Tensor Kernels Research Articles

Urban Noise Inference Model Based on Multiple Views and Kernel Tensor Decomposition

Prediction of urban noise is becoming more significant for tackling noise pollution and protecting human mental health. However, the existing noise prediction algorithms neglected not only the correlation between noise regions, but also the nonlinearity and sparsity of the data, which resulted in low accuracy of filling in the missing entries of data. In this paper, we propose a model based on multiple views and kernel-matrix tensor decomposition to predict the noise situation at different times of day in each region. We first construct a kernel tensor decomposition model by using kernel mapping in order to speed decomposition rate and realize stable estimate the prediction system. Then, we analyze and compute the cause of the noise from multiple views including computing the similarity of regions and the correlation between noise categories by kernel distance, which improves the credibility to infer the noise situation and the categories of regions. Finally, we devise a prediction algorithm based on the kernel-matrix tensor factorization model. We evaluate our method with a real dataset, and the experiments to verify the advantages of our method compared with other existing baselines.

Nonlinear fault detection for batch processes via improved chordal kernel tensor locality preserving projections

The quality and stability of products are seriously influenced by the process conditions. A large number of modern production processes can be considered as batch processes, with nonlinear relationships between the process variables. How to troubleshoot batch processes has attracted considerable attention in the literature. The research object of batch processes is expressed as the third-order tensor data of batch × variable × time. The traditional methods convert the tensors into second-order forms through matrix expansion. A novel method named improved chordal kernel tensor locality preserving projections (ICK-TLPP) is proposed for fault detection of batch processes. First, the chordal distance is introduced as a measurement of the similarity of matrix, and an improved method is proposed for describing the variation of time series data. Then, the chordal kernel function is introduced to preserve the spatial structure of the tensor data without the information loss caused by vectorization, and describe the nonlinear correlation during the multivariate control system. Next, the locality preserving projections algorithm is applied to detect the intrinsic manifold structure. Parallel analysis is applied to optimize the hyper-parameters in the model. Finally, Granger causality analysis is performed to locate the root cause of the process fault. The proposed method is validated on two datasets, penicillin fermentation process and the hot strip rolling process. The best results of false alarm rate and fault detection rate are 16% and 94% respectively. The proposed method performs better compared with the traditional algorithms.

Generalized support vector regression: Duality and tensor-kernel representation

In this paper, we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel–Rockafellar duality theory, we give an explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving the problem which relies on a tensor-kernel representation. This analysis overcomes the typical difficulties connected to learning in Banach spaces. We finally present a large class of tensor-kernels to which our theory fully applies: power series tensor kernels. This type of kernels describes Banach spaces of analytic functions and includes generalizations of the exponential and polynomial kernels as well as, in the complex case, generalizations of the Szegö and Bergman kernels.

A multi-dimensional Morton-ordered block storage for mode-oblivious tensor computations

Computation on tensors, treated as multidimensional arrays, revolve around generalized basic linear algebra subroutines (BLAS). We propose a novel data structure in which tensors are blocked and blocks are stored in an order determined by Morton order. This is not only proposed for efficiency reasons, but also to induce efficient performance regardless of which mode a generalized BLAS call is invoked for; we coin the term mode-oblivious to describe data structures and algorithms that induce such behavior. Experiments on one of the most bandwidth-bound generalized BLAS kernel, the tensor–vector multiplication, not only demonstrate superior performance over two state-of-the-art variants by up to 18%, but additionally show that the proposed data structure induces a 71% less sample standard deviation for tensor–vector multiplication across d modes, where d varies from 2 to 10. Finally, we show our data structure naturally expands to other tensor kernels and demonstrate up to 38% higher performance for the higher-order power method.

Galerkin Boundary Elements for a Computation of the Surface Tractions in Exterior Stokes Flows

In the computation of a three–dimensional steady creeping flow around a rigid body, the total body force and torque are well predicted using a boundary integral equation (BIE) with a single concentrated pair Stokeslet- Rotlet located at an interior point of the body. However, the distribution of surface tractions are seldom considered. Then, a completed indirect velocity BIE of Fredholm type and second-kind is employed for the computation of the pointwise tractions, and it is numerically solved by using either collocation or Galerkin weighting procedures over flat triangles. In the Galerkin case, a full numerical quadrature is proposed in order to handle the weak singularity of the tensor kernels, which is an extension for fluid engineering of a general framework (Taylor, 2003, “Accurate and Efficient Numerical Integration of Weakly Singulars Integrals in Galerkin EFIE Solutions,” IEEE Trans. on Antennas and Propag., 51(7), pp. 1630–1637). Several numerical simulations of steady creeping flow around closed bodies are presented, where results compare well with semianalytical and finite-element solutions, showing the ability of the method for obtaining the viscous drag and capturing the singular behavior of the surface tractions close to edges and corners. Also, deliberately intricate geometries are considered.

A Tensor-Variate Gaussian Process for Classification of Multidimensional Structured Data

As tensors provide a natural and efficient representation of multidimensional structured data, in this paper, we consider probabilistic multinomial probit classification for tensor-variate inputs with Gaussian processes (GP) priors placed over the latent function. In order to take into account the underlying multimodes structure information within the model, we propose a framework of probabilistic product kernels for tensorial data based on a generative model assumption. More specifically, it can be interpreted as mapping tensors to probability density function space and measuring similarity by an information divergence. Since tensor kernels enable us to model input tensor observations, the proposed tensor-variate GP is considered as both a generative and discriminative model. Furthermore, a fully variational Bayesian treatment for multiclass GP classification with multinomial probit likelihood is employed to estimate the hyperparameters and infer the predictive distributions. Simulation results on both synthetic data and a real world application of human action recognition in videos demonstrate the effectiveness and advantages of the proposed approach for classification of multiway tensor data, especially in the case that the underlying structure information among multimodes is discriminative for the classification task.

Tensor kernels for simultaneous fiber model estimation and tractography.

This paper proposes a novel framework for joint orientation distribution function estimation and tractography based on a new class of tensor kernels. Existing techniques estimate the local fiber orientation at each voxel independently so there is no running knowledge of confidence in the measured signal or estimated fiber orientation. In this work, fiber tracking is formulated as recursive estimation: at each step of tracing the fiber, the current estimate of the orientation distribution function is guided by the previous. To do this, second-and higher-order tensor-based kernels are employed. A weighted mixture of these tensor kernels is used for representing crossing and branching fiber structures. While tracing a fiber, the parameters of the mixture model are estimated based on the orientation distribution function at that location and a smoothness term that penalizes deviation from the previous estimate along the fiber direction. This ensures smooth estimation along the direction of propagation of the fiber. In synthetic experiments, using a mixture of two and three components it is shown that this approach improves the angular resolution at crossings. In vivo experiments using two and three components examine the corpus callosum and corticospinal tract and confirm the ability to trace through regions known to contain such crossing and branching.

Text Image Restoration using Adaptive Fuzzy Median Based on 3D Tensors and Iterative Voting

This paper addresses the problem of efficient and effective restoration of text images, by formulating the problem as inferring the surface from a sparse and noisy point set in a 3D structure tensor space. Given a set of noisy data correspondence in corrupted images, the proposed method extracts good matches and rejects the noisy elements. The methodology is unconventional, since, unlike most other methods, it optimizes certain scalar, objective functions. Also, as the proposed approach does not involve initialization, or any iterative search in the parameter space, it is free from the problems of identifying only local optima or having poor convergence properties. Subject to the general restoration of natural images, the removal and restoration of corrupted regions is performed by 3D tensor voting based on a fuzzy median filter. In essence, the input set of matches is first transformed into a sparse 3D point set so that 3D tensor kernels can then be used to vote for the most salient surface that captures all inliers inherent in the input. Lastly the density estimation for detecting the center modes is performed as well as a clustering algorithm for segmenting the values according to the color components in the restored image. Experimental results are presented which show that the proposed approach is efficient and robust in terms of restoring and segmenting corrupted text images.

Long-wavelength behavior of the dynamical spin-resolved local-field factor in a two-dimensional electron liquid

The high frequency limits of the singular component $A(\omega)$ of the small wavevector expansion of the longitudinal (L) and transverse (T) components of the spin-resolved exchange-correlation kernel tensor $f_{xc,\sigma \sigma'}^{L,T}(q,\omega)=-v(q)G_{\sigma \sigma'}^{L,T}(q,\omega)$ in a two-dimensional isotropic electron liquid with arbitrary spin polarization are studied. Here $G_{\sigma \sigma'}^{L,T}(q,\omega)$ is the spin-resolved local field factor, $v(q)$ is the Coulomb interaction in momentum space, and $\sigma$ denotes spin. Particularly, the real part of $A(\omega)$ is found to be logarithmically divergent at large $\omega$. the large wavevetor structure of the corresponding spin-resolved static structure factor is also established.

A particulate basis for a lattice-gas model of amphiphilic fluids

We show that the flux-field expansion derived by Boghosian & Coveney for the Rothman-Keller immiscible fluid model can be derived in a simpler and more general way in terms of the completely symmetric tensor kernels introduced by those authors. Using this generalized flux-field expansion we show that the more complex amphiphilic model of Boghosian, Coveney & Emerton can also be derived from an underlying model of particle interactions. The consequences of this derivation are discussed in the context of previous equilibrium Ising-like lattice models and other non-equilibrium mesoscale models.

Diatom aggregate formation and fluxes: a modeling analysis under different size-resolution schemes and with empirically determined aggregation kernels

A hierarchical analysis is presented for evaluating the accuracy of different formulations to simulate diatom aggregate formation and diagnose its associated carbon fluxes. We find that, for diagnostic purposes, a two-class arrangement of sizes (small particles and aggregates) is an adequate compromise between the accuracy of estimated fluxes and the level of resolution implemented for particle sizes. Ignoring the existence of aggregates severely underestimates fluxes, whereas increasing the size resolution (up to 7 size classes) yields a small improvement in accuracy. We demonstrate that this two-size-class arrangement is also the minimum resolution for accurately modeling the formation of aggregates. This arrangement requires only one aggregation kernel, which we estimate from experimental data without imposing any assumption for the interaction between particles. Although the kernel is robust to changes in the nature of the particles, the simulations obtained with this approach are very sensitive to the initial concentration of aggregates implemented to run the model. This sensitivity to initial conditions does not appear in a model with higher size resolution whose kernel tensor for aggregation has also been derived from experimental mesocosm data through an inverse modeling procedure. Although the level of accuracy achieved in a simulation with this kernel tensor seems promising, its robustness requires further tests in conditions other than a mesocosm experimental design.

N-dimensional tensor voting and application to epipolar geometry estimation

We address the problem of epipolar geometry estimation by formulating it as one of hyperplane inference from a sparse and noisy point set in an 8D space. Given a set of noisy point correspondences in two images of a static scene without correspondences, even in the presence of moving objects, our method extracts good matches and rejects outliers. The methodology is novel and unconventional, since, unlike most other methods optimizing certain scalar, objective functions, our approach does not involve initialization or any iterative search in the parameter space. Therefore, it is free of the problem of local optima or poor convergence. Further, since no search is involved, it is unnecessary to impose simplifying assumption to the scene being analyzed for reducing the search complexity. Subject to the general epipolar constraint only, we detect wrong matches by a computation scheme, 8D tensor voting, which is an instance of the more general N-dimensional tensor voting framework. In essence, the input set of matches is first transformed into a sparse 8D point set. Dense, 8D tensor kernels are then used to vote for the most salient hyperplane that captures all inliers inherent in the input. With this filtered set of matches, the normalized eight-point algorithm can be used to estimate the fundamental matrix accurately. By making use of efficient data structure and locality, our method is both time and space efficient despite the higher dimensionality. We demonstrate the general usefulness of our method using example image pairs for aerial image analysis, with widely different views, and from nonstatic 3D scenes. Each example contains a considerable number of wrong matches.

Three‐dimensional crack analysis using singular boundary elements

AbstractA multi‐domain method of solving three‐dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed. The displacement and traction behaviours near a crack front are incorporated in special crack elements. The elimination of singularities arising from the term combined with Kelvin's kernel for displacement in the integrals is discussed in detail. Stress intensity factors of modes I, II and III are obtained directly from crack‐front nodal values, without any extrapolation as in some other methods. No differentiation of conventional boundary integral equations (with Kelvin's tensor kernels) is necessary in the current approach. This method is applicable to cracks of arbitrary shape. Infinite bodies are modelled precisely as such, not approximated as large finite bodies. Numerical solutions of stress intensity factors are given for several problems involving a penny‐shaped crack.