An apocalypse-free first-order low-rank optimization algorithm

We consider the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the real determinantal variety, and present a first-order algorithm designed to find stationary points of that problem. This algorithm applies steps of steepest descent with backtracking line search on the variety, as proposed by Schneider and Uschmajew (2015), but by taking the numerical rank into account to perform suitable rank reductions. We prove that this algorithm produces sequences of iterates the accumulation points of which are stationary, and therefore does not follow the so-called apocalypses described by Levin, Kileel, and Boumal (2021).