论文标题
大尺寸变形矩形矩阵的边缘统计
Edge statistics of large dimensional deformed rectangular matrices
论文作者
论文摘要
我们考虑$ y_t = y+\ sqrt {t} x，$ y $ as a $ p \ times n $确定信号矩阵的边缘变形矩形矩阵的边缘统计数据，其排名与$ n $相当，$ n $，$ x $ a $ x $是$ p \ times n $ p \ times n $ p \ times n $ nosis noise i.i.i.i.i.i.i.d.具有方差$ n^{ - 1} $的条目和$ t> 0 $给出了噪声级别。该模型被称为属于所谓的Signal-Plus-noise模型类别的大量多输入多输出（MIMO）系统的研究中的干扰加上噪声矩阵。对于$ t = 1 $的情况，在文献中已经在一定程度上研究了该模型的频谱统计数据。在本文中，我们研究了在更硬的制度$ n^{ - 2/3} \ ll t \ ll 1 $中，在奇异价值频谱中的奇异值和奇异矢量统计量。该体制比$ t = 1 $案例要难，因为一方面，$ yy^\ top $的经验频谱分布（ESD）的边缘行为对$ y_ty_t^\ top $ top $ y_ty_t^\ top $具有很强的影响，因为$ y_t \ ll 1 $是“小”，而另一方面，$ y_t $ y y y y y y y y y y y y y y y y y y y y y y y y y y y y ynemers of y y ynecy of y y y yncemastion of y ynmerm y of y yncamestiasy的$ y $很高。 n^{ - 2/3} $是“大”。在$ y上的某些规律性假设下，$我们证明了边缘普遍性，矩阵的特征值刚度和特征向量定位化$ y_ty_t^\ top $和$ y_t^\ y_t^\ top y_t $。这些结果可用于估计和推断大量的MIMO系统。为了证明主要结果，我们以$ y_ty_t^\ top $分析了渐近ESD的边缘行为，并在$ y_ty_t^\ top $的分解上建立一些尖锐的本地法律。这些结果可能具有独立的兴趣，并用作许多有关$ y_t $的光谱统计数据的其他问题的有用输入。
We consider the edge statistics of large dimensional deformed rectangular matrices of the form $Y_t=Y+\sqrt{t}X,$ where $Y$ is a $p \times n$ deterministic signal matrix whose rank is comparable to $n$, $X$ is a $p\times n$ random noise matrix with centered i.i.d. entries with variance $n^{-1}$, and $t>0$ gives the noise level. This model is referred to as the interference-plus-noise matrix in the study of massive multiple-input multiple-output (MIMO) system, which belongs to the category of the so-called signal-plus-noise model. For the case $t=1$, the spectral statistics of this model have been studied to a certain extent in the literature. In this paper, we study the singular value and singular vector statistics of $Y_t$ around the right-most edge of the singular value spectrum in the harder regime $n^{-2/3}\ll t \ll 1$. This regime is harder than the $t=1$ case, because on one hand, the edge behavior of the empirical spectral distribution (ESD) of $YY^\top$ has a strong effect on the edge statistics of $Y_tY_t^\top$ since $t\ll 1$ is "small", while on the other hand, the edge statistics of $Y_t$ is also not merely a perturbation of those of $Y$ since $t\gg n^{-2/3}$ is "large". Under certain regularity assumptions on $Y,$ we prove the edge universality, eigenvalues rigidity and eigenvector delocalization for the matrices $Y_tY_t^\top$ and $Y_t^\top Y_t$. These results can be used to estimate and infer the massive MIMO system. To prove the main results, we analyze the edge behavior of the asymptotic ESD for $Y_tY_t^\top$, and establish some sharp local laws on the resolvent of $Y_tY_t^\top$. These results can be of independent interest, and used as useful inputs for many other problems regarding the spectral statistics of $Y_t$.
