106,050 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
Sparse Support Recovery with Non-smooth Loss Functions
In this paper, we study the support recovery guarantees of underdetermined
sparse regression using the -norm as a regularizer and a non-smooth
loss function for data fidelity. More precisely, we focus in detail on the
cases of and losses, and contrast them with the usual
loss. While these losses are routinely used to account for either
sparse ( loss) or uniform ( loss) noise models, a
theoretical analysis of their performance is still lacking. In this article, we
extend the existing theory from the smooth case to these non-smooth
cases. We derive a sharp condition which ensures that the support of the vector
to recover is stable to small additive noise in the observations, as long as
the loss constraint size is tuned proportionally to the noise level. A
distinctive feature of our theory is that it also explains what happens when
the support is unstable. While the support is not stable anymore, we identify
an "extended support" and show that this extended support is stable to small
additive noise. To exemplify the usefulness of our theory, we give a detailed
numerical analysis of the support stability/instability of compressed sensing
recovery with these different losses. This highlights different parameter
regimes, ranging from total support stability to progressively increasing
support instability.Comment: in Proc. NIPS 201
Lorentzian Iterative Hard Thresholding: Robust Compressed Sensing with Prior Information
Commonly employed reconstruction algorithms in compressed sensing (CS) use
the norm as the metric for the residual error. However, it is well-known
that least squares (LS) based estimators are highly sensitive to outliers
present in the measurement vector leading to a poor performance when the noise
no longer follows the Gaussian assumption but, instead, is better characterized
by heavier-than-Gaussian tailed distributions. In this paper, we propose a
robust iterative hard Thresholding (IHT) algorithm for reconstructing sparse
signals in the presence of impulsive noise. To address this problem, we use a
Lorentzian cost function instead of the cost function employed by the
traditional IHT algorithm. We also modify the algorithm to incorporate prior
signal information in the recovery process. Specifically, we study the case of
CS with partially known support. The proposed algorithm is a fast method with
computational load comparable to the LS based IHT, whilst having the advantage
of robustness against heavy-tailed impulsive noise. Sufficient conditions for
stability are studied and a reconstruction error bound is derived. We also
derive sufficient conditions for stable sparse signal recovery with partially
known support. Theoretical analysis shows that including prior support
information relaxes the conditions for successful reconstruction. Simulation
results demonstrate that the Lorentzian-based IHT algorithm significantly
outperform commonly employed sparse reconstruction techniques in impulsive
environments, while providing comparable performance in less demanding,
light-tailed environments. Numerical results also demonstrate that the
partially known support inclusion improves the performance of the proposed
algorithm, thereby requiring fewer samples to yield an approximate
reconstruction.Comment: 28 pages, 9 figures, accepted in IEEE Transactions on Signal
Processin
Successful Recovery Performance Guarantees of SOMP Under the L2-norm of Noise
The simultaneous orthogonal matching pursuit (SOMP) is a popular, greedy
approach for common support recovery of a row-sparse matrix. However, compared
to the noiseless scenario, the performance analysis of noisy SOMP is still
nascent, especially in the scenario of unbounded noise. In this paper, we
present a new study based on the mutual incoherence property (MIP) for
performance analysis of noisy SOMP. Specifically, when noise is bounded, we
provide the condition on which the exact support recovery is guaranteed in
terms of the MIP. When noise is unbounded, we instead derive a bound on the
successful recovery probability (SRP) that depends on the specific distribution
of the -norm of the noise matrix. Then we focus on the common case when
noise is random Gaussian and show that the lower bound of SRP follows
Tracy-Widom law distribution. The analysis reveals the number of measurements,
noise level, the number of sparse vectors, and the value of mutual coherence
that are required to guarantee a predefined recovery performance.
Theoretically, we show that the mutual coherence of the measurement matrix must
decrease proportionally to the noise standard deviation, and the number of
sparse vectors needs to grow proportionally to the noise variance. Finally, we
extensively validate the derived analysis through numerical simulations
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