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.

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Foundations of Computational Mathematics
Approximation Algorithms, Quantum Information and Semidefinite Optimization , Progress in quantum computing:Algorithms, communication, and applications
Networks and Optimization

Gribling, S., de Laat, D., & Laurent, M. (2019). Lower bounds on matrix factorization ranks via noncommutative polynomial optimization. Foundations of Computational Mathematics, 19, 1013–1070. doi:10.1007/s10208-018-09410-y