Abstract
We consider the problem of recovering a signal consisting of a superposition of point sources from lowresolution data with a cutoff frequency fc. If the distance between the sources is under 1/fc, this problem is not well posed in the sense that the low-pass data corresponding to two different signals may be practically the same. We show that minimizing a continuous version of the 1-norm achieves exact recovery as long as the sources are separated by at least 1.26/fc. The proof is based on the construction of a dual certificate for the optimization problem, which can be used to establish that the procedure is stable to noise. Finally, we illustrate the flexibility of our optimization-based framework by describing extensions to the demixing of sines and spikes and to the estimation of point sources that share a common support.
Original language | English (US) |
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Pages (from-to) | 251-303 |
Number of pages | 53 |
Journal | Information and Inference |
Volume | 5 |
Issue number | 3 |
DOIs | |
State | Published - Sep 2016 |
Keywords
- Convex optimization
- Dual certificates
- Group sparsity
- Line-spectra estimation
- Multiple measurements
- Overcomplete dictionaries
- Sparse recovery
- Super-resolution
ASJC Scopus subject areas
- Analysis
- Statistics and Probability
- Numerical Analysis
- Computational Theory and Mathematics
- Applied Mathematics