  We study the projected gradient descent method on low-rank matrix problems
with a strongly convex objective. We use the Burer-Monteiro factorization
approach to implicitly enforce low-rankness; such factorization introduces
non-convexity in the objective. We focus on constraint sets that include both
positive semi-definite (PSD) constraints and specific matrix norm-constraints.
Such criteria appear in quantum state tomography and phase retrieval
applications.
  We show that non-convex projected gradient descent favors local linear
convergence in the factored space. We build our theory on a novel descent
lemma, that non-trivially extends recent results on the unconstrained problem.
The resulting algorithm is Projected Factored Gradient Descent, abbreviated as
ProjFGD, and shows superior performance compared to state of the art on quantum
state tomography and sparse phase retrieval applications.
