[Paper Review] New Low Rank Optimization Model and Convex Approach for Robust Spectral Compressed Sensing

Zai Yang, Xunmeng Wu|arXiv (Cornell University)|Jan 16, 2021

Sparse and Compressive Sensing Techniques41 references4 citations

TL;DR

This paper proposes a novel low-rank double Hankel model for robust spectral compressed sensing that explicitly enforces spectral poles to lie on the unit circle, overcoming a fundamental limitation of traditional Hankel-based methods. The method, called doubly enhanced matrix completion (DEMaC), enables provably accurate and robust recovery of spectrally sparse signals from partial samples, even under bounded and sparse noise, with theoretical guarantees and improved resolution over state-of-the-art approaches.

ABSTRACT

This paper investigates recovery of an undamped spectrally sparse signal and its spectral components from a set of regularly spaced samples within the framework of spectral compressed sensing and super-resolution. We show that the existing Hankel-based optimization methods suffer from the fundamental limitation that the prior of undampedness cannot be exploited. We propose a new low rank optimization model partially inspired by forward-backward processing for line spectral estimation and show its capability in restricting the spectral poles on the unit circle. We present convex relaxation approaches with the model and show their provable accuracy and robustness to bounded and sparse noise. All our results are generalized from the 1-D to arbitrary-dimensional spectral compressed sensing. Numerical simulations are provided that corroborate our analysis and show efficiency of our model and advantageous performance of our approach in improved accuracy and resolution as compared to the state-of-the-art Hankel and atomic norm methods.

Motivation & Objective

• Address the fundamental limitation of existing Hankel-based methods in spectral compressed sensing, which fail to exploit the on-unit-circle prior for undamped signals.
• Develop a new low-rank optimization model that explicitly enforces spectral poles to lie on the unit circle, improving physical interpretability and performance.
• Propose a convex relaxation framework—doubly enhanced matrix completion (DEMaC)—for robust recovery of spectrally sparse signals from partial, noisy samples.
• Establish theoretical guarantees for DEMaC, including accuracy and robustness to bounded and sparse noise, in both full and compressive sampling regimes.
• Generalize the proposed model and method from 1-D to arbitrary-dimensional spectral compressed sensing.

Proposed method

• Introduce a new low-rank double Hankel model by combining two Hankel matrices to better capture the structure of undamped spectrally sparse signals.
• Formulate a convex optimization problem based on the double Hankel structure, enabling efficient and stable recovery via nuclear norm minimization.
• Employ the golfing scheme to construct a dual certificate that verifies the exact recovery condition under incoherence and sampling assumptions.
• Utilize a modified incoherence condition and probabilistic analysis to show that DEMaC recovers the true signal with high probability when the number of samples scales as $ M = O( u K ext{polylog}(N)) $, where $ u $ is a coherence parameter.
• Adapt the framework to handle both bounded and sparse noise by incorporating robust optimization techniques into the convex relaxation.
• Generalize the model and algorithm from 1-D to multidimensional spectral compressed sensing using Kronecker product structure and tensor-based formulations.

Experimental results

Research questions

• RQ1 Can a low-rank Hankel-based model be modified to explicitly enforce spectral poles on the unit circle, improving physical consistency and performance?
• RQ2 Does the proposed double Hankel model significantly reduce the solution space and enhance the ability to recover true spectral components?
• RQ3 Can a convex relaxation of the new model achieve provable recovery accuracy and robustness under partial sampling and bounded/sparse noise?
• RQ4 What is the required sampling rate for exact recovery using the new model, and how does it compare to existing methods in terms of sample complexity?
• RQ5 Can the proposed method be extended to multidimensional spectral compressed sensing while preserving theoretical guarantees and performance gains?

Key findings

• The proposed double Hankel model restricts the solution space such that spectral poles are naturally constrained to the unit circle, addressing a key limitation of standard Hankel-based methods.
• DEMaC achieves exact recovery with high probability when the number of samples satisfies $ M = O( u K ext{polylog}(N)) $, where $ u $ is a coherence parameter related to the signal structure.
• The method is provably robust to both bounded and sparse noise, with theoretical error bounds derived under the new model framework.
• Numerical simulations demonstrate that DEMaC outperforms state-of-the-art Hankel and atomic norm methods in terms of accuracy and resolution, especially in low-sample regimes.
• The theoretical sample complexity of DEMaC matches the order of magnitude of existing methods but with improved constants due to enhanced structure exploitation, and the dual certificate construction confirms recovery conditions.
• The method generalizes naturally to multidimensional spectral compressed sensing, maintaining theoretical guarantees and performance advantages.