607 research outputs found
Nonlinear Basis Pursuit
In compressive sensing, the basis pursuit algorithm aims to find the sparsest
solution to an underdetermined linear equation system. In this paper, we
generalize basis pursuit to finding the sparsest solution to higher order
nonlinear systems of equations, called nonlinear basis pursuit. In contrast to
the existing nonlinear compressive sensing methods, the new algorithm that
solves the nonlinear basis pursuit problem is convex and not greedy. The novel
algorithm enables the compressive sensing approach to be used for a broader
range of applications where there are nonlinear relationships between the
measurements and the unknowns
Computational Methods for Sparse Solution of Linear Inverse Problems
The goal of the sparse approximation problem is to approximate a target signal using a linear combination of a few elementary signals drawn from a fixed collection. This paper surveys the major practical algorithms for sparse approximation. Specific attention is paid to computational issues, to the circumstances in which individual methods tend to perform well, and to the theoretical guarantees available. Many fundamental questions in electrical engineering, statistics, and applied mathematics can be posed as sparse approximation problems, making these algorithms versatile and relevant to a plethora of applications
New Coherence and RIP Analysis for Weak Orthogonal Matching Pursuit
In this paper we define a new coherence index, named the global 2-coherence,
of a given dictionary and study its relationship with the traditional mutual
coherence and the restricted isometry constant. By exploring this relationship,
we obtain more general results on sparse signal reconstruction using greedy
algorithms in the compressive sensing (CS) framework. In particular, we obtain
an improved bound over the best known results on the restricted isometry
constant for successful recovery of sparse signals using orthogonal matching
pursuit (OMP).Comment: arXiv admin note: substantial text overlap with arXiv:1307.194
Compressive Phase Retrieval From Squared Output Measurements Via Semidefinite Programming
Given a linear system in a real or complex domain, linear regression aims to
recover the model parameters from a set of observations. Recent studies in
compressive sensing have successfully shown that under certain conditions, a
linear program, namely, l1-minimization, guarantees recovery of sparse
parameter signals even when the system is underdetermined. In this paper, we
consider a more challenging problem: when the phase of the output measurements
from a linear system is omitted. Using a lifting technique, we show that even
though the phase information is missing, the sparse signal can be recovered
exactly by solving a simple semidefinite program when the sampling rate is
sufficiently high, albeit the exact solutions to both sparse signal recovery
and phase retrieval are combinatorial. The results extend the type of
applications that compressive sensing can be applied to those where only output
magnitudes can be observed. We demonstrate the accuracy of the algorithms
through theoretical analysis, extensive simulations and a practical experiment.Comment: Parts of the derivations have submitted to the 16th IFAC Symposium on
System Identification, SYSID 2012, and parts to the 51st IEEE Conference on
Decision and Control, CDC 201
Spectral Compressive Sensing with Model Selection
The performance of existing approaches to the recovery of frequency-sparse
signals from compressed measurements is limited by the coherence of required
sparsity dictionaries and the discretization of frequency parameter space. In
this paper, we adopt a parametric joint recovery-estimation method based on
model selection in spectral compressive sensing. Numerical experiments show
that our approach outperforms most state-of-the-art spectral CS recovery
approaches in fidelity, tolerance to noise and computation efficiency.Comment: 5 pages, 2 figures, 1 table, published in ICASSP 201
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