ Y $跨越的子空间与$ S $奇异向量跨越的子空间之间的距离分布。事实证明,极限分布取决于$ S $的奇异向量的结构和$ X $的分布,因此它们是非通用的。还讨论了我们的结果在奇异向量和奇异子空间推论中的统计应用。
更新日期:2021-01-29
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