Robust Spectral Compressed Sensing via Structured Matrix Completion
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The paper explores the problem of spectral compressed sensing, which aims to recover a spectrally sparse signal from a small random subset of its n time domain samples. The signal of interest is assumed to be a superposition of r multi-dimensional complex sinusoids, while the underlying frequencies can assume any continuous values in the normalized frequency domain. Conventional compressed sensing paradigms suffer from the basis mismatch issue when imposing a discrete dictionary on the Fourier representation. To address this issue, we develop a novel algorithm, called Enhanced Matrix Completion (EMaC), based on structured matrix completion that does not require prior knowledge of the model order. The algorithm starts by arranging the data into a low-rank enhanced form exhibiting multi-fold Hankel structure, and then attempts recovery via nuclear norm minimization. Under mild incoherence conditions, EMaC allows perfect recovery as soon as the number of samples exceeds the order of r^4n, and is stable against bounded noise. Even if a constant portion of samples are corrupted with arbitrary magnitude, EMaC still allows exact recovery, provided that the sample complexity exceeds the order of r^2^3n. Along the way, our results demonstrate the power of convex relaxation in completing a low-rank multi-fold Hankel or Toeplitz matrix from minimal observed entries. The performance of our algorithm and its applicability to super resolution are further validated by numerical experiments.
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