Tensor Decomposition

The scope of this paper is to introduce multiway analysis into structural engineering research and to outline methodology used in tensor decomposition of finite element analysis (FEA) data. More specifically, the example evaluated herein evaluates the stress distribution of two different highway bridge structural components (girders and cross frames), being subjected to incrementally increasing forces. Additionally, the paper shows potential advantages of using multiway methods in interpretation of FEA data and makes recommendations for future investigations on the use of multiway methods in FEA post-processing of structural engineering data.

Background model initialization is commonly the first step of the background subtraction process. In practice, several challenges appear and perturb this process, such as dynamic background, bootstrapping, illumination changes, noise image, etc. In this context, we investigate the background model initialization as a reconstruction problem from missing data. This problem can be formulated as a matrix or tensor completion task where the image sequence (or video) is revealed as partially observed data. In this paper , the missing entries are induced from the moving regions through a simple joint motion-detection and frame-selection operation. The redundant frames are eliminated, and the moving regions are represented by zeros in our observation model. The second stage involves evaluating twenty-three state-of-the-art algorithms comprising of thirteen matrix completion and ten tensor completion algorithms. These algorithms aim to recover the low-rank component (or background model) from partially observed data. The Scene Background Initialization data set was selected in order to evaluate this proposal with respect to the background model challenges. Our experimental results show the good performance of LRGeomCG method over its direct competitors.

Traditionally, extending the Singular Value Decomposition (SVD) to third-order tensors (multiway arrays) has involved a representation using the outer product of vectors. These outer products can be written in terms of the n-mode product, which can also be used to describe a type of multiplication between two tensors. In this paper, we present a different type of third-order generalization of

—Many critical EDA problems suffer from the curse of dimensionality, i.e. the very fast-scaling computational burden produced by large number of parameters and/or unknown variables. This phenomenon may be caused by multiple spatial or temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit simulation), nonlinearity of devices and circuits, large number of design or optimization parameters (e.g. full-chip routing/placement and circuit sizing), or extensive process variations (e.g. variability/reliability analysis and design for manufacturability). The computational challenges generated by such high dimensional problems are generally hard to handle efficiently with traditional EDA core algorithms that are based on matrix and vector computation. This paper presents " tensor computation " as an alternative general framework for the development of efficient EDA algorithms and tools. A tensor is a high-dimensional generalization of a matrix and a vector, and is a natural choice for both storing and solving efficiently high-dimensional EDA problems. This paper gives a basic tutorial on tensors, demonstrates some recent examples of EDA applications (e.g., nonlinear circuit modeling and high-dimensional uncertainty quantification), and suggests further open EDA problems where the use of tensor computation could be of advantage.

The Semi-Algebraic framework for the approximate Canonical Polyadic (CP) decomposition via SImultaneaous matrix diagonalization (SECSI) is an efficient tool for the computation of the CP decomposition. The SECSI framework reformulates the CP decomposition into a set of joint eigenvalue decomposition (JEVD) problems. Solving all JEVDs, we obtain multiple estimates of the factor matrices and the best estimate is chosen in a subsequent step by using an exhaustive search or some heuristic strategy that reduces the computational complexity. Moreover, the SECSI framework retains the option of choosing the number of JEVDs to be solved, thus providing an adjustable complexity-accuracy trade-off. In this work, we provide an analytical performance analysis of the SECSI framework for the computation of the approximate CP decomposition of a noise corrupted low-rank tensor, where we derive closed-form expressions of the relative mean square error for each of the estimated factor matrices. These expressions are obtained using a first-order perturbation analysis and are formulated in terms of the second-order moments of the noise, such that apart from a zero mean, no assumptions on the noise statistics are required. Simulation results exhibit an excellent match between the obtained closed-form expressions and the empirical results. Moreover, we propose a new Performance Analysis based Selection (PAS) scheme to choose the final factor matrix estimate. The results show that the proposed PAS scheme outperforms the existing heuristics, especially in the high SNR regime.

Most of the available models for reference evapo-transpiration (ET 0) estimation are based upon only an empirical equation for ET 0. Thus, one of the main issues in ET 0 estimation is the appropriate integration of time information and different empirical ET 0 equations to determine ET 0 and boost the precision. The FAO-56 Penman–Monteith, adjusted Hargreaves, Blaney–Criddle, Priestley–Taylor, and Jensen– Haise equations were utilized in this study for estimating ET 0 for two stations of Belgrade and Nis in Serbia using collected data for the period of 1980 to 2010. Three-order tensor is used to capture three-way correlations among months, years, and ET 0 information. Afterward, the latent correlations among ET 0 parameters were found by the multiway analysis to enhance the quality of the prediction. The suggested method is valuable as it takes into account simultaneous relations between elements, boosts the prediction precision , and determines latent associations. Models are compared with respect to coefficient of determination (R 2), mean absolute error (MAE), and root-mean-square error (RMSE). The proposed tensor approach has a R 2 value of greater than 0.9 for all selected ET 0 methods at both selected stations, which is acceptable for the ET 0 prediction. RMSE is ranged between 0.247 and 0.485 mm day −1 at Nis station and between 0.277 and 0.451 mm day −1 at Belgrade station, while MAE is between 0.140 and 0.337 mm day −1 at Nis and between 0.208 and 0.360 mm day −1 at Belgrade station. The best performances are achieved by Priestley–Taylor model at Nis s t a t i o n (R 2 = 0. 9 8 5 , M A E = 0. 1 4 0 m m d a y − 1 , RMSE = 0.247 mm day −1) and FAO-56 Penman–Monteith model at Belgrade station (MAE = 0.208 mm day −1 , RMSE = 0.277 mm day −1 , R 2 = 0.975).

This paper focuses on the curse of dimensionality in the numerical solution of the stationary Fokker-Planck equation for systems with state-independent excitation. A tensor decomposition approach is combined with Chebyshev spectral differentiation to drastically reduce the number of degrees of freedom required to maintain accuracy as dimensionality increases. Following the enforcement of the normality condition via a penalty method, the discretized system is solved using alternating least squares algorithm. Numerical results for a variety of systems, including separable/non-separable systems, linear/non-linear systems and systems with/without closed-form stationary solutions up-to ten dimensional state-space are presented to illustrate the effectiveness of the proposed method.

We present a regularized method for solving an inverse problem in Diffusion Tensor Magnetic Resonance Imaging (DT-MRI) data. In the case of brain images, DT-MR imagery technique produces a tensor field that indicates the local orientation of nerve bundles. Now days, the spatial resolution of this technique is limited by the partial volume effect produced in voxels that contain fiber crossings or bifurcations. In this paper, we proposed a method for recovering the intra-voxel information and inferring the brain connectivity. We assume that the observed tensor is a linear combination of a given tensor basis, therefore, the aim of our approach is the computation of the unknown coefficients of this linear combination. By regularizing the problem, we introduce the needed prior information about the piecewise smoothness of nerve bundles orientation. As a result, we recover a multi-tensor field. Moreover, for estimating the nerve bundles trajectory, we propose a method based on stochastic walks of particles through the computed multi-tensor field. The performance of the method is demonstrated by experiments in both synthetic and real data.

We present a family of novel methods for embedding knowledge graphs into real-valued tensors. These tensor-based embeddings capture the ordered relations that are typical in the knowledge graphs represented by semantic web languages like RDF. Unlike many previous models, our methods can easily use prior background knowledge provided by users or extracted automatically from existing knowledge graphs. In addition to providing more robust methods for knowledge graph embedding, we provide a provably-convergent, linear tensor factorization algorithm. We demonstrate the efficacy of our models for the task of predicting new facts across eight different knowledge graphs, achieving between 5% and 50% relative improvement over existing state-of-the-art knowledge graph embedding techniques. Our empirical evaluation shows that all of the tensor decomposition models perform well when the average degree of an entity in a graph is high, with constraint-based models doing better on graphs with a small number of highly similar relations and regularization-based models dominating for graphs with relations of varying degrees of similarity.

Two-Dimension Linear Discriminant Analysis (2DLDA) becomes a popular technique for face recognition due to its effectiveness in both accuracy and computational cost. Furthermore, there has been shown that 2DLDA reduces only the row direction of the data. This gives a rise to a new technique, (2D)2LDA. (2D)2LDA performs 2DLDA on the row direction and conducts Alternate 2DLDA on the column direction of the data. Although the eigenvalues associated with eigenvectors simply show the discriminative power of the subspace spanned by the corresponding eigenvectors, there are some evidences indicate the eigenvector with high eigenvalue may correspond to noise signal such as pose, illumination or expression and the eigenvector with high discriminative power may have a low eigenvalue due to its closeness to the null space of the training data. By these reasons, we may improve the performace of 2DLDA-based techniques by properly reordering the importance of their eigenvectors. In this paper, we propose a technique to solve this problem; we use the Subspace Scoring with the Fisher Criterion to rerank the discriminative power of the subspace spanned by certain eigenvectors. The experimental results show that our method makes an improvement to 2DLDA and (2D)2LDA in accuracy. We also combine our proposed method with the wrapper method to determine the target dimension for further use.

The tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border

In this paper, we propose an efficient method for R-peak detection in noninvasive fetal electrocardiogram (ECG) signals which are acquired from multiple electrodes on mother's abdomen. The proposed method is performed in two steps: first, we employ a robust tensor decomposition-based method for fetal ECG extraction, assuming different heart rates for mother and fetal ECG; then a method based on extended Kalman filter (EKF) in which the ECG beat is modeled by 3 state equations (P, QRS and T), is used for fetal R-peak detection. The results show that the proposed method is efficiently able to estimate the location of R-peaks of fetal ECG signals. The obtained average scores of event 4 and 5 on the set B of “Physionet Challenge 2013” data are 1326.21 and 45.06, respectively, which are better than the average score for “sample submission physionet2013.m” (available at PhysioNet) on set B which were 3258.56 and 102.75.

In this paper, a tensor decomposition approach combined with Chebyshev spectral differentiation is presented to solve the high dimensional transient Fokker-Planck equations (FPE) arising in the simulation of polymeric fluids via multi-bead-spring (MBS) model. Generalizing the authors' previous work on the stationary FPE, the transient solution is obtained in a single CANDECOMP/PARAFAC decomposition (CPD) form for all times via the alternating least squares algorithm. This is accomplished by treating the temporal dimension in the same manner as all other spatial dimensions, thereby decoupling it from them. As a result, the transient solution is obtained without resorting to expensive time stepping schemes. A new, relaxed approach for imposing the vanishing boundary conditions is proposed, improving the quality of the approximation. The asymptotic behavior of the temporal basis functions is studied. The proposed solver scales very well with the dimensionality of the MBS model. Numerical results for systems up to 14 dimensional state space are successfully obtained on a regular personal computer and compared with the corresponding matrix Riccati differential equation (for linear models) or Monte Carlo simulations (for nonlinear models)

— A tensor decomposition approach combined with Chebyshev spectral differentiation is developed to solve the transient Fokker-Planck equation (FPE) in high dimensional cases. This method drastically reduces the degrees of freedom required to maintain accuracy in the approximation as dimensionality increases. The transient solution is sought in a single CANDECOMP/PARAFAC decomposition form for all times by the alternating least squares algorithm. This is accomplished by decoupling the spatial dimensions as well as the separation of temporal domain from spatial domain. As a result, the transient solution is obtained without resorting to expensive time-stepping schemes. In addition, a new approach for imposing the vanishing boundary condition in the tensor product framework is proposed, improving the quality of the approximation. Numerical transient solutions for systems up to 14 dimensional state space are successfully obtained on a regular personal computer to demonstrate the advantages of the proposed method.

The Kronecker product is a key algorithm and is ubiquitous across the physical, biological, and computation social sciences. Thus considerations of optimal implementation are important. The need to have high performance and computational reproducibility is paramount. Moreover, due to the need to compose multiple Kronecker products, issues related to data structures, layout and indexing algebra require a new look at an old problem. This paper discusses the outer product/tensor product and a special case of the tensor product: the Kronecker product, along with optimal implementation when composed, and mapped to complex processor/memory hierarchies. We discuss how the use of ``A Mathematics of Arrays" (MoA), and the psi-Calculus, (a calculus of indexing with shapes), provides optimal, verifiable, reproducible, scalable, and portable implementations of both hardware and software.