In this study, the authors propose a semidefinite programming (SDP)-based localisation and tracking algorithm, which mitigates the non-line-of-sight (NLOS) error of range measurement and calibrates the accumulative er...
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In this study, the authors propose a semidefinite programming (SDP)-based localisation and tracking algorithm, which mitigates the non-line-of-sight (NLOS) error of range measurement and calibrates the accumulative error within the inertial sensing data. Both the range measurement in a mixed line-of-sight/NLOS environment and the step length estimated from inertial sensing information are approximated parametrically using Gaussian mixture modelling, and a maximum-likelihood estimator (MLE) is formulated to obtain the optimal position estimation. Since the Gaussian mixture models are non-linear functions of positions, the MLE is a non-convex problem, which global optimum is difficult to attain. Then, the non-convex MLE is transformed into an SDP-based localisation and tracking problem, relying on Jensen's inequality and semidefinite relaxation. Thus, a sub-optimal solution to the original MLE can be achieved. Moreover, the Cramer-Rao lower bound is also derived to serve as a performance indicator for localisation errors. The simulation and experimental results demonstrate the performance of the proposed algorithm. Compared with the existing algorithms, the proposed algorithm owns the best localisation accuracy, and can achieve a sub-metre level accuracy to a root mean square error of 0.46m in the real deployments.
Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and ...
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Nonnegative matrix factorization (NMF) under the separability assumption can provably be solved efficiently, even in the presence of noise, and has been shown to be a powerful technique in document classification and hyperspectral unmixing. This problem is referred to as near-separable NMF and requires that there exists a cone spanned by a small subset of the columns of the input nonnegative matrix approximately containing all columns. In this paper, we propose a preconditioning based on the minimum volume ellipsoid and semidefinite programming making the input matrix well-conditioned. This in turn can improve significantly the performance of near-separable NMF algorithms which is illustrated on the popular successive projection algorithm (SPA). The new preconditioned SPA is provably more robust to noise, and outperforms SPA on several synthetic data sets. We also show how an active-set method allows us to apply the preconditioning on large-scale real-world hyperspectral images.
We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[H(D circle times E)], maximized with respect to semidefinite constraints on D and E. Applied t...
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We give asymptotically converging semidefinite programming hierarchies of outer bounds on bilinear programs of the form Tr[H(D circle times E)], maximized with respect to semidefinite constraints on D and E. Applied to the problem of approximate error correction in quantum information theory, this gives hierarchies of efficiently computable outer bounds on the success probability of approximate quantum error correction codes in any dimension. The first level of our hierarchies corresponds to a previously studied relaxation (Leung and Matthews in IEEE Trans Inf Theory 61(8):4486, 2015) and positive partial transpose constraints can be added to give a sufficient criterion for the exact convergence at a given level of the hierarchy. To quantify the worst case convergence speed of our sum-of-squares hierarchies, we derive novel quantum de Finetti theorems that allow imposing linear constraints on the approximating state. In particular, we give finite de Finetti theorems for quantum channels, quantifying closeness to the convex hull of product channels as well as closeness to local operations and classical forward communication assisted channels. As a special case this constitutes a finite version of Fuchs-Schack-Scudo's asymptotic de Finetti theorem for quantum channels. Finally, our proof methods answer a question of Brandao and Harrow (Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC'12, p 307, 2012) by improving the approximation factor of de Finetti theorems with no symmetry from O(d(k/2)) tp poly (d, k), where d denotes local dimension and k the number of copies.
For nonnegative integers n, d, and w, let A(n, d, w) be the maximum size of a code C subset of F-n 2 with a constant weight w and minimum distance at least d(2). We consider two semidefinite programs based on quadrupl...
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For nonnegative integers n, d, and w, let A(n, d, w) be the maximum size of a code C subset of F-n 2 with a constant weight w and minimum distance at least d(2). We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on A(n, d, w). The new upper bounds imply that A(22, 8, 10) = 616 and A(22, 8, 11) = 672. Lower bounds on A(22, 8, 10) and A(22, 8, 11) are obtained from the (n, d) = (22, 7) shortened Golay code of size 2048. It can be concluded that the shortened Golay code is a union of constant-weight w codes of sizes A(22, 8, w).
Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality (LMI) constraints. From convex optimi...
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Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs, as well as dual optimization problems, can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems.
semidefinite programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detai...
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semidefinite programs have recently turned out to be a powerful tool for approximating integer problems. To survey the development in this area over the last few years, the following topics are addressed in some detail. First, we investigate ways to derive semidefinite programs from discrete optimization problems. The duality theory for semidefinite programs is the key to understand algorithms to solve them. The relevant duality results are therefore summarized. The second part of the paper deals with the approximation of integer problems both in a theoretical setting, and from a computational point of view. (C) 1999 Elsevier Science B.V. and IMACS, All rights reserved.
An SDP relaxation based method is developed to solve the localization problem in sensor networks using incomplete and inaccurate distance information. The problem is set up to find a set of sensor positions such that ...
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An SDP relaxation based method is developed to solve the localization problem in sensor networks using incomplete and inaccurate distance information. The problem is set up to find a set of sensor positions such that given distance constraints are satisfied. The nonconvex constraints in the formulation are then relaxed in order to yield a semidefinite program that can be solved efficiently. The basic model is extended in order to account for noisy distance information. In particular, a maximum likelihood based formulation and an interval based formulation are discussed. The SDP solution can then also be used as a starting point for steepest descent based local optimization techniques that can further refine the SDP solution. We also describe the extension of the basic method to develop an iterative distributed SDP method for solving very large scale semidefinite programs that arise out of localization problems for large dense networks and are intractable using centralized methods. The performance evaluation of the technique with regard to estimation accuracy and computation time is also presented by the means of extensive simulations. Our SDP scheme also seems to be applicable to solving other Euclidean geometry problems where points are locally connected.
We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive one-shot semidefinite programming (SDP) converse bo...
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We derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one-shot and asymptotic regime. First, we derive one-shot semidefinite programming (SDP) converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel, which improve the previous bound from [Tomamichel/Berta/Renes, Nat. Commun. 7, 2016]. As applications, we study quantum communication over depolarizing channels and amplitude damping channels with finite resources. Second, we find an SDP-strong converse bound for the quantum capacity of an arbitrary quantum channel, which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases. Furthermore, we prove that the SDP-strong converse bound improves the partial transposition bound introduced by Holevo and Werner. Third, we prove that this SDP strong converse bound is equal to the so-called max-Rains information, which is an analog to the Rains information introduced in [Tomamichel/Wilde/Winter, IEEE Trans. Inf. Theory 63:715, 2017]. Our SDP strong converse bound is weaker than the Rains information, but it is efficiently computable for general quantum channels.
A common technique for passive source localization is to utilize the range-difference (RD) measurements between the source and several spatially separated sensors. The RD information defines a set of hyperbolic equati...
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A common technique for passive source localization is to utilize the range-difference (RD) measurements between the source and several spatially separated sensors. The RD information defines a set of hyperbolic equations from which the source position can be calculated with the knowledge of the sensor positions. Under the standard assumption of Gaussian distributed RD measurement errors, it is well known that the maximum-likelihood (ML) position estimation is achieved by minimizing a multimodal cost function which corresponds to a difficult task. In this correspondence, we propose to approximate the nonconvex ML optimization by relaxing it to a convex optimization problem using semidefinite programming. A semidefinite relaxation RD-based positioning algorithm, which makes use of the admissible source position information, is proposed and its estimation performance is contrasted with the two-step weighted least squares method and nonlinear least squares estimator as well as Cramer-Rao lower bound.
作者：
Parrilo, PAETH
Swiss Fed Inst Technol Automat Control Lab CH-8092 Zurich Switzerland CALTECH
Control & Dynam Syst Dept Pasadena CA 91125 USA
A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polyn...
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A hierarchy of convex relaxations for semialgebraic problems is introduced. For questions reducible to a finite number of polynomial equalities and inequalities, it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility. The main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials, and some results from real algebraic geometry. The techniques provide a constructive approach for finding bounded degree solutions to the Positivstellensatz, and are illustrated with examples from diverse application fields.
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