Robust Spectral Compressed Sensing via Structured Matrix Completion

Abstract

Abstract-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 multidimensional 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 log 4 n, 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 log 3 n. 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.

Compressive recovery of 2-D off-grid frequencies

“…It is worth noting that the atomic norm for spectrally-sparse signals is equivalent to the totalvariation norm studied in [18], [19]. On the other hand, the current authors approached the harmonic retrieval problem via structured matrix completion [21], [22], by performing nuclear norm minimization over multi-fold Hankel matrices. This approach requires slightly larger sample size (O(r log 2 n)) to guarantee perfect recovery, provided that the signal model enjoys a few incoherence properties.…”

A fast algorithm for reconstruction of spectrally sparse signals in super-resolution

“…13 As we have argued, different from existing convex optimization based methods such as EMaC, our proposed algorithm is able to work with high-dimensional spectrally sparse signals. In Table 1, the elapsed time for signals of different dimensions are listed.…”

“…In 12 and, 13 the atomic norm 14 and the nuclear norm of Hankel matrices are minimized respectively to recover spectrally sparse signals with continuous-valued frequencies from nonuniform samples. Though robust signal recovery is guaranteed theoretically, these convex optimization based methods in [11][12][13] are not computationally efficient. They are all implemented by semi-definite programming (SDP) whose variables are matrices containing O(N 2 ) entries with N the dimension of the signal.…”

A General Framework for Compressed Sensing and Parallel MRI Using Annihilating Filter Based Low-Rank Hankel Matrix

“…In Theorem II.3 of our companion paper [25], we also considered the recovery ofŷ from its partial entries with noise, and derived a more improved version of the stability results than that of Chen and Chi [31]. Note that the stability result in [25] can be used to show the compressibility of the proposed structured low-rank matrix completion method.…”