Provable Burer-Monteiro factorization for a class of norm-constrained matrix problems
This work addresses matrix optimization challenges in quantum state tomography and phase retrieval, offering a provable algorithm with practical gains, though it is incremental as it extends existing factorization approaches to constrained settings.
The paper tackles low-rank matrix problems with strongly convex objectives and constraints like PSD and norm constraints, common in quantum state tomography and phase retrieval, by proposing Projected Factored Gradient Descent (ProjFGD), which achieves local linear convergence and shows superior performance compared to state-of-the-art methods.
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.