Approximate optimal designs for multivariate polynomial regression

Castro, Yohann de; Gamboa, Fabrice; Henrion, Didier; Hess, Roxana; Lasserre, Jean-Bernard

doi:10.1214/18-aos1683

Cited by 40 publications

(37 citation statements)

References 21 publications

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“…In Theorem 3, we further reveal that the positive polynomial f (b) ∈ ∑ R[b] 2 for the DFT beamspace manifold. Based on this theorem, we develop the lower dimensional SDP implementation in Equations (26) and (27), which is our main contribution.…”

Section: Low Dimensional Sdp Implementation For Bs-anmmentioning

confidence: 99%

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Low Complexity Beamspace Super Resolution for DOA Estimation of Linear Array

2020

Sensors

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Beamspace processing has become much attractive in recent radar and wireless communication applications, since the advantages of complexity reduction and of performance improvements in array signal processing. In this paper, we concentrate on the beamspace DOA estimation of linear array via atomic norm minimization (ANM). The existed generalized linear spectrum estimation based ANM approaches suffer from the high computational complexity for large scale array, since their complexity depends upon the number of sensors. To deal with this problem, we develop a low dimensional semidefinite programming (SDP) implementation of beamspace atomic norm minimization (BS-ANM) approach for DFT beamspace based on the super resolution theory on the semi-algebraic set. Then, a computational efficient iteration algorithm is proposed based on alternating direction method of multipliers (ADMM) approach. We develop the covariance based DOA estimation methods via BS-ANM and apply the BS-ANM based DOA estimation method to the channel estimation problem for massive MIMO systems. Simulation results demonstrate that the proposed methods exhibit the superior performance compared to the state-of-the-art counterparts.

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Section: Low Dimensional Sdp Implementation For Bs-anmmentioning

confidence: 99%

“…(26) is based on the sum-of-squares relaxation [26], and thus is an approximate algorithm of Equation (15). It can be regarded as the optimal polynomial designing to regression the dual polynomial in Equation (16) under the polynomial order constraint as shown in [27].…”

Section: Remarkmentioning

confidence: 99%

Low Complexity Beamspace Super Resolution for DOA Estimation of Linear Array

2020

Sensors

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“…We have tested this two-steps method on several non-trivial numerical experiments (in particular with highly non-convex design spaces X ) and in all cases we were able to obtain a design. For more details the interested reader is referred to [7].…”

Section: Optimal Design In Statisticsmentioning

confidence: 99%

The Moment-Sos Hierarchy

Lasserre¹

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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The Moment-SOS hierarchy initially introduced in optimization in 2000, is based on the theory of the K-moment problem and its dual counterpart, polynomials that are positive on K. It turns out that this methodology can be also applied to solve problems with positivity constraints "f (x) ≥ 0 for all x ∈ K" and/or linear constraints on Borel measures. Such problems can be viewed as specific instances of the "Generalized Problem of Moments" (GPM) whose list of important applications in various domains is endless. We describe this methodology and outline some of its applications in various domains.1991 Mathematics Subject Classification. 90C26 90C22 90C27 65K05 14P10 44A60.

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“…In this section, we consider five different settings (Castro et al, 2019) to evaluate the performance of the proposed algorithms for obtaining approximate D-and A-optimal designs. We use these numerical examples to illustrate that the proposed algorithms can efficiently identify the optimal design.…”

Section: Numerical Illustrationsmentioning

confidence: 99%

“…Although these existing methods converge and some of them converge monotonically, these algorithms tend to be slow. Recently, Castro, et al (2018) use the moment-sum-of-squares hierarchy of semi-definite programming problems to solve the approximate optimal design problem numerically. Harman, Filová and Richtárik (2018) proposed the randomized exchange method (REX), which is a simple batchrandomized exchange algorithm, for the optimal design problem.…”

Section: Introductionmentioning

confidence: 99%

Efficient computational algorithm for optimal continuous experimental designs

Duan

Gao

2019

Journal of Computational and Applied Mathematics

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A simple yet efficient computational algorithm for computing the continuous optimal experimental design for linear models is proposed. An alternative proof the monotonic convergence for D-optimal criterion on continuous design spaces are provided. We further show that the proposed algorithm converges to the D-optimal design. We also provide an algorithm for the A-optimality and conjecture that the algorithm convergence monotonically on continuous design spaces. Different numerical examples are used to demonstrated the usefulness and performance of the proposed algorithms.

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Approximate optimal designs for multivariate polynomial regression

Cited by 40 publications

References 21 publications

Low Complexity Beamspace Super Resolution for DOA Estimation of Linear Array

Low Complexity Beamspace Super Resolution for DOA Estimation of Linear Array

The Moment-Sos Hierarchy

Efficient computational algorithm for optimal continuous experimental designs

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