Atomic Norm Denoising With Applications to Line Spectral Estimation

By Badri Narayan Bhaskar, Gongguo Tang, B. Recht. IEEE Transactions on Signal Processing, 2012

640 citations recorded by the source corpus.

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.

Subfields: Biomedical Engineering, Computational Mechanics, Computer Networks and Communications

View Paper: https://www.semanticscholar.org/paper/38e696b0fa508ee6e3dda713f136caa7b1febb7d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomedical Engineering

1. Lihua Xie
2. Zai Yang
3. B. Recht
4. Moritz Hardt
5. Gongguo Tang
6. E.J. Candes
7. Jian Li
8. P. Stoica

In Computational Mechanics

1. Lihua Xie
2. Zai Yang
3. B. Recht
4. Tengyu Ma
5. Moritz Hardt
6. Jian Li
7. P. Stoica
8. Gongguo Tang

In Computer Networks and Communications

1. B. Recht
2. Lihua Xie
3. Zai Yang
4. Moritz Hardt
5. Jian Li
6. P. Stoica
7. Parikshit M. Shah
8. Tengyu Ma

Citations in the corpus

• On Gridless Sparse Methods for Line Spectral Estimation From Complete and Incomplete Data (2014, positive): “GLS is obtained based on our recent work [39] where the focus is on the spatial spectral analysis (a.k.a. array processing) as opposed to the temporal spectral analysis considered here.”
• Super-Resolution from Noisy Data (2012, neutral): “Above, z is a noise term which can either be stochastic or deterministic, and Q lo is a bandlimiting operator with a frequency cut-off equal to f lo = 1 /λ lo .”
• Compressed Sensing Off the Grid (2012, methodology): “Thus, proofs based on Bochner’s Theorem [29], the bounded real lemma [30], [31], or spectral factorization [32] would also provide a tight characterization.”
• Gradient Descent Learns Linear Dynamical Systems (2016, result): “One can check that this provides a tight, polyhedral approximation to the set B α , following an argument similar to Appendix C of Bhaskar et al [BTR13].”
• Sparse Methods for Direction-of-Arrival Estimation (2016, methodology): “Gaussian noise, this choice has been studied in [81,128,130].”
• Towards a Mathematical Theory of Super‐resolution (2012, background): “This formulation is similar to that in [3], which concerns a related infinite-dimensional convex program.”