Trace Moments of the Sample Covariance Matrix with Graph-Coloring

TL;DR

This paper provides an explicit expansion of trace moments and covariances of the sample covariance matrix using graph coloring and decomposition, applicable to both high- and low-dimensional data.

Contribution

It introduces a novel graph-coloring and decomposition approach to analyze trace moments of sample covariance matrices, enhancing understanding of their structure.

Findings

Explicit formulas for trace moments and covariances.

Graph coloring simplifies the analysis of complex graph structures.

Decomposition into seed graphs aids in deriving elegant formulas.

Abstract

Let $S_{p,n}$ denote the sample covariance matrix based on $n$ independent identically distributed $p$-dimensional random vectors in the null-case. The main result of this paper is an explicit expansion of trace moments and power-trace covariances of $S_{p,n}$ simultaneously for both high- and low-dimensional data. To this end we expand a well-known ansatz of describing trace moments as weighted sums over routes or graphs. The novelty to our approach is an inherent coloring of the examined graphs and a decomposition of graphs into their tree-structure and their \textit{seed graphs}, which allows for some elegant formulas explaining the effect of the tree structures on the number of Euler-tours. The weighted sums over graphs become weighted sums over the possible seed graphs, which in turn are much easier to analyze.