  The Frank-Wolfe (FW) algorithm has been widely used in solving nuclear norm
constrained problems, since it does not require projections. However, FW often
yields high rank intermediate iterates, which can be very expensive in time and
space costs for large problems. To address this issue, we propose a rank-drop
method for nuclear norm constrained problems. The goal is to generate descent
steps that lead to rank decreases, maintaining low-rank solutions throughout
the algorithm. Moreover, the optimization problems are constrained to ensure
that the rank-drop step is also feasible and can be readily incorporated into a
projection-free minimization method, e.g., Frank-Wolfe. We demonstrate that by
incorporating rank-drop steps into the Frank-Wolfe algorithm, the rank of the
solution is greatly reduced compared to the original Frank-Wolfe or its common
variants.
