9,276 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
Mismatch and resolution in compressive imaging
Highly coherent sensing matrices arise in discretization of continuum
problems such as radar and medical imaging when the grid spacing is below the
Rayleigh threshold as well as in using highly coherent, redundant dictionaries
as sparsifying operators. Algorithms (BOMP, BLOOMP) based on techniques of band
exclusion and local optimization are proposed to enhance Orthogonal Matching
Pursuit (OMP) and deal with such coherent sensing matrices. BOMP and BLOOMP
have provably performance guarantee of reconstructing sparse, widely separated
objects {\em independent} of the redundancy and have a sparsity constraint and
computational cost similar to OMP's. Numerical study demonstrates the
effectiveness of BLOOMP for compressed sensing with highly coherent, redundant
sensing matrices.Comment: Figure 5 revise
Critical nonlocal systems with concave-convex powers
By using the fibering method jointly with Nehari manifold techniques, we
obtain the existence of multiple solutions to a fractional -Laplacian system
involving critical concave-convex nonlinearities provided that a suitable
smallness condition on the parameters involved is assumed. The result is
obtained despite there is no general classification for the optimizers of the
critical fractional Sobolev embedding.Comment: 22 page
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