  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.
