  The affine rank minimization (ARM) problem is well known for both its
applications and the fact that it is NP-hard. One of the most successful
approaches, yet arguably underrepresented, is iteratively reweighted least
squares (IRLS), more specifically $\mathrm{IRLS}$-$0$. Despite comprehensive
empirical evidence that it overall outperforms nuclear norm minimization and
related methods, it is still not understood to a satisfying degree. In
particular, the significance of a slow decrease of the therein appearing
regularization parameter denoted $\gamma$ poses interesting questions. While
commonly equated to matrix recovery, we here consider the ARM independently. We
investigate the particular structure and global convergence property behind the
asymptotic minimization of the log-det objective function on which
$\mathrm{IRLS}$-$0$ is based. We expand on local convergence theorems, now with
an emphasis on the decline of $\gamma$, and provide representative examples as
well as counterexamples such as a diverging $\mathrm{IRLS}$-$0$ sequence that
clarify theoretical limits. We present a data sparse, alternating realization
$\mathrm{AIRLS}$-$p$ (related to prior work under the name $\mathrm{SALSA}$)
that, along with the rest of this work, serves as basis and introduction to the
more general tensor setting. In conclusion, numerical sensitivity experiments
are carried out that reconfirm the success of $\mathrm{IRLS}$-$0$ and
demonstrate that in surprisingly many cases, a slower decay of $\gamma$ will
yet lead to a solution of the ARM problem, up to the point that the exact
theoretical phase transition for generic recoverability can be observed.
Likewise, this suggests that non-convexity is less substantial and problematic
for the log-det approach than it might initially appear.
