Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

Lou, Yifei; Yin, Pengbin; Xin, Jack

doi:10.1007/s10915-016-0169-x

Cited by 50 publications

(37 citation statements)

References 36 publications

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“…The scale-invariant L 1 , formulated as the ratio of L 1 and L 2 , was discussed in [9,26]. Other nonconvex L 1 variants include transformed L 1 [35], sorted L 1 [12], and capped L 1 [21]. It is demonstrated in a series of papers [19,20,34] that the difference of the L 1 and L 2 norms, denoted as L 1 -L 2 , outperforms L 1 and L p in terms of promoting sparsity when sensing matrix A is highly coherent.…”

Section: Introductionmentioning

confidence: 99%

Fast L1–L2 Minimization via a Proximal Operator

Lou¹,

Yan

2017

J Sci Comput

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145

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This paper aims to develop new and fast algorithms for recovering a sparse vector from a small number of measurements, which is a fundamental problem in the field of compressive sensing (CS). Currently, CS favors incoherent systems, in which any two measurements are as little correlated as possible. In reality, however, many problems are coherent, and conventional methods such as L 1 minimization do not work well. Recently, the difference of the L 1 and L 2 norms, denoted as L 1 -L 2 , is shown to have superior performance over the classic L 1 method, but it is computationally expensive. We derive an analytical solution for the proximal operator of the L 1 -L 2 metric, and it makes some fast L 1 solvers such as forward-backward splitting (FBS) and alternating direction method of multipliers (ADMM) applicable for L 1 -L 2 . We describe in details how to incorporate the proximal operator into FBS and ADMM and show that the resulting algorithms are convergent under mild conditions. Both algorithms are shown to be much more efficient than the original implementation of L 1 -L 2 based on a difference-of-convex approach in the numerical experiments.Keywords Compressive sensing · proximal operator · forward-backward splitting · alternating direction method of multipliers · difference-of-convex Mathematics Subject Classification (2000) 90C26 · 65K10 · 49M29

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Section: Introductionmentioning

confidence: 99%

Fast L1–L2 Minimization via a Proximal Operator

Lou¹,

Yan

2017

J Sci Comput

Self Cite

145

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show abstract

“…Several numerical examples in [32,55] have demonstrated that the ℓ 1 − ℓ 2 minimization consistently outperforms the ℓ 1 minimization and the ℓ p minimization in [27] when the measurement matrix A is highly coherent. In addition, the metric ℓ 1 − ℓ 2 has shown advantages in various applications such as signal processing [24,28,48], point source super-resolution [33], image restoration [23, 28,34], matrix completion [36], uncertainty quantification [26,54] and phase retrieval [52,56].…”

Section: Contributionsmentioning

confidence: 99%

The Dantzig selector: Recovery of Signal via $\ell_1-α\ell_2$ Minimization

Ge,

2021

Preprint

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In the paper, we proposed the Dantzig selector based on the ℓ 1 − αℓ 2 (0 < α ≤ 1) minimization for the signal recovery. In the Dantzig selector, the constraint A ⊤ (b − Ax) ∞ ≤ η for some small constant η > 0 means the columns of A has very weakly correlated with the error vector e = Ax − b. First, recovery guarantees based on the restricted isometry property (RIP) are established for signals. Next, we propose the effective algorithm to solve the proposed Dantzig selector. Last, we illustrate the proposed model and algorithm by extensive numerical experiments for the recovery of signals in the cases of Gaussian, impulsive and uniform noise. And the performance of the proposed Dantzig selector is better than that of the existing methods.

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“…In applications where sensing hardwares cannot be modified or upgraded, a robust recovery algorithm is a valuable tool for information retrieval. An example is super-resolution where sparse signals are recovered from low frequency measurements within the hardware resolution limit [7,22].…”

Section: Over-sampled Dctmentioning

confidence: 99%

Minimization of transformed $$L_1$$ L 1 penalty: theory, difference of convex function algorithm, and robust application in compressed sensing

2018

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We study the minimization problem of a non-convex sparsity promoting penalty function, the transformed l 1 (TL1), and its application in compressed sensing (CS). The TL1 penalty interpolates l 0 and l 1 norms through a nonnegative parameter a ∈ (0, +∞), similar to lp with p ∈ (0, 1], and is known to satisfy unbiasedness, sparsity and Lipschitz continuity properties. We first consider the constrained minimization problem, and discuss the exact recovery of l 0 norm minimal solution based on the null space property (NSP). We then prove the stable recovery of l 0 norm minimal solution if the sensing matrix A satisfies a restricted isometry property (RIP). We formulated a normalized problem to overcome the lack of scaling property of the TL1 penalty function. For a general sensing matrix A, we show that the support set of a local minimizer corresponds to linearly independent columns of A. Next, we present difference of convex algorithms for TL1 (DCATL1) in computing TL1-regularized constrained and unconstrained problems in CS. The DCATL1 algorithm involves outer and inner loops of iterations, one time matrix inversion, repeated shrinkage operations and matrix-vector multiplications. The inner loop concerns an l 1 minimization problem on which we employ the Alternating Direction Method of Multipliers (ADMM). For the unconstrained problem, we prove convergence of DCATL1 to a stationary point satisfying the first order optimality condition. In numerical experiments, we identify the optimal value a = 1, and compare DCATL1 with other CS algorithms on two classes of sensing matrices: Gaussian random matrices and over-sampled discrete cosine transform matrices (DCT). Among existing algorithms, the iterated reweighted least squares method based on l 1/2 norm is the best in sparse recovery for Gaussian matrices, and the DCA algorithm based on l 1 minus l 2 penalty is the best for over-sampled DCT matrices. We find that for both classes of sensing matrices, the performance of DCATL1 algorithm (initiated with l 1 minimization) always ranks near the top (if not the top), and is the most robust choice insensitive to the conditioning of the sensing matrix A. DCATL1 is also competitive in comparison with DCA on other non-convex penalty functions commonly used in statistics with two hyperparameters.

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Point Source Super-resolution Via Non-convex $$L_1$$ Based Methods

Cited by 50 publications

References 36 publications

Fast L1–L2 Minimization via a Proximal Operator

Fast L1–L2 Minimization via a Proximal Operator

The Dantzig selector: Recovery of Signal via $\ell_1-α\ell_2$ Minimization

Minimization of transformed $$L_1$$ L 1 penalty: theory, difference of convex function algorithm, and robust application in compressed sensing

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