Robust 1-bit Compressive Sensing Using Adaptive Outlier Pursuit

Yan, Ming; Yang, Yi; Osher, Stanley

doi:10.1109/tsp.2012.2193397

Cited by 159 publications

(120 citation statements)

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“…Specifically in this experiment, when n = 1000, m = 1000, the average computational times are: 6.7ms (Passive), 8 Similar observations can be found in Fig. 2, where different noise levels are considered.…”

Section: Larizations Section 3 Gives Several Such Kinds Of Algorithmsupporting

confidence: 81%

“…If the underlying signal is sparse, then sparsity pursuit techniques can help signal recovery, which is similar to the regular compressive sensing. Therefore, since its proposal by [6], 1bit-CS has attracted much attention in both the signal processing society ( [7,8,9,10]) and the machine learning society ( [11,12,13,14]). Because the one-bit information has no capability to specify the magnitude of the original signal, we assume x 2 = 1 without loss of generality (there is also some work on norm estimation, see, e.g., [15]), and 1bit-CS can be explained as finding the sparest vector on the unit sphere that coincides with the observed signs, i.e, minimize x∈R n x 0 , subject to y i = sgn(u ⊤ i x), ∀i = 1, 2, .…”

Section: Introductionmentioning

confidence: 99%

“…One way to deal with noise and sign flips is to replace the constraint y i = sgn(u ⊤ i x) by a loss function. For example, the one-sided ℓ 1 loss and the one-sided ℓ 2 loss are considered in [18] and [8]; the linear loss is used in [9] and [11]. It is reported that the linear loss generally outperforms the one-sided ℓ 1 /ℓ 2 loss.…”

Section: Introductionmentioning

confidence: 99%

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Nonconvex penalties with analytical solutions for one-bit compressive sensing

Huang

Yan

2018

Signal Processing

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One-bit measurements widely exist in the real world and can be used to recover sparse signals. This task is known as one-bit compressive sensing (1bit-CS). In this paper, we propose novel algorithms based on both convex and nonconvex sparsity-inducing penalties for robust 1bit-CS. We consider the dual problem, which has only one variable and provides a sufficient condition to verify whether a solution is globally optimal or not. For positive homogeneous penalties, a globally optimal solution can be obtained in two steps: a proximal operator and a normalization step. For other penalties, we solve the dual problem, and it needs to evaluate the proximal operators for many times. Then we provide fast algorithms for finding analytical solutions for three penalties: minimax concave penalty (MCP), ℓ 0 norm, and sorted ℓ 1 penalty. Specifically, our algorithm is more than 200 times faster than the existing algorithm for MCP. Its efficiency is comparable to the algorithm for the ℓ 1 penalty in time, while its performance is much better than ℓ 1 . Among these penalties, sorted ℓ 1 is most robust to noise in different settings.

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Section: Larizations Section 3 Gives Several Such Kinds Of Algorithmsupporting

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonconvex penalties with analytical solutions for one-bit compressive sensing

Huang

Yan

2018

Signal Processing

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“…Extensions include recovering the norm of the target [32, 3] and non-Gaussian measurement settings [1]. Many first order methods [6,34,49,14] and greedy methods [35,5,29] are developed to minimize the sparsity promoting nonconvex objected function arising from either the unit sphere constraint or the nonconvex regularizers. To address…”

mentioning

confidence: 99%

“…Now we compare our proposed model(1.2) and PDASC algorithm with several state-of-the-art methods such as BIHT[28] (http://perso. uclouvain.be/laurent.jacques/index.php/Main/BIHTDemo), AOP[49] and PBAOP[24] (both AOP and PBAOP available at http://www.esat.kuleuven.be/stadius/ADB/huang/downloads/ 1bitCSLab.zip) and linear projection (LP)[47,42]. BIHT, AOP, LP and PBAOP are all required to specify the true sparsity level s. Both AOP and PBAOP also need to required to specify thesign flips probability q.…”

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confidence: 99%

Robust Decoding from 1-Bit Compressive Sampling with Ordinary and Regularized Least Squares

Huang¹,

Jiao²,

Lu³

et al. 2018

SIAM J. Sci. Comput.

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In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: y = η ⊙ sign(Ψx * + ǫ), where x * ∈ R n , y ∈ R m , Ψ ∈ R m×n , and ǫ is the random error before quantization and η ∈ R n is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For m > n, we show that, up to a constant c, with high probability, the least squares solution x ls approximates x * with precision δ as long as m ≥ O( n δ 2 ). For m < n, we prove that, up to a constant c, with high probability, the ℓ 1 -regularized least-squares solution x ℓ 1 lies in the ball with center x * and radius δ provided that m ≥ O( s log n δ 2 ) and x * 0 := s < m. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.Keywords: 1-bit compressive sensing, ℓ 1 -regularized least squares, primal dual active set algorithm, one step convergence, continuation 1. Introduction. Compressive sensing (CS) is an important approach to acquiring low dimension signals from noisy under-determined measurements [8,16,19,20]. For storage and transmission, the infinite-precision measurements are often quantized, [6] considered recovering the signals from the 1-bit compressive sensing (1-bit CS) where measurements are coded into a single bit, i.e., their signs. The 1-bit CS is superior to the CS in terms of inexpensive hardware implementation and storage. However, it is much more challenging to decode from nonlinear, noisy and sign-flipped 1-bit measurements.1.1. Previous work. Since the seminal work of [6], much effort has been devoted to studying the theoretical and computational challenges of the 1-bit CS. Sample complexity was analyzed for support and vector recovery with and without noise [21,28,40,23,29,22,23,41,50]. Existing works indicate that, m > O(s log n) is adequate for both support and vector recovery. The sample size required here has the same order as that required in the standard CS setting. These results have also been refined by adaptive sampling [22,14,4]. Extensions include recovering the norm of the target [32, 3] and non-Gaussian measurement settings [1]. Many first order methods [6,34,49,14] and greedy methods [35,5,29] are developed to minimize the sparsity promoti...

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Algorithms for Sparsity-Constrained Optimization

Bahmani

2014

Springer Theses

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Robust 1-bit Compressive Sensing Using Adaptive Outlier Pursuit

Cited by 159 publications

References 16 publications

Nonconvex penalties with analytical solutions for one-bit compressive sensing

Nonconvex penalties with analytical solutions for one-bit compressive sensing

Robust Decoding from 1-Bit Compressive Sampling with Ordinary and Regularized Least Squares

Algorithms for Sparsity-Constrained Optimization

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