Weiyu Xu

University of Iowa, Iowa City, Iowa, United States

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Publications (84)70.91 Total impact

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    ABSTRACT: We study the problem of recovering an $n$-dimensional vector of $\{\pm1\}^n$ (BPSK) signals from $m$ noise corrupted measurements $\mathbf{y}=\mathbf{A}\mathbf{x}_0+\mathbf{z}$. In particular, we consider the box relaxation method which relaxes the discrete set $\{\pm1\}^n$ to the convex set $[-1,1]^n$ to obtain a convex optimization algorithm followed by hard thresholding. When the noise $\mathbf{z}$ and measurement matrix $\mathbf{A}$ have iid standard normal entries, we obtain an exact expression for the bit-wise probability of error $P_e$ in the limit of $n$ and $m$ growing and $\frac{m}{n}$ fixed. At high SNR our result shows that the $P_e$ of box relaxation is within 3dB of the matched filter bound MFB for square systems, and that it approaches MFB as $m $ grows large compared to $n$. Our results also indicates that as $m,n\rightarrow\infty$, for any fixed set of size $k$, the error events of the corresponding $k$ bits in the box relaxation method are independent.
    No preview · Article · Oct 2015
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    ABSTRACT: Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as $\ell_1$ minimization and nuclear norm minimization are well understood through recent years' research. However, rigorously characterizing the phase transition of total variation (TV) minimization in recovering sparse-gradient signal is still open. In this paper, we fully characterize the phase transition curve of the TV minimization. Our proof builds on Donoho, Johnstone and Montanari's conjectured phase transition curve for the TV approximate message passing algorithm (AMP), together with the linkage between the minmax Mean Square Error of a denoising problem and the high-dimensional convex geometry for TV minimization.
    No preview · Article · Sep 2015
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    DESCRIPTION: In this paper, we consider uplink channel estimation in massive MIMO-OFDM systems with frequency selective channels. With increased number of antennas, the channel estimation problem becomes very challenging as exceptionally large number of channel parameters have to be estimated. We propose an efficient distributed ...
    Full-text · Research · Aug 2015
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    ABSTRACT: We propose novel algorithms that enhance the performance of recovering unknown continuous-valued frequencies from undersampled signals. Our iterative reweighted frequency recovery algorithms employ the support knowledge gained from earlier steps of our algorithms as block prior information to enhance frequency recovery. Our methods improve the performance of the atomic norm minimization which is a useful heuristic in recovering continuous-valued frequency contents. Numerical results demonstrate that our block iterative reweighted methods provide both better recovery performance and faster speed than other known methods.
    Full-text · Article · Jul 2015 · IEEE Signal Processing Letters
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    ABSTRACT: Massive MIMO communication systems, by virtue of utilizing very large number of antennas, have a potential to yield higher spectral and energy efficiency in comparison with the conventional MIMO systems. In this paper, we consider uplink channel estimation in massive MIMO-OFDM systems with frequency selective channels. With increased number of antennas, the channel estimation problem becomes very challenging as exceptionally large number of channel parameters have to be estimated. We propose an efficient distributed linear minimum mean square error (LMMSE) algorithm that can achieve near optimal channel estimates at very low complexity by exploiting the strong spatial correlations and symmetry of large antenna array elements. The proposed method involves solving a (fixed) reduced dimensional LMMSE problem at each antenna followed by a repetitive sharing of information through collaboration among neighboring antenna elements. To further enhance the channel estimates and/or reduce the number of reserved pilot tones, we propose a data-aided estimation technique that relies on finding a set of most reliable data carriers. We also analyse the effect of pilot contamination on the mean square error (MSE) performance of different channel estimation techniques. Unlike the conventional approaches, we use stochastic geometry to obtain analytical expression for interference variance (or power) across OFDM frequency tones and use it to derive the MSE expressions for different algorithms under both noise and pilot contaminated regimes. Simulation results validate our analysis and the near optimal MSE performance of proposed estimation algorithms.
    Full-text · Article · Jul 2015
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    Jian-Feng Cai · Suhui Liu · Weiyu Xu
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    ABSTRACT: This paper considers reconstructing a spectrally sparse signal from a small number of randomly observed time-domain samples. The signal of interest is a linear combination of complex sinusoids at $R$ distinct frequencies. The frequencies can assume any continuous values in the normalized frequency domain $[0,1)$. After converting the spectrally sparse signal recovery into a low rank structured matrix completion problem, we propose an efficient feasible point approach, named projected Wirtinger gradient descent (PWGD) algorithm, to efficiently solve this structured matrix completion problem. We further accelerate our proposed algorithm by a scheme inspired by FISTA. We give the convergence analysis of our proposed algorithms. Extensive numerical experiments are provided to illustrate the efficiency of our proposed algorithm. Different from earlier approaches, our algorithm can solve problems of very large dimensions very efficiently.
    Preview · Article · Jul 2015
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    Jian-Feng Cai · Xiaobo Qu · Weiyu Xu · Gui-Bo Ye
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    ABSTRACT: This paper explores robust recovery of a superposition of $R$ distinct complex exponential functions from a few random Gaussian projections. We assume that the signal of interest is of $2N-1$ dimensional and $R<<2N-1$. This framework covers a large class of signals arising from real applications in biology, automation, imaging science, etc. To reconstruct such a signal, our algorithm is to seek a low-rank Hankel matrix of the signal by minimizing its nuclear norm subject to the consistency on the sampled data. Our theoretical results show that a robust recovery is possible as long as the number of projections exceeds $O(R\ln^2N)$. No incoherence or separation condition is required in our proof. Our method can be applied to spectral compressed sensing where the signal of interest is a superposition of $R$ complex sinusoids. Compared to existing results, our result here does not need any separation condition on the frequencies, while achieving better or comparable bounds on the number of measurements. Furthermore, our method provides theoretical guidance on how many samples are required in the state-of-the-art non-uniform sampling in NMR spectroscopy. The performance of our algorithm is further demonstrated by numerical experiments.
    Full-text · Article · Mar 2015

  • No preview · Article · Jan 2015
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    ABSTRACT: Massive MIMO systems have made significant progress in increasing spectral and energy efficiency over traditional MIMO systems by exploiting large antenna arrays. In this paper we consider the joint maximum likelihood (ML) channel estimation and data detection problem for massive SIMO (single input multiple output) wireless systems. Despite the large number of unknown channel coefficients for massive SIMO systems, we improve an algorithm to achieve the exact ML non-coherent data detection with a low expected complexity. We show that the expected computational complexity of this algorithm is linear in the number of receive antennas and polynomial in channel coherence time. Simulation results show the performance gain of the optimal non-coherent data detection with a low computational complexity.
    Full-text · Article · Nov 2014
  • Weiyu Xu · Er-Wei Bai · Myung Cho
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    ABSTRACT: In this paper, we consider robust system identification of FIR systems when both sparse outliers and random noises are present. We reduce this problem of system identification to a sparse error correcting problem using a Toeplitz structured real-numbered coding matrix and prove the performance guarantee. Thresholds on the percentage of correctable errors for Toeplitz structured matrices are established. When both outliers and observation noise are present, we have shown that the estimation error goes to asymptotically as long as the probability density function for observation noise is not “vanishing” around origin. No probabilistic assumptions are imposed on the outliers.
    No preview · Article · Oct 2014 · Automatica
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    ABSTRACT: We formulate a nonlinear distributed control law that guides the motion a group of sensors to achieve a configuration that permits them to optimally localize a hazardous source they must keep a prescribed distance from. Earlier work shows that such a configuration involves the sensors being placed in an equispaced manner on a prescribed circle. The nonlinear control law we propose assumes that each sensor resides and moves on the prescribed circle, by accessing only the states of its two immediate clockwise and counterclockwise neighbors. We theoretically prove and verify through simulations, that the law allows the sensors to achieve the desired configuration while avoiding collisions.
    No preview · Conference Paper · Sep 2014
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    ABSTRACT: We address the problem of super-resolution frequency recovery using prior knowledge of the structure of a spectrally sparse, undersampled signal. In many applications of interest, some structure information about the signal spectrum is often known. The prior information might be simply knowing precisely some signal frequencies or the likelihood of a particular frequency component in the signal. We devise a general semidefinite program to recover these frequencies using theories of positive trigonometric polynomials. Our theoretical analysis shows that, given sufficient prior information, perfect signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies. Numerical experiments demonstrate great performance enhancements using our method. We show that the nominal resolution necessary for the grid-free results can be improved if prior information is suitably employed.
    Full-text · Article · Sep 2014 · IEEE Transactions on Signal Processing
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    ABSTRACT: We address the problem of super-resolution line spectrum estimation of an undersampled signal with block prior information. The component frequencies of the signal are assumed to take arbitrary continuous values in known frequency blocks. We formulate a general semidefinite program to recover these continuous-valued frequencies using theories of positive trigonometric polynomials. The proposed semidefinite program achieves super-resolution frequency recovery by taking advantage of known structures of frequency blocks. Numerical experiments show great performance enhancements using our method.
    Full-text · Article · Apr 2014
  • Er-wei Bai · Kang Li · Wenxiao Zhao · Weiyu Xu
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    ABSTRACT: In this paper, we consider the variable selection problem for a nonlinear non-parametric system. Two approaches are proposed, one top-down approach and one bottom-up approach. The top-down algorithm selects a variable by detecting if the corresponding partial derivative is zero or not at the point of interest. The algorithm is shown to have not only the parameter but also the set convergence. This is critical because the variable selection problem is binary, a variable is either selected or not selected. The bottom-up approach is based on the forward/backward stepwise selection which is designed to work if the data length is limited. Both approaches determine the most important variables locally and allow the unknown non-parametric nonlinear system to have different local dimensions at different points of interest. Further, two potential applications along with numerical simulations are provided to illustrate the usefulness of the proposed algorithms.
    No preview · Article · Jan 2014 · Automatica
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    ABSTRACT: Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In particular, atomic norm minimization was proposed in \cite{tang2012csotg} to recover $1$-dimensional spectrally sparse signal. However, in spite of existing research efforts \cite{chi2013compressive}, it was still an open problem how to formulate an equivalent positive semidefinite program for atomic norm minimization in recovering signals with $d$-dimensional ($d\geq 2$) off-the-grid frequencies. In this paper, we settle this problem by proposing equivalent semidefinite programming formulations of atomic norm minimization to recover signals with $d$-dimensional ($d\geq 2$) off-the-grid frequencies.
    Full-text · Article · Dec 2013
  • Er-wei Bai · Kang Li · Wenxiao Zhao · Weiyu Xu
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    ABSTRACT: In this paper, we consider the variable selection problem for a nonlinear non-parametric system. The approach proposed selects a variable by detecting if the corresponding partial derivative is zero or not at the point of interest. The algorithm is shown to have not only the parameter but also the set convergence. This is critical because the variable selection problem is binary, a variable is either selected or not selected. The approach allows the unknown non-parametric nonlinear system to have different local dimensions at different points of interest.
    No preview · Conference Paper · Dec 2013
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    ABSTRACT: Recent research in off-the-grid compressed sensing (CS) has demonstrated that, under certain conditions, one can successfully recover a spectrally sparse signal from a few time-domain samples even though the dictionary is continuous. In this paper, we extend off-the-grid CS to applications where some prior information about spectrally sparse signal is known. We specifically consider cases where a few contributing frequencies or poles, but not their amplitudes or phases, are known a priori. Our results show that equipping off-the-grid CS with the known-poles algorithm can increase the probability of recovering all the frequency components.
    Full-text · Article · Nov 2013 · Proceedings - ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing
  • Myung Cho · Weiyu Xu
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    ABSTRACT: The null space condition is a condition under which k-sparse signal can be recovered uniquely in compressed sensing (CS) problems. In this paper, new efficient algorithms are introduced to verify the null space condition for l1 minimization in compressed sensing. Suppose A is an (n - m) × n (m > 0) sensing matrix, we can verify whether the sensing matrix A satisfies the null space condition or not for k-sparse signals by computing αk = max{z: Az=0, z≠0} max{K:|K|≤k} ||zK||1/||z||1, where K represents subsets of {1, 2,..., n}, and |K| is the cardinality of K. However, computing αk is known to be extremely challenging because of high computational complexity. In this paper, a series of new polynomial-time algorithms are proposed to compute upper bounds on αk. Based on these new polynomial-time algorithms, we further design new algorithm, which is called the sandwiching algorithm, to compute the exact αk with much lower complexity than exhaustive search.
    No preview · Conference Paper · Nov 2013
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    ABSTRACT: In this paper we introduce an optimized Markov Chain Monte Carlo (MCMC) technique for solving the integer least-squares (ILS) problems, which include Maximum Likelihood (ML) detection in Multiple-Input Multiple-Output (MIMO) systems. Two factors contribute to the speed of finding the optimal solution by the MCMC detector: the probability of the optimal solution in the stationary distribution, and the mixing time of the MCMC detector. Firstly, we compute the optimal value of the "temperature" parameter, in the sense that the temperature has the desirable property that once the Markov chain has mixed to its stationary distribution, there is polynomially small probability ($1/\mbox{poly}(N)$, instead of exponentially small) of encountering the optimal solution. This temperature is shown to be at most $O(\sqrt{SNR}/\ln(N))$, where $SNR$ is the signal-to-noise ratio, and $N$ is the problem dimension. Secondly, we study the mixing time of the underlying Markov chain of the proposed MCMC detector. We find that, the mixing time of MCMC is closely related to whether there is a local minimum in the lattice structures of ILS problems. For some lattices without local minima, the mixing time of the Markov chain is independent of $SNR$, and grows polynomially in the problem dimension; for lattices with local minima, the mixing time grows unboundedly as $SNR$ grows, when the temperature is set, as in conventional wisdom, to be the standard deviation of noises. Our results suggest that, to ensure fast mixing for a fixed dimension $N$, the temperature for MCMC should instead be set as $\Omega(\sqrt{SNR})$ in general. Simulation results show that the optimized MCMC detector efficiently achieves approximately ML detection in MIMO systems having a huge number of transmit and receive dimensions.
    Full-text · Article · Oct 2013 · IEEE Transactions on Signal Processing
  • Meng Wang · Weiyu Xu · R. Calderbank
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    ABSTRACT: Compressed sensing (CS) theory promises one can recover real-valued sparse signal from a small number of linear measurements. Motivated by network monitoring with link failures, we for the first time consider the problem of recovering signals that contain both real-valued entries and corruptions, where the real entries represent transmission delays on normal links and the corruptions represent failed links. Unlike conventional CS, here a measurement is real-valued only if it does not include a failed link, and it is corrupted otherwise. We prove that O((d + 1)max(d, k) log n) nonadaptive measurements are enough to recover all n-dimensional signals that contain k nonzero real entries and d corruptions. We provide explicit constructions of measurements and recovery algorithms. We also analyze the performance of signal recovery when the measurements contain errors.
    No preview · Conference Paper · Oct 2013

Publication Stats

1k Citations
70.91 Total Impact Points

Institutions

  • 2012-2015
    • University of Iowa
      • Department of Electrical and Computer Engineering
      Iowa City, Iowa, United States
  • 2013
    • Duke University
      Durham, North Carolina, United States
  • 2010-2012
    • Cornell University
      • Department of Electrical and Computer Engineering
      Итак, New York, United States
  • 2007-2011
    • California Institute of Technology
      • Department of Electrical Engineering
      Pasadena, California, United States