A Dimensional Reduction Approach Based on Essential Constraints in Linear Programming

This paper presents a new dimension reduction strategy for medium and large-scale linear programming problems. The proposed method uses a subset of the original constraints and combines two algorithms: the weighted average and the cosine simplex algorithm. The first approach identifies binding constraints by using the weighted average of each constraint, whereas the second algorithm is based on the cosine similarity between the vector of the objective function and the constraints. These two approaches are complementary, and when used together, they locate the essential subset of initial constraints required for solving medium and large-scale linear programming problems. After reducing the dimension of the linear programming problem using the subset of the essential constraints, the solution method can be chosen from any suitable method for linear programming. The proposed approach was applied to a set of well-known benchmarks as well as more than 2000 random medium and large-scale linear programming problems. The results are promising, indicating that the new approach contributes to the reduction of both the size of the problems and the total number of iterations required. A tree-based classification model also confirmed the need for combining the two approaches. A detailed numerical example, the general numerical results, and the statistical analysis for the decision tree procedure are presented.

A New Dynamics Analysis Model for Five-Axis Machining of Curved Surface Based on Dimension Reduction and Mapping

The equipment used in various fields contains an increasing number of parts with curved surfaces of increasing size.Five-axis computer numerical control(CNC)milling is the main parts machining method,while dynamics analysis has always been a research hotspot.The cutting conditions determined by the cutter axis,tool path,and workpiece geometry are complex and changeable,which has made dynamics research a major challenge.For this reason,this paper introduces the innovative idea of applying dimension reduction and mapping to the five-axis machining of curved surfaces,and proposes an efficient dynamics analysis model.To simplify the research object,the cutter position points along the tool path were discretized into inclined plane five-axis machining.The cutter dip angle and feed deflection angle were used to define the spatial position relationship in five-axis machining.These were then taken as the new base variables to construct an abstract two-dimensional space and establish the mapping relationship between the cutter position point and space point sets to further simplify the dimensions of the research object.Based on the in-cut cutting edge solved by the space limitation method,the dynamics of the inclined plane five-axis machining unit were studied,and the results were uniformly stored in the abstract space to produce a database.Finally,the prediction of the milling force and vibration state along the tool path became a data extraction process that significantly improved efficiency.Two experiments were also conducted which proved the accuracy and efficiency of the proposed dynamics analysis model.This study has great potential for the online synchronization of intelligent machining of large surfaces.

Multi-surrogate framework with an adaptive selection mechanism for production optimization

Data-driven surrogate models that assist with efficient evolutionary algorithms to find the optimal development scheme have been widely used to solve reservoir production optimization problems.However,existing research suggests that the effectiveness of a surrogate model can vary depending on the complexity of the design problem.A surrogate model that has demonstrated success in one scenario may not perform as well in others.In the absence of prior knowledge,finding a promising surrogate model that performs well for an unknown reservoir is challenging.Moreover,the optimization process often relies on a single evolutionary algorithm,which can yield varying results across different cases.To address these limitations,this paper introduces a novel approach called the multi-surrogate framework with an adaptive selection mechanism(MSFASM)to tackle production optimization problems.MSFASM consists of two stages.In the first stage,a reduced-dimensional broad learning system(BLS)is used to adaptively select the evolutionary algorithm with the best performance during the current optimization period.In the second stage,the multi-objective algorithm,non-dominated sorting genetic algorithm II(NSGA-II),is used as an optimizer to find a set of Pareto solutions with good performance on multiple surrogate models.A novel optimal point criterion is utilized in this stage to select the Pareto solutions,thereby obtaining the desired development schemes without increasing the computational load of the numerical simulator.The two stages are combined using sequential transfer learning.From the two most important perspectives of an evolutionary algorithm and a surrogate model,the proposed method improves adaptability to optimization problems of various reservoir types.To verify the effectiveness of the proposed method,four 100-dimensional benchmark functions and two reservoir models are tested,and the results are compared with those obtained by six other surrogate-model-based methods.The results demonstrate that our approach can obtain the maximum net present value(NPV)of the target production optimization problems.

Prediction of collapse process and tipping points for mutualistic and competitive networks with k-core method

Ecosystems generally have the self-adapting ability to resist various external pressures or disturbances,which is always called resilience.However,once the external disturbances exceed the tipping points of the system resilience,the consequences would be catastrophic,and eventually lead the ecosystem to complete collapse.We capture the collapse process of ecosystems represented by plant-pollinator networks with the k-core nested structural method,and find that a sufficiently weak interaction strength or a sufficiently large competition weight can cause the structure of the ecosystem to collapse from its smallest k-core towards its largest k-core.Then we give the tipping points of structure and dynamic collapse of the entire system from the one-dimensional dynamic function of the ecosystem.Our work provides an intuitive and precise description of the dynamic process of ecosystem collapse under multiple interactions,and provides theoretical insights into further avoiding the occurrence of ecosystem collapse.

A Power Data Anomaly Detection Model Based on Deep Learning with Adaptive Feature Fusion

With the popularisation of intelligent power,power devices have different shapes,numbers and specifications.This means that the power data has distributional variability,the model learning process cannot achieve sufficient extraction of data features,which seriously affects the accuracy and performance of anomaly detection.Therefore,this paper proposes a deep learning-based anomaly detection model for power data,which integrates a data alignment enhancement technique based on random sampling and an adaptive feature fusion method leveraging dimension reduction.Aiming at the distribution variability of power data,this paper developed a sliding window-based data adjustment method for this model,which solves the problem of high-dimensional feature noise and low-dimensional missing data.To address the problem of insufficient feature fusion,an adaptive feature fusion method based on feature dimension reduction and dictionary learning is proposed to improve the anomaly data detection accuracy of the model.In order to verify the effectiveness of the proposed method,we conducted effectiveness comparisons through elimination experiments.The experimental results show that compared with the traditional anomaly detection methods,the method proposed in this paper not only has an advantage in model accuracy,but also reduces the amount of parameter calculation of the model in the process of feature matching and improves the detection speed.

Model-based Predictive Control for Spatially-distributed Systems Using Dimensional Reduction Models

In this paper, a low-dimensional multiple-input and multiple-output （MIMO） model predictive control （MPC） configuration is presented for partial differential equation （PDE） unknown spatially-distributed systems （SDSs）. First, the dimension reduction with principal component analysis （PCA） is used to transform the high-dimensional spatio-temporal data into a low-dimensional time domain. The MPC strategy is proposed based on the online correction low-dimensional models, where the state of the system at a previous time is used to correct the output of low-dimensional models. Sufficient conditions for closed-loop stability are presented and proven. Simulations demonstrate the accuracy and efficiency of the proposed methodologies.

Optimizing progress variable definition in flamelet-based dimension reduction in combustion

An automated method to optimize the definition of the progress variables in the flamelet-based dimension reduction is proposed. The performance of these optimized progress variables in coupling the flamelets and flow solver is presented. In the proposed method, the progress variables are defined according to the first two principal components （PCs） from the principal component analysis （PCA） or kernel-density-weighted PCA （KEDPCA） of a set of flamelets. These flamelets can then be mapped to these new progress variables instead of the mixture fraction/conventional progress variables. Thus, a new chemistry look-up table is constructed. A priori validation of these optimized progress variables and the new chemistry table is implemented in a CH4/N2/air lift-off flame. The reconstruction of the lift-off flame shows that the optimized progress variables perform better than the conventional ones, especially in the high temperature area. The coefficient determinations （R2 statistics） show that the KEDPCA performs slightly better than the PCA except for some minor species. The main advantage of the KEDPCA is that it is less sensitive to the database. Meanwhile, the criteria for the optimization are proposed and discussed. The constraint that the progress variables should monotonically evolve from fresh gas to burnt gas is analyzed in detail.

Adaptive subspace detection based on two-step dimension reduction in the underwater waveguide

In the underwater waveguide,the conventional adaptive subspace detector(ASD),derived by using the generalized likelihood ratio test(GLRT)theory,suffers from a significant degradation in detection performance when the samplings of training data are deficient.This paper proposes a dimension-reduced approach to alleviate this problem.The dimension reduction includes two steps:firstly,the full array is divided into several subarrays;secondly,the test data and the training data at each subarray are transformed into the modal domain from the hydrophone domain.Then the modal-domain test data and training data at each subarray are processed to formulate the subarray statistic by using the GLRT theory.The final test statistic of the dimension-reduced ASD(DR-ASD)is obtained by summing all the subarray statistics.After the dimension reduction,the unknown parameters can be estimated more accurately so the DR-ASD achieves a better detection performance than the ASD.In order to achieve the optimal detection performance,the processing gain of the DR-ASD is deduced to choose a proper number of subarrays.Simulation experiments verify the improved detection performance of the DR-ASD compared with the ASD.

A study on sustainable development capacity of China's coastal areas using indices dimension reduction method

Sustainable Development Capacity (SDC) is a comprehensive concept. In order to obtain a relatively objective evaluation of it, many indices of various aspects are often used in assessing index systems. However, the overlapping information of indices is a frequent source deviating the result from the truth. In this paper, 48 indices are selected as original variables in assessing SDC of China's coastal areas. The mathematical method of dimension reducing treatment is used for eliminating the overlapping information in 48 variables. Five new comprehensive indices are extracted bearing efficient messages of original indices. On the base of new indices values, the sequencing of 12 coastal areas SDC is gained, and five patterns of sustainable development regions are sorted. Then, the leading factors and their relations of SDC in these patterns are analyzed. The gains of research are discussed in the end.

Multi-state Information Dimension Reduction Based on Particle Swarm Optimization-Kernel Independent Component Analysis

The precision of the kernel independent component analysis( KICA) algorithm depends on the type and parameter values of kernel function. Therefore,it's of great significance to study the choice method of KICA's kernel parameters for improving its feature dimension reduction result. In this paper, a fitness function was established by use of the ideal of Fisher discrimination function firstly. Then the global optimal solution of fitness function was searched by particle swarm optimization( PSO) algorithm and a multi-state information dimension reduction algorithm based on PSO-KICA was established. Finally,the validity of this algorithm to enhance the precision of feature dimension reduction has been proven.

A Dimension Reduction Subdivision Scheme Based on Proper Parameterization

In our previous work, we have given an algorithm for segmenting a simplex in the n-dimensional space into rt n＋ 1 polyhedrons and provided map F which maps the n-dimensional unit cube to these polyhedrons. In this paper, we prove that the map F is a one to one correspondence at least in lower dimensional spaces （n _〈 3）. Moreover, we propose the approximating subdivision and the interpolatory subdivision schemes and the estimation of computational complexity for triangular Bézier patches on a 2-dimensional space. Finally, we compare our schemes with Goldman＇s in computational complexity and speed.

Review of Dimension Reduction Methods

Purpose: This study sought to review the characteristics, strengths, weaknesses variants, applications areas and data types applied on the various Dimension Reduction techniques. Methodology: The most commonly used databases employed to search for the papers were ScienceDirect, Scopus, Google Scholar, IEEE Xplore and Mendeley. An integrative review was used for the study where 341 papers were reviewed. Results: The linear techniques considered were Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA), Singular Value Decomposition (SVD), Latent Semantic Analysis (LSA), Locality Preserving Projections (LPP), Independent Component Analysis (ICA) and Project Pursuit (PP). The non-linear techniques which were developed to work with applications that have complex non-linear structures considered were Kernel Principal Component Analysis (KPCA), Multi-dimensional Scaling (MDS), Isomap, Locally Linear Embedding (LLE), Self-Organizing Map (SOM), Latent Vector Quantization (LVQ), t-Stochastic neighbor embedding (t-SNE) and Uniform Manifold Approximation and Projection (UMAP). DR techniques can further be categorized into supervised, unsupervised and more recently semi-supervised learning methods. The supervised versions are the LDA and LVQ. All the other techniques are unsupervised. Supervised variants of PCA, LPP, KPCA and MDS have been developed. Supervised and semi-supervised variants of PP and t-SNE have also been developed and a semi supervised version of the LDA has been developed. Conclusion: The various application areas, strengths, weaknesses and variants of the DR techniques were explored. The different data types that have been applied on the various DR techniques were also explored.

Adaptive Metric Learning for Dimensionality Reduction

Finding a suitable space is one of the most critical problems for dimensionality reduction. Each space corresponds to a distance metric defined on the sample attributes, and thus finding a suitable space can be converted to develop an effective distance metric. Most existing dimensionality reduction methods use a fixed pre-specified distance metric. However, this easy treatment has some limitations in practice due to the fact the pre-specified metric is not going to warranty that the closest samples are the truly similar ones. In this paper, we present an adaptive metric learning method for dimensionality reduction, called AML. The adaptive metric learning model is developed by maximizing the difference of the distances between the data pairs in cannot-links and those in must-links. Different from many existing papers that use the traditional Euclidean distance, we use the more generalized l2,p-norm distance to reduce sensitivity to noise and outliers, which incorporates additional flexibility and adaptability due to the selection of appropriate p-values for different data sets. Moreover, considering traditional metric learning methods usually project samples into a linear subspace, which is overstrict. We extend the basic linear method to a more powerful nonlinear kernel case so that well capturing complex nonlinear relationship between data. To solve our objective, we have derived an efficient iterative algorithm. Extensive experiments for dimensionality reduction are provided to demonstrate the superiority of our method over state-of-the-art approaches.

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.

Tensor Train Random Projection

This work proposes a Tensor Train Random Projection(TTRP)method for dimension reduction,where pairwise distances can be approximately preserved.Our TTRP is systematically constructed through a Tensor Train(TT)representation with TT-ranks equal to one.Based on the tensor train format,this random projection method can speed up the dimension reduction procedure for high-dimensional datasets and requires fewer storage costs with little loss in accuracy,comparedwith existingmethods.We provide a theoretical analysis of the bias and the variance of TTRP,which shows that this approach is an expected isometric projectionwith bounded variance,and we show that the scaling Rademacher variable is an optimal choice for generating the corresponding TT-cores.Detailed numerical experiments with synthetic datasets and theMNIST dataset are conducted to demonstrate the efficiency of TTRP.

An Economical Approach to Four-dimensional Variational Data Assimilation

Four-dimensional variational data assimilation （4DVar） is one of the most promising methods to provide optimal analysis for numerical weather prediction （NWP）. Five national NWP centers in the world have successfully applied 4DVar methods in their global NWPs, thanks to the increment method and adjoint technique. However, the application of 4DVar is still limited by the computer resources available at many NWP centers and research institutes. It is essential, therefore, to further reduce the computational cost of 4DVar. Here, an economical approach to implement 4DVar is proposed, using the technique of dimension- reduced projection （DRP）, which is called ＂DRP-4DVar.＂ The proposed approach is based on dimension reduction using an ensemble of historical samples to define a subspace. It directly obtains an optimal solution in the reduced space by fitting observations with historical time series generated by the model to form consistent forecast states, and therefore does not require implementation of the adjoint of tangent linear approximation. To evaluate the performance of the DRP-4DVar on assimilating different types of mesoscale observations, some observing system simulation experiments are conducted using MM5 and a comparison is made between adjoint-based 4DVar and DRP-4DVar using a 6-hour assimilation window.

Total Variation Constrained Non-Negative Matrix Factorization for Medical Image Registration

This paper presents a novel medical image registration algorithm named total variation constrained graphregularization for non-negative matrix factorization(TV-GNMF).The method utilizes non-negative matrix factorization by total variation constraint and graph regularization.The main contributions of our work are the following.First,total variation is incorporated into NMF to control the diffusion speed.The purpose is to denoise in smooth regions and preserve features or details of the data in edge regions by using a diffusion coefficient based on gradient information.Second,we add graph regularization into NMF to reveal intrinsic geometry and structure information of features to enhance the discrimination power.Third,the multiplicative update rules and proof of convergence of the TV-GNMF algorithm are given.Experiments conducted on datasets show that the proposed TV-GNMF method outperforms other state-of-the-art algorithms.

A Bi-population Cooperative Optimization Algorithm Assisted by an Autoencoder for Medium-scale Expensive Problems

This study presents an autoencoder-embedded optimization(AEO)algorithm which involves a bi-population cooperative strategy for medium-scale expensive problems(MEPs).A huge search space can be compressed to an informative lowdimensional space by using an autoencoder as a dimension reduction tool.The search operation conducted in this low space facilitates the population with fast convergence towards the optima.To strike the balance between exploration and exploitation during optimization,two phases of a tailored teaching-learning-based optimization(TTLBO)are adopted to coevolve solutions in a distributed fashion,wherein one is assisted by an autoencoder and the other undergoes a regular evolutionary process.Also,a dynamic size adjustment scheme according to problem dimension and evolutionary progress is proposed to promote information exchange between these two phases and accelerate evolutionary convergence speed.The proposed algorithm is validated by testing benchmark functions with dimensions varying from 50 to 200.As indicated in our experiments,TTLBO is suitable for dealing with medium-scale problems and thus incorporated into the AEO framework as a base optimizer.Compared with the state-of-the-art algorithms for MEPs,AEO shows extraordinarily high efficiency for these challenging problems,t hus opening new directions for various evolutionary algorithms under AEO to tackle MEPs and greatly advancing the field of medium-scale computationally expensive optimization.

Study on Mach stems induced by interaction of planar shock waves on two intersecting wedges

The properties of Mach stems in hypersonic corner flow induced by Mach interaction over 3D intersecting wedges were studied theoretically and numerically.A new method called ＂spatial dimension reduction＂ was used to analyze theoretically the location and Mach number behind Mach stems. By using this approach, the problem of 3D steady shock/shock interaction over 3D intersecting wedges was transformed into a 2D moving one on cross sections, which can be solved by shock-polar theory and shock dynamics theory. The properties of Mach interaction over 3D intersecting wedges can be analyzed with the new method,including pressure, temperature, density in the vicinity of triple points, location, and Mach number behind Mach stems.Theoretical results were compared with numerical results,and good agreement was obtained. Also, the influence of Mach number and wedge angle on the properties of a 3D Mach stem was studied.

A review on the coordinative structure of human walking and the application of principal component analysis

Walking is a complex task which includes hundreds of muscles, bones and joints working together to deliver smooth movements. With the complexity, walking has been widely investigated in order to identify the pattern of multi-segment movement and reveal the control mechanism. The degree of freedom and dimensional properties provide a view of the coordinative structure during walking, which has been extensively studied by using dimension reduction technique. In this paper, the studies related to the coordinative structure, dimensions detection and pattern reorganization during walking have been reviewed. Principal component analysis, as a popular technique, is widely used in the processing of human movement data. Both the principle and the outcomes of principal component analysis were introduced in this paper. This technique has been reported to successfully reduce the redundancy within the original data, identify the physical meaning represented by the extracted principal components and discriminate the different patterns. The coordinative structure during walking assessed by this technique could provide further information of the body control mechanism and correlate walking pattern with injury.