# Sparse Noncommutative Polynomial Optimization

arXiv: https://arxiv.org/abs/1909.00569 · HTML: https://mlsanity.com/paper/1909.00569/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Sep 2019.

## Built on by (most cited first, 2 of 2)

- [Fairness in Forecasting of Observations of Linear Dynamical Systems](https://mlsanity.com/paper/2209.05274/) (2022)
- [Joint Problems in Learning Multiple Dynamical Systems](https://mlsanity.com/paper/2311.02181/) (2023)

## Abstract (arXiv)

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre [SIAM J. Optim. 17(3) (2006), pp. 822--843] and Waki et al. [SIAM J. Optim. 17(1) (2006), pp. 218--242]. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue and trace of noncommutative polynomials from the literature.