Strong convexity of affine phase retrieval

TL;DR

This paper proves that the least squares formulation for affine phase retrieval is strongly convex under certain conditions, enabling guaranteed linear convergence of gradient descent from arbitrary initial points.

Contribution

It establishes strong convexity of the least squares approach for affine phase retrieval with complex Gaussian measurements, unlike classical phase retrieval.

Findings

Strong convexity holds when measurements are complex Gaussian and m ≳ d log d.

Gradient descent converges linearly to the solution from any starting point.

Affine phase retrieval differs fundamentally from classical phase retrieval in convexity properties.

Abstract

The recovery of a signal from the intensity measurements with some entries being known in advance is termed as {\em affine phase retrieval}. In this paper, we prove that a natural least squares formulation for the affine phase retrieval is strongly convex on the entire space under some mild conditions, provided the measurements are complex Gaussian random vecotrs and the measurement number $m \gtrsim d \log d$ where $d$ is the dimension of signals. Based on the result, we prove that the simple gradient descent method for the affine phase retrieval converges linearly to the target solution with high probability from an arbitrary initial point. These results show an essential difference between the affine phase retrieval and the classical phase retrieval, where the least squares formulations for the classical phase retrieval are non-convex.