Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 094, 22 pages      arXiv:2402.08663      https://doi.org/10.3842/SIGMA.2024.094
Contribution to the Special Issue on Asymptotics and Applications of Special Functions in Memory of Richard Paris

Complete Asymptotic Expansions for the Normalizing Constants of High-Dimensional Matrix Bingham and Matrix Langevin Distributions

Armine Bagyan and Donald Richards
Department of Statistics, Pennsylvania State University, University Park, PA 16802, USA

Received February 28, 2024, in final form October 10, 2024; Published online October 21, 2024

Abstract
For positive integers $d$ and $p$ such that $d \ge p$, let $\mathbb{R}^{d \times p}$ denote the set of $d \times p$ real matrices, $I_p$ be the identity matrix of order $p$, and $V_{d,p} = \bigl\{x \in \mathbb{R}^{d \times p} \mid x'x = I_p\bigr\}$ be the Stiefel manifold in $\mathbb{R}^{d \times p}$. Complete asymptotic expansions as $d \to \infty$ are obtained for the normalizing constants of the matrix Bingham and matrix Langevin probability distributions on $V_{d,p}$. The accuracy of each truncated expansion is strictly increasing in $d$; also, for sufficiently large $d$, the accuracy is strictly increasing in $m$, the number of terms in the truncated expansion. Lower bounds are obtained for the truncated expansions when the matrix parameters of the matrix Bingham distribution are positive definite and when the matrix parameter of the matrix Langevin distribution is of full rank. These results are applied to obtain the rates of convergence of the asymptotic expansions as both $d \to \infty$ and $p \to \infty$. Values of $d$ and $p$ arising in numerous data sets are used to illustrate the rate of convergence of the truncated approximations as $d$ or $m$ increases. These results extend recently-obtained asymptotic expansions for the normalizing constants of the high-dimensional Bingham distributions.

Key words: Frobenius norm; generalized hypergeometric function of matrix argument; Grassmann manifold; Hadamard's inequality; hippocampus; neural spike activity; Stiefel manifold; Super Chris (the rat); symmetric cone; zonal polynomial.

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