论文标题
关于单位模块的预计梯度下降的局部线性收敛性
On Local Linear Convergence of Projected Gradient Descent for Unit-Modulus Least Squares
论文作者
论文摘要
单位模式最小二乘（UMLS）问题在信号处理中具有广泛的应用，例如，仅相位波束成形，相位检索，雷达代码设计和传感器网络定位。最近已经研究了可扩展的一阶方法，例如投影梯度下降（PGD），作为解决UMLS问题的一种简单而有效的方法。 UMLS PGD收敛的现有结果通常集中在全球融合到固定点上。作为一个非凸问题，仅建立了均方根收敛率。但是，这些结果不能解释在实践中经常观察到的PGD的快速收敛。该手稿对UMLS的PGD收敛性进行了新的分析，证明了溶液附近算法的线性收敛行为是合理的。通过利用目标函数和约束集的局部结构，我们建立了收敛速率的精确表达式，并表征线性收敛的条件。模拟表明，我们的理论分析证实了数值示例。此外，根据我们的收敛分析中揭示的新见解，提出了具有自适应步骤大小的PGD的变体。这些变体在实践中显示出很大的加速度。
The unit-modulus least squares (UMLS) problem has a wide spectrum of applications in signal processing, e.g., phase-only beamforming, phase retrieval, radar code design, and sensor network localization. Scalable first-order methods such as projected gradient descent (PGD) have recently been studied as a simple yet efficient approach to solving the UMLS problem. Existing results on the convergence of PGD for UMLS often focus on global convergence to stationary points. As a non-convex problem, only a sublinear convergence rate has been established. However, these results do not explain the fast convergence of PGD frequently observed in practice. This manuscript presents a novel analysis of convergence of PGD for UMLS, justifying the linear convergence behavior of the algorithm near the solution. By exploiting the local structure of the objective function and the constraint set, we establish an exact expression for the convergence rate and characterize the conditions for linear convergence. Simulations show that our theoretical analysis corroborates numerical examples. Furthermore, variants of PGD with adaptive step sizes are proposed based on the new insight revealed in our convergence analysis. The variants show substantial acceleration in practice.
