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Let $A,B$ be two symmetric tensors of same rank $m$; and orders $k$ and $\ell$, respectively. In particular, assume that $A,B$ admits the following structure: There exists $v_1,\dots,v_m\in\mathbb{R}^n$, such that, $$ A=\sum_{j=1}^m \underbrace{v_j\otimes v_j\otimes \cdots \otimes v_j}_{k\text{ times}} \quad\text{and}\quad B=\sum_{j=1}^m \underbrace{v_j\otimes v_j\otimes \cdots \otimes v_j}_{\ell\text{ times}}. $$ My question is about characterizing these objects.

1) Let $A=\sum_{j=1}^{N_1}\omega_j\otimes\cdots\otimes \omega_j$ and $B=\sum_{j=1}^{N_2}\xi_j\otimes\cdots\otimes \xi_j$ be two decompositions of $A$ and $B$. What can we say about the collections $\{\omega_j\}$ and $\{\xi_j\}$? That is, I know essentially that $A$ and $B$ are two symmetric tensors of same rank, different orders; constructed from the same family. Then, does this say anything about any decomposition of them?

2) In general, how can we encode this knowledge into a constraint involving $A$ and $B$ (such as convex or polyhedral constraint)?

Some Progress

a) If I assume there exists a $\lambda$ such that $\langle v_j,e\rangle =\lambda$ for every $j$ (where $e$ is a vector of all ones), then I can construct a linear relation between $A$ and $B$ (that is, a set of polyhedral constraints, relating each element of $B$ as sum of several elements of $A$).

b) In case of $k=2$ and $\ell=1$, we do not have general identifiability: The tensors $A$ and $B$ are essentially $V^T V$ and $e^T V$ where $V$ is a $m\times n$ matrix whose rows are $v_j$, and $e$ is a vector of all ones. It is easy to cook up examples $V_1,V_2$ where $V_1^T V_1 = V_2^T V_2$, but $e^T V_1\neq e^T V_2$. This makes me think some further condition must be assumed in order to say more. Am I right on this?

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    $\begingroup$ It sounds a little like a moment reconstruction problem. I don’t know that literature as well, but searching for moment reconstruction or blind source separation might turn up something. $\endgroup$ – Zach Teitler Aug 21 '19 at 18:51

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