Recently, recovering an unknown signal from quadratic measurements has gained popularity because it includes many interesting applications as special cases such as phase retrieval, fusion frame phase retrieval, and positive operator-valued measure. In this paper, by employing the least squares approach to reconstruct the signal, we establish the non-asymptotic statistical property showing that the gap between the estimator and the true signal is vanished in the noiseless case and is bounded in the noisy case by an error rate of $O(\sqrt{p\log(1+2n)/n})$, where $n$ and $p$ are the number of measurements and the dimension of the signal, respectively. We develop a gradient regularized Newton method (GRNM) to solve the least squares problem and prove that it converges to a unique local minimum at a superlinear rate under certain mild conditions. In addition to the deterministic results, GRNM can reconstruct the true signal exactly for the noiseless case and achieve the above error rate with a high probability for the noisy case. Numerical experiments demonstrate the GRNM performs nicely in terms of high order of recovery accuracy, faster computational speed, and strong recovery capability.

翻译：最近，从二次测量中恢复未知信号的方法受到广泛关注，因为其涵盖了许多有趣的应用作为特例，例如相位恢复、融合框架相位恢复以及正算子值测度。本文采用最小二乘法重建信号，建立了非渐近统计性质，表明在无噪声情况下估计量与真实信号之间的差距消失，而在有噪声情况下该差距以$O(\sqrt{p\log(1+2n)/n})$的错误率被界住，其中$n$和$p$分别表示测量次数和信号维度。我们提出了一种梯度正则化牛顿法（GRNM）来求解最小二乘问题，并证明在特定温和条件下该方法以超线性速率收敛到唯一局部最小值。除确定性结果外，GRNM可在无噪声情况下精确重建真实信号，并在有噪声情况下以高概率达到上述错误率。数值实验表明，GRNM在恢复精度高阶性、计算速度更快和恢复能力强方面表现优异。