Precise semidefinite programming formulation of atomic norm minimization for recovering d-dimensional (D ≥ 2) off-the-grid frequencies

Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In particular, atomic norm minimization was proposed in [1] to recover 1-dimensional spectrally sparse signal. However, in spite of existing research efforts [2], it was still an open problem how to formulate an equivalent positive semidefinite program for atomic norm minimization in recovering signals with d-dimensional (d ≥ 2) off-the-grid frequencies. In this paper, we settle this problem by proposing equivalent semidefinite programming formulations of atomic norm minimization to recover signals with d-dimensional (d ≥ 2) off-the-grid frequencies.

[1]  Yuxin Chen,et al.  Compressive recovery of 2-D off-grid frequencies , 2013, 2013 Asilomar Conference on Signals, Systems and Computers.

[2]  Musheng Wei,et al.  A new theoretical approach for Prony's method☆ , 1990 .

[3]  Emmanuel J. Cand.  Towards a Mathematical Theory of Super-Resolution , 2012 .

[4]  A. Robert Calderbank,et al.  Sensitivity to Basis Mismatch in Compressed Sensing , 2011, IEEE Trans. Signal Process..

[5]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[6]  Yuxin Chen,et al.  Robust Spectral Compressed Sensing via Structured Matrix Completion , 2013, IEEE Transactions on Information Theory.

[7]  Gitta Kutyniok,et al.  Theory and applications of compressed sensing , 2012, 1203.3815.

[8]  C. Carathéodory.  Über den variabilitätsbereich der fourier’schen konstanten von positiven harmonischen funktionen , 1911 .

[9]  Marco F. Duarte,et al.  Spectral compressive sensing , 2013 .

[10]  Gongguo Tang,et al.  Sparse recovery over continuous dictionaries-just discretize , 2013, 2013 Asilomar Conference on Signals, Systems and Computers.

[11]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[12]  Pablo A. Parrilo,et al.  The Convex Geometry of Linear Inverse Problems , 2010, Foundations of Computational Mathematics.

[13]  Parikshit Shah,et al.  Compressed Sensing Off the Grid , 2012, IEEE Transactions on Information Theory.

[14]  B. Dumitrescu.  Positive Trigonometric Polynomials and Signal Processing Applications , 2007 .

[15]  G. Majda,et al.  A simple procedure to eliminate known poles from a time series , 1989 .