High-dimensional Structure Research Articles

Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships.

One of the main goals of current systems neuroscience is to understand how neuronal populations integrate sensory information to inform behavior. However, estimating stimulus or behavioral information that is encoded in high-dimensional neuronal populations is challenging. We propose a method based on parametric copulas which allows modeling joint distributions of neuronal and behavioral variables characterized by different statistics and timescales. To account for temporal or spatial changes in dependencies between variables, we model varying copula parameters by means of Gaussian Processes (GP). We validate the resulting Copula-GP framework on synthetic data and on neuronal and behavioral recordings obtained in awake mice. We show that the use of a parametric description of the high-dimensional dependence structure in our method provides better accuracy in mutual information estimation in higher dimensions compared to other non-parametric methods. Moreover, by quantifying the redundancy between neuronal and behavioral variables, our model exposed the location of the reward zone in an unsupervised manner (i.e., without using any explicit cues about the task structure). These results demonstrate that the Copula-GP framework is particularly useful for the analysis of complex multidimensional relationships between neuronal, sensory and behavioral variables.

Analysis of the Trends Between Indoor Carbon Dioxide Concentration and Plug-Level Electricity Usage Through Topological Data Analysis

Sensors, such as environmental sensors and vital sensors collect various timeseries data, however, we hardly analyze all of them because of difficulty in visualization at once. We propose a method to bring out the events of interest from multi-dimensional time-series data, which is difficult to recognize everything, and to facilitate the analysis of factors, and verify its effectiveness. As a case study, we analyze multidimensional timeseries data collected from many sensors equipped in an active-learning space in a university library using topological data analysis (TDA), in particular the Mapper. Furthermore, we propose a method to examine the mapper graph visually and to interpret results for taking advantage of sensing data. The proposed method visualizes the relationship between environment data consists of temperature, humidity, illuminance, human motion, and electricity usages and observes indoor carbon dioxide (CO2) concentration data, extracting the features of environment data as the factors contributing to the increase in CO2. This method, which relies on TDA-Mapper, clarifies the high dimensional data structure and expresses the correspondence with the original data, which is effective for the visualization and application of multidimensional time-series data. Correlations were found between CO2 levels, energy consumption, and the motions of people. This method is useful as a simple way to interpret the features of large data sets acquired from many sensors.

The flow of the Berry curvature vector field

The concept of Berry phase and Berry curvature has become ubiquitous in solid state physics as it relates to variety of phenomena, such as topological insulators, polarization, and various Hall effects. It is well known that large Berry curvatures arise from close proximity of hybridizing bands, however, the vectorial nature of the Berry curvature is not utilized in current research. On bulk bcc Fe, we demonstrate the flow of the Berry curvature vector field which features not only monopoles but also higher dimensional structures with its own topological features. They can provide a novel unique view on the electronic structure in all three dimensions. This knowledge is also used to quantify particular contributions to the intrinsic anomalous Hall effect in a simple analytical form.

Ground Truth-Free 3-D Seismic Random Noise Attenuation via Deep Tensor Convolutional Neural Networks in the Time-Frequency Domain

The inherent challenge of 3-D seismic noise attenuation is determining how to uncover high-dimensional concise structures that only exist in true signals to eliminate random noise. The prevailing deep learning (DL) denoising methods have achieved promising performance in revealing the compact structures underlying contaminated seismic data. However, as clean ground-truth seismic data are generally unavailable in real-world settings, most existing matrix-based DL denoising schemes fail to automatically describe this type of high-dimensional structure in an unsupervised manner, potentially rendering them unable to effectively perform 3-D seismic data denoising tasks. To tackle this challenge, this article presents a tensor convolutional neural network (TCNN)-based data denoising scheme using Stein’s unbiased risk estimate (SURE) (called SURE-TCNN) to learn intrinsic high-dimensional structures without ground-truth seismic data. Considering that SURE provides an almost unbiased estimate of the mean squared error (MSE), SURE-TCNN has the potential to provide similar results to those of the supervised MSE-based TCNN with ground-truth data. For ease of implementation, the properties of a transform-based tensor-tensor product (t-product) are followed to establish a solid theoretical connection between the SURE-TCNN tensor and matrix. Derived from this connection, the SURE-TCNN weight parameters are determined by implementing matrix-based SURE-convolutional neural networks (CNNs) on each frontal slice in the time-frequency domain (e.g., the wavelet domain). Synthetic and field data examples demonstrate the superior performances of the proposed model against three state-of-the-art (SOTA) methods.

Autonomous Driving using Linear Model Predictive Control with a Koopman Operator based Bilinear Vehicle Model

This paper presents a real-time Model Predictive Control (MPC) formulation for autonomous driving based on a lifted bilinear vehicle model developed using the Koopman operator. Koopman operator based models can closely mimic the original nonlinear behaviors with a higher dimensional linear structure, which is attractive for computationally efficient linear MPC formulations for controlling nonlinear systems. However, current linear models based on linear Koopman realizations cannot capture the control-affine dynamics in nonlinear systems. This may result in large discrepancies between the original nonlinear system and the data-driven linear model, hindering its use in MPC. To address this gap, first, a novel Koopman bilinear vehicle model that takes control-affine dynamics into consideration is constructed and tested in open-loop simulations. This bilinear Koopman model is then linearized to serve as a prediction model in MPC, and is shown to have higher accuracy compared to the state-of-the-art linear models. The model is then used to develop a linear MPC formulation for simultaneous planning and control of an autonomous vehicle. The formulation is tested on lane change scenarios with obstacles against the nonlinear MPC and standard linear MPC benchmarks. The results show that the new formulation can achieve a lane change performance closer to the nonlinear MPC with a computational performance similar to the standard linear MPC. The new formulation is observed to be successful in handling high speeds where the standard linear MPC fails.

Spatial-Aware Collaboration–Competition Preserving Graph Embedding for Hyperspectral Image Classification

Recently, graph-based discriminant analysis has drawn much attention in representing a high-dimensional hyperspectral data set using a low-dimensional subspace by defining high-dimensional data structure on a graph. Obtaining optimal representation coefficients for classification purposes are the key in such methods. A closed form solution can be found to solve the problem related to collaborative representation using labeled samples, which offers computational efficiency. There exists an unsupervised approach of collaboration preserving graph embedding (CPGE) for dimensionality reduction (DR), and its performance is further enhanced by imposing locality-preserving constraint in the method called collaboration–competition preserving graph embedding (CCPGE). In this letter, we introduce spatial-aware collaboration–competitive preserving graph embedding with Tikhonov (SaCCPGT) by imposing a spatial regularization term in the objective function of CCPGE with Tikhonov regularization. In this way, spectral and spatial information can be utilized in a closed form solution in the proposed method. Experimental results on different hyperspectral data sets demonstrate the superior performance of the proposed SaCCPGT in comparison to state-of-the-art graph-based discriminant analysis approaches for DR.

Neighborhood hypergraph model for topological data analysis

Abstract Hypergraph, as a generalization of the notions of graph and simplicial complex, has gained a lot of attention in many fields. It is a relatively new mathematical model to describe the high-dimensional structure and geometric shapes of data sets. In this paper,we introduce the neighborhood hypergraph model for graphs and combine the neighborhood hypergraph model with the persistent (embedded) homology of hypergraphs. Given a graph,we can obtain a neighborhood complex introduced by L. Lovász and a filtration of hypergraphs parameterized by aweight function on the power set of the vertex set of the graph. Theweight function can be obtained by the construction fromthe geometric structure of graphs or theweights on the vertices of the graph. We show the persistent theory of such filtrations of hypergraphs. One typical application of the persistent neighborhood hypergraph is to distinguish the planar square structure of cisplatin and transplatin. Another application of persistent neighborhood hypergraph is to describe the structure of small fullerenes such as C20. The bond length and the number of adjacent carbon atoms of a carbon atom can be derived from the persistence diagram. Moreover, our method gives a highly matched stability prediction (with a correlation coefficient 0.9976) of small fullerene molecules.

Spatial Information Regularized Tensor Decomposition Framework for Super-Resolution Reconstruction of Medical MRI and Radiographs

Super-resolution reconstruction of medical images effectually enhances visual qualities to provide clear visions on anatomical structures. However, the spatial information, which abounds in low-resolution images, has received scant attention in the task of super-resolution, resulting in suppressed reconstruction qualities. This paper brings the spatial information into effective action by virtue of high-dimensional structures of tensor-format data, and presents a method named <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Tensor Decomposition based medical Image Super-resolution reconstruction</i> (TDIS). In particular, TDIS employs tensors to preserve rich image spatial information and effectually processes the spatial information contained in image tensors by reformulating transposed convolutional layers using tensor decomposition. Moreover, TDIS provides tensor based error terms to capture spatial differences between generated and target images to reduce visual contrasts between them. Experimental results about reconstructing a range of medical images empirically demonstrate the competence of TDIS compared to the state-of-the-art methods.

Electricity‐heat‐gas integrated demand response dependency assessment based on BOXCOX‐Pair Copula model

With the continuous development of Regional Integrated Energy System (RIES), demand response (DR) is composed of diversified loads including electric load, heat load and gas load. Their cross-dependencies reflecting the nonlinear coupling complementary relationship between each load type are one of the key factors to improve multi-energy flow optimisation modelling and utilisation efficiency on the demand side. Accordingly, this paper proposes a DR dependency assessment method considering electricity-heat-gas loads based on the BOXCOX-Pair Copula model. BOXCOX transformation is introduced to convert various probability distribution statistics of electricity-heat-gas loads into Gaussian distributed variables. C-vine Pair Copula is used to characterise an ensemble-of-trees of high-dimensional dependency structure among multi-energy demand modalities. Then the combined model of BOXCOX transformation and C-vine Pair Copula can be employed to determine the complex coupling dependency among electricity-heat-gas loads according to different DR statistics. Some metrics between the original empirical distribution and BOXCOX-Pair Copula distribution are introduced to assess the dependency evaluation precision of the proposed model. Finally, the novel dependency assessment model is numerically tested utilising the electricity load, heat load and gas load data sequences in a real RIES. The results illustrate that the cross-dependency of electricity-heat-gas integrated DR based on the BOXCOX-C-vine copula model is closer to that of actual sample data, which verify the effectiveness and superiority of the proposed approach.

Extract Metabolomic Information from Mass Spectrometry Images Using Advanced Data Analysis.

Mass spectrometry imaging (MSI) data generally contains large sizes and high-dimensional structures due to their inherent complex chemical and spatial information. A variety of data analysis methods have been developed to comprehensively analyze the MSI experimental results and extract essential information. Here, we describe the protocols of data preprocessing and emerging methods for data analyses, including multivariate analysis, machine learning, and image fusion, that have been applied to the data generated from the Single-probe MSI technique. These strategies and methods can be potentially applied to handling data produced from other MSI techniques.

Constructing higher-order topological states in higher dimensions

Higher-order topological phase as a generalization of Berry phase attracts an enormous amount of research. The current theoretical models supporting higher-order topological phases, however, cannot give the connection between lower and higher-order topological phases when extending the lattice from lower to higher dimensions. Here, we theoretically propose and experimentally demonstrate a topological corner state constructed from the edge states in one dimensional lattice. The two-dimensional square lattice owns independent spatial modulation of coupling in each direction, and the combination of edge states in each direction come up to the higher-order topological corner state in two-dimensional lattice, revealing the connection of topological phase in lower and higher dimensional lattices. Moreover, the topological corner states in two-dimensional lattice can also be viewed as the dimension-reduction from a four-dimensional topological phase characterized by vector Chern number, considering two modulation phases as synthetic dimensions in Aubry-Andre-Harper model discussed as example here. Our work deeps the understanding to topological phases breaking through the lattice dimension, and provides a promising tool constructing higher topological phases in higher dimensional structures.

Elastic Correlation Adjusted Regression (ECAR) scores for high dimensional variable importance measuring

Investigation of the genetic basis of traits or clinical outcomes heavily relies on identifying relevant variables in molecular data. However, characteristics such as high dimensionality and complex correlation structures of these data hinder the development of related methods, resulting in the inclusion of false positives and negatives. We developed a variable importance measure method, termed the ECAR scores, that evaluates the importance of variables in the dataset. Based on this score, ranking and selection of variables can be achieved simultaneously. Unlike most current approaches, the ECAR scores aim to rank the influential variables as high as possible while maintaining the grouping property, instead of selecting the ones that are merely predictive. The ECAR scores’ performance is tested and compared to other methods on simulated, semi-synthetic, and real datasets. Results showed that the ECAR scores improve the CAR scores in terms of accuracy of variable selection and high-rank variables’ predictive power. It also outperforms other classic methods such as lasso and stability selection when there is a high degree of correlation among influential variables. As an application, we used the ECAR scores to analyze genes associated with forced expiratory volume in the first second in patients with lung cancer and reported six associated genes.

Self-Adaptive bagging approach to credit rating

We propose an enhanced bagging strategy based on self adaptive learning. Recent work in computational biology has seen an increasing use of ensemble learning methods due to their unique advantages in dealing with small sample size, high-dimensionality, and complex data structures. The proposed strategy has a unique advantage in dealing with classification tasks (i.e., credit rating in this study) with data on a relatively small sample size but a large number of heterogeneously distributed features. The self-organising learning mechanism makes the traditional bagging strategy more efficient in terms of model structure. Each submodel in the bagging network becomes a self-adapting base learner in credit rating. In order to prove the applicability and to measure the performance of the proposed method, we carried out a validation using three different traditional algorithms over both artificial datasets and real market data. The results of the proposed algorithm are, on average, better in most of the comparative experiments, while maintaining a model structure that is less complex.

Commodity Price Recognition and Simulation of Image Recognition Technology Based on the Nonlinear Dimensionality Reduction Method

Dimensionality reduction of images with high-dimensional nonlinear structure is the key to improving the recognition rate. Although some traditional algorithms have achieved some results in the process of dimensionality reduction, they also expose their respective defects. In order to achieve the ideal effect of high-dimensional nonlinear image recognition, based on the analysis of the traditional dimensionality reduction algorithm and refining its advantages, an image recognition technology based on the nonlinear dimensionality reduction method is proposed. As an effective nonlinear feature extraction method, the nonlinear dimensionality reduction method can find the nonlinear structure of datasets and maintain the intrinsic structure of data. Applying the nonlinear dimensionality reduction method to image recognition is to divide the input image into blocks, take it as a dataset in high-dimensional space, reduce the dimension of its structure, and obtain the low-dimensional expression vector of its eigenstructure so that the problem of image recognition can be carried out in a lower dimension. Thus, the computational complexity can be reduced, the recognition accuracy can be improved, and it is convenient for further processing such as image recognition and search. The defects of traditional algorithms are solved, and the commodity price recognition and simulation experiments are carried out, which verifies the feasibility of image recognition technology based on the nonlinear dimensionality reduction method in commodity price recognition.

Long-term dynamics of the inner planets in the Solar System

Although the discovery of the chaotic motion of the inner planets in the Solar System (Mercury to Mars) was made more than 30 years ago, the secular chaos of their orbits still demands more analytical analyses. In addition to the high-dimensional structure of the motion, this is probably related to the lack of an adequately simple dynamical model. We consider a new secular dynamics for the inner planets, with the aim of retaining a fundamental set of interactions that explains their chaotic behaviour and at the same time is consistent with the predictions of the most precise orbital solutions currently available. We exploit the regularity in the secular motion of the outer planets (Jupiter to Neptune) to predetermine a quasi-periodic solution for their orbits. This reduces the secular phase space to the degrees of freedom dominated by the inner planets. In addition, the low masses of the inner planets and the absence of strong mean-motion resonances permits us to restrict ourselves to first-order secular averaging. The resulting dynamics can be integrated numerically in a very efficient way through Gauss’s method, while computer algebra allows an analytical inspection of planet interactions when the Hamiltonian is truncated at a given total degree in eccentricities and inclinations. The new model matches reference orbital solutions of the Solar System over timescales shorter than or comparable to the Lyapunov time very satisfactorily. It correctly reproduces the maximum Lyapunov exponent of the inner system and the statistics of the high eccentricities of Mercury over the next five billion years. The destabilizing role of the g1 − g5 secular resonance also arises. A numerical experiment, consisting of a thousand orbital solutions over one hundred billion years, reveals the essential properties of the stochastic process driving the destabilization of the inner Solar System and clarifies its current metastable state.

Gene Expression Analysis through Parallel Non-Negative Matrix Factorization

Genetic expression analysis is a principal tool to explain the behavior of genes in an organism when exposed to different experimental conditions. In the state of art, many clustering algorithms have been proposed. It is overwhelming the amount of biological data whose high-dimensional structure exceeds mostly current computational architectures. The computational time and memory consumption optimization actually become decisive factors in choosing clustering algorithms. We propose a clustering algorithm based on Non-negative Matrix Factorization and K-means to reduce data dimensionality but whilst preserving the biological context and prioritizing gene selection, and it is implemented within parallel GPU-based environments through the CUDA library. A well-known dataset is used in our tests and the quality of the results is measured through the Rand and Accuracy Index. The results show an increase in the acceleration of 6.22× compared to the sequential version. The algorithm is competitive in the biological datasets analysis and it is invariant with respect to the classes number and the size of the gene expression matrix.

Inverse Design Framework With Invertible Neural Networks for Passive Vibration Suppression in Phononic Structures

Abstract Automated inverse design methods are critical to the development of metamaterial systems that exhibit special user-demanded properties. While machine learning approaches represent an emerging paradigm in the design of metamaterial structures, the ability to retrieve inverse designs on-demand remains lacking. Such an ability can be useful in accelerating optimization-based inverse design processes. This paper develops an inverse design framework that provides this capability through the novel usage of invertible neural networks (INNs). We exploit an INN architecture that can be trained to perform forward prediction over a set of high-fidelity samples and automatically learns the reverse mapping with guaranteed invertibility. We apply this INN for modeling the frequency response of periodic and aperiodic phononic structures, with the performance demonstrated on vibration suppression of drill pipes. Training and testing samples are generated by employing a transfer matrix method. The INN models provide competitive forward and inverse prediction performance compared to typical deep neural networks (DNNs). These INN models are used to retrieve approximate inverse designs for a queried non-resonant frequency range; the inverse designs are then used to initialize a constrained gradient-based optimization process to find a more accurate inverse design that also minimizes mass. The INN-initialized optimizations are found to be generally superior in terms of the queried property and mass compared to randomly initialized and inverse DNN-initialized optimizations. Particle swarm optimization with INN-derived initial points is then found to provide even better solutions, especially for the higher-dimensional aperiodic structures.

High-dimensional structure learning of sparse vector autoregressive models using fractional marginal pseudo-likelihood

Learning vector autoregressive models from multivariate time series is conventionally approached through least squares or maximum likelihood estimation. These methods typically assume a fully connected model which provides no direct insight to the model structure and may lead to highly noisy estimates of the parameters. Because of these limitations, there has been an increasing interest towards methods that produce sparse estimates through penalized regression. However, such methods are computationally intensive and may become prohibitively time-consuming when the number of variables in the model increases. In this paper we adopt an approximate Bayesian approach to the learning problem by combining fractional marginal likelihood and pseudo-likelihood. We propose a novel method, PLVAR, that is both faster and produces more accurate estimates than the state-of-the-art methods based on penalized regression. We prove the consistency of the PLVAR estimator and demonstrate the attractive performance of the method on both simulated and real-world data.

Support structure representation learning for sequential data clustering

Sequential data clustering is a challenging task in data mining (e.g., motion recognition and video segmentation). For good performance in dealing with complex local correlation and high-dimensional structure of sequential data, representation based methods have become one of the hot topics for sequential data clustering, in which subspace clustering is a representative tool. Subspace clustering methods divide the sequence into disjoint segments according to a locally continuous and connected representation of raw data. Although the subspace clustering methods maintain the successive property of sequential data well, there exist redundant connections in the intersection of two subsequences, which will destroy the integrity of a cluster and easily cause the chained partition of the sequence. So it is necessary to learn a more specific structure representation of a sequence to preserves both sequential information and efficient connections. Besides, the representation that conducive to clustering should have sparsity and connectivity under some assumptions. To this end, we propose a novel method to learn the support structure representation of sequence, which can extract sufficient information about instances and get the compact structure of sequential data. Furthermore, a new subspace clustering method is proposed based on the representation based method. Theoretical analysis and experimental results show the effectiveness of the proposed method.

IXVC: An interactive pipeline for explaining visual clusters in dimensionality reduction visualizations with decision trees

High-dimensional data with many features are usually challenging to represent with standard visualization techniques. Usually, one has to resort to dimensionality reduction techniques such as PCA, MDS or t -SNE to represent such data. Such dimensionality reduction techniques make it possible to highlight the high-dimensional structures of data. In many of such visualizations, comparable instances appear to form visual clusters. However, no feedback is directly given by these techniques to the user about the features that make the instances cluster together in the visualization. As such, the interpretation of which features define a given visual cluster is a complicated task. In this paper, we propose a novel interactive approach (called Interactive eXplanation of Visual Clusters — IXVC) to explain dimensionality reduction visualizations by mapping their clusters to explanations provided by decision trees. The decision trees use features in high-dimensional data to explain two-dimensional clusters, filling the gap between the dimensionality reduction visualization and the original data.