Discrete Constraints Research Articles

Mode separability-based state estimation for uncertain constrained dynamic systems

This paper addresses the state estimation problem for dynamic systems subject to uncertain constraints, i.e., all possible constraints are described by a finite set, and only one constraint is satisfied at each moment. For this typical hybrid system estimation problem with coupled discrete constraint modes and continuous states, we design a mode separability-based state estimation (MSSE) framework. Based on the measurement model over a window, the hypothesis testing is performed to detect whether the mode changes firstly. Then the maximum-likelihood criterion is used to estimate the mode change sequence once detecting the change. Next, to consider the possible impact of the decided modes on the state estimation, a metric of mode separability is proposed to evaluate the separability of the recognized modes, and two different state estimation methods are introduced. Specifically, if the recognized modes are separable from the others, the state estimates are obtained by a recursive mode-based constraint Kalman filter (MCKF) which is proved that the estimation error is bounded in mean square. Otherwise, the estimation results output the fused state estimates (FSE) of the inseparable modes. Finally, simulation results of road-constrained vehicle tracking are provided to demonstrate the effectiveness of the proposed framework.

Treatment of Complex Interfaces for Maxwell’s Equations with Continuous Coefficients Using the Correction Function Method

We propose a high-order FDTD scheme based on the correction function method (CFM) to treat interfaces with complex geometry without significantly increasing the complexity of the numerical approach for constant coefficients. Correction functions are modeled by a system of PDEs based on Maxwell’s equations with interface conditions. To be able to compute approximations of correction functions, a functional that is a square measure of the error associated with the correction functions’ system of PDEs is minimized in a divergence-free discrete functional space. Afterward, approximations of correction functions are used to correct a FDTD scheme in the vicinity of an interface where it is needed. We perform a perturbation analysis on the correction functions’ system of PDEs. The discrete divergence constraint and the consistency of resulting schemes are studied. Numerical experiments are performed for problems with different geometries of the interface. A second-order convergence is obtained for a second-order FDTD scheme corrected using the CFM. High-order convergence is obtained with a corrected fourth-order FDTD scheme. The discontinuities within solutions are accurately captured without spurious oscillations.

Fast discrete cross-modal hashing with semantic consistency

Supervised cross-modal hashing has attracted widespread concentrations for large-scale retrieval task due to its promising retrieval performance. However, most existing works suffer from some of following issues. Firstly, most of them only leverage the pair-wise similarity matrix to learn hash codes, which may result in class information loss. Secondly, the pair-wise similarity matrix generally lead to high computing complexity and memory cost. Thirdly, most of them relax the discrete constraints during optimization, which generally results in large cumulative quantization error and consequent inferior hash codes. To address above problems, we present a Fast Discrete Cross-modal Hashing method in this paper, FDCH for short. Specifically, it firstly leverages both class labels and the pair-wise similarity matrix to learn a sharing Hamming space where the semantic consistency can be better preserved. Then we propose an asymmetric hash codes learning model to avoid the challenging issue of symmetric matrix factorization. Finally, an effective and efficient discrete optimal scheme is designed to generate discrete hash codes directly, and the computing complexity and memory cost caused by the pair-wise similarity matrix are reduced from O(n2) to O(n), where n denotes the size of training set. Extensive experiments conducted on three real world datasets highlight the superiority of FDCH compared with several cross-modal hashing methods and demonstrate its effectiveness and efficiency.

Cross-view hashing via supervised deep discrete matrix factorization

Matrix factorization has been utilized for the task of cross-view hashing, where basis functions are learned to map data from different views to the same hamming embedding. It is possible that the basis functions between the hamming embedding and the original data matrix contain rather complex hierarchical information, which existing work can not capture. In addition, previous work employs relaxation technique in the matrix factorization based hashing which may lead to large quantization error. To address these issues, this paper presents a novel Supervised Discrete Deep Matrix Factorization (SDDMF) for cross-view hashing. We introduce deep matrix factorization so that SDDMF is able to learn a set of hierarchical basis functions and unified binary codes from different views. In addition, a classification error term is incorporated into the objective to learn discriminative binary codes. We then employ a linearization technique to directly optimize the discrete constraints which can significantly reduce the quantization error. Experimental results on three standard datasets with image-text modalities verify that SDDMF significantly outperforms several state-of-the-art methods.

Exploiting Subspace Relation in Semantic Labels for Cross-Modal Hashing

Hashing methods have been extensively applied to efficient multimedia data indexing and retrieval on account of the explosion of multimedia data. Cross-modal hashing usually learns binary codes by mapping multi-modal data into a common Hamming space. Most supervised methods utilize relation information like class labels as pairwise similarities of cross-modal data pair to narrow intra-modal and inter-modal gap. In this paper, we propose a novel supervised cross-modal hashing method dubbed Subspace Relation Learning for Cross-modal Hashing (SRLCH), which exploits relation information of labels in semantic space to make similar data from different modalities closer in the low-dimension Hamming subspace. SRLCH preserves the modality relationships, the discrete constraints and nonlinear structures, while admitting a closed-form binary codes solution, which effectively enhances the training efficiency. An iterative alternative optimization algorithm is developed to simultaneously learn both hash functions and unified binary codes. With these binary codes and hash functions, we can index multimedia data and search them in an efficient way. Evaluations in two cross-modal retrieval tasks on several widely-used datasets show that the proposed SRLCH outperforms most cross-modal hashing methods. Theoretical analysis also illustrates reasons for our method’s promotion in subspace relation learning.

Intelligent Reflecting Surface-Assisted Multiple Access With User Pairing: NOMA or OMA?

The integration of intelligent reflecting surface (IRS) to multiple access networks is a cost-effective solution for boosting spectrum/energy efficiency and enlarging network coverage/connections. However, due to the new capability of IRS in reconfiguring the wireless propagation channels, it is fundamentally unknown which multiple access scheme is superior in the IRS-assisted wireless network. In this letter, we pursue a theoretical performance comparison between non-orthogonal multiple access (NOMA) and orthogonal multiple access (OMA) in the IRS-assisted downlink communication, for which the transmit power minimization problems are formulated under the discrete unit-modulus reflection constraint on each IRS element. We analyze the minimum transmit powers required by different multiple access schemes and compare them numerically, which turn out to not fully comply with the stereotyped superiority of NOMA over OMA in conventional systems without IRS. Moreover, to avoid the exponential complexity of the brute-force search for the optimal discrete IRS phase shifts, we propose a low-complexity solution to achieve near-optimal performance.

Power Consumption Minimization of UAV Relay in NOMA Networks

In this letter, we consider employing a non-orthogonal multiple access (NOMA) unmanned aerial vehicle (UAV) as relay to ferry data from a remote base station to multiple ground users. Our objective is to minimize the power consumption of UAV by jointly optimizing the location and transmission power of UAV. The formulated problem is non-convex with highly coupled binary discrete constraints, which is intractable to solve. Firstly, it is transformed into a more tractable form, where the logarithmic constraints are transformed into polynomial form, and the discrete constraints are transformed into continuous form. Then the non-convex equality constraints are handled by penalty method, while the non-convex inequality constraints are handled by successive convex approximation technique. Based on these measures, we propose a double-loop iterative algorithm to obtain the solution approximately. Numerical results show the effectiveness of our proposed algorithm.

Top Position Sensitive Ordinal Relation Preserving Bitwise Weight for Image Retrieval

In recent years, binary coding methods have become increasingly popular for tasks of searching approximate nearest neighbors (ANNs). High-dimensional data can be quantized into binary codes to give an efficient similarity approximation via a Hamming distance. However, most of existing schemes consider the importance of each binary bit as the same and treat training samples at different positions equally, which causes many data pairs to share the same Hamming distance and a larger retrieval loss at the top position. To handle these problems, we propose a novel method dubbed by the top-position-sensitive ordinal-relation-preserving bitwise weight (TORBW) method. The core idea is to penalize data points without preserving an ordinal relation at the top position of a ranking list more than those at the bottom and assign different weight values to their binary bits according to the distribution of query data. Specifically, we design an iterative optimization mechanism to simultaneously learn binary codes and bitwise weights, which makes their learning processes related to each other. When the iterative procedure converges, the binary codes and bitwise weights are effectively adapted to each other. To reduce the training complexity, we relax the discrete constraints of both the binary codes and the indicator function. Furthermore, we pretrain a tensor ordinal graph to decrease the time consumption of computing a relative similarity relationship among data points. Experimental results on three large-scale ANN search benchmark datasets, i.e., SIFT1M, GIST1M, and Cifar10, show that the proposed TORBW method can achieve superior performance over state-of-the-art approaches.

Graph Structured Sparse Subset Selection

We propose a new method for variable subset selection and regression coefficient estimation in linear regression models that incorporates a graph structure of the predictor variables. The proposed method is based on the cardinality constraint that controls the number of selected variables and the graph structured subset constraint that encourages the predictor variables adjacent in the graph to be simultaneously selected or eliminated from the model. Moreover, we develop an efficient discrete projected gradient descent method to handle the NP-hardness of the problem originating from the discrete constraints. Numerical experiments on simulated and real-world data are conducted to demonstrate the usefulness and applicability of the proposed method by comparing it with existing graph regularization methods in terms of the predictive accuracy and variable selection performance. The results confirm that the proposed method outperforms the existing methods.

Deep Pairwise Hashing for Cold-start Recommendation

Recommendation efficiency and data sparsity problems have been regarded as two main challenges of real-world recommendation systems. Most existing works focus on improving recommendation accuracy instead of efficiency. In this paper, we propose a Deep Pairwise Hashing (DPH) to map users and items to binary vectors in the Hamming space, where a user's preference for an item can be efficiently calculated by the Hamming distance, which significantly improves the efficiency of online recommendation. To alleviate data sparsity and cold-start problems, the item content information exploited and integrated to learn effective representations of items. Specifically, we first pre-train robust item representation from item content data by a robust Denoising Auto-encoder instead of other deterministic deep learning frameworks. Then we fine-tune the entire recommender framework by adding a pairwise loss function with discrete constraints, which is more consistent with the ultimate goal of producing a ranked list of items. Finally, we adopt the alternating optimization method to optimize the proposed model with discrete constraints. Extensive experiments conducted on three different datasets show that DPH can significantly advance the state-of-the-art frameworks regarding data sparsity and cold-start item recommendation.

Cross-Modal Retrieval for CPSS Data

Data generated and collected from Cyber-Physical-Social Systems (CPSS) that usually in the forms of image, audio, video, and text, are complex and heterogeneous. How to deal with the cross-modal retrieval problem for heterogeneous CPSS data has drawn considerable interests recently. The hashing based methods have been widely studied in building bilateral semantic associations of binary codes for cross-model retrieval. However, most existing methods discard binary constraints and learn linear projections as hashing functions. Moreover, none of them consider the cross-modal retrieval application in the scenario of Cloud-Fog-Edge computing. Therefore, how to learn more compact and discriminative binary codes with discrete constraints and nonlinear hashing functions for CPSS data in the Cloud-Fog-Edge architecture is still an open problem. In this paper, we propose a nonlinear discrete cross-modal hashing (NDCMH) method based on concise binary classification for CPSS data which fully investigates the nonlinear relationship embedding, discrete optimization as well as the hashing functions learning. Different from previous methods, our work presents a concise but promising cross-modal hashing method that builds a direct connection between original CPSS data and binary codes, which can alleviate the impact of large quantization loss. Furthermore, we execute the cross-modal retrieval service at cloud and fog. Specifically, hashing functions are deployed at the fog plane to reduce the amount data transfer and storage need on the cloud. Extensive experiments carried out on typical CPSS datasets demonstrate that the proposed NDCMH significantly outperforms other state-of-the-art methods.

Finite element analysis of the constrained Dirichlet boundary control problem governed by the diffusion problem

We study an energy space-based approach for the Dirichlet boundary optimal control problem governed by the Laplace equation with control constraints. The optimality system results in a simplified Signorini type problem for control which is coupled with boundary value problems for state and costate variables. We propose a finite element based numerical method using the linear Lagrange finite element spaces with discrete control constraints at the Lagrange nodes. The analysis is presented in a combination for both the gradient and the L2 cost functional. A priori error estimates of optimal order in the energy norm is derived up to the regularity of the solution for both the cases. Theoretical results are illustrated by some numerical experiments.

Continuous-Domain Signal Reconstruction Using $L_{p}$-Norm Regularization

We focus on the generalized-interpolation problem. There, one reconstructs continuous-domain signals that honor discrete data constraints. This problem is infinite-dimensional and ill-posed. We make it well-posed by imposing that the solution balances data fidelity and some $L_{p}$ -norm regularization. More specifically, we consider $p\geq 1$ and the multi-order derivative regularization operator ${\mathrm{L}}={\mathrm{D}}^{N_{0}}$ . We reformulate the regularized problem exactly as a finite-dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid. Our splines are represented in a B-spline basis, which results in a well-conditioned discretization. For a sufficiently fine grid, our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted. This remarkable property is due to the approximation power of splines. We use the alternating-direction method of multipliers along with a multiresolution strategy to compute our solution. We present numerical results for spatial and Fourier interpolation. Through our experiments, we investigate features induced by the $L_{p}$ -norm regularization, namely, sparsity, regularity, and oscillatory behavior.

Joint Modelling and Calibration of SPX and VIX by Optimal Transport

This paper addresses the joint calibration problem of SPX options and VIX options or futures. We show that the problem can be formulated as a semimartingale optimal transport problem under a finite number of discrete constraints, in the spirit of [arXiv:1906.06478]. We introduce a PDE formulation along with its dual counterpart. The solution, a calibrated diffusion process, can be represented via the solutions of Hamilton-Jacobi-Bellman equations arising from the dual formulation. The method is tested on both simulated data and market data. Numerical examples show that the model can be accurately calibrated to SPX options, VIX options and VIX futures simultaneously.

Local Feature Hashing With Binary Auto-Encoder for Face Recognition

The learning-based hashing has recently made encouraging progress in face recognition. However, most existing hashing methods disregard the discrete constraint during optimization, inducing the accumulated quantization errors. In this work, we develop an effective learning-based hashing model, namely local feature hashing with binary auto-encoder (LFH-BAE), to directly learn local binary descriptors in the Hamming space. It attempts to exploit structure factors to well reconstruct the face image from binary codes. Specifically, we first introduce a binary auto-encoder to learn a hashing function to project each face region into high-quality binary codes. Since the original problem is a tricky combinational function, we then present a softened version to decompose it into separate tractable sub-problems. Next, we propose an effective alternating algorithm based on the augmented Lagrange method (ALM) to solve these sub-problems, which helps to generate strong discriminative and excellent robust binary codes. Moreover, we utilize the discrete cyclic coordinate descent (DCC) method to optimize binary codes to reduce the loss of useful information. Lastly, we cluster and pool the obtained binary codes, and construct a histogram feature as the final face representation for each image. Extensive experimental results on four public datasets including FERET, CAS-PEAL-R1, LFW and PaSC show that our LFH-BAE is superior to most state-of-the-art face recognition algorithms.

Semantic-enhanced discrete matrix factorization hashing for heterogeneous modal matching

The simultaneous hash representation of heterogeneous modalities has shown its excellent performance in the multi-modal learning community. However, the binary discrete constraint of hash brings a great challenge to the common hash representation of heterogeneous modal data. To resolve the issue, this paper proposes a novel discrete hash method, referred to as Semantic enhanced Discrete Matrix Factorization Hashing (SDMFH). SDMFH directly extracts the common discrete hash representation of all modalities from the reconstructed semantic intermodal similarity graph, which makes the hash codes more discriminative. Meanwhile, the semantic labels are regressed to the extracted discrete hash book, so as to further strengthen the discriminating ability of the learned discrete hash book. Moreover, a linear embedding from the kernel space of the original data to the Hamming space is explored to generate the common hash codes of each modality to ensure the consistency of the hash codes of heterogeneous modalities. More importantly, we develop an efficient discrete iterative algorithm based on Stiefel manifold that can directly learn the discrete hash book in a closed form, thus both avoiding quantization loss caused by discrete relaxation and reducing computational complexity. Experimental results on three benchmark data sets show that SDMFH performs better than several state-of-the-art methods for heterogeneous modal matching tasks.

Experimental and numerical investigation on static and dynamic characteristics for curvilinearly stiffened plates using DST–BK model

Curvilinearly stiffened plates can have superior mechanical properties than traditional straight-line stiffened plates. To avoid the shear locking problem in the modeling of stiffened thick plates by Reissner-Mindlin theory when the thickness of plate decrease, discrete shear triangle element was introduced by employing the discrete shear constraints on each side. The discrete shear triangle element is employed for the plate and the Timoshenko beam theory is employed for the curvilinearly stiffeners, respectively. The FE equations are established through displacement interpolation function of plate and the displacement compatibility conditions at the plate-stiffened interfaces. The plate and stiffener are modeled separately in the present method, which allows the nodes of stiffener element not coincide with nodes of the plate's shell-element. Based on the newly proposed modeling technique, the statics, vibration and buckling behavior of curvilinearly stiffened plates are investigated. Numerical and experimental investigations are conducted to verify the mechanical properties of proposed model for stiffened plates. Results from the proposed method show good agreements with the numerical and experimental results. The present method can conduct static and dynamic analysis for both stiffened thick and thin plates and guarantee the accuracy of calculation with less number of elements.

Distributed optimal control of viscous Burgers' equation via a high‐order, linearization, integral, nodal discontinuous Gegenbauer‐Galerkin method

SummaryWe developed a novel direct optimization method to solve distributed optimal control of viscous Burgers' equation over a finite‐time horizon by minimizing the distance between the state function and a desired target state profile along with the energy of the control. Through a novel linearization strategy, well‐conditioned integral reformulations, optimal Gegenbauer barycentric quadratures, and nodal discontinuous Galerkin discretizations, the method reduces such optimal control problems into finite‐dimensional, nonlinear programming problems subject to linear algebraic system of equations and discrete mixed path inequality constraints that can be solved easily using standard optimization software. The proposed method produces “an auxiliary control function” that provides a useful model to explicitly define the optimal controller of the state variable. We present an error analysis of the semidiscretization and full discretization of the weak form of the reduced equality constraint system equations to demonstrate the exponential convergence of the method. The accuracy of the proposed method is examined using two numerical examples for various target state functions in the existence/absence of control bounds. The proposed method is exponentially convergent in both space and time, thus producing highly accurate approximations using a significantly small number of collocation points.

An analysis of multi-item inventory model using particle swarm optimization under discrete delivery orders and limited storage space

This study explores an economic production quantity (EPQ) model designed with the assumptions of discrete delivery orders and storage capacity constraints for a multi-item production inventory system. The main purpose of this study is to determine the optimal order quantity, the optimal number of deliveries and the optimal delivery quantity. First, the developed model as part of this study was analyzed using Genetic Algorithm (GA). Numerical analysis results were compared with those of previous studies and it was found that it is possible to have better results with an increasing number of iteration. The same model was then analyzed using Particle Swarm Optimization (PSO) algorithm. A comparison of the optimization methods showed that PSO gives better results over the GA under the same number of iterations and using the same population. The effects of important model parameters such as number of iterations, population, crossover, mutation rate on the optimal solution were analyzed. The results showed that PSO performs better than the GA with respect to the total cost and the total runtime as the solution of the problem in question.

Supervised discrete cross-modal hashing based on kernel discriminant analysis

Cross-modal hashing methods have drawn considerable attention due to the rapid growth of multi-modal data. To obtain efficient binary codes in a low-dimensional Hamming space, most existing approaches relaxed the discrete constraint, which could cause quantization loss and even result in performance degradation. In order to avoid this bottleneck, some scholars employed iterative discrete cyclic coordinate descent (DCC) to learn hash codes bit by bit, but this was very time-consuming. To counter this problem, a simple yet novel supervised discrete cross-modal hashing framework is represented to directly learn the unified discrete binary codes with a close-form, rather than bit by bit. Furthermore, to preserve label separability, the kernel discriminant analysis is fused into the proposed framework to enrich the discrimination ability of the learned binary codes. The goal of the proposed method is to obtain the common discrete binary codes of different modalities in a shared latent Hamming space so that the different modalities of a sample can be effectively connected. Experimental study shows the encouraging results of the proposed algorithm in comparisons to the state-of-the-art baseline approaches on four real-world datasets. Especially on the LabelMe dataset, the superiority of the proposed method is obvious, with an average improvement of 9% over the best available results.