Asymmetric Factorization for Low-Rank PSD Learning: When Is the Relaxation Exact ?

Low-rank factorization is a standard way to scale optimization in machine learning by replacing large matrix variables with compact factors. For positive semidefinite (PSD) variables, the symmetric Burer--Monteiro factorization (sBMF) writes $Z=XX^\top$ with a single low-rank factor $X$. A recent asymmetric alternative (aBMF) writes $Z=XY^\top$ and adds a quadratic penalty $(\gamma/2)\|X-Y\|_F^2$ to encourage symmetry. This split is attractive because it yields a biconvex objective with alternating convex subproblems, but its practical value depends strongly on how the penalty parameter $\gamma$ is chosen. We study a unified regularized aBMF framework and derive an explicit lower bound on $\gamma$ that guarantees exactness: under mild assumptions, any $\gamma$ above this threshold makes aBMF and sBMF share the same critical points. This gives a principled way to use the asymmetric formulation without altering the critical-point structure of the symmetric problem. In particular, it answers the open question of whether an exact penalty exists for asymmetric relaxation.