Fisher information approximation of random orthogonal matrices by Gaussian matrices

2025/04/15 by Yutong Chen, Yutao Ma, Chen, Yutong +5 · 1 citation
Computer Science · Mathematics · #15B52 #60B20 #62B10 #62E17 #FOS: Mathematics #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2504.10887

openalex publication_date 2025/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ_n be an n× n Haar-invariant orthogonal matrix. Let Z_n be the p× q upper-left submatrix of Γ_n and G_n be a p× q matrix whose pq entries are independent standard normals, where p and q are two positive integers. Let L(√(n) Z_n) and L(G_n) be their joint distribution, respectively. Consider the Fisher information I(L(√(n) Z_n)|L(G_n)) between the distributions of √(n) Z_n and G_n. In this paper, we conclude that I(L(√(n) Z_n)|L(G_n))\longrightarrow 0 as n→∞ if pq=o(n) and it does not tend to zero if c=lim_n→∞(pq)/(n)∈(0, +∞). Precisely, we obtain that I(L(√(n) Z_n)|L(G_n))=(p²q(q+1))/(4n²)(1+o(1)) when p=o(n).

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• Large and Moderate Deviations for Entries of Orthogonal Matrices and the Stiefel Manifold

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