Laplacian Eigenmaps Algorithm Research Articles

Hypergraph analysis based on a compatible tensor product structure

We propose a tensor product structure that is compatible with the hypergraph structure. We define the algebraic connectivity of the (m+1)-uniform hypergraph in this product, and prove the relationship with the vertex connectivity. We introduce some connectivity optimization problem into the hypergraph, and solve them with the algebraic connectivity. We introduce the Laplacian eigenmap algorithm to the hypergraph under our tensor product.

Application of the combination method based on RF and LE in near infrared spectral modeling

The dimensionality of near-infrared (NIR) spectral data is often extremely large. Dimensionality reduction of spectral data can effectively reduce the redundant information and correlation between spectral variables and simplify the model, which is crucial to increasing the model's performance. As a nonlinear feature extraction method, Laplacian Eigenmaps (LE) may preserve the local neighborhood information of the dataset, has high robustness, and is simple to compute. However, when the LE algorithm maps the data from high-dimensional space to low-dimensional space, it is often disturbed by irrelevant information and multicollinearity in the spectral data, which lowers the model's prediction performance. Random Frog (RF) algorithm can eliminate noise and collinearity in the spectrum. Therefore, before using the LE algorithm, we use the RF algorithm to eliminate irrelevant information in the spectrum and reduce the correlation between the spectra variables to increase the efficiency of the LE algorithm. We used the RF + LE algorithm to reduce the dimensionality of two public NIRS datasets (soil datasets and pharmaceutical tablets datasets) and compared it with RF and LE algorithms alone. We utilized Partial Least Squares Regression (PLSR) and Support Vector Regression (SVR) to establish regression models. The experimental findings demonstrate that compared with the RF algorithm and LE algorithm, the RF + LE combination method can reduce the dimension of spectral variables and model complexity, and improve regression models' prediction accuracy and stability. It is an effective dimensionality reduction method for the near-infrared spectrum.

Manifold Learning with Arbitrary Norms

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge with each pair. Existing theory shows that the Laplacian matrix of the graph converges to the Laplace-Beltrami operator of the data manifold, under the assumption that the pairwise affinities are based on the Euclidean norm. In this paper, we determine the limiting differential operator for graph Laplacians constructed using $\textit{any}$ norm. Our proof involves an interplay between the second fundamental form of the manifold and the convex geometry of the given norm's unit ball. To demonstrate the potential benefits of non-Euclidean norms in manifold learning, we consider the task of mapping the motion of large molecules with continuous variability. In a numerical simulation we show that a modified Laplacian eigenmaps algorithm, based on the Earthmover's distance, outperforms the classic Euclidean Laplacian eigenmaps, both in terms of computational cost and the sample size needed to recover the intrinsic geometry.

Dimensionality reduction for classification of object weight from electromyography.

Electromyography (EMG) is a simple, non-invasive, and cost-effective technology for measuring muscle activity. However, multi-muscle EMG is also a noisy, complex, and high-dimensional signal. It has nevertheless been widely used in a host of human-machine-interface applications (electrical wheelchairs, virtual computer mice, prosthesis, robotic fingers, etc.) and, in particular, to measure the reach-and-grasp motions of the human hand. Here, we developed an automated pipeline to predict object weight in a reach-grasp-lift task from an open dataset, relying only on EMG data. In doing so, we shifted the focus from manual feature-engineering to automated feature-extraction by using pre-processed EMG signals and thus letting the algorithms select the features. We further compared intrinsic EMG features, derived from several dimensionality-reduction methods, and then ran several classification algorithms on these low-dimensional representations. We found that the Laplacian Eigenmap algorithm generally outperformed other dimensionality-reduction methods. What is more, optimal classification accuracy was achieved using a combination of Laplacian Eigenmaps (simple-minded) and k-Nearest Neighbors (88% F1 score for 3-way classification). Our results, using EMG alone, are comparable to other researchers’, who used EMG and EEG together, in the literature. A running-window analysis further suggests that our method captures information in the EMG signal quickly and remains stable throughout the time that subjects grasp and move the object.

Spatial correlation based channel compression feedback algorithm for massive MIMO systems

Aiming at the non-linear structure of massive multiple-input multiple-output (MIMO) channel data, this paper proposes a channel state information (CSI) compression feedback algorithm based on Laplacian Eigenmaps (LE) non-linear processing for massive MIMO uniform linear array. The spatial correlation of the channel array determines the Laplacian matrix, and the channel compression matrix is obtained by Laplacian matrix eigenvalue decomposition. The simulation results show that the proposed LE algorithm can reduce the feedback overhead, and its bit error rate (BER) performance is better than that of the discrete cosine transform (DCT) sparse compression algorithm. In addition, the proposed LE algorithm computational complexity is higher than DCT, and lower than principal component analysis (PCA) and Karhunen-Loeve transform (KLT) algorithms, but the LE algorithm can achieve higher feedback accuracy when the feedback overhead is slightly lower than DCT.

Novel Metric Using Laplacian Eigenmaps to Evaluate Ischemic Stress on the Torso Surface.

The underlying pathophysiology of myocardial ischemia is incompletely understood, resulting in persistent difficulty of diagnosis. This limited understanding of underlying mechanisms encourages a data driven approach, which seeks to identify patterns in the ECG data that can be linked statistically to disease states. Laplacian Eigen-maps (LE) is a dimensionality reduction method popularized in machine learning that we have shown in large animal experiments to identify underlying ischemic stress both earlier in an ischemic episode, and more robustly, than typical clinical markers. We have now extended this approach to body surface potential mapping (BSPM) recordings acquired during acute, transient ischemia episodes from animal and human PTCA studies. Our previous studies, suggest that the LE approach is sensitive to the spatiotemporal electrocardiographic consequences of ischemia-induced stress within the heart and on the epicardial surface. In this study, we expand this technique to the body surface of animals and humans. Across 10 episodes of induced ischemia in animals and 200 human recordings during PTCA, the LE algorithm was able to detect ischemic events from BSPM as changes in the morphology of the resulting trajectories while maintaining the superior temporal performance the LE-metric has shown previously.

The correlation analysis of gear tooth broken-pitting compound fault and single fault based on Laplacian Eigenmaps

Gear break and pitting are two common faults in transmission system, when these two faults coexist and form a compound fault, the damage speed and frequency of gear transmission system will be greatly increased. Taking the gear fault-pitting compound fault as the object, the dynamic model of gear single fault and compound fault is established, and the vibration characteristics of gear single fault, pitting single fault and broken tooth-pitting compound fault signal are analyzed. The characteristic manifolds of the intrinsic dimension space in the case of gear single failure and compound fault are extracted by using the Laplacian Eigenmaps algorithm, the evolution trend of single fault and compound fault in the overlapping region of the feature space, the degree of correlation and the curvature of the fault circle core are analyzed and obtained. The study found that with the deepening of the fault severity, the overlapping area of fault circle between compound fault and single fault become smaller gradually, that is, the degree of correlation become weakened, tooth broken single fault and compound fault can be identified in mid-late stage of fault, while the pitting single fault and compound fault are in the late stage. The experimental results of gearbox compound fault correlation show that the conclusion of the simulation analysis is correct and effective, which provides a new idea for the diagnosis of mechanical complex faults.

Fast Spatial Spectral Schroedinger Eigenmaps algorithm for hyperspectral feature extraction

Based on the Laplacian Eigenmaps (LE) algorithm and a potential matrix, the Spatial Spectral Schroedinger Eigenmaps (SSSE) technique has proved a great yield during the hyperspectral dimensionality reduction process. Experimentally, SSSE is in deficiency of high computing time which may hinder its contribution in the remote sensing field. In this paper, a fast variant of the SSSE approach, called Fast SSSE, was proposed. The new suggested method substitutes the quadratic constraint employed during the optimization problem, by a linear constraint. This overhaul preserves the data properties in analogous way to the SSSE technique, but with a fast implementation. Two real hyperspectral data sets were adopted during the experimental process. Experiment analysis exhibited good classification accuracy with a reduced computational effort, compared with the original SSSE approach.

An improved Laplacian Eigenmaps method for machine nonlinear fault feature extraction

In the feature extraction of mechanical fault detection field, manifold learning is one of the effective nonlinear techniques. In this paper, aiming for the situations of noise sensitivity to manifold learning algorithms, an improved Laplacian Eigenmap (I-LapEig) algorithm is proposed and applied to the process of fault feature extraction. The new method takes advantage of local principal component analysis to eliminate the influence of noise points by reconstructing the neighborhood relation amongst the samples, and maintain the global intrinsic manifold structure, which enhances the performance of the feature extraction. To determine the parameters of I-LapEig algorithm, an adaptive neighborhood choose approach is presented. The K-nearest neighbor classifier is also adopted to implement feature classification and recognition. The experimental results on S-curve, rotor bed data, and compressor fault data show that the new method can effectively improve the performance of noise reduction in the feature extraction process when compared with the conventional local linear embedding and Laplacian Eigenmaps.

PolSAR image classification based on Laplacian Eigenmaps and superpixels

This paper proposes a method of polarimetric synthetic aperture radar (PolSAR) image classification using improved superpixel segmentation and manifold learning. Firstly, a 27-dimension polarimetric feature space is extracted by simple arithmetic operations of polarimetric parameters and polarimetric target decomposition. Secondly, Laplacian Eigenmap (LE) algorithm is used to reduce the dimension of the 27-dimension polarimetric features. This algorithm can reduce redundant information in feature space and extract the main information. Then, the paper uses SVM which has the best classification performance to classify the low-dimension PolSAR data for the first time. And then, the superpixel segmentation is obtained by improving SLIC algorithm. At last, the majority voting principle is used to classify the superpixel blocks, which is the second classification and final classification of PolSAR data.

Nonlinear machine fault detection by semi-supervised Laplacian Eigenmaps

A semi-supervised Laplacian Eigenmaps algorithm for machine fault detection is proposed. The purpose of the algorithm is to efficiently extract the manifold geometric characteristics of nonlinear vibration signal samples, and to determine fault classification of operating equipment so that the accuracy of fault detection can be improved. The data acquisition and pre-processing of the vibration signal is firstly implemented from monitoring equipment, then hybrid domain feature is obtained, and the initial sample set can be built. This is followed by implementing the semi-supervised Laplacian Eigenmaps algorithm so that the sensitive nature characteristics of manifold can be obtained from the device initial sample set. In order to establish the intelligent diagnostic model, the Least square Support vector machine (LS-SVM) is then adopted, which fault diagnosis and decisions can be achieved in the feature space of the low-dimensional manifold. The experiment results of using the IRIS data, gearbox and compressor fault data show the proposed method has more advantage when compared with the PCA and Laplacian Eigenmaps on improving the accuracy of fault detection.

Fault identification method for planetary gear based on DT-CWT threshold denoising and LE

Planetary gear is widely used in large-scale complex mechanical systems. However, because of the particularity of planetary gear transmission, serious wear and fatigue crack failures often occur in the sun gear, planet gears, and inner gear ring. In addition, every type of fault will experience different degradation processes. Improving the operation reliability of mechanical equipment through fault diagnosis of planetary gears and monitoring their degradation process is beneficial. This paper proposes a planetary gear fault identification method based on Dual-tree Complex wavelet transform (DT-CWT) threshold denoising and Laplacian eigenmaps (LE). The noise reduction processing of the original signal is achieved by the DT-CWT threshold denoising method, which takes full advantage of DT-CWT and is combined with the wavelet threshold of rigrsure principle. The original high-dimensional feature set, including the time domain features, frequency domain features, permutation entropy, and fractal box dimension of the denoised signal, is constructed from multi-angles. To solve the problems of excessively large feature dimension and the existence of redundant information, the LE algorithm is used to reduce the dimension of the original high-dimensional feature set, and the low-dimensional sensitive features are obtained. Through the above method, the effective identification of different fault states and different degradation states of the planetary gear are achieved.

Automatic defect detection based on segmentation of pulsed thermographic images

Pulsed thermography, widely used as a nondestructive testing method, offers many advantages for material defect detection. However, most existing methods for pulsed thermographic data processing aim to enhance the defect signals in each single thermal image, whereas automatic defect detection is not achieved. Instead, laborious and time-consuming visual inspection of the processed images is required to draw final conclusions. It is usually impossible to visually inspect all images. Therefore, manual selection of a few informative images is often a required step before thermal image processing, probably resulting in the oversight of necessary defect information. To overcome the drawbacks of the existing methods, a hyper-image segmentation method is proposed in this study, which analyzes all thermographic data simultaneously to achieve automatic defect detection and avoid the risk of losing information. Specifically, an iterative defect detection procedure is designed on the basis of the Laplacian eigenmap algorithm. The results of a case study on the carbon-fiber-reinforced plastic (CFRP) materials show the feasibility of the proposed method.

Adaptive Laplacian Eigenmap-Based Dimension Reduction for Ocean Target Discrimination

It is well known that polarimetric synthetic aperture radar (PolSAR) backscattering features are highly influenced by the variation of incidence angle (VIA), which usually hampers the classification of most grazing-angle-sensitive targets, such as land and ocean targets. To relieve this issue, various feature extraction approaches have been suggested to enhance the class discriminability while reducing the observed feature dimensionality. The Laplacian eigenmap-based dimension reduction (DR) has been proven to be an effective way to deal with VIA problems, provided that the manifold parameters [e.g., the heat kernel (HK)] have been optimally sought, which is often difficult in practice. In this letter, an adaptive Laplacian eigenmap-based DR method is presented to find a learned subspace where the local geometry with discriminative prior knowledge is preserved as much as possible while near optimal HK and scale factor parameters are automatically identified. The learned feature representation is then employed for the subsequent classification. The improved Laplacian eigenmap algorithm was validated by three uninhabited-aerial-vehicle-synthetic-aperture-radar L-band PolSAR images from the Gulf Deepwater Horizon oil spill, which were clearly impacted by the VIA phenomenon. The experimental results showed that the proposed algorithm works well in ocean target discrimination compared with the current common methods.

A general framework for evolutionary multiobjective optimization via manifold learning

Under certain mild condition, the Pareto-optimal set (PS) of a continuous multiobjective optimization problem, with m objectives, is a piece-wise continuous (m−1)-dimensional manifold. This regularity property is important, yet has been unfortunately ignored in many evolutionary multiobjective optimization (EMO) studies. The first work that explicitly takes advantages of this regularity property in EMO is the regularity model-based multiobjective estimation of distribution algorithm (RM-MEDA). However, its performance largely depends on its model parameter, which is problem dependent. Manifold learning, also known as nonlinear dimensionality reduction, is able to discover the geometric property of a low-dimensional manifold embedded in the high-dimensional ambient space. This paper presents a general framework that applies advanced manifold learning techniques in EMO. At each generation, we first use a principal curve algorithm to obtain an approximation of the PS manifold. Then, the Laplacian eigenmaps algorithm is employed to find the low-dimensional representation of this PS approximation. Afterwards, we identify the neighborhood relationship in this low-dimensional representation, which is also applicable for the original high-dimensional data. Based on the neighborhood relationship, we can interpolate new candidate solutions that obey the geometric property of the PS manifold. Empirical results validate the effectiveness of our proposal.

A Wide-Area Measurement Systems-Based Adaptive Strategy for Controlled Islanding in Bulk Power Systems

Controlled islanding is the last countermeasure for a bulk power system when it suffers from severe cascading contingencies. The objective of controlled islanding is to maintain the stability of each island and to keep the total loss of loads of the whole system to a minimum. This paper presents a novel integrated wide-area measurement systems (WAMS)-based adaptive controlled islanding strategy, which depends on the dynamic post-fault trajectories under different failure modes. We first utilize an improved Laplacian eigenmap algorithm (ILEA) to identify the coherent generators and use the slow coherency grouping algorithm to guarantee coherent stability within an island. Using the identification result, we then define the minimum coherent generator virtual nodes to reduce the searching space in a graph and utilize the k-way partitioning (KWP) algorithm to obtain a preliminary partition of the simplified graph. Based on the preliminary partition, we consider the direction of power flow and propose a variable neighborhood heuristic searching algorithm to search the optimal separation surfaces so that the net imbalanced power of islands is minimized. Finally, the bidirectional power flow tracing algorithm and PQ decomposition power flow analysis are utilized to determine the corrective controls within each island. The test results with the New England 39-bus system and the IEEE 118-bus system show that the proposed integrated controlled islanding strategy can automatically adapt to different fault modes through generator coherency identification and effectively group the different coherent generators into different islands.

Laplacian Eigenmaps-Based Polarimetric Dimensionality Reduction for SAR Image Classification

In this paper, we propose a novel scheme of polarimetric synthetic aperture radar (PolSAR) image classification. We apply Laplacian eigenmaps (LE), a nonlinear dimensionality reduction (NDR) technique, to a high-dimensional polarimetric feature representation for PolSAR land-cover classification. A wide variety of polarimetric signatures are chosen to construct a high-dimensional polarimetric manifold which can be mapped into the most compact low-dimensional structure by manifold-based dimensionality reduction techniques. This NDR technique is employed to obtain a low-dimensional intrinsic feature vector by the LE algorithm, which is beneficial to PolSAR land-cover classification owing to its local preserving property. The effectiveness of our PolSAR land-cover classification scheme with LE intrinsic feature vector is demonstrated with the RadarSat-2 C-band PolSAR data set and the 38th Research Institute of China Electronics Technology Group Corporation X-band PolInSAR data set. The performance of our method is measured by the separability in the feature space and the accuracy of classification. Comparisons on the feature space show that the LE intrinsic feature vector is more separable than different original feature vectors. Our LE intrinsic feature vector also improves the classification accuracy.

Color image segmentation using Laplacian eigenmaps

The novel technique of Laplacian eigenmaps (LE) is studied as a means of improving the clustering-based segmentation of color images. Taking advantage of the ability of the LE algorithm to learn the actual manifold of the multivariate data, a computationally efficient scheme is introduced. After embedding the local image characteristics, extracted from overlapping regions, in a high-dimensional feature space, the skeleton of the intrinsically low-dimensional manifold is constructed using spectral graph theory. Using the LE-based dimensionality reduction technique, a low-dimensional map is computed in which the variations of the local image characteristics are presented in the context of global image variation. The nonlinear projections on this map serve as inputs to the Fuzzy C-Means (FCM) algorithm, boosting its clustering performance significantly. The final segmentation is produced by a simple labeling scheme. The application of the presented approach to color images is very encouraging and illustrates the effectiveness of the performance over alternative methods.

MODIFIED LAPLACIAN EIGENMAP METHOD FOR FAULT DIAGNOSIS

A novel method based on the improved Laplacian eigenmap algorithm for fault pattern classification is proposed.Via modifying the Laplacian eigenmap algorithm to replace Euclidean distance with kernel-based geometric distance in the neighbor graph construction,the method can preserve the consistency of local neighbor information and effectively extract the low-dimensional manifold features embedded in the high-dimensional nonlinear data sets.A nonlinear dimensionality reduction algorithm based on the improved Laplacian eigenmap is to directly learn high-dimensional fault signals and extract the intrinsic manifold features from them.The method greatly preserves the global geometry structure information embedded in the signals,and obviously improves the classification performance of fault pattern recognition.The experimental results on both simulation and engineering indicate the feasibility and effectiveness of the new method.