We propose the Dantzig selector based on the l_(1-q)(1<q≤2)minimization model for the sparse signal recovery.First,we discuss some properties of l_(1-q)minimization model and give some useful inequalities.Then,we ...We propose the Dantzig selector based on the l_(1-q)(1<q≤2)minimization model for the sparse signal recovery.First,we discuss some properties of l_(1-q)minimization model and give some useful inequalities.Then,we give a sufficient condition based on the restricted isometry property for the stable recovery of signals.The l_(1-2)minimization model of Yin-Lou-He is extended to the l_(1-q)minimization model.展开更多
As two popularly used variable selection methods, the Dantzig selector and the LASSO have been proved asymptotically equivalent in some scenarios. However, it is not the case in general for linear models, as disclosed...As two popularly used variable selection methods, the Dantzig selector and the LASSO have been proved asymptotically equivalent in some scenarios. However, it is not the case in general for linear models, as disclosed in Gai, Zhu and Lin's paper in 2013. In this paper, it is further shown that generally the asymptotic equivalence is not true either for a general single-index model with random design of predictors. To achieve this goal, the authors systematically investigate necessary and sufficient conditions for the consistent model selection of the Dantzig selector. An adaptive Dantzig selector is also recommended for the cases where those conditions are not satisfied. Also, different from existing methods for linear models, no distributional assumption on error term is needed with a trade-off that more stringent condition on the predictor vector is assumed. A small scale simulation is conducted to examine the performances of the Dantzig selector and the adaptive Dantzig selector.展开更多
This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy measurements.We first introduce truncated sparse approximation p...This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy measurements.We first introduce truncated sparse approximation property,a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation property.We also explore the relationship between the restricted isometry property and truncated sparse approximation property.And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k,then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constantsδk andδ2k,respectively.Last,we show that ifδs(k+|T^c|)<√(s-1)/s for some s≥4/3,then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k.It should be pointed out that when Tc=Ф,our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.展开更多
In this paper,we establish the oracle inequalities of highly corrupted linear observationsb=Ax_(0)+f_(0)+e∈R^(m).Here the vectorx_(0)∈R^(m)is a(approximately)sparse signal and∈R^(n)n>>mis a sparse error vecto...In this paper,we establish the oracle inequalities of highly corrupted linear observationsb=Ax_(0)+f_(0)+e∈R^(m).Here the vectorx_(0)∈R^(m)is a(approximately)sparse signal and∈R^(n)n>>mis a sparse error vector with nonzero entries that can be possible infinitely large,erepresents the Gaussian random noise vector.We extend the oracle inequality■for Dantzig selector and Lasso models in[E.J.Candès and T.Tao,Ann.Statist.,35(2007),2313-2351]and[T.T.Cai,L.Wang,and G.Xu,IEEE Trans.Inf.Theory,56(2010),3516-3522]to■for the extended Dantzig selector and Lasso models.Here{|λf_(0)(j)|^(2),σ^(2)}is the solution of the extended model,and■is the balance parameter between■.展开更多
广义Dantzig选择器问题是解决参数估计的有效途径,其中任何范数都可以用于估计.本文采用对偶交替方向乘子法(dual Alternating Direction Method of Multipliers,简称dADMM)求解e_(1)范数,e_(2)范数和e_(∞)范数广义Dantzig选择器问题,...广义Dantzig选择器问题是解决参数估计的有效途径,其中任何范数都可以用于估计.本文采用对偶交替方向乘子法(dual Alternating Direction Method of Multipliers,简称dADMM)求解e_(1)范数,e_(2)范数和e_(∞)范数广义Dantzig选择器问题,并给出了dADMM的全局收敛性和局部线性收敛速度.数值试验验证了dADMM的有效性.展开更多
In this paper,we study compressed data separation(CDS)problem,i.e.,sparse data separation from a few linear random measurements.We propose the nonconvex ℓ_(q)-split analysis with ℓ_(∞)-constraint and 0<q≤1.We cal...In this paper,we study compressed data separation(CDS)problem,i.e.,sparse data separation from a few linear random measurements.We propose the nonconvex ℓ_(q)-split analysis with ℓ_(∞)-constraint and 0<q≤1.We call the algorithm ℓ_(q)-split-analysis Dantzig selector(ℓ_(q)-split-analysis DS).We show that the two distinct subcomponents that are approximately sparse in terms of two different dictionaries could be stably approximated via the ℓ_(q)-split-analysis DS,provided that the measurement matrix satisfies either a classical D-RIP(Restricted Isometry Property with respect to Dictionaries and ℓ_(2) norm)or a relatively new(D,q)-RIP(RIP with respect to Dictionaries and ℓ_(q)-quasi norm)condition and the two different dictionaries satisfy a mutual coherence condition between them.For the Gaussian random measurements,the measurement number needed for the(D,q)-RIP condition is far less than those needed for the D-RIP condition and the(D,1)-RIP condition when q is small enough.展开更多
基金supported by the National Natural Science Foundation of China“Variable exponential function spaces on variable anisotropic Euclidean spaces and their applications”(12261083),“Harmonic analysis on affine symmetric spaces”(12161083).
文摘We propose the Dantzig selector based on the l_(1-q)(1<q≤2)minimization model for the sparse signal recovery.First,we discuss some properties of l_(1-q)minimization model and give some useful inequalities.Then,we give a sufficient condition based on the restricted isometry property for the stable recovery of signals.The l_(1-2)minimization model of Yin-Lou-He is extended to the l_(1-q)minimization model.
基金supported by the National Natural Science Foundation of China under Grant Nos.11501354,11201499,11301309 and 714732802015 Shanghai Young Faculty Training Program under Grant No.A1A-6119-15-003
文摘As two popularly used variable selection methods, the Dantzig selector and the LASSO have been proved asymptotically equivalent in some scenarios. However, it is not the case in general for linear models, as disclosed in Gai, Zhu and Lin's paper in 2013. In this paper, it is further shown that generally the asymptotic equivalence is not true either for a general single-index model with random design of predictors. To achieve this goal, the authors systematically investigate necessary and sufficient conditions for the consistent model selection of the Dantzig selector. An adaptive Dantzig selector is also recommended for the cases where those conditions are not satisfied. Also, different from existing methods for linear models, no distributional assumption on error term is needed with a trade-off that more stringent condition on the predictor vector is assumed. A small scale simulation is conducted to examine the performances of the Dantzig selector and the adaptive Dantzig selector.
基金supported by the National Natural Science Foundation of China(10871013,10871217)the NaturalScience Foundation of Beijing(1072004)Research Fund of Chongqing Technology and Business University(20105609)
基金supported by the National Natural Science Foundation of China(11871109)NSAF(U1830107)the Science Challenge Project(TZ2018001)
文摘This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy measurements.We first introduce truncated sparse approximation property,a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation property.We also explore the relationship between the restricted isometry property and truncated sparse approximation property.And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k,then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constantsδk andδ2k,respectively.Last,we show that ifδs(k+|T^c|)<√(s-1)/s for some s≥4/3,then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k.It should be pointed out that when Tc=Ф,our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.
基金supported by the Natural Science Foundation of China(Grant No.12201268)by the Science and Technology Program of Gansu Province of China(Grant No.21JR7RA511).
文摘In this paper,we establish the oracle inequalities of highly corrupted linear observationsb=Ax_(0)+f_(0)+e∈R^(m).Here the vectorx_(0)∈R^(m)is a(approximately)sparse signal and∈R^(n)n>>mis a sparse error vector with nonzero entries that can be possible infinitely large,erepresents the Gaussian random noise vector.We extend the oracle inequality■for Dantzig selector and Lasso models in[E.J.Candès and T.Tao,Ann.Statist.,35(2007),2313-2351]and[T.T.Cai,L.Wang,and G.Xu,IEEE Trans.Inf.Theory,56(2010),3516-3522]to■for the extended Dantzig selector and Lasso models.Here{|λf_(0)(j)|^(2),σ^(2)}is the solution of the extended model,and■is the balance parameter between■.
文摘广义Dantzig选择器问题是解决参数估计的有效途径,其中任何范数都可以用于估计.本文采用对偶交替方向乘子法(dual Alternating Direction Method of Multipliers,简称dADMM)求解e_(1)范数,e_(2)范数和e_(∞)范数广义Dantzig选择器问题,并给出了dADMM的全局收敛性和局部线性收敛速度.数值试验验证了dADMM的有效性.
基金Supported by the National Key Research and Development Program of China(Grant No.2021YFA1003500)the NSFC(Grant Nos.U21A20426,11971427,12071426 and 11901518)。
文摘In this paper,we study compressed data separation(CDS)problem,i.e.,sparse data separation from a few linear random measurements.We propose the nonconvex ℓ_(q)-split analysis with ℓ_(∞)-constraint and 0<q≤1.We call the algorithm ℓ_(q)-split-analysis Dantzig selector(ℓ_(q)-split-analysis DS).We show that the two distinct subcomponents that are approximately sparse in terms of two different dictionaries could be stably approximated via the ℓ_(q)-split-analysis DS,provided that the measurement matrix satisfies either a classical D-RIP(Restricted Isometry Property with respect to Dictionaries and ℓ_(2) norm)or a relatively new(D,q)-RIP(RIP with respect to Dictionaries and ℓ_(q)-quasi norm)condition and the two different dictionaries satisfy a mutual coherence condition between them.For the Gaussian random measurements,the measurement number needed for the(D,q)-RIP condition is far less than those needed for the D-RIP condition and the(D,1)-RIP condition when q is small enough.
