I am studying the eigenvalues of large random tensors and realise that very little is known about it. I was wondering what is already known and what could be potential leads to find their limiting distribution.

Random symmetric tensor

Let $\mathbf{T}$ be a real fully symmetric tensor of order $3$ and size $N$. Its components can be represented as $T_{ijk}\in \mathbb{R}$ for all $1\leq i,j,k\leq N$. By symmetric I mean that if I permute any indices the value stays the same: \begin{equation} T_{ijk}=T_{ikj}=\dots=T_{jki} \end{equation}


One possible definition of eigenvalues: let $x\in \mathbb{C}^N$ and $\lambda\in\mathbb{C}$ such that: \begin{equation} \sum_{jk}^NT_{ijk}x_kx_j=\lambda x_i \label{eq1} \end{equation} with the constraint that $\sum_i x_i^2=1$.

It has been shown (in [1] for example) that for a tensor of order 3 and size $N$ there are $2^{N}-1$ eigenvalues $\lambda$. Perhaps an intuitive idea on why is to see that the eigenvalue equation is composed of $N$ quadratic equations. Each equation possesses two roots, hence $2^N$ possible combinations of solutions, minus the normalisation constraint.

Possible leads

It would be interesting to find the limiting distribution of eigenvalues of random tensors for large $N$. The eigenvalues are real only if their eigenvectors are real. I conjecture that the purely real eigenvalues scale with $\sqrt{N}$ whereas the general eigenvalues scale with $N$. That is to say, I would expect the real eigenvalues to be distributed on a closed non trivial interval when the elements of the tensor have a variance scaled with $1/\sqrt{N}$.

1. Reducing it to a non-linear matrix problem:

We define the matrix $M$ such that: $M_{ij}=\sum_{k}T_{ijk}x_k$. Therefore the eigenvalue problem would boil down to find the eigenvalues of $M$:

\begin{align} \sum_{jk}^NT_{ijk}x_kx_j&=\lambda x_i\\ \implies \sum_{j}M_{ij}x_j&=\lambda x_i \end{align}

Where the elements of $M$ now also depend on the eigenvectors.

2. Choosing an appropriate basis to uncouple the equations:

If we fix the index $i$, we can represent $T$ as a collection of $N$ matrices: ${M}^{(1)},{M}^{(2)}\dots{M}^{(N)}$ where $T_{ijk}={M_{jk}}^{(i)}$.

Since $T$ is real and fully symmetric, ${M}^{(i)}$ is also real symmetric $(1\leq i\leq N)$. There will also share some elements in common and not be fully independent from one another. For example $T_{ijk}=T_{kij}\implies M_{ij}^{(1)}=M_{1j}^{(i)}$. So matrix $M^{(1)}$ will be composed of the first column of all the other matrices. What is the relation between the eigenvalues of $M^{(i)}$ and $M^{(j)}$? Are they directly linked to the eigenvalues of $T$?

Here were some of the first ideas that came naturally to my mind. If you know a paper or some ideas on how to tackle this problem this would be very interesting.

[1]: Cartwright, D., & Sturmfels, B. (2013). The number of eigenvalues of a tensor. Linear algebra and its applications, 438(2), 942-952.


1 Answer 1


A productive way to approach this problem has been to focus on the real eigenvalues of the symmetric tensor and identify these with the critical points on the unit sphere of a Kostlan polynomial. In this way Paul Breiding was able to find an exact answer for the expected number of real eigenvalues when the symmetric tensor has a Gaussian distribution.

The generalization of the Wigner semicircle law to real symmetric tensors (tensor GOE) was studied by Razvan Gurau.

  • $\begingroup$ Looking at the eigenvalues as the roots of random polynomials is something very new to me, thank you for your reference. On the other hand I studied the work of Razvan Gurau and the spectral density he derives in his paper does not seem to be directly linked with the eigenvalues. I computed the eigenvalues numerically for low ranks and size $\approx 20$ and the distribution and choice of scaling is noticeably different from his spectral density... $\endgroup$
    – Matt
    Apr 20, 2021 at 9:30
  • $\begingroup$ The link between the actual distribution of eigenvalues and his spectral density is rather obscure. $\endgroup$
    – Matt
    Apr 20, 2021 at 9:31
  • $\begingroup$ Indeed, the variational characterization of tensor eigenpairs provides an interesting viewpoint. Connections can be made with the study of the critical points of the random landscape arising from that optimization problem, see e.g. the paper "The landscape of the spiked tensor model" by G. Ben Arous, S. Mei, A. Montanari and M. Nica. Also, this connection between tensor and matrix eigenvalues has been exploited in the study of a rank-one tensor model with Gaussian noise, see " A Random Matrix Perspective on Random Tensors" by myself, R. Couillet and P. Comon. $\endgroup$
    – jhgoulart
    Jan 20 at 13:27

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