On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data (1407.2490v2)

Abstract: This paper is concerned about sparse, continuous frequency estimation in line spectral estimation, and focused on developing gridless sparse methods which overcome grid mismatches and correspond to limiting scenarios of existing grid-based approaches, e.g., $\ell_1$ optimization and SPICE, with an infinitely dense grid. We generalize AST (atomic-norm soft thresholding) to the case of nonconsecutively sampled data (incomplete data) inspired by recent atomic norm based techniques. We present a gridless version of SPICE (gridless SPICE, or GLS), which is applicable to both complete and incomplete data without the knowledge of noise level. We further prove the equivalence between GLS and atomic norm-based techniques under different assumptions of noise. Moreover, we extend GLS to a systematic framework consisting of model order selection and robust frequency estimation, and present feasible algorithms for AST and GLS. Numerical simulations are provided to validate our theoretical analysis and demonstrate performance of our methods compared to existing ones.

• The paper extends the Atomic Norm Soft Thresholding method to accommodate nonconsecutively sampled and incomplete data.
• The paper proposes GLS, a gridless variant of SPICE, eliminating the need for noise level estimation in line spectral analysis.
• The methodology offers improved frequency estimation accuracy and systematic model order selection for applications in communications and radar.

Overview of "On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data"

This paper addresses the problem of line spectral estimation (LSE), crucial in various applications such as communications, radar, and seismology. Traditional methods often rely on discretizing the frequency spectrum, which can lead to grid mismatch errors. The authors aim to mitigate these limitations by exploring gridless sparse methods that estimate continuous frequencies without discretization.

The paper presents novel approaches and theoretical insights into gridless versions of well-known algorithms. The authors focus on atomic norm-based approaches, particularly extending the atomic norm soft thresholding (AST) to incomplete data scenarios. They introduce a gridless version of SPICE, or GLS, which is applicable to both complete and incomplete datasets without prior knowledge of noise levels. This broadens the usability of these methods, potentially leading to more accurate frequency estimation in practical scenarios where complete data is not always available.

Key Contributions

1. Generalization of AST: The paper extends the AST method for nonconsecutively sampled data. It also provides theoretical backing for AST's performance under certain noise assumptions, showing expected mean squared errors in frequency estimations are minimized.
2. Development of GLS: The authors propose GLS, a gridless variant of SPICE, which operates without requiring noise level estimations. They prove the equivalence between GLS and atomic norm methods under various noise conditions.
3. Systematic Framework for LSE: By extending GLS, the paper proposes a model that includes model order selection and robust frequency estimation.
4. Algorithmic Advancements: Feasible algorithms for AST and GLS are developed, utilizing the alternating direction method of multipliers (ADMM) for computational efficiency. Additionally, they provide connections showing that existing grid-based SPICE and $\ell_1$ optimization are approximations of GLS.

Implications and Future Directions

The research advances the field of spectral estimation by reducing reliance on grid-based methods, which can suffer from inaccuracies due to discretization. This is particularly impactful in applications involving sparse signals with missing data, where such approximations of continuous domain methods are advantageous.

The practical implications include more accurate models for physical phenomena, improved communication systems, and enhanced capabilities in radar and sonar technologies, where detecting and estimating signals accurately is paramount.

Theoretical implications suggest that further research could enhance the robustness of these methods against varying noise levels and more complex signal environments. Future developments might also focus on improving computational efficiencies, making these methods more viable for real-time applications or large-scale data.

The paper lays groundwork for future exploration in adapting gridless methods to other signal processing areas, expanding their utility beyond current spectral estimation challenges.

In summary, this work represents a progress in sparse signal estimation, offering methods that demonstrate both theoretical rigor and practical utility in handling incomplete and noisy data, paving the path for more precise LSE techniques.