A Tight Converse to the Spectral Resolution Limit via Convex Programming
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It is now well understood that convex programming can be used to estimate the frequency components of a spectrally sparse signal from m uniform temporal measurements. It is conjectured that a phase transition on the success of the total-variation regularization occurs when the distance between the spectral components of the signal to estimate crosses 1/m. We prove the necessity part of this conjecture by demonstrating that this regularization can fail whenever the spectral distance of the signal of interest is asymptotically equal to 1/m.
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