2,455 research outputs found
Coherence-Based Performance Guarantees of Orthogonal Matching Pursuit
In this paper, we present coherence-based performance guarantees of
Orthogonal Matching Pursuit (OMP) for both support recovery and signal
reconstruction of sparse signals when the measurements are corrupted by noise.
In particular, two variants of OMP either with known sparsity level or with a
stopping rule are analyzed. It is shown that if the measurement matrix
satisfies the strong coherence property, then with
, OMP will recover a -sparse signal with high
probability. In particular, the performance guarantees obtained here separate
the properties required of the measurement matrix from the properties required
of the signal, which depends critically on the minimum signal to noise ratio
rather than the power profiles of the signal. We also provide performance
guarantees for partial support recovery. Comparisons are given with other
performance guarantees for OMP using worst-case analysis and the sorted one
step thresholding algorithm.Comment: appeared at 2012 Allerton conferenc
Relaxed Recovery Conditions for OMP/OLS by Exploiting both Coherence and Decay
We propose extended coherence-based conditions for exact sparse support
recovery using orthogonal matching pursuit (OMP) and orthogonal least squares
(OLS). Unlike standard uniform guarantees, we embed some information about the
decay of the sparse vector coefficients in our conditions. As a result, the
standard condition (where denotes the mutual coherence and
the sparsity level) can be weakened as soon as the non-zero coefficients
obey some decay, both in the noiseless and the bounded-noise scenarios.
Furthermore, the resulting condition is approaching for strongly
decaying sparse signals. Finally, in the noiseless setting, we prove that the
proposed conditions, in particular the bound , are the tightest
achievable guarantees based on mutual coherence
Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection
We study the problem of selecting a subset of k random variables from a large
set, in order to obtain the best linear prediction of another variable of
interest. This problem can be viewed in the context of both feature selection
and sparse approximation. We analyze the performance of widely used greedy
heuristics, using insights from the maximization of submodular functions and
spectral analysis. We introduce the submodularity ratio as a key quantity to
help understand why greedy algorithms perform well even when the variables are
highly correlated. Using our techniques, we obtain the strongest known
approximation guarantees for this problem, both in terms of the submodularity
ratio and the smallest k-sparse eigenvalue of the covariance matrix. We further
demonstrate the wide applicability of our techniques by analyzing greedy
algorithms for the dictionary selection problem, and significantly improve the
previously known guarantees. Our theoretical analysis is complemented by
experiments on real-world and synthetic data sets; the experiments show that
the submodularity ratio is a stronger predictor of the performance of greedy
algorithms than other spectral parameters
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
Exact Recovery Conditions for Sparse Representations with Partial Support Information
We address the exact recovery of a k-sparse vector in the noiseless setting
when some partial information on the support is available. This partial
information takes the form of either a subset of the true support or an
approximate subset including wrong atoms as well. We derive a new sufficient
and worst-case necessary (in some sense) condition for the success of some
procedures based on lp-relaxation, Orthogonal Matching Pursuit (OMP) and
Orthogonal Least Squares (OLS). Our result is based on the coherence "mu" of
the dictionary and relaxes the well-known condition mu<1/(2k-1) ensuring the
recovery of any k-sparse vector in the non-informed setup. It reads
mu<1/(2k-g+b-1) when the informed support is composed of g good atoms and b
wrong atoms. We emphasize that our condition is complementary to some
restricted-isometry based conditions by showing that none of them implies the
other.
Because this mutual coherence condition is common to all procedures, we carry
out a finer analysis based on the Null Space Property (NSP) and the Exact
Recovery Condition (ERC). Connections are established regarding the
characterization of lp-relaxation procedures and OMP in the informed setup.
First, we emphasize that the truncated NSP enjoys an ordering property when p
is decreased. Second, the partial ERC for OMP (ERC-OMP) implies in turn the
truncated NSP for the informed l1 problem, and the truncated NSP for p<1.Comment: arXiv admin note: substantial text overlap with arXiv:1211.728
- âŠ