# The phaseless rank of a matrix

arXiv: https://arxiv.org/abs/1909.02417 · HTML: https://mlsanity.com/paper/1909.02417/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Sep 2019.

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

- [On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization](https://mlsanity.com/paper/2607.04875/) (2026)

## Abstract (arXiv)

We consider the problem of finding the smallest rank of a complex matrix whose absolute values of the entries are given. We call this minimum the phaseless rank of the matrix of the entrywise absolute values. In this paper we study this quantity, extending a classic result of Camion and Hoffman and connecting it to the study of amoebas of determinantal varieties and of semidefinite representations of convex sets. As a consequence, we prove that the set of maximal minors of a matrix of indeterminates form an amoeba basis for the ideal they define, and we attain a new upper bound on the complex semidefinite extension complexity of polytopes, dependent only on their number of vertices and facets. We also highlight the connections between the notion of phaseless rank and the problem of finding large sets of complex equiangular lines or mutually unbiased bases.