# Maximizing Products of Linear Forms, and The Permanent of Positive Semidefinite Matrices

arXiv: https://arxiv.org/abs/2002.04149 · HTML: https://mlsanity.com/paper/2002.04149/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2020.

## Built on by (most cited first, 1 of 1)

- [Fast Minimization of Expected Logarithmic Loss via Stochastic Dual Averaging](https://mlsanity.com/paper/2311.02557/) (2023)

## Abstract (arXiv)

We study the convex relaxation of a polynomial optimization problem, maximizing a product of linear forms over the complex sphere. We show that this convex program is also a relaxation of the permanent of Hermitian positive semidefinite (HPSD) matrices. By analyzing a constructive randomized rounding algorithm, we obtain an improved multiplicative approximation factor to the permanent of HPSD matrices, as well as computationally efficient certificates for this approximation. We also propose an analog of van der Waerden's conjecture for HPSD matrices, where the polynomial optimization problem is interpreted as a relaxation of the permanent.