Stable low-rank matrix recovery from 3-designs
Abstract
We study the recovery of low-rank Hermitian matrices from rank-one measurements obtained by uniform sampling from complex projective 3-designs, using nuclear-norm minimization. This framework includes phase retrieval as a special case via the PhaseLift method. In general, complex projective -designs provide a practical means of partially derandomizing Gaussian measurement models. While near-optimal recovery guarantees are known for -designs, and it is known that -designs do not permit recovery with a subquadratic number of measurements, the case of -designs has remained open. In this work, we close this gap by establishing recovery guarantees for (exact and approximate) -designs that parallel the best-known results for -designs. In particular, we derive bounds on the number of measurements sufficient for stable and robust low-rank recovery via nuclear-norm minimization. Our results are especially relevant in practice, as explicit constructions of -designs are significantly more challenging than those of -designs.
Cite
@article{arxiv.2512.11642,
title = {Stable low-rank matrix recovery from 3-designs},
author = {Timm Gilles},
journal= {arXiv preprint arXiv:2512.11642},
year = {2025}
}