Atomic norm denoising with applications to line spectral estimation (1204.0562v2)

Abstract: Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.

• The paper introduces an innovative atomic norm denoising framework that estimates line spectra without requiring prior model order and provides theoretical MSE guarantees.
• It formulates the estimation as a convex optimization problem solvable by semidefinite programming and efficiently implemented with ADMM and FFT-based Lasso approximations.
• Experimental results demonstrate that the proposed AST and Lasso methods outperform traditional techniques like MUSIC, especially under low SNR conditions.

Atomic Norm Denoising with Applications to Line Spectral Estimation

This paper presents a novel approach to line spectral estimation using atomic norm denoising, providing theoretical guarantees for mean-squared-error (MSE) performance amidst noise, without requiring prior knowledge of the model order. The authors propose an abstract framework for denoising with atomic norms, applying it specifically to estimate frequencies and phases in complex exponential mixtures. The framework poses this estimation as a convex optimization problem, solvable via semidefinite programming (SDP).

Theoretical Contributions

The primary contribution is the development of a general theory for denoising using atomic norms, which are convex penalties that promote specific structural properties, such as sparsity. This theory is extended to the problem of line spectral estimation. The authors derive MSE estimates based on the atomic norm, introducing the Atomic norm Soft Thresholding (AST) algorithm.

Key aspects of the theory include:

• A polynomial-time solution to the convex optimization problem via SDP.
• An approximation of the SDP by an $\ell_1$-regularized least-squares problem, suitable for larger problems while maintaining similar error rates.
• Comparison of SDP and $\ell_1$-based methods with classical line spectral analysis, showing superior MSE performance of SDP across various SNRs.

Practical Implementation and Results

The paper describes how AST can be efficiently implemented using the Alternating Direction Method of Multipliers (ADMM), capable of handling moderately large instances. In scenarios with very large instances, the Lasso approximation on an oversampled grid is employed, leveraging Fast Fourier Transform (FFT) for efficient computation.

Experimental results reveal that both AST and the Lasso method outperform traditional approaches such as MUSIC, Cadzow's, and Matrix Pencil methods, particularly in low SNR environments. The authors demonstrate that their approach does not require exact model order estimation, which is a limitation in many classical methods.

Implications and Future Directions

The implications of this work are significant for fields reliant on precise frequency estimation in noisy environments, such as radar, spectroscopy, and sensor array processing. The ability to perform robust denoising without prior model order information represents a step forward in adaptive spectral analysis.

Future research could explore:

• Fast algorithmic implementations for AST in very large-scale datasets.
• Extensions of this framework to other signal processing problems that could benefit from atomic norm representations.
• Exploration of dynamic and non-uniform sampling patterns in time-varying scenarios.

Conclusion

This paper lays a solid foundation for the use of atomic norms in spectral estimation, with promising theoretical and experimental outcomes. It opens avenues for further advancements in efficient spectral analysis and denoising across various applications in signal processing and communications.