An Oracle Gradient Regularized Newton Method for Quadratic Measurements Regression

TL;DR

This paper introduces a gradient regularized Newton method for quadratic measurement regression, providing theoretical guarantees for exact recovery in noiseless cases and bounded error in noisy scenarios, with demonstrated high accuracy and speed.

Contribution

It develops a novel two-phase Newton-based algorithm with proven convergence properties for quadratic measurement problems, improving recovery accuracy and computational efficiency.

Findings

Perfect recovery in noiseless case

Error bounded by $O(\sqrt{p\log(1+2n)/n})$ in noisy case

Fast convergence and high recovery accuracy

Abstract

Recovering an unknown signal from quadratic measurements has gained popularity due to its wide range of applications, including phase retrieval, fusion frame phase retrieval, and positive operator-valued measures. In this paper, we employ a least squares approach to reconstruct the signal and establish its non-asymptotic statistical properties. Our analysis shows that the estimator perfectly recovers the true signal in the noiseless case, while the error between the estimator and the true signal is bounded by $O(\sqrt{p\log(1+2n)/n})$ in the noisy case, where $n$ is the number of measurements and $p$ is the dimension of the signal. We then develop a two-phase algorithm, gradient regularized Newton method (GRNM), to solve the least squares problem. It is proven that the first phase terminates within finitely many steps, and the sequence generated in the second phase converges to a unique local minimum at a superlinear rate under certain mild conditions. Beyond these deterministic results, GRNM is capable of exactly reconstructing the true signal in the noiseless case and achieving the stated error rate with a high probability in the noisy case. Numerical experiments demonstrate that GRNM offers a high level of recovery capability and accuracy as well as fast computational speed.