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

TL;DR

This paper introduces a new distribution mixing operation via random orthogonal matrices, analyzes its properties, and applies it to understand the spectrum of singular values in complex graph matrices, connecting to non-crossing partitions.

Contribution

It defines and studies a novel distribution operation $ullet_R$, demonstrating its properties and applications to spectral analysis of Z-shaped graph matrices, extending previous work.

Findings

The operation $ullet_R$ is commutative and associative.

Moments of the mixed distribution relate to moments of original distributions.

Application to Z-shaped graph matrices clarifies their spectral behavior.

Abstract

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.