  Affine sum-of-ranks minimization (ASRM) generalizes the affine rank
minimization (ARM) problem from matrices to tensors. Here, the interest lies in
the ranks of a family $\mathcal{K}$ of different matricizations. Transferring
our priorly discussed results on asymptotic log-det rank minimization, we show
that iteratively reweighted least squares with weight strength $p = 0$ remains
a, theoretically and practically, particularly viable method denoted as
$\mathrm{IRLS}$-$0\mathcal{K}$. As in the matrix case, we prove global
convergence of asymptotic minimizers of the log-det sum-of-ranks function to
desired solutions. Further, we show local convergence of
$\mathrm{IRLS}$-$0\mathcal{K}$ in dependence of the rate of decline of the
therein appearing regularization parameter $\gamma \searrow 0$. For
hierarchical families $\mathcal{K}$, we show how an alternating version
($\mathrm{AIRLS}$-$0\mathcal{K}$, related to prior work under the name
$\mathrm{SALSA}$) can be evaluated solely through tensor tree network based
operations. The method can thereby be applied to high dimensions through the
avoidance of exponential computational complexity. Further, the otherwise
crucial rank adaption process becomes essentially superfluous even for
completion problems. In numerical experiments, we show that the therefor
required subspace restrictions and relaxation of the affine constraint cause
only a marginal loss of approximation quality. On the other hand, we
demonstrate that $\mathrm{IRLS}$-$0\mathcal{K}$ allows to observe the
theoretical phase transition also for generic tensor recoverability in
practice. Concludingly, we apply $\mathrm{AIRLS}$-$0\mathcal{K}$ to larger
scale problems.
