non-Euclidean Metrics Research Articles

Correction: Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics

Correction: Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics

Application of non-Euclidean metric in the electric power industry for reduction of measurement uncertainty

The paper proposes the use of the non-Euclidean metric to reduce the uncertainty that occurs when measuring voltage for the tasks of ongoing continuous control of electric power consumption in large, branched high-voltage electric networks. The problem is that for continuous control of electric power consumption, it is necessary to install the active and reactive power measuring equipment in each node of the electric network (at each substation) and to ensure the transmission of measurement information to dispatching control centers. For countries with large electric networks, long distances between electric grid nodes and dispatch control centers, this requires huge capital costs. Therefore, it is advisable to place equipment for measuring electric power and voltage only in individual nodes of the electrical network, and then calculate the parameters of the remaining nodes based on Kirchhoff's laws. But at the same time, there is a significant measurement uncertainty, because the complex value of the voltage is usually not measured, only the modulus of the voltage values is used for the calculation. The use of non-Euclidean metrics provides the reduction of the input data uncertainty, which are necessary to control the consumption of electric power in each node of the electric network.

ДОСЛІДЖЕННЯ ЕФЕКТИВНОСТІ ВИКОРИСТАННЯ PNN ДЛЯ РОЗВ’ЯЗАННЯ ЗАДАЧ МЕДИЧНОЇ ДІАГНОСТИКИ В УМОВАХ АНАЛІЗУ МАЛИХ ДАНИХ ВИСОКОЇ РОЗМІРНОСТІ

Parkinson's disease is one of the illnesses that cause certain difficulties at the stage of diagnosis. Recently, there has been a tendency to increase the popularity of the use of artificial intelligence methods as an auxiliary diagnostic tool. A probabilistic neural network (PNN) has become widely used for solving problems in the field of medicine. Despite the rather high efficiency of its use for certain tasks, some aspects of its functioning remain insufficiently researched in practice. Existing scientific works do not pay due attention to the issue of using the optimal distance as a measure of similarity between objects. Calculating the distance between the current data vector and each reference sample vector is the first step in implementing a PNN. The classification accuracy of a neural network of this type depends on its efficiency. In this work, the effectiveness of using different distances in the algorithm of different implementations of a probabilistic neural network for detecting Parkinson's disease based on biomedical voice indicators in the case of small high-dimensional data was investigated. Experimental modeling of three different variants of PNN implementation was carried out using the following distances: Chebyshov, Manhattan, Minkowski, cosine, and Canberra. The results of the study showed different values of the F1-measure when applying different distances. It was found that the use of the Euclidean metric in the structure of a probabilistic neural network is not always the best option. In particular, the application of non-Euclidean metrics provided a significant increase in accuracy for the analyzed dataset. This indicates the need for correctly selecting this parameter of the probabilistic neural network to obtain the highest accuracy when solving medical diagnosis problems.

Modeling Spatial Anisotropic Relationships Using Gradient-Based Geographically Weighted Regression

Distance and direction play crucial roles in modeling the spatial nonstationarity relationship. Because Euclidean distance ignores the effect of direction, several modified geographically weighted regression (GWR) attempts have been made to model anisotropic relationships using various non-Euclidean distance metrics. These methods, however, adopt uniform parameters to define the non-Euclidean metrics over the whole study area, neglecting the varying numerical features existing in different regions. As a result, they fail to accurately depict spatial anisotropic relationships between variables. To address this issue, we propose a novel method called gradient-based geographically weighted regression (GGWR) that integrates the gradient of spatial relationships into GWR. Additionally, we introduce an l 0-norm regularization technique to achieve the parameter estimation of GGWR. Both simulated and actual data sets were used to validate the proposed method, and the experimental results demonstrate that the gradient field of the spatial relationship obtained by GGWR can effectively characterize the direction and intensity of variable relationships at various locations. Moreover, GGWR outperforms other models, including GWR, directional geographically weighted regression, and Minkowski distance-based geographically weighted regression, in terms of fitting accuracy, coefficient estimation accuracy, and interpretation of coefficient symbols. These findings indicate that the GGWR can be a valuable tool for modeling spatial anisotropic relationships by leveraging the spatial relationship gradient field.

A clustering method based on multi-positive–negative granularity and attenuation-diffusion pattern

As an important part of machine learning, clustering methods have been continuously paid attention to. Current clustering methods divide data objects usually based on Euclidean metric, which is a basic and effective metric method. However, with the high dimensionality of data and the diversification of data representation, the complexity of the spatial structure of real-world data continues to rise. Classical clustering methods face many challenges such as insufficient clustering effectiveness, the sensitivity of clustering method parameters, and lack of stability of clustering results. Aiming at the above problems, this paper designs a non-Euclidean metric and constructs a multi-granularity staged clustering method based on the metric. First of all, this paper uses the sequential relationship of each feature of the data to construct a similarity measure between objects from the perspective of positive and negative granularity to improve the clustering algorithm’s understanding of complex spatial structure data. Secondly, this paper designs the attenuation-diffusion pattern divides and conquers according to the distribution characteristics of data objects in different patterns, and uses the heuristic idea to effectively cluster the data in stages from local to global. Again, based on the above, this paper proposes a clustering method based on multi-positive-negative granularity and attenuation-diffusion pattern, which can effectively deal with the challenges brought by complex spatial structure data to clustering methods. Finally, the effectiveness and robustness of the proposed method and advanced clustering methods are compared and analyzed on UCI real data sets. Experimental results show that the method proposed in this paper has obvious advantages in clustering results on complex spatial structure data. In addition, in the two directions of non-Euclidean metrics and multi-granularity clustering, the method proposed in this paper provides a new perspective for effectively dealing with the design of clustering methods on complex spatial structure data.

Reproducible clustering with non-Euclidean distances: a simulation and case study

AbstractCertain categorical sequence clustering applications require path connectivity, such as the clustering of DNA, click-paths through web-user sessions, or paths of care clustering with sequences of patient medical billing codes. $$K$$ K -means and $$k$$ k -medoids clustering with non-Euclidean distance metrics such as the Jaccard or edit distances maintains such path connectivity. Although $$k$$ k -means and $$k$$ k -medoids clustering with the Jaccard and edit distances have enjoyed success in these domains, the limits of accurate cluster recovery in these conditions have not yet been defined. As a first step in approaching this goal, we performed a simulated study using $$k$$ k -means and $$k$$ k -medoids clustering with non-Euclidean distances and show the performance deteriorates at a certain level of noise and when the number of clusters increases. However, we identify initialization strategies that improve upon cluster recovery in the presence of noise. We employ the use of the Tibshirani and Guenther (J Comput Graph Stat 14(3):511–528, 2005) Prediction Strength method, which creates a hypothesis testing scenario that determines if there is clustering structure to the data (if the clusters are reproducible), with the null hypothesis being there is none. We then applied the framework to perinatal episodes of care and the clusters reproducibly and organically split between Cesarean and vaginal deliveries, which itself is not a clinical finding but sensibly validates the approach. Further visualizations of the clusters did bring insights into subclusters that split along groups of physicians, cost and risk scores, warranting the outlined future work into ways of improving this framework for better resolution.

Probabilistic motion planning for non-Euclidean and multi-vehicle problems

Trajectory planning tasks for non-holonomic or collaborative systems are naturally modeled by state spaces with non-Euclidean metrics. However, existing proofs of convergence for sample-based motion planners only consider the setting of Euclidean state spaces. We resolve this issue by formulating a flexible framework and set of assumptions for which the widely-used PRM*, RRT, and RRT* algorithms remain asymptotically optimal in the non-Euclidean setting. The framework is compatible with collaborative trajectory planning: given a fleet of robotic systems that individually satisfy our assumptions, we show that the corresponding collaborative system again satisfies the assumptions and therefore has guaranteed convergence for the trajectory-finding methods. Our joint state space construction builds in a coupling parameter 1≤p≤∞, which interpolates between a preference for minimizing total energy at one extreme and a preference for minimizing the travel time at the opposite extreme. We illustrate our theory with trajectory planning for simple coupled systems, fleets of Reeds–Shepp vehicles, and a highly non-Euclidean fractal space.

Study on CO2 transport in fractal porous media for a Hausdorff fractal derivative advection-dispersion model

PurposeThe classical advection-dispersion equation (ADE) model cannot accurately depict the gas transport process in natural geological formations. This paper aims to study the behavior of CO2 transport in fractal porous media by using an effective Hausdorff fractal derivative advection-dispersion equation (HFDADE) model.Design/methodology/approachAnomalous dispersion behaviors of CO2 transport are effectively characterized by the investigation of time and space Hausdorff derivatives on non-Euclidean fractal metrics. The numerical simulation has been performed with different Hausdorff fractal dimensions to reveal characteristics of the developed fractal ADE in fractal porous media. Numerical experiments focus on the influence of the time and space fractal dimensions on flow velocity and dispersion coefficient.FindingsThe physical mechanisms of parameters in the Hausdorff fractal derivative model are analyzed clearly. Numerical results demonstrate that the proposed model can well fit the history of gas production data and it can be a powerful technique for depicting the early arrival and long-tailed phenomenon by incorporating a fractal dimension.Originality/valueTo the best of the authors’ knowledge, first time these results are presented.

Development of the theory and schematic solutions of the gyroscope based on superconducting quantum interferometers

Based on the principle of equivalence of the general theory of relativity and the definition of the non-Euclidean metric of space in a non-inertial frame of reference associated with rotation, a geometric (topological) phase is found that occurs when traversing any closed loop. This approach made it possible to establish a deep physical analogy between various wave effects (both classical and quantum) in closed waveguides that arise under the conditions of their rotation. Due to the coherence of the wave function of spinless charge carriers (Cooper pairs of conduction electrons with oppositely directed spins) in superconductors in the ground quantum state (superconducting state), the appearance of a geometric phase in closed loops under rotation conditions can lead to interference effects in the presence of weak bonds in the loop. To register interference effects, it was proposed to use superconducting quantum interferometers placed in the electric field of a cylindrical or spherical capacitor. In accordance with the general theory of the geometric phase of rotation, the study obtained the basic relations between the geometric phase of rotation and the phase of the wave function induced by an external magnetic field, and an estimate of their values was obtained for acceptable values of the angular velocity of rotation, dimensions of superconducting quantum interferometers, and voltages on the capacitor. The errors in measuring the magnetic flux during the rotation of the measuring complex are determined. Taking into account the dependence of the magnitude of the geometric phase of rotation on the voltage across the capacitor and the size of the superconducting quantum interferometers, and also taking into account the specific range of angular velocities of rotation, the sensitivity of superconducting quantum interferometers to the angular velocity can be regulated by a rational choice of these parameters. Based on the results obtained, a new method for measuring magnetic fields was proposed using two superconducting quantum interferometers with different values of areas bounded by closed contours under conditions of rotational movements, which makes it possible to compensate for the disturbing influence of rotations on magnetic measurements, as well as to simultaneously determine the magnetic induction and angular velocity of rotation.

Traffic flow prediction using multi-view graph convolution and masked attention mechanism

Traffic flow prediction is one of the essential technologies in the intelligent transportation system. Graph convolution is naturally suitable for graph structure data mining and is widely used to forecast traffic data. However, most graph convolution methods rely on the Euclidean distance adjacency matrix, which cannot mine richer semantic information. In addition, regardless of static or dynamic graph convolution methods, many graph convolution methods adopt the same adjacency matrix to mine graph information for each convolution layer. In reality, the proportion of abnormal traffic flows such as congestion is slight. Therefore, those networks tend to learn normal traffic patterns. This paper proposes a deep learning model including a dilated Temporal causal convolution module, multi-view diffusion Graph convolution module, and masked multi-head Attention module (TGANet) to address the above issues. The dilated temporal causal convolution is used to mine the temporal correlation of traffic data. Multi-view diffusion graph convolution operation leverages non-euclidean metrics like Jaccard distance and Pearson correlation to mine semantic information. In addition, the masked multi-head attention module is applied to mine fine-grained spatiotemporal dynamics. Experimental results on two public traffic data sets demonstrate that the proposed method outperforms other counterpart methods in most cases.

A Minimax Model for Generalized Penetration Distance Between Convex Sets by Directed Hausdorff Distance

The penetration distance, which characterizes the depth of intersection between two sets, is a fundamental computational geometry problem with a wide variety of applications in image processing, robotics, and circuit design. However, the penetration distance (even in Euclidean metric) is typically challenging to handle because it is essentially a nonconvex nonlinear optimization involving the projection onto the complement of a convex set. In this letter, we explore the penetration distance in non-Euclidean metrics (e.g., Manhattan, Chebyshev, and hexagon distances) by deploying the gauge function in convex analysis, and reformulate it as a nonlinear equation by virtue of the directed Hausdorff distance. More precisely, the generalized penetration distance amounts to seeking the largest zero point of a nondecreasing convex “black-box” function, whose function values can be obtained by the recent algorithmic advances in minimax optimization. We develop a solver for calculating generalized penetration distance between compact convex sets (possibly non-polyhedron) in arbitrary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> dimension. Numerical experiments on variant types of compact convex sets demonstrate that the proposed solver is compelling to render solutions with high accuracy.

Experimental Closed-Loop Excitation of Nonlinear Normal Modes on an Elastic Industrial Robot

Adding elastic elements to the mechanical structure should enable robots to perform efficient oscillatory tasks. Still, even characterizing natural oscillations in nonlinear systems is a challenge in itself, which nonlinear modal theory promises to solve. Therein eigenmanifolds generalize eigenspaces to mechanical systems with non-Euclidean metrics and thus characterize families of oscillations that are autonomous evolutions of the robot. Eigenmanifolds likewise provide a framework for deriving feedback controllers to excite and sustain these oscillations. Nevertheless, these results have been so far essentially theoretical. They have been applied on relatively low dimensional systems and almost exclusively in simulation. We aim to bridge the theory to the real-world gap with the present work and show that we can excite nonlinear modes in complex systems. To this end, we propose control strategies that can simultaneously stabilize numerically evaluated eigenmanifolds and sustain oscillations in the presence of dissipation. We then focus on the KUKA iiwa with simulated parallel springs as an example of the highly nonlinear and articulated system. We calculate all the nonlinear modes of the system, and we use the proposed strategies to excite the associated natural oscillations.

A non-Euclidean approach to source localization in an ocean waveguide

In recent years, there has been significant interest in the natural extension of signal processing techniques based on Euclidean distances to non-Euclidean metrics. Covariance or cross-spectral density matrices (CSDMs), often the basis for localization and detection processors, can be interpreted as “points” in a Riemannian manifold on which a distance metric can be defined. This mathematical structure has proven quite useful in a number of processing schemes involving detection, source localization and classification of data. In this presentation, I consider a fully geometric, non-Euclidean approach to source localization in a simulated shallow water ocean waveguide where the sound speed field is stochastic due to the presence of internal gravity waves. A number of different CSDM estimators are constructed by formulas describing the mean or median of a set of sample dyads obtained from Green function realizations of the propagating field from a source point to a vertical array of acoustic sensors. These CSDM estimates are not derived by statistics, but through the geometric structure of the CSDM. The resulting estimates are incorporated into matched-field localization processors whose measure of similarity or distance between CSDMs is determined on the manifold. Work supported by funds from the Office of Naval Research.

Exploiting Spatial Information to Enhance DTI Segmentations via Spatial Fuzzy c-Means with Covariance Matrix Data and Non-Euclidean Metrics

A diffusion tensor models the covariance of the Brownian motion of water at a voxel and is required to be symmetric and positive semi-definite. Therefore, image processing approaches, designed for linear entities, are not effective for diffusion tensor data manipulation, and the existence of artefacts in diffusion tensor imaging acquisition makes diffusion tensor data segmentation even more challenging. In this study, we develop a spatial fuzzy c-means clustering method for diffusion tensor data that effectively segments diffusion tensor images by accounting for the noise, partial voluming, magnetic field inhomogeneity, and other imaging artefacts. To retain the symmetry and positive semi-definiteness of diffusion tensors, the log and root Euclidean metrics are used to estimate the mean diffusion tensor for each cluster. The method exploits spatial contextual information and provides uncertainty information in segmentation decisions by calculating the membership values for assigning a diffusion tensor at one voxel to different clusters. A regularisation model that allows the user to integrate their prior knowledge into the segmentation scheme or to highlight and segment local structures is also proposed. Experiments on simulated images and real brain datasets from healthy and Spinocerebellar ataxia 2 subjects showed that the new method was more effective than conventional segmentation methods.

The role of feature space in atomistic learning

Efficient, physically-inspired descriptors of the structure and composition of molecules and materials play a key role in the application of machine-learning techniques to atomistic simulations. The proliferation of approaches, as well as the fact that each choice of features can lead to very different behavior depending on how they are used, e.g. by introducing non-linear kernels and non-Euclidean metrics to manipulate them, makes it difficult to objectively compare different methods, and to address fundamental questions on how one feature space is related to another. In this work we introduce a framework to compare different sets of descriptors, and different ways of transforming them by means of metrics and kernels, in terms of the structure of the feature space that they induce. We define diagnostic tools to determine whether alternative feature spaces contain equivalent amounts of information, and whether the common information is substantially distorted when going from one feature space to another. We compare, in particular, representations that are built in terms of n-body correlations of the atom density, quantitatively assessing the information loss associated with the use of low-order features. We also investigate the impact of different choices of basis functions and hyperparameters of the widely used SOAP and Behler–Parrinello features, and investigate how the use of non-linear kernels, and of a Wasserstein-type metric, change the structure of the feature space in comparison to a simpler linear feature space.

Sensitivity of superconducting quantum interferometers to rotational motions

Based on the principle of equivalence of the general theory of relativity and the definition of the non-Euclidean metric of space in a non-inertial frame of reference associated with rotation, a geometric (topological) phase is found that occurs when traversing any closed loop. This approach made it possible to establish a deep physical analogy between various wave effects (both classical and quantum) in closed waveguides that arise under the conditions of their rotation. Due to the coherence of the wave function of spinless charge carriers (Cooper pairs of conduction electrons with oppositely directed spins) in superconductors in the ground quantum state (superconducting state), the appearance of a geometric phase in closed loops under rotation conditions can lead to interference effects in the presence of weak bonds in the loop. To register interference effects, it was proposed to use superconducting quantum interferometers placed in the electric field of a cylindrical or spherical capacitor. In accordance with the general theory of the geometric phase of rotation, the study obtained the basic relations between the geometric phase of rotation and the phase of the wave function induced by an external magnetic field, and an estimate of their values was obtained for acceptable values of the angular velocity of rotation, dimensions of superconducting quantum interferometers, and voltages on the capacitor. The errors in measuring the magnetic flux during the rotation of the measuring complex are determined. Taking into account the dependence of the magnitude of the geometric phase of rotation on the voltage across the capacitor and the size of the superconducting quantum interferometers, and also taking into account the specific range of angular velocities of rotation, the sensitivity of superconducting quantum interferometers to the angular velocity can be regulated by a rational choice of these parameters. Based on the results obtained, a new method for measuring magnetic fields was proposed using two superconducting quantum interferometers with different values of areas bounded by closed contours under conditions of rotational movements, which makes it possible to compensate for the disturbing influence of rotations on magnetic measurements, as well as to simultaneously determine the magnetic induction and angular velocity of rotation.

Fixed Point Strategies in Data Science

The goal of this paper is to promote the use of fixed point strategies in data science by showing that they provide a simplifying and unifying framework to model, analyze, and solve a great variety of problems. They are seen to constitute a natural environment to explain the behavior of advanced convex optimization methods as well as of recent nonlinear methods in data science which are formulated in terms of paradigms that go beyond minimization concepts and involve constructs such as Nash equilibria or monotone inclusions. We review the pertinent tools of fixed point theory and describe the main state-of-the-art algorithms for provably convergent fixed point construction. We also incorporate additional ingredients such as stochasticity, block-implementations, and non-Euclidean metrics, which provide further enhancements. Applications to signal and image processing, machine learning, statistics, neural networks, and inverse problems are discussed.

Morphogenesis through elastic phase separation in a pneumatic surface

We report a phenomenon of phase separation that relates in many aspects to Yves Couder's work: an inflatable architectured elastomer plate, expected to expand homogeneously in its plane, buckles instead widely out-of-plane into very complex shape when internal pressure is applied. We show that this morphogenetic pattern formation is due to a two-dimensional elastic phase separation, which induces incompatible patchy non-Euclidean reference metric.

Vector Field Guided RRT* Based on Motion Primitives for Quadrotor Kinodynamic Planning

The intelligent drone is the key device of the future Internet of Drone, and its safe and robust flight in complicated environments still faces challenges. In this paper, we present a sampling-based kinodynamic planning algorithm for quadrotors, which plans a dynamically feasible trajectory in a complicated environment. We have designed a method to constrain the sampling state by using the vector field to construct a cone in the sampling stage of RRT*, so that the generated trajectory is connected as smoothly as possible to other states in the reachable set. The motion primitives are then generated by solving an optimal control problem and an explicit solution of the optimal duration for the motion primitives is given to optimally connect any pair of states. In addition, we have tried a new method to determine the neighbor radius for the non-Euclidean metrics of this paper. Finally, the planned trajectory is applied to the simulated quadrotor, which verifies the dynamic feasibility of the trajectory. Simulation results show that compared with the existing kinodynamic RRT* under the same number of iterations, the proposed algorithm explores more states with a shorter execution time and generates a smoother trajectory.

Transport Inequalities on Euclidean Spaces for Non-Euclidean Metrics

We explore upper bounds on Kantorovich transport distances between probability measures on the Euclidean spaces in terms of their Fourier-Stieltjes transforms, with focus on non-Euclidean metrics. The results are illustrated on empirical measures in the optimal matching problem on the real line.