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Compressed Sensing Off the Grid

Gongguo Tang(gtang5***at***wisc.edu)
Badri Narayan Bhaskar(bnbhaskar***at***wisc.edu)
Parikshit Shah(pshah***at***discovery.wisc.edu)
Benjamin Recht(brecht***at***cs.wisc.edu))

Abstract: We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples. We reformulate this atomic norm minimization as an exact semidefinite program. Even with this continuous dictionary, we show that most sampling sets of size O(s log s log n) are sufficient to guarantee the exact frequency estimation with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method.

Keywords: atomic norm, basis mismatch, compressed sensing, continuous dictionary, line spectral estimation, nuclear norm relaxation, Prony’s method, sparsity

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Applications -- Science and Engineering (Data-Mining )

Category 3: Linear, Cone and Semidefinite Programming (Semi-definite Programming )

Citation:

Download: [PDF]

Entry Submitted: 09/13/2012
Entry Accepted: 09/13/2012
Entry Last Modified: 09/13/2012

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