234 research outputs found
Super-resolution, Extremal Functions and the Condition Number of Vandermonde Matrices
Super-resolution is a fundamental task in imaging, where the goal is to
extract fine-grained structure from coarse-grained measurements. Here we are
interested in a popular mathematical abstraction of this problem that has been
widely studied in the statistics, signal processing and machine learning
communities. We exactly resolve the threshold at which noisy super-resolution
is possible. In particular, we establish a sharp phase transition for the
relationship between the cutoff frequency () and the separation ().
If , our estimator converges to the true values at an inverse
polynomial rate in terms of the magnitude of the noise. And when no estimator can distinguish between a particular pair of
-separated signals even if the magnitude of the noise is exponentially
small.
Our results involve making novel connections between {\em extremal functions}
and the spectral properties of Vandermonde matrices. We establish a sharp phase
transition for their condition number which in turn allows us to give the first
noise tolerance bounds for the matrix pencil method. Moreover we show that our
methods can be interpreted as giving preconditioners for Vandermonde matrices,
and we use this observation to design faster algorithms for super-resolution.
We believe that these ideas may have other applications in designing faster
algorithms for other basic tasks in signal processing.Comment: 19 page
Precise Semidefinite Programming Formulation of Atomic Norm Minimization for Recovering d-Dimensional () Off-the-Grid Frequencies
Recent research in off-the-grid compressed sensing (CS) has demonstrated
that, under certain conditions, one can successfully recover a spectrally
sparse signal from a few time-domain samples even though the dictionary is
continuous. In particular, atomic norm minimization was proposed in
\cite{tang2012csotg} to recover -dimensional spectrally sparse signal.
However, in spite of existing research efforts \cite{chi2013compressive}, it
was still an open problem how to formulate an equivalent positive semidefinite
program for atomic norm minimization in recovering signals with -dimensional
() off-the-grid frequencies. In this paper, we settle this problem by
proposing equivalent semidefinite programming formulations of atomic norm
minimization to recover signals with -dimensional () off-the-grid
frequencies.Comment: 4 pages, double-column,1 Figur
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
Off-The-Grid Spectral Compressed Sensing With Prior Information
Recent research in off-the-grid compressed sensing (CS) has demonstrated
that, under certain conditions, one can successfully recover a spectrally
sparse signal from a few time-domain samples even though the dictionary is
continuous. In this paper, we extend off-the-grid CS to applications where some
prior information about spectrally sparse signal is known. We specifically
consider cases where a few contributing frequencies or poles, but not their
amplitudes or phases, are known a priori. Our results show that equipping
off-the-grid CS with the known-poles algorithm can increase the probability of
recovering all the frequency components.Comment: 5 pages, 4 figure
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