  We compare two first-order low-rank optimization algorithms, namely
$\text{P}^2\text{GD}$ (Schneider and Uschmajew, 2015), which has been proven to
be apocalypse-prone (Levin et al., 2021), and its apocalypse-free version
$\text{P}^2\text{GDR}$ obtained by equipping $\text{P}^2\text{GD}$ with a
suitable rank reduction mechanism (Olikier et al., 2022). Here an apocalypse
refers to the situation where the stationarity measure goes to zero along a
convergent sequence whereas it is nonzero at the limit. The comparison is
conducted on two simple examples of apocalypses, the original one (Levin et
al., 2021) and a new one. We also present a potential side effect of the rank
reduction mechanism of $\text{P}^2\text{GDR}$ and discuss the choice of the
rank reduction parameter.
