Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

Abstract

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 analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples.

51 pages, 2 figures. The source file includes two implementations of all bounds constructed in this paper, one in Matlab and one in Julia

References in corpus (5)

• Introduction to Nonnegative Matrix Factorization
• A universality theorem for nonnegative matrix factorizations
• Algorithms for Positive Semidefinite Factorization
• Non-closure of the set of quantum correlations via graphs
• Correlation matrices, Clifford algebras, and completely positive semidefinite rank

Cited by in corpus (1)

• Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization