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18 pages, 2609 KiB

Open AccessArticle

Robust AOA-Based Target Localization for Uniformly Distributed Noise via ℓ_p-ℓ₁ Optimization

by Yanping Chen, Chunmei Wang and Qingli Yan

Entropy 2022, 24(9), 1259; https://doi.org/10.3390/e24091259 - 7 Sep 2022

Cited by 2 | Viewed by 1825

This paper addresses the problem of robust angle of arrival (AOA) target localization in the presence of uniformly distributed noise which is modeled as the mixture of Laplacian distribution and uniform distribution. Motivated by the distribution of noise, we develop a localization model [...] Read more.

ℓ_{p}

-norm with

0 \leq p < 2

as the measurement error and the

ℓ_{1}

-norm as the regularization term. Then, an estimator for introducing the proximal operator into the framework of the alternating direction method of multipliers (POADMM) is derived to solve the convex optimization problem when

1 \leq p < 2

. However, when

0 \leq p < 1

, the corresponding optimization problem is nonconvex and nonsmoothed. To derive a convergent method for this nonconvex and nonsmooth target localization problem, we propose a smoothed POADMM estimator (SPOADMM) by introducing the smoothing strategy into the optimization model. Eventually, the proposed algorithms are compared with some state-of-the-art robust algorithms via numerical simulations, and their effectiveness in uniformly distributed noise is discussed from the perspective of root-mean-squared error (RMSE). The experimental results verify that the proposed method has more robustness against outliers and is less sensitive to the selected parameters, especially the variance of the measurement noise. Full article

(This article belongs to the Section Signal and Data Analysis)

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24 pages, 1168 KiB

Open AccessArticle

Faster Provable Sieving Algorithms for the Shortest Vector Problem and the Closest Vector Problem on Lattices in ℓ_p Norm

by Priyanka Mukhopadhyay

Algorithms 2021, 14(12), 362; https://doi.org/10.3390/a14120362 - 13 Dec 2021

Cited by 4 | Viewed by 3511

Abstract

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in

ℓ_{p}

norm (

1 \leq p \leq \infty

). The running time we obtain is better than existing provable [...] Read more.

In this work, we give provable sieving algorithms for the Shortest Vector Problem (SVP) and the Closest Vector Problem (CVP) on lattices in

ℓ_{p}

norm (

1 \leq p \leq \infty

). The running time we obtain is better than existing provable sieving algorithms. We give a new linear sieving procedure that works for all

ℓ_{p}

norm (

1 \leq p \leq \infty

). The main idea is to divide the space into hypercubes such that each vector can be mapped efficiently to a sub-region. We achieve a time complexity of

2^{2.751 n + o (n)}

, which is much less than the

2^{3.849 n + o (n)}

complexity of the previous best algorithm. We also introduce a mixed sieving procedure, where a point is mapped to a hypercube within a ball and then a quadratic sieve is performed within each hypercube. This improves the running time, especially in the

ℓ_{2}

norm, where we achieve a time complexity of

2^{2.25 n + o (n)}

, while the List Sieve Birthday algorithm has a running time of

2^{2.465 n + o (n)}

. We adopt our sieving techniques to approximation algorithms for SVP and CVP in

ℓ_{p}

norm (

1 \leq p \leq \infty

) and show that our algorithm has a running time of

2^{2.001 n + o (n)}

, while previous algorithms have a time complexity of

2^{3.169 n + o (n)}

. Full article

(This article belongs to the Section Randomized, Online, and Approximation Algorithms)

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16 pages, 419 KiB

Open AccessArticle

Solving High-Dimensional Problems in Statistical Modelling: A Comparative Study

by Stamatis Choudalakis, Marilena Mitrouli, Athanasios Polychronou and Paraskevi Roupa

Mathematics 2021, 9(15), 1806; https://doi.org/10.3390/math9151806 - 30 Jul 2021

Cited by 4 | Viewed by 1908

Abstract

In this work, we present numerical methods appropriate for parameter estimation in high-dimensional statistical modelling. The solution of these problems is not unique and a crucial question arises regarding the way that a solution can be found. A common choice is to keep the corresponding solution with the minimum norm. There are cases in which this solution is not adequate and regularisation techniques have to be considered. We classify specific cases for which regularisation is required or not. We present a thorough comparison among existing methods for both estimating the coefficients of the model which corresponds to design matrices with correlated covariates and for variable selection for supersaturated designs. An extensive analysis for the properties of design matrices with correlated covariates is given. Numerical results for simulated and real data are presented. Full article

(This article belongs to the Special Issue Numerical Linear Algebra and the Applications)

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13 pages, 3488 KiB

Open AccessArticle

ℓ_p-STFT: A Robust Parameter Estimator of a Frequency Hopping Signal for Impulsive Noise

by Yang Su, Lina Wang, Yuan Chen and Xiaolong Yang

Electronics 2019, 8(9), 1017; https://doi.org/10.3390/electronics8091017 - 11 Sep 2019

Cited by 6 | Viewed by 3083

Abstract

Impulsive noise is commonly present in many applications of actual communication networks, leading to algorithms based on the Gaussian model no longer being applicable. A robust parameter estimator of frequency-hopping (FH) signals suitable for various impulsive noise environments, referred to as ℓ_p-STFT, is proposed. The ℓ_p-STFT estimator replaces the ℓ₂-norm by using the generalized version ℓ_p-norm where 1 < p < 2 for the derivation of the short-time Fourier transform (STFT) as an objective function. It combines impulsive noise processing with any time-frequency analysis algorithm based on STFT. Considering the accuracy of parameter estimation, the double-window spectrogram difference (DWSD) algorithm is used to illustrate the suitability of ℓ_p-STFT. Computer simulations are mainly conducted in α-stable noise to compare the performance of ℓ_p-STFT with STFT and fractional low-order STFT (FLOSTFT), Cauchy noise, and Gaussian mixture noise as supplements of different background noises to better demonstrate the robustness and accuracy of ℓ_p-STFT. Full article

(This article belongs to the Section Circuit and Signal Processing)

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11 pages, 2283 KiB

Open AccessArticle

Sparse Coding Algorithm with Negentropy and Weighted ℓ₁-Norm for Signal Reconstruction

by Yingxin Zhao, Zhiyang Liu, Yuanyuan Wang, Hong Wu and Shuxue Ding

Entropy 2017, 19(11), 599; https://doi.org/10.3390/e19110599 - 8 Nov 2017

Cited by 7 | Viewed by 3965

Abstract

Compressive sensing theory has attracted widespread attention in recent years and sparse signal reconstruction has been widely used in signal processing and communication. This paper addresses the problem of sparse signal recovery especially with non-Gaussian noise. The main contribution of this paper is the proposal of an algorithm where the negentropy and reweighted schemes represent the core of an approach to the solution of the problem. The signal reconstruction problem is formalized as a constrained minimization problem, where the objective function is the sum of a measurement of error statistical characteristic term, the negentropy, and a sparse regularization term, ℓ_p-norm, for 0 < p < 1. The ℓ_p-norm, however, leads to a non-convex optimization problem which is difficult to solve efficiently. Herein we treat the ℓ_p -norm as a serious of weighted ℓ₁-norms so that the sub-problems become convex. We propose an optimized algorithm that combines forward-backward splitting. The algorithm is fast and succeeds in exactly recovering sparse signals with Gaussian and non-Gaussian noise. Several numerical experiments and comparisons demonstrate the superiority of the proposed algorithm. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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22 pages, 2868 KiB

Open AccessArticle

Reconstruction of Self-Sparse 2D NMR Spectra from Undersampled Data in the Indirect Dimension

by Xiaobo Qu, Di Guo, Xue Cao, Shuhui Cai and Zhong Chen

Sensors 2011, 11(9), 8888-8909; https://doi.org/10.3390/s110908888 - 15 Sep 2011

Cited by 38 | Viewed by 11923

Abstract

Reducing the acquisition time for two-dimensional nuclear magnetic resonance (2D NMR) spectra is important. One way to achieve this goal is reducing the acquired data. In this paper, within the framework of compressed sensing, we proposed to undersample the data in the indirect dimension for a type of self-sparse 2D NMR spectra, that is, only a few meaningful spectral peaks occupy partial locations, while the rest of locations have very small or even no peaks. The spectrum is reconstructed by enforcing its sparsity in an identity matrix domain with ℓ_p (p = 0.5) norm optimization algorithm. Both theoretical analysis and simulation results show that the proposed method can reduce the reconstruction errors compared with the wavelet-based ℓ₁ norm optimization. Full article

(This article belongs to the Section Chemical Sensors)

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