Current spectral compressed sensing methods via Hankel matrix completion employ symmetric factorization to demonstrate the low-rank property of the Hankel matrix. However, previous non-convex gradient methods only utilize asymmetric factorization to achieve spectral compressed sensing. In this paper, we propose a novel nonconvex projected gradient descent method for spectral compressed sensing via symmetric factorization named Symmetric Hankel Projected Gradient Descent (SHGD), which updates only one matrix and avoids a balancing regularization term. SHGD reduces about half of the computation and storage costs compared to the prior gradient method based on asymmetric factorization. {Besides, the symmetric factorization employed in our work is completely novel to the prior low-rank factorization model, introducing a new factorization ambiguity under complex orthogonal transformation}. Novel distance metrics are designed for our factorization method and a linear convergence guarantee to the desired signal is established with $O(r^2\log(n))$ observations. Numerical simulations demonstrate the superior performance of the proposed SHGD method in phase transitions and computation efficiency compared to state-of-the-art methods.

翻译：现有基于Hankel矩阵完备化的谱压缩感知方法采用对称分解来展示Hankel矩阵的低秩特性。然而，先前的非凸梯度方法仅利用非对称分解实现谱压缩感知。本文提出一种新颖的基于对称分解的非凸投影梯度下降谱压缩感知方法——对称Hankel投影梯度下降法(SHGD)，该方法仅更新一个矩阵并避免平衡正则化项。与基于非对称分解的先前梯度方法相比，SHGD可减少约一半的计算与存储开销。此外，本文采用的对称分解对先前的低秩分解模型而言具有完全新颖性，在复正交变换下引入了一种新的分解歧义性。针对所提出的分解方法，我们设计了新的距离度量，并证明了在线性收敛条件下，仅需$O(r^2\log(n))$次观测即可保证恢复目标信号。数值仿真表明，与现有最优方法相比，SHGD方法在相变图与计算效率方面均展现出更优性能。