# On symmetric tensors with same rank, different orders

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?

• 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. – Zach Teitler Aug 21 '19 at 18:51