We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogs: the completely positive rank and the completely positive semidefinite rank. We study convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples.

Completely positive rank, Completely positive semidefinite rank, Matrix factorization ranks, Noncommutative polynomial optimization, Nonnegative rank, Positive semidefinite rank
Foundations of Computational Mathematics
Networks and Optimization

Gribling, S.J, de Laat, D, & Laurent, M. (2019). Lower bounds on matrix factorization ranks via noncommutative polynomial optimization. Foundations of Computational Mathematics, 1–58. doi:10.1007/s10208-018-09410-y