77 research outputs found
Joint Sparsity Recovery for Spectral Compressed Sensing
Compressed Sensing (CS) is an effective approach to reduce the required
number of samples for reconstructing a sparse signal in an a priori basis, but
may suffer severely from the issue of basis mismatch. In this paper we study
the problem of simultaneously recovering multiple spectrally-sparse signals
that are supported on the same frequencies lying arbitrarily on the unit
circle. We propose an atomic norm minimization problem, which can be regarded
as a continuous counterpart of the discrete CS formulation and be solved
efficiently via semidefinite programming. Through numerical experiments, we
show that the number of samples per signal may be further reduced by harnessing
the joint sparsity pattern of multiple signals
Super-Resolution of Mutually Interfering Signals
We consider simultaneously identifying the membership and locations of point
sources that are convolved with different low-pass point spread functions, from
the observation of their superpositions. This problem arises in
three-dimensional super-resolution single-molecule imaging, neural spike
sorting, multi-user channel identification, among others. We propose a novel
algorithm, based on convex programming, and establish its near-optimal
performance guarantee for exact recovery by exploiting the sparsity of the
point source model as well as incoherence between the point spread functions.
Numerical examples are provided to demonstrate the effectiveness of the
proposed approach.Comment: ISIT 201
Off-the-Grid Line Spectrum Denoising and Estimation with Multiple Measurement Vectors
Compressed Sensing suggests that the required number of samples for
reconstructing a signal can be greatly reduced if it is sparse in a known
discrete basis, yet many real-world signals are sparse in a continuous
dictionary. One example is the spectrally-sparse signal, which is composed of a
small number of spectral atoms with arbitrary frequencies on the unit interval.
In this paper we study the problem of line spectrum denoising and estimation
with an ensemble of spectrally-sparse signals composed of the same set of
continuous-valued frequencies from their partial and noisy observations. Two
approaches are developed based on atomic norm minimization and structured
covariance estimation, both of which can be solved efficiently via semidefinite
programming. The first approach aims to estimate and denoise the set of signals
from their partial and noisy observations via atomic norm minimization, and
recover the frequencies via examining the dual polynomial of the convex
program. We characterize the optimality condition of the proposed algorithm and
derive the expected convergence rate for denoising, demonstrating the benefit
of including multiple measurement vectors. The second approach aims to recover
the population covariance matrix from the partially observed sample covariance
matrix by motivating its low-rank Toeplitz structure without recovering the
signal ensemble. Performance guarantee is derived with a finite number of
measurement vectors. The frequencies can be recovered via conventional spectrum
estimation methods such as MUSIC from the estimated covariance matrix. Finally,
numerical examples are provided to validate the favorable performance of the
proposed algorithms, with comparisons against several existing approaches.Comment: 14 pages, 10 figure
Coherence-Based Performance Guarantees of Orthogonal Matching Pursuit
In this paper, we present coherence-based performance guarantees of
Orthogonal Matching Pursuit (OMP) for both support recovery and signal
reconstruction of sparse signals when the measurements are corrupted by noise.
In particular, two variants of OMP either with known sparsity level or with a
stopping rule are analyzed. It is shown that if the measurement matrix
satisfies the strong coherence property, then with
, OMP will recover a -sparse signal with high
probability. In particular, the performance guarantees obtained here separate
the properties required of the measurement matrix from the properties required
of the signal, which depends critically on the minimum signal to noise ratio
rather than the power profiles of the signal. We also provide performance
guarantees for partial support recovery. Comparisons are given with other
performance guarantees for OMP using worst-case analysis and the sorted one
step thresholding algorithm.Comment: appeared at 2012 Allerton conferenc
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