论文标题
通过随机正交矩阵和多Z形矩阵的奇异值的频谱进行混合分布
On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices
论文作者
论文摘要
在本文中，我们通过随机正交矩阵介绍和分析了一个新操作$ \ Circ_ {r} $，该$ \ Circ_ {r} $混合了两个分布$ω$和$ω'$。特别是，我们将$ω\ circ_rω'$作为限制为$ n \ to \ infty $ $ drd' $的分布，其中$ d $ and $ d $和$ d $是$ n \ times n $ n $对角矩阵的对角线矩阵，其对角线的矩阵的分布分布$ω$和$Ω'$ $ $ $ $ $ $ n是$ n是$ n \ n是$ n \ tiese tiee tiee tiee tiese tiee tiee tiee tiese tiese tiese tiese tiese tiese tiese tiese tiese \ tiese tiese tiese \ tiese。我们证明$ \ circ_r $具有多个不错的属性。我们首先观察到$ \ circ_r $是交换性和关联性的，并在$ω$和$ω'$的矩时计算$ω\Circ_rΩ$的矩。然后，我们证明$ \ circ_r $与Z形和多Z形的图形矩阵的奇异值相互作用。这使我们可以回答以前的论文提出的问题，即当输入分布不是$ \ { - 1,1 \} $时，如何描述Z形和多Z形的图形矩阵的奇异值的频谱。在我们的分析中，我们表明，分布的时刻与非交叉分区密切相关，并证明了可能具有独立利益的非交叉分区的许多新结果。
In this paper, we introduce and analyze a new operation $\circ_{R}$ which mixes two distributions $Ω$ and $Ω'$ via a random orthogonal matrix. In particular, we take $Ω\circ_R Ω'$ to be the limit as $n \to \infty$ of the distribution of singular values of $DRD'$ where $D$ and $D'$ are $n \times n$ diagonal matrices whose diagonal entries have distributions $Ω$ and $Ω'$ respectively and $R$ is a random $n \times n$ orthogonal matrix. We show that $\circ_R$ has several nice properties. We first observe that $\circ_R$ is commutative and associative and compute the moments of $Ω\circ_R Ω'$ in terms of the moments of $Ω$ and $Ω'$. We then show that $\circ_R$ interacts very nicely with the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices. This allows us to answer the question posed by our previous paper of how to describe the spectrum of the singular values of Z-shaped and multi-Z-shaped graph matrices when the input distribution is not $\{-1,1\}$. In our analysis, we show that the moments of our distributions are closely connected to non-crossing partitions and prove a number of new results on non-crossing partitions which may be of independent interest.
