Tracking a maneuvering target is an important technology. Due to complex environment and diversity of sensors, errors need to be optimized with respect to various motion states during the tracking ...process. In this paper, we first propose how to unify the coordinate system and data preprocessing in case of tracking using multiple sensors. We then combine fuzzy sets with a novel trace optimization method based on extended Kalman filter (EKF) with nested probabilistic-numerical linguistic information (NPN-EKFTO). We present a case study of trace optimization of an unknown maneuvering target in Sichuan province in China. We solve the case by using both the proposed method and the traditional EKF and offer comparative analysis to validate the proposed approach.
This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue ...and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre (SIAM J Optim 17(3):822–843, 2006) and Waki et al. (SIAM J Optim 17(1):218–242, 2006). The Gelfand–Naimark–Segal construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue of noncommutative polynomials from the literature.
Many systems in nature and society, ranging from types of cancers to engineering, can be modeled as temporal networks. Identification of dynamic communities is fundamental for investigating the ...mechanisms of systems. Compared with community detection in static networks, dynamic community detection is challenging because it simultaneously leverages clustering accuracy with clustering drift. Current methods employ the temporal smoothness framework to balance these two items, but the inaccuracy and complexity of these methods have been criticized. To overcome these issues, a novel algorithm called NE2NMF for dynamic community detection is proposed by fusing Network Embedding based Evolutionary Nonnegative Matrix Factorization. Specifically, we use third-order smoothness to characterize clustering drift by fusing snapshots of previous, current, and subsequent time steps, thereby improving the accuracy of the characterization of dynamic communities. Moreover, to reduce time complexity, we decompose the network embedding matrix, rather than the adjacency matrix of each snapshot on the basis of the equivalence between evolutionary non-negative matrix factorization (ENMF) and network embedding. Results demonstrate that NE2NMF outperforms state-of-the-art approaches in terms of accuracy and robustness.
•We prove the equivalence between network embedding and NMF, providing the theoretical foundation for algorithms.•We propose a novel NE2NMF algorithm for dynamic community in temporal networks.•The experimental results demonstrate that NE2NMF not only outperforms state-of-the-art approaches but also reduces the running times of the algorithm.
We provide a new hierarchy of semidefinite programming relaxations, called
NCTSSOS
, to solve large-scale sparse noncommutative polynomial optimization problems. This hierarchy features the ...exploitation of
term sparsity
hidden in the input data for eigenvalue and trace optimization problems. NCTSSOS complements the recent work that exploits
correlative sparsity
for noncommutative optimization problems by Klep et al. (MP, 2021), and is the noncommutative analogue of the TSSOS framework by Wang et al. (SIAMJO 31: 114–141, 2021, SIAMJO 31: 30–58, 2021). We also propose an extension exploiting simultaneously correlative and term sparsity, as done previously in the commutative case (Wang in CS-TSSOS: Correlative and term sparsity for large-scale polynomial optimization, 2020). Under certain conditions, we prove that the optima of the NCTSSOS hierarchy converge to the optimum of the corresponding dense semidefinite programming relaxation. We illustrate the efficiency and scalability of NCTSSOS by solving eigenvalue/trace optimization problems from the literature as well as randomly generated examples involving up to several thousand variables.
Optimization Over Trace Polynomials Klep, Igor; Magron, Victor; Volčič, Jurij
Annales Henri Poincaré,
01/2022, Letnik:
23, Številka:
1
Journal Article
Recenzirano
Odprti dostop
Motivated by recent progress in quantum information theory, this article aims at optimizing trace polynomials, i.e., polynomials in noncommuting variables and traces of their products. A novel ...Positivstellensatz certifying positivity of trace polynomials subject to trace constraints is presented, and a hierarchy of semidefinite relaxations converging monotonically to the optimum of a trace polynomial subject to tracial constraints is provided. This hierarchy can be seen as a tracial analog of the Pironio, Navascués and Acín scheme (Pironio et al. in New J. Phys. 10(7):073013, 2008) for optimization of noncommutative polynomials. The Gelfand–Naimark–Segal (GNS) construction is applied to extract optimizers of the trace optimization problem if flatness and extremality conditions are satisfied. These conditions are sufficient to obtain finite convergence of our hierarchy. The results obtained are applied to violations of polynomial Bell inequalities in quantum information theory. The main techniques used in this paper are inspired by real algebraic geometry, operator theory, and noncommutative algebra.
The minimum cost control problem is one of the most important issues in controlling complex networks. Different from the previous works, in this paper, we consider the minimum cost control problem ...with selectable inputs by adopting the cost function summed over both quadratic terms of system input and system state with a weighting factor. To address such an issue, the orthonormal-constraint-based projected gradient method is proposed to determine the input matrix iteratively. Convergence of the proposed algorithm is established. Extensive simulation results are carried out to show the effectiveness of the proposed algorithm. We also investigate what kinds of nodes are most important for minimizing average control cost in directed stems/circles and small networks through simulation studies. The presented results in this paper bring meaningful physical insights in controlling the directed networks from an energy point of view.
Discriminant analysis has been used for decades to extract features that preserve class separability. It is commonly defined as an optimization problem involving covariance matrices that represent ...the scatter within and between clusters. The requirement that one of these matrices be nonsingular limits its application to data sets with certain relative dimensions. We examine a number of optimization criteria, and extend their applicability by using the generalized singular value decomposition to circumvent the nonsingularity requirement. The result is a generalization of discriminant analysis that can be applied even when the sample size is smaller than the dimension of the sample data. We use classification results from the reduced representation to compare the effectiveness of this approach with some alternatives, and conclude with a discussion of their relative merits.
This paper considers the problem of optimizing the ratio f TrVT AV/Tr VT BV over all unitary matrices V with p columns, where A,B are two positive definite matrices. This problem is common in ...supervised learning techniques. However, because its numerical solution is typically expensive it is often replaced by the simpler optimization problem which consists of optimizing TrVT AV under the constraint that VT BV = I, the identity matrix. The goal of this paper is to examine this trace ratio optimization problem in detail, to consider different algorithms for solving it, and to illustrate the use of these algorithms for dimensionality reduction.
The hybrid precoding problem in point-to-point millimeter wave (mmWave) multiple input multiple output (MIMO) for narrowband channel has been established as a Frobenius norm minimization problem. It ...is translated into trace minimization problem, and two algorithms are proposed to solve it. In the first method based on alternating minimization, we alternately determine the analog precoder and digital precoder, keeping the other constant to minimize the trace. The analog precoding subproblem with the fixed digital precoder is converted into a semi-definite programming (SDP) problem and solved by block coordinate descent (BCD) algorithm with suitable modifications. In the second method, we segregate the analog precoding and digital precoding subproblems by considering orthogonality of analog precoder. The analog precoding is further rephrased as a trace maximization problem and solved by an iterative power method by enforcing orthogonality constraint on the analog precoder. The adapted form of modified Gramm-Schmidt orthogonalization procedure is employed to impose orthogonality on the analog precoder. The proposed methods are extended for wideband channel by considering orthogonal frequency division multiplexing (OFDM). The proposed methods not only exhibit a good performance but also come with lower computational complexity when compared to existing methods with comparable performances.
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