[Paper Review] Spectral Compressed Sensing via Projected Gradient Descent

Jian‐Feng Cai, Tianming Wang+1 more|arXiv (Cornell University)|Jul 31, 2017

Sparse and Compressive Sensing Techniques38 references3 citations

TL;DR

This paper proposes a non-convex projected gradient descent algorithm for spectral compressed sensing that leverages the low-rank structure of Hankel matrices formed from spectrally sparse signals. It achieves exact recovery with O(r² log n) random samples under the sampling-with-replacement model, demonstrating competitive performance in phase transitions and computational efficiency compared to state-of-the-art methods.

ABSTRACT

Let $x\in\mathbb{C}^n$ be a spectrally sparse signal consisting of $r$ complex sinusoids with or without damping. We consider the spectral compressed sensing problem, which is about reconstructing $x$ from its partial revealed entries. By utilizing the low rank structure of the Hankel matrix corresponding to $x$, we develop a computationally efficient algorithm for this problem. The algorithm starts from an initial guess computed via one-step hard thresholding followed by projection, and then proceeds by applying projected gradient descent iterations to a non-convex functional. Based on the sampling with replacement model, we prove that $O(r^2\log(n))$ observed entries are sufficient for our algorithm to achieve the successful recovery of a spectrally sparse signal. Moreover, extensive empirical performance comparisons show that our algorithm is competitive with other state-of-the-art spectral compressed sensing algorithms in terms of phase transitions and overall computational time.

Motivation & Objective

• To develop a computationally efficient algorithm for reconstructing spectrally sparse signals from partial, non-uniform time-domain samples.
• To exploit the low-rank structure of Hankel matrices associated with spectrally sparse signals to enable robust recovery from few observations.
• To establish theoretical recovery guarantees for a non-convex optimization approach under the sampling-with-replacement model.
• To demonstrate superior empirical performance in phase transitions and computational time compared to existing spectral compressed sensing algorithms.

Proposed method

• The algorithm begins with an initial guess obtained via one-step hard thresholding followed by projection onto the low-rank Hankel matrix manifold.
• It applies projected gradient descent iterations to a non-convex functional that promotes low-rank Hankel structure in the signal reconstruction.
• The method leverages the Vandermonde decomposition, which ensures that a spectrally sparse signal of rank r yields a low-rank Hankel matrix.
• Theoretical analysis relies on incoherence assumptions and probabilistic bounds on sampling operators to control gradient and Hessian behavior.
• Key technical components include the use of the nuclear norm and structured matrix perturbation analysis to bound the gradient of the objective function.
• The algorithm is designed to handle both undamped and damped complex sinusoids in the spectral model.

Experimental results

Research questions

• RQ1 Can a non-convex projected gradient descent method achieve exact recovery of spectrally sparse signals with a suboptimal number of samples?
• RQ2 Is O(r² log n) sampling sufficient for successful recovery under the sampling-with-replacement model?
• RQ3 How does the proposed algorithm compare to state-of-the-art methods in terms of phase transition and computational efficiency?
• RQ4 Does the geometric landscape of the non-convex objective function support stable convergence from random initialization?

Key findings

• The proposed algorithm achieves exact recovery of spectrally sparse signals with O(r² log n) randomly sampled entries under the sampling-with-replacement model.
• Theoretical analysis shows that the gradient of the objective function is well-controlled under incoherence and sampling assumptions, enabling convergence guarantees.
• Empirical results show that the algorithm outperforms FIHT in phase transitions when the number of observations is small.
• The algorithm is competitive with other state-of-the-art spectral compressed sensing methods in both recovery accuracy and computational runtime.
• Preliminary numerical results suggest the objective function may have a favorable geometric landscape, with no spurious local minima, supporting robust convergence from random initialization.