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.

中文翻译：

---

通过非交换多项式优化确定矩阵分解阶的下界

我们使用（种族非交换）多项式优化中的技术来制定矩阵因式分解秩上的半定规划下界的层次结构。特别地，我们考虑非负秩，正半定秩及其对称类似物：完全正秩和完全正半定秩。我们研究了层次结构的收敛属性，将它们与已知的下限进行了广泛比较，并提供了一些（数字）示例。