Fisher information approximation of random orthogonal matrices by Gaussian matrices
Yutong Chen, Yutao Ma, Shuhong Xie, Zhuoya Yao
TL;DR
This paper analyzes how well the distribution of the top-left $p\times q$ submatrix ${Z}_n$ of a Haar-invariant orthogonal matrix, scaled by $\sqrt{n}$, is approximated by a standard Gaussian matrix in the sense of Fisher information. The authors express the Fisher information between $\mathcal{L}(\sqrt{n}{Z}_n)$ and $\mathcal{L}(G_n)$ as a functional of the eigenvalues of $Z_n'Z_n$, identify their joint law as a Jacobi ensemble, and compute high-precision asymptotics for key moments using beta-Jacobi/tridiagonal representations. They prove that $I(\mathcal{L}(\sqrt{n}{Z}_n)|\mathcal{L}(G_n))\to 0$ if $pq= o(n)$, and that in the regime $p=o(n)$ the asymptotic rate is $I(\cdot)=\frac{p^2 q (q+1)}{4 n^2}(1+o(1))$, with non-vanishing information when $pq/n$ converges to a positive constant. This provides a precise Fisher-information threshold for Gaussian approximation of random orthogonal submatrices and complements existing KL-divergence and total-variation results by identifying the exact scaling governing information-based convergence.
Abstract
Let $Γ_n$ be an $n\times n$ Haar-invariant orthogonal matrix. Let ${ Z}_n$ be the $p\times q$ upper-left submatrix of $Γ_n$ and ${G}_n$ be a $p\times q$ matrix whose $pq$ entries are independent standard normals, where $p$ and $q$ are two positive integers. Let $\mathcal{L}(\sqrt{n} {Z}_n)$ and $\mathcal{L}({G}_n)$ be their joint distribution, respectively. Consider the Fisher information $I(\mathcal{L}(\sqrt{n} { Z}_n)|\mathcal{L}(G_n))$ between the distributions of $\sqrt{n} {Z}_n$ and ${ G}_n.$ In this paper, we conclude that $$I(\mathcal{L}(\sqrt{n} {Z}_n)|\mathcal{L}(G_n))\longrightarrow 0 $$ as $n\to\infty$ if $pq=o(n)$ and it does not tend to zero if $c=\lim\limits_{n\to\infty}\frac{pq}{n}\in(0, +\infty).$ Precisely, we obtain that $$I(\mathcal{L}(\sqrt{n} {Z}_n)|\mathcal{L}(G_n))=\frac{p^2q(q+1)}{4n^2}(1+o(1))$$ when $p=o(n).$