Nonlinear Space Research Articles

Local Manifold-Based Sparse Discriminant Learning for Feature Extraction of Hyperspectral Image.

Hyperspectral image (HSI) generally contains a complex manifold structure and strong sparse correlation in its nonlinear high-dimensional data space. However, the existing manifold learning and sparse learning methods usually consider the manifold structure and sparse relationship separately rather than combining manifold and sparse properties to discover the intrinsic information in the original data. To simultaneously reveal the complex sparse relation and manifold structure of HSI, a novel feature extraction (FE) method, called local manifold-based sparse discriminant learning (LMSDL), has been proposed on the basis of manifold learning and sparse representation (SR). The LMSDL method first designs a new sparse optimization model called local manifold-based SR (LMSR) to reveal the local manifold-based sparse structure of data. Then, two geometrical sparse graphs are constructed to represent the discriminant relationship between samples and the geometrical and sparse neighbors. An objective function is constructed via geometrical sparse graphs and reconstruction points to learn a projection matrix for FE. The LMSDL effectively reveals the complex sparse relation and manifold structure in high-dimensional data, and it enhances the representation ability of extracted features for HSI classification significantly. The experimental results on the three real HSI datasets show that the proposed LMSDL algorithm possesses better performance in comparison with some state-of-the-art FE methods.

Designing Dynamic Machines With Large-Scale Root Finding

Achieving high-performance dynamic behavior in a robot requires careful design of morphology. However, searching for a global optimum morphology in an intensely nonlinear design space is difficult, especially if stochastic seeding is used. In contrast to optimization, we encode design requirements into a polynomial system with a huge number of isolated roots. Each root describes an alternate robot morphology in the design space. Following this, the computation of nearly all isolated roots constitutes design space exploration. Previously, these systems were intractable, due to the heavy burden of degenerate roots. We relieve this burden by using the finite root generation (FRG) method to enable the discovery of nearly all isolated roots for a certain six-bar design problem for the first time. The FRG synthesis method enables the design of a transmission function from motor dynamics to a loaded end effector to influence the overall dynamic behavior. In an example, we formulate synthesis equations which were previously intractable, obtain 1 528 608 isolated roots (estimated 99.0%), and find 3764 physical designs. Design options are compared according to their sensitivity to joint errors.

Nonlinear Dynamic Modeling of Urban Water Consumption Using Chaotic Approach (Case Study: City of Kelowna)

This study investigated urban water consumption complexity using chaos theory to improve forecasting performance to help optimize system management, reduce costs and improve reliability. The objectives of this study were to (1) investigate urban water distribution consumption complexity and its role in forecasting technique performance, (2) evaluate forecasting models by periodicity and lead time, and (3) propose a suitable forecasting technique based on operator applications and performance through various time scales. An urban consumption dataset obtained from the City of Kelowna (British Columbia, Canada) was used as a test case to forecast future consumption values using varying lead times under different temporal scales to identify models which may improve forecasting performance. Chaos theory techniques were employed to inform model optimization. This study attempted to address the paucity of studies on chaos theory applications in water consumption forecasting. This was accomplished by applying non-linear approximation, dynamic investigation, and phase space reconstruction for input variables, to improve the accuracy in various periodicity and lead time. To reconstruct the phase space, lag time was calculated using average mutual information for daily resolution as 17 days to reconstruct the phase space. The optimum embedding dimension and correlation exponent for the phase space were 18 and 3.5, respectively. Comparing the results, the non-linear local approximation model provided the best performance. The forecasting horizon for the models was 122 days. Moreover, phase space reconstruction improved the accuracy of the models for the different lead times. The findings of this study may improve forecasting performance and provide evidence to support further investigation of the chaotic behaviour of water consumption values over different time scales.

Convex integration solutions for the geometrically nonlinear two-well problem with higher Sobolev regularity

In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential inclusion [Formula: see text] subject to suitable affine boundary conditions for [Formula: see text] with [Formula: see text] such that the associated deformation gradients [Formula: see text] enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where [Formula: see text], and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the nonlinear matrix space geometry, it is possible to deal with the geometrically nonlinear two-well problem within the framework outlined in [A. Rüland, C. Zillinger and B. Zwicknagl, Higher Sobolev regularity of convex integration solutions in elasticity: The Dirichlet problem with affine data in int[Formula: see text], SIAM J. Math. Anal. 50 (2018) 3791–3841]. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures.

Nonlinear contributions to angular power spectra

Future galaxy clustering surveys will probe small scales where non-linearities become important. Since the number of modes accessible on intermediate to small scales is very high, having a precise model at these scales is important especially in the context of discriminating alternative cosmological models from the standard one. In the mildly non-linear regime, such models typically differ from each other, and galaxy clustering data will become very precise on these scales in the near future. As the observable quantity is the angular power spectrum in redshift space, it is important to study the effects of non-linear density and redshift space distortion (RSD) in the angular power spectrum. We compute non-linear contributions to the angular power spectrum using a flat-sky approximation that we introduce in this work, and compare the results of different perturbative approaches with $N$-body simulations. We find that the TNS perturbative approach is significantly closer to the $N$-body result than Eulerian or Lagrangian 1-loop approximations, effective field theory of large scale structure or a halofit-inspired model. However, none of these prescriptions is accurate enough to model the angular power spectrum well into the non-linear regime. In addition, for narrow redshift bins, $\Delta z \lesssim 0.01$, the angular power spectrum acquires non-linear contributions on all scales, right down to $\ell=2$, and is hence not a reliable tool at this time. To overcome this problem, we need to model non-linear RSD terms, for example as TNS does, but for a matter power spectrum that remains reasonably accurate well into the deeply non-linear regime, such as halofit.

Finding characteristically rich nonlinear solution space: a statistical mechanics approach

AbstractIn this work, the modally equivalent perturbed system (MEPS) which was originally developed for finding the parametrically rich solution space of linear time‐invariant systems is modified for time‐varying cases and applied to find the characteristically rich nonlinear solution space given arbitrary initial or boundary conditions, or system inputs. An integral form of the non‐Hamiltonian Liouville equation is derived such that a rich ensemble average of its solutions covers a broad range of the modal space when a maximum uncertainty is present in the solutions. The MEPS degenerates the integrated Liouville equation into a linear differential equation with the Gauge Modal Invariance, a newly found field property that allows extending the application beyond the initial conditions or impulse inputs, making it possible to calculate the rich set of basis modes by taking snapshots of the linear responses at a considerably low computational cost. The proposed theory and algorithm are demonstrated using a computational model of a two‐dimensional incompressible, viscous flow at low Reynolds numbers. It is shown that the basis modes obtained herein, when used in conjunction with a low dimensional modeling, reproduce time simulation results very accurately for a wide range of Reynolds numbers and boundary conditions.

Noval advance non-linear descriptor and characteristic equivalency to predict CT from MRI image

Attenuation correction designed for PET/MR hybrid imaging frameworks along with portion making arrangements used for MR-based radiation treatment remain testing because of lacking high-energy photon weakening data. We present a new method so as to uses the learned nonlinear neighborhood descriptors also highlight coordinating toward foresee pseudo-CT pictures starting T1w along with T2w MRI information. The nonlinear neighborhood descriptors are acquired through anticipating the direct descriptors interested in the nonlinear high-dimensional space utilizing an unequivocal constituent guide also low-position guess through regulated complex regularization. The nearby neighbors of every near descriptor inside the data MR pictures are looked during an obliged spatial extent of the MR pictures among the training dataset. By that point, the pseudo-CT patches are evaluated through k-closest neighbor relapse. The planned procedure designed for pseudo-CT forecast is quantitatively broke downward on top of a dataset comprising of coordinated mind MRI along with CT pictures on or after 13 subjects.

Approximation of a Sideways Problem for Nonlinear Space Fractional Heat Equation

Approximation of a Sideways Problem for Nonlinear Space Fractional Heat Equation

Improved image registration scheme based on multiscale nonlinear Harris and Gaussian–Krawtchouk moment

Image registration is an indispensable operation in image processing. To solve the problem of nonlinear distortion which easily causes low registration property, a registration scheme including a nonlinear scale space, a new discrete orthogonal Gaussian–Krawtchouk (GK) descriptor vector, and a dual similarity measurement of distance-cosine is proposed. Nonlinear scale space is constructed to preserve a lot of important structures for increasing the quantity of the salient keypoints. Meanwhile, considering both blur stability and simple computation, multiscale Harris algorithm on the basis of nonlinear scale space is applied to extract the keypoints. Moreover, according to different scales, a descriptor vector based on multiscale GK invariant moments is presented and applied to the following image matching. Finally, a dual similarity measurement of the initial Euclidean distance and the fine space cosine is proposed for the purpose of enhancing the matching accuracy. The effectiveness of the proposed algorithm has been tested on the standard images with different types of variations. The experimental results show that the performance of feature extraction and the correct matching rate of the proposed scheme are superior to those of other classical methods. The proposed scheme has a better robustness and a higher registration accuracy.

Electronic Tongue Recognition with Feature Specificity Enhancement.

As a kind of intelligent instrument, an electronic tongue (E-tongue) realizes liquid analysis with an electrode-sensor array and certain machine learning methods. The large amplitude pulse voltammetry (LAPV) is a regular E-tongue type that prefers to collect a large amount of response data at a high sampling frequency within a short time. Therefore, a fast and effective feature extraction method is necessary for machine learning methods. Considering the fact that massive common-mode components (high correlated signals) in the sensor-array responses would depress the recognition performance of the machine learning models, we have proposed an alternative feature extraction method named feature specificity enhancement (FSE) for feature specificity enhancement and feature dimension reduction. The proposed FSE method highlights the specificity signals by eliminating the common mode signals on paired sensor responses. Meanwhile, the radial basis function is utilized to project the original features into a nonlinear space. Furthermore, we selected the kernel extreme learning machine (KELM) as the recognition part owing to its fast speed and excellent flexibility. Two datasets from LAPV E-tongues have been adopted for the evaluation of the machine-learning models. One is collected by a designed E-tongue for beverage identification and the other one is a public benchmark. For performance comparison, we introduced several machine-learning models consisting of different combinations of feature extraction and recognition methods. The experimental results show that the proposed FSE coupled with KELM demonstrates obvious superiority to other models in accuracy, time consumption and memory cost. Additionally, low parameter sensitivity of the proposed model has been demonstrated as well.

Legendre Collocation Method for the Nonlinear Space–Time Fractional Partial Differential Equations

In this study, nonlinear space–time fractional partial differential equations with variable coefficients are considered. The fractional derivatives are described in the conformable sense and Caputo sense. The numerical approach is based on the shifted Legendre polynomials and the finite difference method. Several illustrative examples are given to observe the validity, effectiveness and accuracy of the present numerical method.

Improving malicious URLs detection via feature engineering: Linear and nonlinear space transformation methods

In malicious URLs detection, traditional classifiers are challenged because the data volume is huge, patterns are changing over time, and the correlations among features are complicated. Feature engineering plays an important role in addressing these problems. To better represent the underlying problem and improve the performances of classifiers in identifying malicious URLs, this paper proposed a combination of linear and non-linear space transformation methods. For linear transformation, a two-stage distance metric learning approach was developed: first, singular value decomposition was performed to get an orthogonal space, and then a linear programming was used to solve an optimal distance metric. For nonlinear transformation, we introduced Nyström method for kernel approximation and used the revised distance metric for its radial basis function such that the merits of both linear and non-linear transformations can be utilized. 33,1622 URLs with 62 features were collected to validate the proposed feature engineering methods. The results showed that the proposed methods significantly improved the efficiency and performance of certain classifiers, such as k-Nearest Neighbor, Support Vector Machine, and neural networks. The malicious URLs’ identification rate of k-Nearest Neighbor was increased from 68% to 86%, the rate of linear Support Vector Machine was increased from 58% to 81%, and the rate of Multi-Layer Perceptron was increased from 63% to 82%. We also developed a website to demonstrate a malicious URLs detection system which uses the methods proposed in this paper. The system can be accessed at: http://url.jspfans.com.

Regular and Irregular Vegetation Pattern Formation in Semiarid Regions: A Study on Discrete Klausmeier Model

The research on regular and irregular vegetation pattern formation in semiarid regions is an important field in ecology. Applying the framework of coupled map lattice, a novel nonlinear space- and time-discrete model is developed based on discretizing the classical Klausmeier model and the vegetation pattern formation in semiarid regions is restudied in this research. Through analysis of Turing-type instability for the discrete model, the conditions for vegetation pattern formation are determined. The discrete model is verified by Klausmeier’s results with the same parametric data, and shows advantages in quantitatively describing diverse vegetation patterns in semiarid regions, such as the patterns of regular mosaicirregular patches, stripes, fractured stripesspots, and stripes-spots, in comparing with former theoretical models. Moreover, the discrete model predicts variations of rainfall and vegetation types can cause transitions of vegetation patterns. This research demonstrates that the nonlinear mechanism of the discrete model better captures the diversity and complexity of vegetation pattern formation in semiarid regions.

Probability Distribution-Based Dimensionality Reduction on Riemannian Manifold of SPD Matrices

Representing images and videos with Symmetric Positive Definite (SPD) matrices and utilizing the intrinsic Riemannian geometry of the resulting manifold has proved successful in many computer vision tasks. Since SPD matrices lie in a nonlinear space known as a Riemannian manifold, researchers have recently shown a growing interest in learning discriminative SPD matrices with appropriate Riemannian metrics. However, the computational complexity of analyzing high-dimensional SPD matrices is nonnegligible in practical applications. Inspired by the theory of nonparametric estimation, we propose a probability distribution-based approach to overcome this drawback by learning a mapping from the manifold of high-dimensional SPD matrices to a manifold of lower-dimension, which can be expressed as an optimization problem on the Grassmann manifold. Specifically, we perform the dimensionality reduction for high-dimensional SPD matrices with popular Riemannian metrics and an affinity matrix constructed using an estimated probability distribution function (PDF) to achieve maximum class separability. The evaluation of several classification tasks shows the competitiveness of the proposed approach compared with state-of-the-art methods.

Hierarchical two-pathway autoencoders neural networks for skyline context conceptualisation

In this paper, we proposed a novel hierarchical two-pathway autoencoders architecture to transform a local information based on skyline scene representation, into nonlinear space. The first pathway is intended for the transformation of the geometric features extracted from the horizon line. The second pathway is applied after the first one to joint the colour information under the skyline to the transformed geometric features, and to get the landscape context conceptualisation. To evaluate our suggested system, we constructed the SKYLINEScene database containing 2,000 images of rural and urban landscapes, with a high degree of diversity. In order to investigate the performance of our HTANN-Skyline, many experiments were carried out using this new database. Our approach shows its robustness in skyline context conceptualisation and enhances the classification rates by 1% compared to the AlexNet architecture; and by more than 10% compared to the hand-crafted approaches based on global and local features.

An investigation of new quicker implicit iterations in hyperbolic spaces

The present paper investigates the convergence of some new implicit iterations of coupled fixed point for non-linear contractive like functions on W-hyperbolic metric spaces. Moreover, we provide a theoretical comparison of our new iterations to illustrate the fastest iteration the coupled fixed point.

An investigation of new quicker implicit iterations in hyperbolic spaces

The present paper investigates the convergence of some new implicit iterations of coupled fixed point for non-linear contractive like functions on W-hyperbolic metric spaces. Moreover, we provide a theoretical comparison of our new iterations to illustrate the fastest iteration the coupled fixed point.

An Optimal Monitoring Model of Desertification in Naiman Banner Based on Feature Space Utilizing Landsat8 Oli Image

Current feature space models of desertification were almost linear, which ignored the complicated and non-linear relationships among variables for monitoring desertification. Fully considering the influencing factors of the desertification process in Naiman Banner, four sensitive indices including MSAVI, NDVI, TGSI, and Albedo have been selected to construct five feature spaces. Then, the precisions of different feature space models for monitoring desertification information (including non-linear and linear models) have been compared and analyzed. The non-linear Albedo-MSAVI feature space model for Naiman Banner has higher efficiency with the overall precision of 90.1%, while that of Albedo-TGSI had the worst precision with 0.69. Overall, the feature space model (non-linear) of Albedo-MSAVI has the highest applicability for monitoring the desertification information in Naiman Banner.

Analysis of damping ratio on the optimization of geometrically nonlinear truss structures subjected to dynamic loading

The objective of this paper is to present the formulation for optimizing truss structures with geometric nonlinearity under dynamic loads, provide pertinent case studies and investigate the influence of damping on the final result. The type of optimization studied herein aims to determine the cross-sectional areas that will minimize the weight of a given structural system, by imposing constraints on nodal displacements and axial stresses. The analyses are carried out using Sequential Quadratic Programming (SQP), available in MATLAB’s Optimization Toolbox™. The nonlinear finite space truss element is defined with an updated Lagrangian formulation, and the geometrically nonlinear dynamic analysis performed herein combines the Newmark method with Newton-Raphson iterations. The dynamic analysis approach was validated by comparing the results obtained with solutions available in the literature as well as with numerical models developed with ANSYS® 18.2. A number of optimization examples of planar and space trusses under dynamic loading with geometric nonlinearity are presented. Results indicate that the consideration of damping effects may lead to a significant reduction in structural weight and that such weight reduction is proportional to increases in damping ratio.

Tunable Adaptive Target Detection With Kernels in Colocated MIMO Radar

We introduce a framework for exploring adaptive target detection in colocated MIMO radar in non-linear feature space by exploiting the theory of kernel. The kernel theory inspires us to replace the inner products of test statistics with that of nonlinear mapped data in the feature space or that of their kernel tricks to achieve better detection performance. We apply this framework to the problem of tunable adaptive target detection in colocated MIMO radar, according to the principle of the generalized likelihood ratio (GLR), Rao and Wald (RaW) tests, and propose several new detectors under two new tunable detector forms, namely kernel tunable GLR-based and kernel tunable Raw-based detectors. The proposed tunable detectors include most of the prior detectors in colocated MIMO radar as special cases. One capability of the proposed detectors is that their robustness or selectivity (RoS) to steering vector mismatch (SVM) can be tuned flexibility through an RoS tuning parameter. In addition, we are able to incorporate the prior distribution (PD) of the disturbance covariance matrix, if available, through a PD tuning parameter. Therefore, the proposed detectors can be tuned based on the tuning RoS parameter to achieve robust or selective performance in the presence of SVM as well as to switch between the Bayesian or non-Bayesian based detectors through the PD parameter. For practical situations, we show that the proposed detectors possess CFAR property against disturbance covariance matrix by resorting to the invariance principle. Extensive Monte Carlo simulation results are provided to indicate that the proposed detectors have better detection performance than their counterparts in both single-target and multi-target scenarios.