Projected Gradient Descent for Spectral Compressed Sensing via Symmetric Hankel Factorization

TL;DR

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.

Abstract

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.

Figures (7)

• Figure 1: Left: Asymmetric factorization and update rule of PGD. Right: Symmetric factorization and update rule of the proposed SHGD. The loss function, $f(\cdot)$, is defined separately in the PGD and SHGD approaches, while the regularization function $g(\cdot)$ is specific to PGD. SHGD does not need a balancing regularization term and updates one single factor at a time compared to PGD, reducing at least half of operations and storage costs.
• Figure 2: The comparisons of different algorithms in terms of phase transitions under frequencies without separation. The red curve is the 90% success rate curve.
• Figure 3: The comparisons of different algorithms in terms of phase transitions under frequencies with separation. The red curve is the 90% success rate curve.
• Figure 4: The comparisons of computation time versus relative error. Left: Frequencies without separation. Right: Frequencies with separation.
• Figure 5: The comparisons of computation time versus different problem scales. Left: The average computation time of three algorithms SHGD, PGD, and FIHT. Right: The average computation time ratio between SHGD and PGD (SHGD/PGD).

Theorems & Definitions (27)

• Definition 1
• Remark 1
• Example 1
• proof
• Lemma 1
• proof
• Theorem 1
• Remark 2
• Remark 3
• Remark 4