Lower bounds on matrix factorization ranks via noncommutative polynomial optimization
Optimization and Control
2018-11-06 v3
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.
Cite
@article{arxiv.1708.01573,
title = {Lower bounds on matrix factorization ranks via noncommutative polynomial optimization},
author = {Sander Gribling and David de Laat and Monique Laurent},
journal= {arXiv preprint arXiv:1708.01573},
year = {2018}
}
Comments
51 pages, 2 figures. The source file includes two implementations of all bounds constructed in this paper, one in Matlab and one in Julia