A Scale-Invariant Approach for Sparse Signal Recovery

Rahimi, Yaghoub; Wang, Chao; Dong, Hongbo; Lou, Yifei

doi:10.1137/18m123147x

Cited by 91 publications

(91 citation statements)

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“…For the definition of 2,1 norm, see (5). Our method generalizes the 1 / 2 minimization method studied in [12,17,21,29,36]. We demonstrate in our numerical examples that it outperforms the 2,1 minimization commonly used to solve the joint sparse recovery problem in the sense that much fewer measurements are required for accurate sparse reconstructions.…”

Section: Introductionmentioning

confidence: 76%

Reconstruction of jointly sparse vectors via manifold optimization

Petrosyan

Tran

Webster

2019

Applied Numerical Mathematics

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In this paper, we consider the challenge of reconstructing jointly sparse vectors from linear measurements. Firstly, we show that by utilizing the rank of the output data matrix we can reduce the problem to a full column rank case. This result reveals a reduction in the computational complexity of the original problem and enables a simple implementation of joint sparse recovery algorithms for full-rank setting. Secondly, we propose a new method for joint sparse recovery in the form of a non-convex optimization problem on a non-compact Stiefel manifold. In our numerical experiments our method outperforms the commonly used 2,1 minimization in the sense that much fewer measurements are required for accurate sparse reconstructions. We postulate this approach possesses the desirable rank aware property, that is, being able to take advantage of the rank of the unknown matrix to improve the recovery.

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Section: Introductionmentioning

confidence: 76%

Reconstruction of jointly sparse vectors via manifold optimization

Petrosyan

Tran

Webster

2019

Applied Numerical Mathematics

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“…From Fig.1, we [70], where they using toy examples to illustrate the advantages of ℓ p and ℓ 1 /ℓ 2 , respectively, we also use a similar example to show that with some special data sets (A, b), the R (x) − R (x s ) tends to select a sparser solution.…”

Section: B Comparing With Other Regularizationmentioning

confidence: 99%

Sparse optimization problem with s-difference regularization

et al. 2020

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In this paper, a s-difference type regularization for sparse recovery problem is proposed, which is the difference of the normal penalty function R (x) and its corresponding struncated function R (x s ). First, we show the equivalent conditions between the ℓ0 constrained problem and the unconstrained s-difference penalty regularized problem. Next, we choose the forward-backward splitting (FBS) method to solve the nonconvex regularizes function and further derive some closed-form solutions for the proximal mapping of the s-difference regularization with some commonly used R (x), which makes the FBS easy and fast. We also show that any cluster point of the sequence generated by the proposed algorithm converges to a stationary point. Numerical experiments demonstrate the efficiency of the proposed s-difference regularization in comparison with some other existing penalty functions.

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“…For one-sparse signal case, the L 1 /L 2 is the same as the L 0 norm. Recently, L 1 /L 2 minimization model has been empirically verified its efficiency when the sensing matrix is coherent and redundant; see, e.g., [8,[11][12][13][14][15][16][17]. More specifically, two types of L 1 /L 2 models are commonly used: the constrained and the penalized/unconstrained:…”

Section: Introductionmentioning

confidence: 99%

“…where the compressing matrix A ∈ R m×n and the observation of b ∈ R n . The constrained model (2) has been widely used in sparse recovery and MRI reconstruction [12,14]. However, since the penalized/unconstrained model (3) can tackle both noisy and noiseless observations while (2) can only deal with noiseless data, it turns to be more meaningful to develop efficient and convergent algorithm to solve (3).…”

Section: Introductionmentioning

confidence: 99%

Unified Analysis on L1 over L2 Minimization for signal recovery

Tao¹,

Zhang²

2023

Preprint

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<div>In this paper, we carry out a unified study for L_1 over L_2 sparsity promoting models, which are widely used in the regime of coherent dictionaries for recovering sparse nonnegative/arbitrary signal. First, we provide the exact recovery condition on both the constrained and the unconstrained models for a broad set of signals. Next, we prove the solution existence of these L_{1}/L_{2} models under the assumption that the null space of the measurement matrix satisfies the $s$-spherical section property. Then by deriving an analytical solution for the proximal operator of the L_{1} / L_{2} with nonnegative constraint, we develop a new alternating direction method of multipliers based method (ADMM$_p^+$) to solve the unconstrained model. We establish its global convergence to a d-stationary solution (sharpest stationary) and its local linear convergence under certain conditions. Numerical simulations on two specific applications confirm the superior of ADMM$_p^+$ over the state-of-the-art methods in sparse recovery. ADMM$_p^+$ reduces computational time by about $95\%\sim99\%$ while achieving a much higher accuracy compared to commonly used scaled gradient projection method for wavelength misalignment problem.</div>

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A Scale-Invariant Approach for Sparse Signal Recovery

Cited by 91 publications

References 50 publications

Reconstruction of jointly sparse vectors via manifold optimization

Reconstruction of jointly sparse vectors via manifold optimization

Sparse optimization problem with s-difference regularization

Unified Analysis on L1 over L2 Minimization for signal recovery

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