Solving Quadratic Equations via PhaseLift when There Are About As Many Equations As Unknowns

TL;DR

This paper demonstrates that a complex vector can be exactly recovered from approximately as many quadratic equations as unknowns using PhaseLift, improving bounds and providing sharper results in noisy scenarios.

Contribution

It improves the theoretical bounds for exact recovery of vectors from quadratic equations using PhaseLift and extends results to noisy measurements with sharper guarantees.

Findings

Exact recovery with about n equations, improving previous n log n bounds.

Optimal recovery results in noisy settings.

Sharper theoretical guarantees for PhaseLift performance.

Abstract

This note shows that we can recover a complex vector x in C^n exactly from on the order of n quadratic equations of the form |<a_i, x>|^2 = b_i, i = 1, ..., m, by using a semidefinite program known as PhaseLift. This improves upon earlier bounds in [3], which required the number of equations to be at least on the order of n log n. We also demonstrate optimal recovery results from noisy quadratic measurements; these results are much sharper than previously known results.