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  • 1.
    Leffler, Klara
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Department of Statistics, Zhejiang University City College, China.
    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 noise2019In: Journal of Computational Mathematics, ISSN 0254-9409, E-ISSN 1991-7139Article in journal (Refereed)
    Abstract [en]

    We study the recovery conditions of weighted mixed ℓ2/ℓp minimization for block sparse signal reconstruction from compressed measurements when partial block supportinformation is available. We show theoretically that the extended block restricted isometry property can ensure robust recovery when the data fidelity constraint is expressed in terms of an ℓq norm of the residual error, thus establishing a setting wherein we arenot restricted to Gaussian measurement noise. We illustrate the results with a series of numerical experiments.

  • 2.
    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.
    Bayesian sparsity estimation in compressive sensing with application to MR images2019In: Communications in Statistics: Case Studies, Data Analysis and Applications, ISSN 2373-7484Article in journal (Refereed)
    Abstract [en]

    The theory of compressive sensing (CS) asserts that an unknownsignal x ∈ CN can be 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. In the simulation study and real data study, magnetic resonance image data is used as input signal, which becomes sparse after sparsified transformation. The results from the simulation study confirm the theoretical properties of the estimator. In practice, the estimate from a real MR image can be used for recovering future MR images under the framework of CS if they are believed to have the same sparsity level after sparsification.

  • 3.
    Wang, Jianfeng
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Department of Statistics, Zhejiang University City College, Hangzhou, China.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Enhanced block sparse signal recovery based on q-ratio block constrained minimal singular values2019Manuscript (preprint) (Other academic)
    Abstract [en]

    In this paper we introduce theq-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed ℓ2/ℓq and the mixed ℓ2/ℓ1 norms of the reconstruction errors for stable and robust recovery using block Basis Pursuit (BBP), the block Dantzig selector (BDS) and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, theq-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV based error bounds are tighter than the block restricted isotropic constant based bounds.

  • 4.
    Wang, Jianfeng
    et al.
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Zhou, Zhiyong
    Department of Statistics, Zhejiang University City College, China.
    Yu, Jun
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Error bounds of block sparse signal recovery based on q-ratio block constrained minimal singular values2019In: EURASIP Journal on Advances in Signal Processing, ISSN 1687-6172, E-ISSN 1687-6180, Vol. 57Article in journal (Refereed)
    Abstract [en]

    In this paper, we introduce the q-ratio block constrained minimal singular values (BCMSV) as a new measure of measurement matrix in compressive sensing of block sparse/compressive signals and present an algorithm for computing this new measure. Both the mixed ℓ2/ℓq and the mixed ℓ2/ℓ1 norms of the reconstruction errors for stable and robust recovery using block basis pursuit (BBP), the block Dantzig selector (BDS), and the group lasso in terms of the q-ratio BCMSV are investigated. We establish a sufficient condition based on the q-ratio block sparsity for the exact recovery from the noise-free BBP and developed a convex-concave procedure to solve the corresponding non-convex problem in the condition. Furthermore, we prove that for sub-Gaussian random matrices, the q-ratio BCMSV is bounded away from zero with high probability when the number of measurements is reasonably large. Numerical experiments are implemented to illustrate the theoretical results. In addition, we demonstrate that the q-ratio BCMSV-based error bounds are tighter than the block-restricted isotropic constant-based bounds.

  • 5.
    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.

  • 6.
    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.

  • 7. 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.

  • 8.
    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.

1 - 8 of 8
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