I’m studying the paper Expressive power of tensor-network factorizations for probabilistic modeling by Glasser et al. In equation S6 (page 2 of supplementary material, excerpt of paper figure below) it is claimed that a 2-local unitary $\rm U$ may be split into a contraction of two 1-local gate tensors through a singular value decomposition (SVD). How this is done, however, is not clear to me.
In tensor notation, the entries of a unitary $\rm U$ acting on two qudits may be written $U^{ij}_{kl}$, with the first upper and lower index referring to the first qudit, and the second upper and lower index referring to the second qudit. Splitting $\rm U$ into a contraction of two 1-local gate tensors $\rm A$ and $\rm B$ would mean that $$U^{ij}_{kl}=\sum_\alpha A^{i\alpha}_kB^j_{l\alpha}\tag{1}$$ or, equivalently $${\rm U}=\sum_\alpha{\rm A}^{\kern-0.085em\alpha}\otimes{\rm B}_\alpha\tag{2}$$ with $\rm A$ acting on the first qudit and $\rm B$ acting on the second. Now, when $\rm U$ is decomposed into $\rm C\Sigma\rm D^\dagger$, we have in tensor notation that $$U^{ij}_{kl}=\sum_{\gamma\delta}\sigma_{\gamma\delta}C^{ij}_{\gamma\delta}(D^\dagger)^{\gamma\delta}_{kl}.\tag{3}$$ This is much different from the first decomposition, as $\rm C$ and $\rm D$ still act on both qudits.
Can somebody explain how the decomposition should be carried out so as to arrive at eq. 1 or 2?