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  • 1.
    Bayisa, Fekadu
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Cronie, Ottmar
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Adaptive algorithm for sparse signal recovery2019In: Digital signal processing (Print), ISSN 1051-2004, E-ISSN 1095-4333, Vol. 87, p. 16p. 10-18Article in journal (Refereed)
    Abstract [en]

    The development of compressive sensing in recent years has given much attention to sparse signal recovery. In sparse signal recovery, spike and slab priors are playing a key role in inducing sparsity. The use of such priors, however, results in non-convex and mixed integer programming problems. Most of the existing algorithms to solve non-convex and mixed integer programming problems involve either simplifying assumptions, relaxations or high computational expenses. In this paper, we propose a new adaptive alternating direction method of multipliers (AADMM) algorithm to directly solve the suggested non-convex and mixed integer programming problem. The algorithm is based on the one-to-one mapping property of the support and non-zero element of the signal. At each step of the algorithm, we update the support by either adding an index to it or removing an index from it and use the alternating direction method of multipliers to recover the signal corresponding to the updated support. Moreover, as opposed to the competing “adaptive sparsity matching pursuit” and “alternating direction method of multipliers” methods our algorithm can solve non-convex problems directly. Experiments on synthetic data and real-world images demonstrated that the proposed AADMM algorithm provides superior performance and is computationally cheaper than the recently developed iterative convex refinement (ICR) and adaptive matching pursuit (AMP) algorithms.

  • 2.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    An Extended Block Restricted Isometry Property for Sparse Recovery with Non-Gaussian Noise2018Conference paper (Refereed)
  • 3.
    Wang, Jianfeng
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Garpebring, Anders
    Umeå University, Faculty of Medicine, Department of Radiation Sciences.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sparsity estimation in compressive sensing with application to MR images2017Manuscript (preprint) (Other academic)
    Abstract [en]

    The theory of compressive sensing (CS) asserts that an unknown signal x in C^N canbe accurately recovered from m measurements with m << N provided that x is sparse. Most of the recovery algorithms need the sparsity s = ||x||_0 as an input. However,generally s is unknown, and directly estimating the sparsity has been an open problem.In this study, an estimator of sparsity is proposed by using Bayesian hierarchical model. Its statistical properties such as unbiasedness and asymptotic normality are proved. Inthe simulation study and real data study, magnetic resonance image data is used asinput signal, which becomes sparse after sparsified transformation. The results fromthe simulation study confirm the theoretical properties of the estimator. In practice, theestimate from a real MR image can be used for recovering future MR images under theframework of CS if they are believed to have the same sparsity level after sparsification.

  • 4.
    Wang, Jianfeng
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics. Department of Statistics, Zhejiang University City College, China.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Statistical inference for block sparsity of complex signals2019Manuscript (preprint) (Other academic)
    Abstract [en]

    Block sparsity is an important parameter in many algorithms to successfully recover block sparse signals under the framework of compressive sensing. However, it is often unknown and needs to beestimated. Recently there emerges a few research work about how to estimate block sparsity of real-valued signals, while there is, to the best of our knowledge, no investigation that has been conductedfor complex-valued signals. In this paper, we propose a new method to estimate the block sparsity of complex-valued signal. Its statistical properties are obtained and verified by simulations. In addition,we demonstrate the importance of accurately estimating the block sparsity in signal recovery through asensitivity analysis.

  • 5.
    Yu, Jun
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Stable and robust ℓp-constrained compressive sensing recovery via robust width property2019In: Journal of the Korean Mathematical Society, ISSN 0304-9914, E-ISSN 2234-3008, Vol. 56, no 3, p. 689-701Article in journal (Refereed)
    Abstract [en]

    We study the recovery results of ℓp-constrained compressive sensing (CS) with p≥1 via robust width property and determine conditions on the number of measurements for standard Gaussian matrices under which the property holds with high probability. Our paper extendsthe existing results in Cahill and Mixon [2] from ℓ2-constrained CS to ℓp-constrained case with p≥1 and complements the recovery analysisfor robust CS with ℓp loss function.

  • 6.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Lin, Zhengyan
    School of Mathematical Sciences, Zhejiang University.
    Varying coefficient partially nonlinear models with nonstationary regressors2018In: Journal of Statistical Planning and Inference, ISSN 0378-3758, E-ISSN 1873-1171, Vol. 194, p. 47-64Article in journal (Refereed)
    Abstract [en]

    We study a varying coefficient partially nonlinear model in which the regressors are generated by the multivariate unit root processes. A profile nonlinear least squares estimation procedure is applied to estimate the parameter vector and the functional coefficients. Under some mild conditions, the asymptotic distribution theory for the resulting estimators is established. The rate of convergence for the parameter vector estimator depends on the properties of the nonlinear regression function. However, the rate of convergence for the functional coefficients estimator is the same and enjoys the super-consistency property. Furthermore, a simulation study is conducted to investigate the finite sample performance of the proposed estimating procedures.

  • 7.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    A New Nonconvex Sparse Recovery Method for Compressive Sensing2019In: Frontiers in Applied Mathematics and Statistics, ISSN 2297-4687, Vol. 5, p. 1-11, article id 14Article in journal (Refereed)
    Abstract [en]

    As an extension of the widely used ℓr-minimization with 0 < r ≤ 1, a new non-convex weighted ℓr − ℓ1 minimization method is proposed for compressive sensing. The theoretical recovery results based on restricted isometry property and q-ratio constrained minimal singular values are established. An algorithm that integrates the iteratively reweighted least squares algorithm and the difference of convex functions algorithmis given to approximately solve this non-convex problem. Numerical experiments are presented to illustrate our results.

  • 8.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Adaptive estimation for varying coefficient modelswith nonstationary covariates2019In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 48, no 16, p. 4034-4050Article in journal (Refereed)
    Abstract [en]

    In this paper, the adaptive estimation for varying coefficient models proposed by Chen, Wang, and Yao (2015) is extended to allowing for nonstationary covariates. The asymptotic properties of the estimator are obtained, showing different convergence rates for the integrated covariates and stationary covariates. The nonparametric estimator of the functional coefficient with integrated covariates has a faster convergence rate than the estimator with stationary covariates, and its asymptotic distribution is mixed normal. Moreover, the adaptive estimation is more efficient than the least square estimation for non normal errors. A simulation study is conducted to illustrate our theoretical results.

  • 9.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Estimation of block sparsity in compressive sensing2017Manuscript (preprint) (Other academic)
    Abstract [en]

    In this paper, we consider a soft measure of block sparsity, k_α(x)=(∥x∥2,α/∥x∥2,1)^α/(1−α),α∈[0,∞] and propose a procedure to estimate it by using multivariate isotropic symmetric α-stable random projections without sparsity or block sparsity assumptions. The limiting distribution of the estimator is given. Some simulations are conducted to illustrate our theoretical results.

  • 10.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Estimation of block sparsity in compressive sensing2017Conference paper (Refereed)
  • 11. Zhou, Zhiyong
    et al.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    On q-ratio CMSV for sparse recovery2019In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 165, p. 128-132Article in journal (Refereed)
    Abstract [en]

    As a kind of computable incoherence measure of the measurement matrix, q-ratio constrained minimal singular values (CMSV) was proposed in Zhou and Yu (2019) to derive the performance bounds for sparse recovery. In this paper, we study the geometrical properties of the q-ratio CMSV, based on which we establish new sufficient conditions for signal recovery involving both sparsity defect and measurement error. The ℓ1-truncated set q-width of the measurement matrix is developed as the geometrical characterization of q-ratio CMSV. In addition, we show that the q-ratio CMSVs of a class of structured random matrices are bounded away from zero with high probability as long as the number of measurements is large enough, therefore these structured random matrices satisfy those established sufficient conditions. Overall, our results generalize the results in Zhang and Cheng (2012) from q=2 to any q ∈ (1, ∞] and complement the arguments of q-ratio CMSV from a geometrical view.

  • 12.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Phaseless compressive sensing using partial support information2017Manuscript (preprint) (Other academic)
    Abstract [en]

    We study the recovery conditions of weighted l_1 minimization for real signal reconstruction fromphaseless compressed sensing measurements when partial support information is available. A Strong Restricted Isometry Property (SRIP) condition is provided to ensure the stable recovery. Moreover,we present the weighted null space property as the sucient and necessary condition for the success of k-sparse phase retrieval via weighted l_1 minimization

  • 13.
    Zhou, Zhiyong
    et al.
    Department of Statistics, Zhejiang University City College, China.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Phaseless compressive sensing using partial support information2019In: Optimization Letters, ISSN 1862-4472, E-ISSN 1862-4480Article in journal (Refereed)
    Abstract [en]

    We study the recovery conditions of weighted ℓ1 minimization for real-valued signal reconstruction from phaseless compressive sensing measurements when partial support information is available. A strong restricted isometry property condition is provided to ensure the stable recovery. Moreover, we present the weighted null space property as the sufficient and necessary condition for the success of k-sparse phaseless recovery via weighted ℓ1 minimization. Numerical experiments are conducted to illustrate our results.

  • 14.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Recovery analysis for weighted mixed ℓ2/ℓp minimization with 0 < p ≤ 12019In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 352, p. 12p. 210-222Article in journal (Refereed)
    Abstract [en]

    We study the recovery conditions of weighted mixed ℓ2/ℓp (0 < p ≤ 1) minimization for block sparse signal reconstruction from compressed measurements when partial block support information is available. We show that the block p-restricted isometry property (RIP) can ensure the robust recovery. Moreover, we present the sufficient and necessary condition for the recovery by using weighted block p-null space property. The relationship between the block p-RIP and the weighted block p-null space property has been established. Finally, we illustrate our results with a series of numerical experiments.

  • 15.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sparse recovery based on q-ratio constrained minimal singular values2018Manuscript (preprint) (Other academic)
    Abstract [en]

    We study verifiable sufficient conditions and computable performance bounds for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector and the Lasso estimator, in terms of a newly defined family of quality measures for the measurement matrices. With high probability, the developed measures for subgaussian random matrices are bounded away from zero as long as the number of measurements is reasonably large. Comparing to the restricted isotropic constant based performance analysis, the arguments in this paper are much more concise and the obtained bounds are tighter. Numerical experiments are presented to illustrate our theoretical results.

  • 16.
    Zhou, Zhiyong
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Sparse recovery based on q-ratio constrained minimal singular values2019In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 155, p. 247-258Article in journal (Refereed)
    Abstract [en]

    We study verifiable sufficient conditions and computable performance bounds for sparse recovery algorithms such as the Basis Pursuit, the Dantzig selector and the Lasso estimator, in terms of a newly defined family of quality measures for the measurement matrices. With high probability, the developed measures for subgaussian random matrices are bounded away from zero as long as the number of measurements is reasonably large. Comparing to the restricted isotropic constant based performance analysis, the arguments in this paper are much more concise and the obtained bounds are tighter. Numerical experiments are presented to illustrate our theoretical results.

  • 17.
    Zhou, Zhiyong
    et al.
    Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China.
    Zhengyan, Lin
    Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China.
    Asymptotic normality of locally modelled regression estimator for functional data2016In: Journal of nonparametric statistics (Print), ISSN 1048-5252, E-ISSN 1029-0311, Vol. 28, no 1, p. 116-131Article in journal (Refereed)
    Abstract [en]

    We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As analternative to the well-known Nadaraya-Watson estimator for regression function in this framework, thelocally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu,P. (2010), ‘Locally Modelled Regression and Functional Data’,Journal of Nonparametric Statistics, 22,617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functionaldata are considered. The mean-squared convergence as well as asymptotic normality for the estimator areestablished. We also adapt the empirical likelihood method to construct the point-wise confidence intervalsfor the regression function and derive the Wilk’s phenomenon for the empirical likelihood inference.Furthermore, a simulation study is presented to illustrate our theoretical results.

  • 18.
    Zhou, Zhiyong
    et al.
    Department of Mathematics, Zhejiang University, Hangzhou, China.
    Zhengyan, Lin
    Department of Mathematics, Zhejiang University, Hangzhou, China.
    Limit theory for random coefficient autoregressive process under possibly infinite variance error sequence2016In: Communications in Statistics - Theory and Methods, ISSN 0361-0926, E-ISSN 1532-415X, Vol. 45, no 12, p. 3562-3576Article in journal (Refereed)
1 - 18 of 18
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