1,587 research outputs found
Stable super-resolution limit and smallest singular value of restricted Fourier matrices
Super-resolution refers to the process of recovering the locations and
amplitudes of a collection of point sources, represented as a discrete measure,
given of its noisy low-frequency Fourier coefficients. The recovery
process is highly sensitive to noise whenever the distance between the
two closest point sources is less than . This paper studies the {\it
fundamental difficulty of super-resolution} and the {\it performance guarantees
of a subspace method called MUSIC} in the regime that .
The most important quantity in our theory is the minimum singular value of
the Vandermonde matrix whose nodes are specified by the source locations. Under
the assumption that the nodes are closely spaced within several well-separated
clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our
estimate is given as a weighted sum, where each term only depends on
the configuration of each individual clump. This implies that, as the noise
increases, the super-resolution capability of MUSIC degrades according to a
power law where the exponent depends on the cardinality of the largest clump.
Numerical experiments validate our theoretical bounds for the minimum singular
value and the resolution limit of MUSIC.
When there are point sources located on a grid with spacing , the
fundamental difficulty of super-resolution can be quantitatively characterized
by a min-max error, which is the reconstruction error incurred by the best
possible algorithm in the worst-case scenario. We show that the min-max error
is closely related to the minimum singular value of Vandermonde matrices, and
we provide a non-asymptotic and sharp estimate for the min-max error, where the
dominant term is .Comment: 47 pages, 8 figure
Superfast Line Spectral Estimation
A number of recent works have proposed to solve the line spectral estimation
problem by applying off-the-grid extensions of sparse estimation techniques.
These methods are preferable over classical line spectral estimation algorithms
because they inherently estimate the model order. However, they all have
computation times which grow at least cubically in the problem size, thus
limiting their practical applicability in cases with large dimensions. To
alleviate this issue, we propose a low-complexity method for line spectral
estimation, which also draws on ideas from sparse estimation. Our method is
based on a Bayesian view of the problem. The signal covariance matrix is shown
to have Toeplitz structure, allowing superfast Toeplitz inversion to be used.
We demonstrate that our method achieves estimation accuracy at least as good as
current methods and that it does so while being orders of magnitudes faster.Comment: 16 pages, 7 figures, accepted for IEEE Transactions on Signal
Processin
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