Convex recovery from interferometric measurements

TL;DR

This paper establishes deterministic conditions for exact and stable recovery of vectors from interferometric quadratic measurements using semidefinite programming, with performance depending on graph spectral properties.

Contribution

It provides a new recovery guarantee for interferometric measurements formulated as a semidefinite program, linking stability to graph spectral gaps.

Findings

Exact recovery under well-connected graph conditions

Stable recovery in noisy scenarios with spectral gap dependence

Application relevance to phase retrieval and interferometric inversion

Abstract

This note formulates a deterministic recovery result for vectors $x$ from quadratic measurements of the form $(Ax)_i \overline{(Ax)_j}$ for some left-invertible $A$. Recovery is exact, or stable in the noisy case, when the couples $(i,j)$ are chosen as edges of a well-connected graph. One possible way of obtaining the solution is as a feasible point of a simple semidefinite program. Furthermore, we show how the proportionality constant in the error estimate depends on the spectral gap of a data-weighted graph Laplacian. Such quadratic measurements have found applications in phase retrieval, angular synchronization, and more recently interferometric waveform inversion.