Suppose tensor $\mathcal{A}$ is a symmetric real tensor of order $k$. Then, symmetric outer product decomposition of $\mathcal{A}$ is
$$ \mathcal{A} = \sum_{i=1}^p \lambda_i v_i^{\bigotimes k}, $$
where $v_i^{\bigotimes k}$ is the k-times outer product of $v_i \in \mathbb{R}^n$ and $p$ is the symmetric rank of $\mathcal{A}$.
Alternatively, symmetric multilinear decomposition of the same tensor $\mathcal{A}$ is
$$ \mathcal{A} = (U, U, ..., U) \ \cdot \ \mathcal{C} $$ where $r$ is the multilinear rank of $\mathcal{A}$, matrix $U \in \mathbb{R}^{n \times r}$ and $\mathcal{C}$ is the core tensor of shape $r \times r \times \ldots r \ (k$ times).
My Questions are
- Is there any relationship between multilinear rank $r$ and symmetric rank $p$?
- Can we write symmetric multilinear decomposition in the form of outer product decomposition? To be specific, are there any relationship between matrix $U$ and matrix $V$ ($V$ consists of all vectors $v_i$)?
Any ideas are welcome. Thank you in advance.