On Invariant Conjugate Symmetric Statistical Structures on the Space of Zero-Mean Multivariate Normal Distributions
Hikozo Kobayashi, Takayuki Okuda
TL;DR
The paper investigates invariant conjugate symmetric statistical structures on the zero-mean $n$-variate normal family $\mathcal{N}_0$, revealing that GL$(n,\mathbb{R})$-invariance (with respect to the Fisher metric) is in one-to-one correspondence with homogeneous cubic real symmetric polynomials in $n$ variables. It shows that for $n\ge 2$ this invariance does not uniquely determine the Amari–Chentsov $α$-tensors on $\mathcal{N}_0$, contrasting with the full family where affine invariance yields a unique characterization. The approach leverages the identification $\mathcal{N}_0 \cong \mathrm{Sym}^+(n,\mathbb{R})$, the $GL(n,\mathbb{R})$-action with isotropy $O(n)$, and invariant theory of cubic polynomials, establishing a 3-dimensional invariant CS space for general $n$ (dim$(S^3(\mathrm{Sym}(n,\mathbb{R})^*)^K)=3$ for $n\ge 3$) that is spanned by $C_1,C_2,C_3$ and maps to $α\sum_i x_i^3$ under the natural correspondence. A key theoretical ingredient is the demonstration that invariant structures on any Riemannian symmetric space are conjugate symmetric, reinforcing the structural rigidity of homogeneous statistical manifolds. These results illuminate the space of invariant information-geometric tensors on symmetric spaces and provide explicit bases and correspondences for cubic invariants in terms of trace functionals.
Abstract
By the results of Furuhata--Inoguchi--Kobayashi [Inf. Geom. (2021)] and Kobayashi--Ohno [Osaka Math. J. (2025)], the Amari--Chentsov $α$-connections on the space $\mathcal{N}$ of all $n$-variate normal distributions are uniquely characterized by the invariance under the transitive action of the affine transformation group among all conjugate symmetric statistical connections with respect to the Fisher metric. In this paper, we investigate the Amari--Chentsov $α$-connections on the submanifold $\mathcal{N}_0$ consisting of zero-mean $n$-variate normal distributions. It is known that $\mathcal{N}_0$ admits a natural transitive action of the general linear group $GL(n,\mathbb{R})$. We establish a one-to-one correspondence between the set of $GL(n,\mathbb{R})$-invariant conjugate symmetric statistical connections on $\mathcal{N}_0$ with respect to the Fisher metric and the space of homogeneous cubic real symmetric polynomials in $n$ variables. As a consequence, if $n \geq 2$, we show that the Amari--Chentsov $α$-connections on $\mathcal{N}_0$ are not uniquely characterized by the invariance under the $GL(n,\mathbb{R})$-action among all conjugate symmetric statistical connections with respect to the Fisher metric. Furthermore, we show that any invariant statistical structure on a Riemannian symmetric space is necessarily conjugate symmetric.