# Can I decompose a tensor in this way?

I found several "generalizations" of the singular value decomposition (with some overlap), here is a short list to some references: Tucker, Tensor Train, Tensor Train rank-1, CANDECOMP/PARAFAC (CP), Tensor rank, Kruskal.

None of these helped me find the following: given a tensor $G_{abcd}$ of shape $(r,s,t,t)$, I'd like to find a decomposition in the following form:

$$G = \sum_i \sigma_i\, w_iz_i^\dagger\otimes T_i$$

where $\mathrm{tr}(w_i^\dagger w_j)=\mathrm{tr}(z_i^\dagger z_j)=\mathrm{tr}(T_i^\dagger T_j)=\delta_{ij}$ and the $T_i$ matrices (which are $t\times t$) are rank-2. How can I do this? If so, can I make sure I use the least number of terms?

I would be happy even if I could only extract from $G$ the first element of such a decomposition.