On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices

The talk is based on the paper, On Mixing Distributions Via Random Orthogonal Matrices and the Spectrum of the Singular Values of Multi-Z Shaped Graph Matrices by Wenjun Cai, and, Aaron Potechin. arXiv link.

Abstract (from the paper)

In this paper, we introduce and analyze a new operation $\circ_R$ which mixes two distributions $\Omega$ and $\Omega’$ via a random orthogonal matrix. In particular, we take $\Omega\circ_R\Omega’$ 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 $\Omega$ and $\Omega’$ 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 $\Omega\circ_R\Omega’$ in terms of the moments of $\Omega$ and $\Omega’$. 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.