2,409 research outputs found
Sharp RIP Bound for Sparse Signal and Low-Rank Matrix Recovery
This paper establishes a sharp condition on the restricted isometry property
(RIP) for both the sparse signal recovery and low-rank matrix recovery. It is
shown that if the measurement matrix satisfies the RIP condition
, then all -sparse signals can be recovered exactly
via the constrained minimization based on . Similarly, if
the linear map satisfies the RIP condition ,
then all matrices of rank at most can be recovered exactly via the
constrained nuclear norm minimization based on . Furthermore, in
both cases it is not possible to do so in general when the condition does not
hold. In addition, noisy cases are considered and oracle inequalities are given
under the sharp RIP condition.Comment: to appear in Applied and Computational Harmonic Analysis (2012
Noisy Signal Recovery via Iterative Reweighted L1-Minimization
Compressed sensing has shown that it is possible to reconstruct sparse high
dimensional signals from few linear measurements. In many cases, the solution
can be obtained by solving an L1-minimization problem, and this method is
accurate even in the presence of noise. Recent a modified version of this
method, reweighted L1-minimization, has been suggested. Although no provable
results have yet been attained, empirical studies have suggested the reweighted
version outperforms the standard method. Here we analyze the reweighted
L1-minimization method in the noisy case, and provide provable results showing
an improvement in the error bound over the standard bounds
Sparse Representation of a Polytope and Recovery of Sparse Signals and Low-rank Matrices
This paper considers compressed sensing and affine rank minimization in both
noiseless and noisy cases and establishes sharp restricted isometry conditions
for sparse signal and low-rank matrix recovery. The analysis relies on a key
technical tool which represents points in a polytope by convex combinations of
sparse vectors. The technique is elementary while leads to sharp results.
It is shown that for any given constant , in compressed sensing
guarantees the exact recovery of all
sparse signals in the noiseless case through the constrained
minimization, and similarly in affine rank minimization
ensures the exact reconstruction of
all matrices with rank at most in the noiseless case via the constrained
nuclear norm minimization. Moreover, for any ,
is not sufficient to guarantee
the exact recovery of all -sparse signals for large . Similar result also
holds for matrix recovery. In addition, the conditions and are also shown to
be sufficient respectively for stable recovery of approximately sparse signals
and low-rank matrices in the noisy case.Comment: to appear in IEEE Transactions on Information Theor
TV-min and Greedy Pursuit for Constrained Joint Sparsity and Application to Inverse Scattering
This paper proposes a general framework for compressed sensing of constrained
joint sparsity (CJS) which includes total variation minimization (TV-min) as an
example. TV- and 2-norm error bounds, independent of the ambient dimension, are
derived for the CJS version of Basis Pursuit and Orthogonal Matching Pursuit.
As an application the results extend Cand`es, Romberg and Tao's proof of exact
recovery of piecewise constant objects with noiseless incomplete Fourier data
to the case of noisy data.Comment: Mathematics and Mechanics of Complex Systems (2013
Uniform Uncertainty Principle and signal recovery via Regularized Orthogonal Matching Pursuit
This paper seeks to bridge the two major algorithmic approaches to sparse
signal recovery from an incomplete set of linear measurements --
L_1-minimization methods and iterative methods (Matching Pursuits). We find a
simple regularized version of the Orthogonal Matching Pursuit (ROMP) which has
advantages of both approaches: the speed and transparency of OMP and the strong
uniform guarantees of the L_1-minimization. Our algorithm ROMP reconstructs a
sparse signal in a number of iterations linear in the sparsity (in practice
even logarithmic), and the reconstruction is exact provided the linear
measurements satisfy the Uniform Uncertainty Principle.Comment: This is the final version of the paper, including referee suggestion
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