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Characterization of Gaussian Tensor Ensembles

Remi Bonnin

TL;DR

The paper extends Maxwell's isotropy characterization of Gaussian laws from vectors and matrices to real, complex, and self-dual tensor ensembles of any order $p\ge 1$ under orthogonal, unitary, or symplectic invariance. It proves that independence of entries up to symmetry and invariance forces the tensor law to be Gaussian, with density either $f(H) \propto \exp(- \|H - \beta \mathcal{I}\|_F^2 / \gamma)$ or equivalently $f(H) \propto \exp(- a \|H\|_F^2 + b \mathrm{Tr}^{\bullet\bullet}(H) + c)$, where only two trace invariants (the Frobenius norm and the paired trace when $p$ is even) can appear. The work constructs a complete basis of invariants via $p$-regular multigraphs (trace invariants) and uses Weingarten calculus to relate polynomial invariants to these graphs, thereby unifying vector/matrix results and enabling higher-order tensor analysis. It also discusses the Letac isotropy extension in the tensor setting and situates the results within the broader context of random tensor theory and tensor networks. Overall, the findings provide a principled, invariant-driven classification of Gaussian tensor laws and elucidate the role of a minimal set of trace invariants in high-order random tensors.

Abstract

The starting point of this work is a theorem due to Maxwell characterizing the distribution of a Gaussian vector with at least two coordinates. We define the Gaussian Orthogonal, Unitary and Symplectic Tensor Ensembles for notions of real symmetric, hermitian and self-dual hermitian tensors which recover the classical vector and matrix Gaussian Ensembles when the order is one and two. We give a basis of invariant polynomials for orthogonal, unitary and symplectic transformations and we prove a Maxwell-type theorem for these Gaussian tensor distributions unifying and extending the ones known for vectors and matrices.

Characterization of Gaussian Tensor Ensembles

TL;DR

The paper extends Maxwell's isotropy characterization of Gaussian laws from vectors and matrices to real, complex, and self-dual tensor ensembles of any order under orthogonal, unitary, or symplectic invariance. It proves that independence of entries up to symmetry and invariance forces the tensor law to be Gaussian, with density either or equivalently , where only two trace invariants (the Frobenius norm and the paired trace when is even) can appear. The work constructs a complete basis of invariants via -regular multigraphs (trace invariants) and uses Weingarten calculus to relate polynomial invariants to these graphs, thereby unifying vector/matrix results and enabling higher-order tensor analysis. It also discusses the Letac isotropy extension in the tensor setting and situates the results within the broader context of random tensor theory and tensor networks. Overall, the findings provide a principled, invariant-driven classification of Gaussian tensor laws and elucidate the role of a minimal set of trace invariants in high-order random tensors.

Abstract

The starting point of this work is a theorem due to Maxwell characterizing the distribution of a Gaussian vector with at least two coordinates. We define the Gaussian Orthogonal, Unitary and Symplectic Tensor Ensembles for notions of real symmetric, hermitian and self-dual hermitian tensors which recover the classical vector and matrix Gaussian Ensembles when the order is one and two. We give a basis of invariant polynomials for orthogonal, unitary and symplectic transformations and we prove a Maxwell-type theorem for these Gaussian tensor distributions unifying and extending the ones known for vectors and matrices.
Paper Structure (26 sections, 6 theorems, 93 equations, 2 figures)

This paper contains 26 sections, 6 theorems, 93 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Characterization of Gaussian distributions
  3. Real and complex tensors
  4. Real symmetric and antisymmetric tensors.
  5. Hermitian tensors.
  6. Self-dual hermitian tensors.
  7. Similarity transformations.
  8. Gaussian Tensor Ensembles
  9. Gaussian Orthogonal Tensor Ensemble.
  10. Gaussian Unitary Tensor Ensemble.
  11. Gaussian Symplectic Tensor Ensemble.
  12. Trace invariants
  13. Rank one and two.
  14. Maxwell Theorem
  15. Letac extension
  16. ...and 11 more sections

Key Result

Theorem 1

For any $p\geq 1$ and $N\geq 2$, a real symmetric (resp. hermitian, resp. self-dual hermitian) random tensor $H$ of order $p$ and dimension $N$ (realcomplextens) has : if and only if it belongs to the Gaussian Orthogonal (resp. Unitary, resp. Symplectic) Tensor Ensemble (GTE), that is it has a law with a density of the form where $(\alpha,\beta,\gamma) \in \mathbb{R}^*_+ \times \mathbb{R} \times

Figures (2)

  • Figure 1: Bouquet graph ($p=10$), melon graph ($p=5$), another trace invariant of rank $2$ ($p=5$), respectively $m_G(H)=\mathrm{Tr}^{\bullet \bullet}(H)$, $m_G(H)=\| H \|^2$, $m_G(H)=\sum_{a,b,c,d,e} H_{aabcd} H_{ebcde}$.
  • Figure 2: Sketch of proof, unitary case.

Theorems & Definitions (16)

  • Theorem : Main
  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem : Main
  • Proposition 1
  • Lemma 1
  • ...and 6 more