On Hadamard powers of Random Wishart matrices

TL;DR

This paper investigates the positive semi-definiteness of Hadamard powers of random Wishart matrices, revealing a phase transition at a critical exponent related to the matrix dimensions.

Contribution

It extends Horn and Fitzgerald's deterministic results to random Wishart matrices, establishing probabilistic thresholds for positivity of entrywise powers.

Findings

For square Wishart matrices, positivity holds with high probability when  > 1.

Positivity fails with high probability when  < 1.

Transition point for rectangular matrices depends on the aspect ratio s.

Abstract

A famous result of Horn and Fitzgerald is that the $\beta$-th Hadamard power of any $n\times n$ positive semi-definite (p.s.d) matrix with non-negative entries is p.s.d $\forall \beta\geq n-2$ and is not necessarliy p.s.d for $\beta< n-2,$ with $\ \beta\notin \mathbb{N}$. In this article, we study this question for random Wishart matrix $A_n:={X_nX_n^T}$, where $X_n$ is $n\times n$ matrix with i.i.d. Gaussians. It is shown that applying $x\rightarrow |x|^{\alpha}$ entrywise to $A_n$, the resulting matrix is p.s.d, with high probability, for $\alpha>1$ and is not p.s.d, with high probability, for $\alpha<1$. It is also shown that if $X_n$ are $\lfloor n^{s}\rfloor\times n$ matrices, for any $s<1$, the transition of positivity occurs at the exponent $\alpha=s$.