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,从二次测量中恢复一个未知的信号越来越受欢迎,因为它包括许多令人感兴趣的应用,例如阶段检索、聚变框架阶段检索和积极的操作员评价度等特殊案例。在本文中,我们通过使用最小方块方法重建信号,建立了非被动统计属性,表明无噪音情况下天线与真实信号之间的差距消失,在噪音情况下,在噪音情况下,它与真实信号的距离被一个错误率($O) (sqrt{p\log(1+2n)/n}) 捆绑在一起,美元和美元分别为美元和美元,分别是测量数量和信号的尺寸。我们开发了一个梯度固定式牛顿方法(GRNM)以解决最小方块问题,并证明它在某些温和条件下,在超线速速度上与独特的当地最低点一致。除了确定性结果外,GNM可以完全为无噪音案件重建真实信号,并以高概率实现上述错误率,噪音实验显示GNM在高精确度、快速的恢复能力方面表现良好。