845 research outputs found
Signal Recovery from Inaccurate and Incomplete Measurements via Regularized Orthogonal Matching Pursuit
We demonstrate a simple greedy algorithm that can reliably recover a vector v ?? ??d from incomplete and inaccurate measurements x = ??v + e. Here, ?? is a N x d measurement matrix with Nv with O(n) nonzeros from its inaccurate measurements x in at most n iterations, where each iteration amounts to solving a least squares problem. The noise level of the recovery is proportional to ??{logn} ||e||2. In particular, if the error term e vanishes the reconstruction is exact
Signal Recovery from Incomplete and Inaccurate Measurements via Regularized Orthogonal Matching Pursuit
We demonstrate a simple greedy algorithm that can reliably recover a
d-dimensional vector v from incomplete and inaccurate measurements x. Here our
measurement matrix is an N by d matrix with N much smaller than d. Our
algorithm, Regularized Orthogonal Matching Pursuit (ROMP), seeks to close the
gap between two major approaches to sparse recovery. It combines the speed and
ease of implementation of the greedy methods with the strong guarantees of the
convex programming methods. For any measurement matrix that satisfies a Uniform
Uncertainty Principle, ROMP recovers a signal with O(n) nonzeros from its
inaccurate measurements x in at most n iterations, where each iteration amounts
to solving a Least Squares Problem. The noise level of the recovery is
proportional to the norm of the error, up to a log factor. In particular, if
the error vanishes the reconstruction is exact. This stability result extends
naturally to the very accurate recovery of approximately sparse signals
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
Proof of Convergence and Performance Analysis for Sparse Recovery via Zero-point Attracting Projection
A recursive algorithm named Zero-point Attracting Projection (ZAP) is
proposed recently for sparse signal reconstruction. Compared with the reference
algorithms, ZAP demonstrates rather good performance in recovery precision and
robustness. However, any theoretical analysis about the mentioned algorithm,
even a proof on its convergence, is not available. In this work, a strict proof
on the convergence of ZAP is provided and the condition of convergence is put
forward. Based on the theoretical analysis, it is further proved that ZAP is
non-biased and can approach the sparse solution to any extent, with the proper
choice of step-size. Furthermore, the case of inaccurate measurements in noisy
scenario is also discussed. It is proved that disturbance power linearly
reduces the recovery precision, which is predictable but not preventable. The
reconstruction deviation of -compressible signal is also provided. Finally,
numerical simulations are performed to verify the theoretical analysis.Comment: 29 pages, 6 figure
On Probability of Support Recovery for Orthogonal Matching Pursuit Using Mutual Coherence
In this paper we present a new coherence-based performance guarantee for the
Orthogonal Matching Pursuit (OMP) algorithm. A lower bound for the probability
of correctly identifying the support of a sparse signal with additive white
Gaussian noise is derived. Compared to previous work, the new bound takes into
account the signal parameters such as dynamic range, noise variance, and
sparsity. Numerical simulations show significant improvements over previous
work and a closer match to empirically obtained results of the OMP algorithm.Comment: Submitted to IEEE Signal Processing Letters. arXiv admin note:
substantial text overlap with arXiv:1608.0038
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