non-Euclidean Space Research Articles

Multireceptive Field Graph Convolutional Networks for Machine Fault Diagnosis

Deep learning (DL) based methods have swept the field of mechanical fault diagnosis, because of the powerful ability of feature representation. However, many of existing DL methods fail in relationship mining between signals explicitly. Unlike those deep neural networks, graph convolutional networks (GCNs) taking graph data with topological structure as input is more efficient for data relationship mining, making GCN to be powerful for feature representation from graph data in non-Euclidean space. Nevertheless, existing GCNs have two limitations. First, most GCNs are constructed on unweighted graphs, considering importance of neighbors as the same, which is not in line with reality. Second, the receptive field of GCNs is fixed, which limits the effectiveness of GCNs for feature representation. To address these issues, a multireceptive field graph convolutional network (MRF-GCN) is proposed for effective intelligent fault diagnosis. In MRF-GCN, data samples are converted into weighted graphs to indicate differences in relationship of data samples. Moreover, MRF-GCN learns not only features from different receptive field, but also fuses learned features as an enhanced feature representation. To verify the efficacy of MRF-GCN for machine fault diagnosis, case studies are implemented, and the results show that MRF-GCN can achieve superior performance even under imbalanced dataset.

Dirac fermions in graphene using the position-dependent translation operator formalism

Within the position-dependent translation operator formalism for quantum systems, we obtain analytical expressions for the eigenstates and the Landau level spectrum of Dirac fermions in graphene in the presence of a perpendicularly applied magnetic field and, as a consequence of the formalism, with a generalized form of the momentum operator. Moreover, we explore the behavior of wave packet dynamics in such a system by considering different initial pseudospin polarizations and metric parameters. Our findings show that the Landau levels, the wave packet trajectories, and velocities are significantly affected by the choice of the metric in the non-Euclidean space of the deformed momentum operator, exhibiting a tunable energy level spacing. In the dynamics analysis, one observes an enhancement of the oscillation amplitude of the average positions for all investigated pseudospin polarizations due to the nonsymmetric evolution of the wave packet induced by the different metrics in the system. The present formalism is shown to be a theoretical platform to describe the effects of two scenarios due to (i) a lattice deformation in graphene, giving rise to a natural Fermi velocity renormalization, or even (ii) a nonuniform mass term, induced by a specific substrate, that varies on a length scale much greater than the magnetic field length.

Pauli matrices immersion

Spin polarization, a major consideration in materials science, has a three-dimensional Euclidean background, but quantum spin has a non-Euclidean ambient space of the Dirac spinor. This paper bridges the gap by the linear algebra of complexification to cast imaginary numbers in a two-dimensional real space. As real and imaginary eigenvalues together describe motions in the physical space, we immerse Pauli matrices in a three-dimensional ambient space. Thereupon we present an alternative explanation of the anti-matter asymmetry, with implications on spin polarization in electronic engineering and materials science.

GNEA: A Graph Neural Network with ELM Aggregator for Brain Network Classification

Brain networks provide essential insights into the diagnosis of functional brain disorders, such as Alzheimer’s disease (AD). Many machine learning methods have been applied to learn from brain images or networks in Euclidean space. However, it is still challenging to learn complex network structures and the connectivity of brain regions in non-Euclidean space. To address this problem, in this paper, we exploit the study of brain network classification from the perspective of graph learning. We propose an aggregator based on extreme learning machine (ELM) that boosts the aggregation ability and efficiency of graph convolution without iterative tuning. Then, we design a graph neural network named GNEA (Graph Neural Network with ELM Aggregator) for the graph classification task. Extensive experiments are conducted using a real-world AD detection dataset to evaluate and compare the graph learning performances of GNEA and state-of-the-art graph learning methods. The results indicate that GNEA achieves excellent learning performance with the best graph representation ability in brain network classification applications.

Co-Embedding of Nodes and Edges With Graph Neural Networks.

Graph, as an important data representation, is ubiquitous in many real world applications ranging from social network analysis to biology. How to correctly and effectively learn and extract information from graph is essential for a large number of machine learning tasks. Graph embedding is a way to transform and encode the data structure in high dimensional and non-euclidean feature space to a low dimensional and structural space, which is easily exploited by other machine learning algorithms. We have witnessed a huge surge of such embedding methods, from statistical approaches to recent deep learning methods such as the graph convolutional networks (GCN). Deep learning approaches usually outperform the traditional methods in most graph learning benchmarks by building an end-to-end learning framework to optimize the loss function directly. However, most of the existing GCN methods can only perform convolution operations with node features, while ignoring the handy information in edge features, such as relations in knowledge graphs. To address this problem, we present CensNet, Convolution with Edge-Node Switching graph neural network, for learning tasks in graph-structured data with both node and edge features. CensNet is a general graph embedding framework, which embeds both nodes and edges to a latent feature space. By using line graph of the original undirected graph, the role of nodes and edges are switched, and two novel graph convolution operations are proposed for feature propagation. Experimental results on real-world academic citation networks and quantum chemistry graphs show that our approach achieves or matches the state-of-the-art performance in four graph learning tasks, including semi-supervised node classification, multi-task graph classification, graph regression, and link prediction.

Resilient Range-Based d-Dimensional Localization for Mobile Sensor Networks

Knowledge of node locations is essential to Wireless Sensor Networks (WSNs) in a wide range of potential applications and their function-dependent network protocols. A number of localization approaches have already been proposed to fulfill this requirement, but few of them can be applicable to mobile sensor networks, due to their low-dimensional embeddings, Euclidean distance representation limitations, frequent node mobility and additional measurement overhead in the network. In this paper, a resilient range-based $d$ -dimensional localization (RRDL) approach is proposed for mobile WSNs to resolve the issues. RRDL distinguishes itself from previous work with three remarkable characteristics: (1) it works for mobile networks embedded in $d$ -dimensional Non-Euclidean space; (2) it allows static ordinary nodes with pre-known locations to act as the alternative anchor nodes, thus tolerating the motion of the original anchor nodes to ensure that other ordinary nodes can obtain their locations in an efficient manner; and (3) it introduces an efficient path-learning approach, with the knowledge of the existing paths, to represent the real network distances as far as possible, thereby eliminating additional measurement overhead and tolerating node mobility in localization. With these characteristics, RRDL exploits the iterative factorization of the random distance matrix, formed by the distances to and from a set of $k$ -hop static neighbors, to assign each current node $d$ -dimensional Non-Euclidean coordinate in a distributed manner. Simulation results demonstrate that RRDL achieves higher localization accuracy with a moderate communication cost in mobile sensor networks.

SCENE CLASSIFICATION BASED ON THE INTRINSIC MEAN OF LIE GROUP

Abstract. Remote Sensing scene classification aims to identify semantic objects with similar characteristics from high resolution images. Even though existing methods have achieved satisfactory performance, the features used for classification modeling are still limited to some kinds of vector representation within a Euclidean space. As a result, their models are not robust to reflect the essential scene characteristics, hardly to promote classification accuracy higher. In this study, we propose a novel scene classification method based on the intrinsic mean on a Lie Group manifold. By introducing Lie Group machine learning into scene classification, the new method uses the geodesic distance on the Lie Group manifold, instead of Euclidean distance, solving the problem that non-euclidean space samples could not be calculated by Euclidean distance directly. The experiments show that our method produces satisfactory performance on two public and challenging remote sensing scene datasets, UC Merced and SIRI-WHU, respectively.

Using Graph Convolutional Networks Skeleton-Based Pedestrian Intention Estimation Models for Trajectory Prediction

In autonomous driving scenarios, pedestrian trajectory prediction is an important research direction. Based on the spatio-temporal graph convolutional neural network, we propose a new pedestrian trajectory prediction algorithm. The new algorithm constructs a series of new models around pedestrian intention estimation. The construction of the estimation algorithm considers the following aspects: the contextual information of pedestrians and the surrounding environment, the “pedestrian ego-vehicle” interaction combined with the vehicle speed estimation, the pedestrian’s own skeletal structure information and body language estimation, which includes head joints and the relative structural relationship of the torso joints, including whether it is out of the same plane, is rotated, and so on. Skeleton information feature extraction and construction adopts the method of graph convolutional neural network to structure pedestrians into joints in the form of graphs in non-Euclidean space, and further adopts spatial temporal graph convolutional network for feature extraction and learning. The new method is named a “head-torso”-based spatial temporal graph convolutional network (HT-STGCN). On the dataset PID, the novel method achieves substantial improvements over mainstream methods. Experimental results show that combining HT-STGCN with observed action can improve trajectory prediction.

Representation and reconstruction of covariance operators in linear inverse problems

We introduce a framework for the reconstruction and representation of functions in a setting where these objects cannot be directly observed, but only indirect and noisy measurements are available, namely an inverse problem setting. The proposed methodology can be applied either to the analysis of indirectly observed functional images or to the associated covariance operators, representing second-order information, and thus lying on a non-Euclidean space. To deal with the ill-posedness of the inverse problem, we exploit the spatial structure of the sample data by introducing a flexible regularizing term embedded in the model. Thanks to its efficiency, the proposed model is applied to MEG data, leading to a novel approach to the investigation of functional connectivity.

A New Single-Image Super-Resolution Using Efficient Feature Fusion and Patch Similarity in Non-Euclidean Space

Efficient trade-off between the reconstruction qualities and the processing time of any single-image super-resolution reconstruction (SISRR) approach is critically influenced by two major aspects. These aspects are (i) appropriate representation of image patch in feature space and (ii) effective searching of candidate patches from the pool of training patches or learned dictionary. This paper proposes a neighbor embedding-based SISRR method. Novelties of our work include integration of (i) efficient feature mapping scheme which fuses multiple correlated features naturally, (ii) faster searching of candidate patches by measuring the patch correlation in non-Euclidean space and (iii) adaptive selection of neighborhood size using patch characteristic. Correlation among features is modeled via global covariance matrix, and the fusion process enables to preserve sufficient structural, spatial correlation among patches. Distance functions based on notion of generalized eigenvalue are used for measuring patch similarity which support faster searching of candidate patches. Performance analysis of the suggested method is compared with some of the competent state-of-the-art methodologies. From the simulated result analysis, proposed work is found to be outperforming in terms of sharpened image details with diminished effect of artifacts at a reasonable computational burden.

Robust Joint Representation of Intrinsic Mean and Kernel Function of Lie Group for Remote Sensing Scene Classification

Remote sensing scene classification is used to label specific semantic categories for images. The current methods have achieved competitive performances, but they are only for Euclidean space samples. As a result, their representations are not robust for non-Euclidean space samples, which affects the classification accuracy. In this letter, we introduce the Lie group manifold into the traditional feature representation method and propose a novel intrinsic mean representation method within the Lie group. At the same time, the kernel function based on the sample of the Lie group is designed to further improve the robustness and accuracy of classification. In addition, our method achieves satisfactory performance on two public and challenging remote sensing data sets of UC Merced and NWPU-RESISC45.

Intrinsic Wavelet Regression for Curves of Hermitian Positive Definite Matrices

Intrinsic wavelet transforms and wavelet estimation methods are introduced for curves in the non-Euclidean space of Hermitian positive definite matrices, with in mind the application to Fourier spectral estimation of multivariate stationary time series. The main focus is on intrinsic average-interpolation wavelet transforms in the space of positive definite matrices equipped with an affine-invariant Riemannian metric, and convergence rates of linear wavelet thresholding are derived for intrinsically smooth curves of Hermitian positive definite matrices. In the context of multivariate Fourier spectral estimation, intrinsic wavelet thresholding is equivariant under a change of basis of the time series, and nonlinear wavelet thresholding is able to capture localized features in the spectral density matrix across frequency, always guaranteeing positive definite estimates. The finite-sample performance of intrinsic wavelet thresholding is assessed by means of simulated data and compared to several benchmark estimators in the Riemannian manifold. Further illustrations are provided by examining the multivariate spectra of trial-replicated brain signal time series recorded during a learning experiment. Supplementary materials for this article are available online.

A Multi-Task Representation Learning Architecture for Enhanced Graph Classification

Composed of nodes and edges, graph structured data are organized in the non-Euclidean geometric space and ubiquitous especially in chemical compounds, proteins, etc. They usually contain rich structure information, and how to effectively extract inherent features of them is of great significance on the determination of function or traits in medicine and biology. Recently, there is a growing interest in learning graph-level representations for graph classification. Existing graph classification strategies based on graph neural networks broadly follow a single-task learning framework and manage to learn graph-level representations through aggregating node-level representations. However, they lack the efficient utilization of labels of nodes in a graph. In this paper, we propose a novel multi-task representation learning architecture coupled with the task of supervised node classification for enhanced graph classification. Specifically, the node classification task enforces node-level representations to take full advantage of node labels available in the graph and the graph classification task allows for learning graph-level representations in an end-to-end manner. Experimental results on multiple benchmark datasets demonstrate that the proposed architecture performs significantly better than various single-task graph neural network methods for graph classification.

Transformation of Non-Euclidean Space to Euclidean Space for Efficient Learning of Singular Vectors

Singular value decomposition (SVD) is a popular technique to extract essential information by reducing the dimension of a feature set. SVD is able to analyze a vast matrix in spite of a relatively low computational cost. However, singular vectors produced by SVD have been seldom used in convolutional neural networks (CNNs). This is because the inherent properties of singular vectors such as sign ambiguity and manifold features make CNNs difficult to learn singular vectors. In order to overcome the limitations, this paper analyzes the undesirable properties of singular vectors and presents the transformation of singular vectors into Euclidean space as a smart solution. If the singular vectors are transformed to follow Euclidean geometry, SVD can be used for pooling to maintain the feature information well, which is called singular vector pooling (SVP). Since SVP can extract essential information from a feature map, it is robust against adversarial attacks in comparison to global average pooling. Thus, SVP shows a quantitative performance improvement of about 36% for the CIFAR10 dataset. In addition, we applied SVP to a knowledge distillation scheme that uses singular vectors in a restricted manner. As a result, SVP improved the performance by up to 1.7% for the CIFAR100 dataset.

Czy Bruno Latour testuje Rema Koolhaasa i co z tego dla nowej humanistyki wynika?

Is Bruno Latour testing Rem Koolhaas and what are the results for the new humanities?One of today’s most famous architects, Rem Koolhaas, tackles the problem of architectural objects as networked and transformable processes — not only structural, but also social and politi­cal ones. Bruno Latour, in turn, who is a sociologist, is constructing this time a theory — a weapon/ a new tool which will help architects move objects which so far have been static “Give me a gun and I will make all the buildings move” and design them in a non-Euclidean space. Theoretically, Latour — by “tapping lightly on Koolhaas’s architecture with a blind person’s cane” — proves once again that we have never been modern and, on the other hand, proposes to look at the design process in such a manner as to see the public character of things, discover the agency of non-human actors.What is the result of Latour’s collaboration with Koolhaas? Will the considerations of a prac­titioner and a theoretician, an architect and a sociologist prove to be an important contribution to the revision of cultural expressions? In my article, in the spirit of critical reflection on the myth of a lonely genius, I shall follow the demands of treating design as a collective process, networking the work of helpers and collaborators, data collectors and seekers of contexts. Are they really imple­mented in architectural practice? Is it really possible to talk about experimental methods of produc­tion and presentation of knowledge in this case?

Light path averages in spacetimes with nonvanishing average spatial curvature

Effects of inhomogeneities on observations have been vastly studied using both perturbative methods, N-body simulations and Swiss cheese solutions to the Einstein equations. In nearly all cases, such studied setups assume vanishing spatial background curvature. While a spatially flat Friedmann-Lemaitre-Robertson-Walker model is in accordance with observations, a non-vanishing curvature is not ruled out. It is therefore important to note that, as has been pointed out in the literature, 1 dimensional averages might not converge to volume averages in non-Euclidean space. If this is indeed the case, it will affect the interpretation of observations in spacetimes with non-vanishing average spatial curvature. This possibility is therefore studied here by computing the integrated expansion rate and shear, the accumulated density contrast, and fluctuations in the redshift-distance relation in Swiss cheese models with different background curvatures. It is found that differences in mean and dispersion of these quantities in the different models are small and naturally attributable to differences in background expansion rate and density contrasts. Thus, the study does not yield an indication that the relationship between 1 dimensional spatial averages and volume averages depends significantly on background curvature.

Variant Grassmann Manifolds

In classification tasks, classifiers trained with finite examples might generalize poorly to new data with unknown variance. For this issue, data augmentation is a successful solution where numerous artificial examples are added to training sets. In this article, we focus on the data augmentation for improving the accuracy of action recognition, where action videos are modeled by linear dynamical systems and approximately represented as linear subspaces. These subspace representations lie in a non-Euclidean space, named Grassmann manifold, containing points as orthonormal matrixes. It is our concern that poor generalization may result from the variance of manifolds when data come from different sources or classes. Thus, we introduce infinitely many variant Grassmann manifolds (VGM) subject to a known distribution, then represent each action video as different Grassmann points leading to augmented representations. Furthermore, a prior based on the stability of subspace bases is introduced, so the manifold distribution can be adaptively determined, balancing discrimination and representation. Experimental results of multi-class and multi-source classification show that VGM softmax classifiers achieve lower test error rates compared to methods with a single manifold.

Convergences in cognitive science, social network analysis, pattern recognition and machine intelligence as dynamic processes in non-Euclidean space

Students of human cognitive and cultural processes, social networks, pattern recognition and machine intelligence often find that the coordinate systems resulting from commonly used measurement and analysis tools yield non-Euclidean configurations. Typically, researchers consider this unfortunate, and seek methods to return the spaces to Euclidean configurations. This article details all the known methods of such transformations, but presents evidence from multiple fields of inquiry that shows the non-Euclidean nature of the space is meaningful, and that all transformations to Euclidean form produce serious distortions to measured values. The article further presents methods for describing processes in the non-Euclidean spaces along with empirical examples of such uses.

Development and Evaluation of Geostatistical Methods for Non-Euclidean-Based Spatial Covariance Matrices.

Customary and routine practice of geostatistical modeling assumes that inter-point distances are a Euclidean metric (i.e., as the crow flies) when characterizing spatial variation. There are many real-world settings, however, in which the use of a non-Euclidean distance is more appropriate, for example in complex bodies of water. However, if such a distance is used with current semivariogram functions, the resulting spatial covariance matrices are no longer guaranteed to be positive-definite. Previous attempts to address this issue for geostatistical prediction (i.e., kriging) models transform the non-Euclidean space into a Euclidean metric, such as through multi-dimensional scaling (MDS). However, these attempts estimate spatial covariances only after distances are scaled. An alternative method is proposed to re-estimate a spatial covariance structure originally based on a non-Euclidean distance metric to ensure validity. This method is compared to the standard use of Euclidean distance, as well as a previously utilized MDS method. All methods are evaluated using cross-validation assessments on both simulated and real-world experiments. Results show a high level of bias in prediction variance for the previously developed MDS method that has not been highlighted previously. Conversely, the proposed method offers a preferred tradeoff between prediction accuracy and prediction variance and at times outperforms the existing methods for both sets of metrics. Overall results indicate that this proposed method can provide improved geostatistical predictions while ensuring valid results when the use of non-Euclidean distances is warranted.

Optimal Energy Trajectory Generation for a Quadrotor UAV Using Geometrically Exact Computations on SE(3)

A quadrotor UAV consumes considerable proportions of energy in its rotors resulting in reduced flight endurance. This letter discusses an energy optimal trajectory generation scheme for the quadrotor UAVs using geometrically exact computation techniques. The dynamics of the UAV is represented using a non-Euclidean space known as special Euclidean space, SE(3) which employs rotation matrix and thus overcomes the limitations introduced by other attitude representations. The goal is to maneuver the quadrotor from its initial states to the desired states utilizing minimum energy. By adopting the lie group variational integrator technique, possible inaccuracies that can be caused by the commonly used numerical integration techniques are eliminated. A numerical backstepping approach is used to estimate the initial values of co-states to meet the desired terminal conditions.