Tensor Framework Research Articles

A massively parallel tensor contraction framework for coupled-cluster computations

Precise calculation of molecular electronic wavefunctions by methods such as coupled-cluster requires the computation of tensor contractions, the cost of which has polynomial computational scaling with respect to the system and basis set sizes. Each contraction may be executed via matrix multiplication on a properly ordered and structured tensor. However, data transpositions are often needed to reorder the tensors for each contraction. Writing and optimizing distributed-memory kernels for each transposition and contraction is tedious since the number of contractions scales combinatorially with the number of tensor indices. We present a distributed-memory numerical library (Cyclops Tensor Framework (CTF)) that automatically manages tensor blocking and redistribution to perform any user-specified contractions. CTF serves as the distributed-memory contraction engine in Aquarius, a new program designed for high-accuracy and massively-parallel quantum chemical computations. Aquarius implements a range of coupled-cluster and related methods such as CCSD and CCSDT by writing the equations on top of a C++ templated domain-specific language. This DSL calls CTF directly to manage the data and perform the contractions. Our CCSD and CCSDT implementations achieve high parallel scalability on the BlueGene/Q and Cray XC30 supercomputer architectures showing that accurate electronic structure calculations can be effectively carried out on top of general distributed-memory tensor primitives.

Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis

The widespread use of multi-sensor technology and the emergence of big datasets has highlighted the limitations of standard flat-view matrix models and the necessity to move towards more versatile data analysis tools. We show that higher-order tensors (i.e., multiway arrays) enable such a fundamental paradigm shift towards models that are essentially polynomial and whose uniqueness, unlike the matrix methods, is guaranteed under verymild and natural conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical backbone, data analysis techniques using tensor decompositions are shown to have great flexibility in the choice of constraints that match data properties, and to find more general latent components in the data than matrix-based methods. A comprehensive introduction to tensor decompositions is provided from a signal processing perspective, starting from the algebraic foundations, via basic Canonical Polyadic and Tucker models, through to advanced cause-effect and multi-view data analysis schemes. We show that tensor decompositions enable natural generalizations of some commonly used signal processing paradigms, such as canonical correlation and subspace techniques, signal separation, linear regression, feature extraction and classification. We also cover computational aspects, and point out how ideas from compressed sensing and scientific computing may be used for addressing the otherwise unmanageable storage and manipulation problems associated with big datasets. The concepts are supported by illustrative real world case studies illuminating the benefits of the tensor framework, as efficient and promising tools for modern signal processing, data analysis and machine learning applications; these benefits also extend to vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker decomposition, HOSVD, tensor networks, Tensor Train.

Multilinear Approximation of Nonlinear State Space Models

There are numerical tools for representation, editing and simulation of continuous-time state space models, e.g. numerous block-based tools like SIMULINK. Algorithms for numerical and symbolical linearization are available but structurally fail to compute models where nonlinear effects are essential. The paper presents a numerical method to find a multilinear model as approximation which is valid in appropriate regions usually given by the engineering application. The approximate model belongs to the class of Multilinear Time Invariant systems (MTI systems) which can be represented within a tensor framework and thus, tensor decomposition methods are applicable. The proposed approximation algorithm is applied to a test case of a chemical system, which is inherently nonlinear and simulation results show the improvement of the approximation of the behaviour compared to a linear model.

Extraction of Surface-feature Lines on Meshes Using Normal Tensor Framework

Computer-Aided Design and Applications is an international journal on the applications of CAD and CAM. It publishes papers in the general domain of CAD plus in emerging fields like bio-CAD, nano-CAD, soft-CAD, garment-CAD, PLM, PDM, CAD data mining, CAD and the internet, CAD education, genetic algorithms and CAD engines. The journal is aimed at all developers and users of CAD technology to ptovide CAD solutions for various stages of design and manufacturing. The journal publishes all about Computer-Aided Design and Computer-Aided technologies.

Determining the bending modulus of a lipid membrane by simulating buckling

The force needed to buckle a thin elastic surface is proportional to its bending rigidity. This fact suggests using a buckling setup to measure the bending modulus of lipid membranes. Extending the work of Noguchi [Phys. Rev. E 83, 061919 (2011)], we systematically derive highly accurate analytical expressions for the forces along and perpendicular to the buckle, and we elucidate some of their counterintuitive properties using the framework of a surface stress tensor. Furthermore, we estimate the corrections to buckling forces due to thermal fluctuations and find them significant only for stresses along the ridges. We then apply this buckling protocol to four different lipid membrane models, which widely differ in their level of resolution and the treatment of solvent, and show that in all cases buckling is a reliable and accurate means for measuring their rigidity. Finally, we show that monitoring both stresses and energies during a simulation offers additional insights into the thermodynamics of curvature elasticity and permits one to predict the bending rigidity for a range of temperatures around the actual simulation temperature.

White matter fiber tractography: why we need to move beyond DTI

Diffusion-based MRI tractography is an imaging tool increasingly used in neurosurgical procedures to generate 3D maps of white matter pathways as an aid to identifying safe margins of resection. The majority of white matter fiber tractography software packages currently available to clinicians rely on a fundamentally flawed framework to generate fiber orientations from diffusion-weighted data, namely diffusion tensor imaging (DTI). This work provides the first extensive and systematic exploration of the practical limitations of DTI-based tractography and investigates whether the higher-order tractography model constrained spherical deconvolution provides a reasonable solution to these problems within a clinically feasible timeframe. Comparison of tractography methodologies in visualizing the corticospinal tracts was made using the diffusion-weighted data sets from 45 healthy controls and 10 patients undergoing presurgical imaging assessment. Tensor-based and constrained spherical deconvolution-based tractography methodologies were applied to both patients and controls. Diffusion tensor imaging-based tractography methods (using both deterministic and probabilistic tractography algorithms) substantially underestimated the extent of tracks connecting to the sensorimotor cortex in all participants in the control group. In contrast, the constrained spherical deconvolution tractography method consistently produced the biologically expected fan-shaped configuration of tracks. In the clinical cases, in which tractography was performed to visualize the corticospinal pathways in patients with concomitant risk of neurological deficit following neurosurgical resection, the constrained spherical deconvolution-based and tensor-based tractography methodologies indicated very different apparent safe margins of resection; the constrained spherical deconvolution-based method identified corticospinal tracts extending to the entire sensorimotor cortex, while the tensor-based method only identified a narrow subset of tracts extending medially to the vertex. This comprehensive study shows that the most widely used clinical tractography method (diffusion tensor imaging-based tractography) results in systematically unreliable and clinically misleading information. The higher-order tractography model, using the same diffusion-weighted data, clearly demonstrates fiber tracts more accurately, providing improved estimates of safety margins that may be useful in neurosurgical procedures. We therefore need to move beyond the diffusion tensor framework if we are to begin to provide neurosurgeons with biologically reliable tractography information.

Facial Recognition Using Tensor-Tensor Decompositions

A tensor is a multidimensional array. First-order tensors and second-order tensors can be viewed as vectors and matrices, respectively. Tensors of higher order, with the ability to include more information, appear more frequently nowadays in image and signal processing, data mining, biomedical engineering, and so on. With the recent work of Kilmer and Martin, familiar matrix-based factorizations in linear algebra can be extended in a straightforward way to third-order tensors based on their new tensor multiplication and concepts. Our method has an advantage over a popular tensor-based face recognition algorithm called TensorFaces, which is based on the higher-order SVD of an arrangement of the images in a database, in that it does not require a least squares solve for the coefficients. In this paper, we give a brief introduction to the new tensor framework and apply the induced tensor decompositions to the application of facial recognition. In the numerical results, we compare our new approaches with the traditional principal component analysis method based on matrix decomposition (also known as eigenfaces) and with TensorFaces.

Tensor-based methods for numerical homogenization from high-resolution images

We present a complete numerical strategy based on tensor approximation techniques for the solution of numerical homogenization problems with geometrical data coming from high resolution images. We first introduce specific numerical treatments for the translation of image-based homogenization problems into a tensor framework. It includes the tensor approximations in suitable tensor formats of fields of material properties or indicator functions of multiple material phases recovered from segmented images. We then introduce some variants of proper generalized decomposition (PGD) methods for the construction of tensor decompositions in different tensor formats of the solution of boundary value problems. A new definition of PGD is introduced which allows the progressive construction of a Tucker decomposition of the solution. This tensor format is well adapted to the present application and improves convergence properties of tensor decompositions. Finally, we use a dual-based error estimator on quantities of interest which was recently introduced in the context of PGD. We exhibit its specificities when it is used for assessing the error on the homogenized properties of the heterogeneous material. We also provide a complete goal-oriented adaptive strategy for the progressive construction of tensor decompositions (of primal and dual solutions) yielding to predictions of homogenized quantities with a prescribed accuracy.

Surface mesh denoising with normal tensor framework

In this paper, we propose a novel method for feature-preserving mesh denoising based on the normal tensor framework. We utilize the normal tensor voting directly for the mesh denoising whose eigenvalues and eigenvectors are used for detecting saliency, and introduce an algorithm that updates a vertex by the Laplacian of curvature which minimizes a difference of the curvature in one neighborhood. By connecting the feature saliency with a distance metric in the normal tensor space, our algorithm preserves sharp features more robustly and clearly for noisy mesh data. Comparing our method with the existing ones, we demonstrate the effectiveness of our algorithm against some synthetic noisy data and real-world scanned data.

A Tensor Framework for Nonunitary Joint Block Diagonalization

This paper introduces a tensor framework to solve the problem of nonunitary joint block diagonalization (JBD) of a set of real or complex valued matrices. We show that JBD can be seen as a particular case of the block-component-decomposition (BCD) of a third-order tensor. The resulting tensor model fitting problem does not require the block-diagonalizer to be a square matrix: the over- and underdetermined cases can be handled. To compute the tensor decomposition, we build an efficient nonlinear conjugate gradient (NCG) algorithm. In the over- and exactly determined cases, we show that exact JBD can be computed by a closed-form solution based on eigenvalue analysis. In approximate JBD problems, this solution can be used to efficiently initialize any iterative JBD algorithm such as NCG. Finally, we illustrate the performance of our technique in the context of independent subspace analysis (ISA) based on second-order statistics (SOS).

A new internally contracted multi-reference configuration interaction method

We present a new internally contracted multi-reference configuration interaction (MRCI) method which, at the same time, efficiently handles large active orbital spaces, long configuration expansions, and many closed-shell orbitals in the reference function. This is achieved by treating the closed-shell orbitals explicitly, so that all required coupling coefficients and density matrices only depend on active orbital labels. As a result, closed-shell orbitals are handled as efficiently as in a closed-shell single-reference program, and this opens up the possibility to perform high-accuracy MRCI calculations for much larger molecules than before. The enormously complex equations are derived using a new domain-specific computer algebra system and semi-automatically implemented using a newly developed integrated tensor framework. The accuracy and efficiency of the MRCI method is demonstrated with applications to dioxygen-copper complexes with different ligands, some of which involve more than 30 atoms, and to spin-state splittings of ferrocene.

Spectra of symmetric tensors and m-root Finsler models

The present work studies spectral properties of multilinear forms attached to the Berwald-Moor, Chernov and Bogoslovsky locally Minkowski Finsler geometric structures of m-root type. We determine eigenvalues and the corresponding eigenvectors (of type Z, H and E) of these forms, in the framework of symmetric tensors and multivariate homogeneous polynomials. The geometric relevance of the spectral data is emphasized, and the existent relations between spectra, polyangles and Riesz-type associated 1-forms of the corresponding geometric models, are described. As well, the best rank-one approximation for the 4-dimensional Berwald-Moor and Chernov cases, is derived.

TWave: High-order analysis of functional MRI

The traditional approach to functional image analysis models images as matrices of raw voxel intensity values. Although such a representation is widely utilized and heavily entrenched both within neuroimaging and in the wider data mining community, the strong interactions among space, time, and categorical modes such as subject and experimental task inherent in functional imaging yield a dataset with "high-order" structure, which matrix models are incapable of exploiting. Reasoning across all of these modes of data concurrently requires a high-order model capable of representing relationships between all modes of the data in tandem. We thus propose to model functional MRI data using tensors, which are high-order generalizations of matrices equivalent to multidimensional arrays or data cubes. However, several unique challenges exist in the high-order analysis of functional medical data: naïve tensor models are incapable of exploiting spatiotemporal locality patterns, standard tensor analysis techniques exhibit poor efficiency, and mixtures of numeric and categorical modes of data are very often present in neuroimaging experiments. Formulating the problem of image clustering as a form of Latent Semantic Analysis and using the WaveCluster algorithm as a baseline, we propose a comprehensive hybrid tensor and wavelet framework for clustering, concept discovery, and compression of functional medical images which successfully addresses these challenges. Our approach reduced runtime and dataset size on a 9.3GB finger opposition motor task fMRI dataset by up to 98% while exhibiting improved spatiotemporal coherence relative to standard tensor, wavelet, and voxel-based approaches. Our clustering technique was capable of automatically differentiating between the frontal areas of the brain responsible for task-related habituation and the motor regions responsible for executing the motor task, in contrast to a widely used fMRI analysis program, SPM, which only detected the latter region. Furthermore, our approach discovered latent concepts suggestive of subject handedness nearly 100× faster than standard approaches. These results suggest that a high-order model is an integral component to accurate scalable functional neuroimaging.

Robust tensor subspace learning for anomaly detection

Background modeling plays an important role in many applications of computer vision such as anomaly detection and visual tracking. Most existing algorithms for learning appearance model are vector-based methods without maintaining the 2D spatial structure information of objects in an image. To this end, a robust tensor subspace learning algorithm is developed for background modeling which can capture the appearance changes through adaptively updating the tensor subspace. In the tensor framework, the spatial structure information is maintained and utilized for feature extraction of objects. Then by incorporating the robust scheme, we can weight individual pixel of an image to reduce the influence of outliers on background modeling. Furthermore an incremental algorithm for the robust tensor subspace learning is proposed to adapt to the variation of appearance model. The experimental results illustrate the effectiveness of the proposed robust learning algorithm for anomaly detection.

Longitudinal detection of ferromagnetic resonance using x-ray transmission measurements

We describe a setup for the x-ray detection of ferromagnetic resonance in the longitudinal geometry using element-specific transmission measurements. Thin magnetic film samples are placed in a static magnetic field collinear with the propagation direction of a polarized soft x-ray beam and driven to ferromagnetic resonance by a continuous wave microwave magnetic field perpendicular to it. The transmitted photon flux is measured both as a function of the x-ray photon energy and as a function of the applied static magnetic field. We report experiments performed on a 15 nm film of doped Permalloy (Ni(73)Fe(18)Gd(7)Co(2)) at the L(3)/L(2)-edges of Fe, Co, and Ni. The achieved ferromagnetic resonance sensitivity is about 0.1 monolayers/square root(Hz). The obtained results are interpreted in the framework of a conductivity tensor based formalism. The factors limiting the sensitivity as well as different approaches for the x-ray detection of ferromagnetic resonance are discussed.

Tensor-Based Transductive Learning for Multimodality Video Semantic Concept Detection

Interaction and integration of multimodality media types such as visual, audio, and textual data in video are the essence of video semantic analysis. Contextual information propagation is useful for both intra- and inter-shot correlations. However, the traditional concatenated vector representation of videos weakens the power of the propagation and compensation among the multiple modalities. In this paper, we introduce a higher-order tensor framework for video analysis. We represent image frame, audio, and text in video shots as data points by the 3rd-order tensor. Then we propose a novel dimension reduction algorithm which explicitly considers the manifold structure of the tensor space from contextual temporal associated cooccurring multimodal media data. Our algorithm inherently preserves the intrinsic structure of the sub- manifold where tensorshots are sampled and is also able to map out-of-sample data points directly. We propose a new transductive support tensor machines algorithm to train effective classifier using large amount of unlabeled data together with the labeled data. Experiment results on TREVID 2005 data set show that our method improves the performance of video semantic concept detection.

Transductive Multi-Modality Video Semantic Concept Detection with Tensor Representation

提出了一种基于高阶张量表示的视频语义分析与理解框架.在此框架中,视频镜头首先被表示成由视频中所包含的文本、视觉和听觉等多模态数据构成的三阶张量;其次,基于此三阶张量表达及视频的时序关联共生特性设计了一种子空间嵌入降维方法,称为张量镜头;由于直推式学习从已知样本出发能对特定的未知样本进行学习和识别,最后在这个框架中提出了一种基于张量镜头的直推式支持张量机算法,它不仅保持了张量镜头所在的流形空间的本征结构,而且能够将训练集合外数据直接映射到流形子空间,同时充分利用未标记样本改善分类器的学习性能.实验结果表明,该方法能够有效地进行视频镜头的语义概念检测.;A higher-order tensor framework for video analysis and understanding is proposed in this paper. In this framework, image frame, audio and text are represented, which are the three modalities in video shots as data points by the 3rd-order tensor. Then a subspace embedding and dimension reduction method is proposed, which explicitly considers the manifold structure of the tensor space from temporal-sequenced associated co-occurring multimodal media data in video. It is called TensorShot approach. Transductive learning uses a large amount of unlabeled data together with the labeled data to build better classifiers. A transductive support tensor machines algorithm is proposed to train effective classifier. This algorithm preserves the intrinsic structure of the submanifold where tensorshots are sampled, and is also able to map out-of-sample data points directly. Moreover, the utilization of unlabeled data improves classification ability. Experimental results show that this method improves the performance of video semantic concept detection.

A unified tensor framework for face recognition

In this paper we propose a new optimization framework that unites some of the existing tensor based methods for face recognition on a common mathematical basis. Tensor based approaches rely on the ability to decompose an image into its constituent factors (i.e. person, lighting, viewpoint, etc.) and then utilizing these factor spaces for recognition. We first develop a multilinear optimization problem relating an image to its constituent factors and then develop our framework by formulating a set of strategies that can be followed to solve this optimization problem. The novelty of our research is that the proposed framework offers an effective methodology for explicit non-empirical comparison of the different tensor methods as well as providing a way to determine the applicability of these methods in respect to different recognition scenarios. Importantly, the framework allows the comparative analysis on the basis of quality of solutions offered by these methods. Our theoretical contribution has been validated by extensive experimental results using four benchmark datasets which we present along with a detailed discussion.

Tensor Framework and Combined Symmetry for Hypertext Mining

We have made a case here for utilizing tensor framework for hypertext mining. Tensor is a generalization of vector and tensor framework discussed here is a generalization of vector space model which is widely used in the information retrieval and web mining literature. Most hypertext documents have an inherent internal tag structure and external link structure that render the desirable use of multidimensional representations such as those offered by tensor objects. We have focused on the advantages of Tensor Space Model, in which documents are represented using sixth-order tensors. We have exploited the local-structure and neighborhood recommendation encapsulated by the proposed representation. We have defined a similarity measure for tensor objects corresponding to hypertext documents, and evaluated the proposed measure for mining tasks. The superior performance of the proposed methodology for clustering and classification tasks of hypertext documents have been demonstrated here. The experiment using different types of similarity measure in the different components of hypertext documents provides the main advantage of the proposed model. It has been shown theoretically that, the computational complexity of an algorithm performing on tensor framework using tensor similarity measure as distance is at most the computational complexity of the same algorithmperforming on vector space model using vector similarity measure as distance.

A Continuous STAPLE for Scalar, Vector, and Tensor Images: An Application to DTI Analysis

The comparison of images of a patient to a reference standard may enable the identification of structural brain changes. These comparisons may involve the use of vector or tensor images (i.e., 3-D images for which each voxel can be represented as an RN vector) such as diffusion tensor images (DTI) or transformations. The recent introduction of the Log-Euclidean framework for diffeomorphisms and tensors has greatly simplified the use of these images by allowing all the computations to be performed on a vector-space. However, many sources can result in a bias in the images, including disease or imaging artifacts. In order to estimate and compensate for these sources of variability, we developed a new algorithm, called continuous STAPLE, that estimates the reference standard underlying a set of vector images. This method, based on an expectation-maximization method similar in principle to the validation method STAPLE, also estimates for each image a set of parameters characterizing their bias and variance with respect to the reference standard. We demonstrate how to use these parameters for the detection of atypical images or outliers in the population under study. We identified significant differences between the tensors of diffusion images of multiple sclerosis patients and those of control subjects in the vicinity of lesions.