Parallel Tensor Research Articles

Compact locally homogeneous manifolds with parallel Weyl tensor

Abstract We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak. Previously known examples of compact ECS manifolds, in every dimension greater than 4, were all of rank 1, geodesically complete, and none of them was locally homogeneous. In contrast, our new examples — all of them geodesically incomplete — realize all odd dimensions starting from 5 and are of rank 2, as well as locally homogeneous.

Characterization of Bach and Cotton Tensors on a Class of Lorentzian Manifolds

In this work, we aim to investigate the characteristics of the Bach and Cotton tensors on Lorentzian manifolds, particularly those admitting a semi-symmetric metric ω-connection. First, we prove that a Lorentzian manifold admitting a semi-symmetric metric ω-connection with a parallel Cotton tensor is quasi-Einstein and Bach flat. Next, we show that any quasi-Einstein Lorentzian manifold admitting a semi-symmetric metric ω-connection is Bach flat.

Tensor approximation of functional differential equations.

Functional differential equations(FDEs) play a fundamental role in many areas of mathematical physics, including fluid dynamics (Hopf characteristic functional equation), quantum field theory (Schwinger-Dyson equations), and statistical physics. Despite their significance, computing solutions to FDEs remains a longstanding challenge in mathematical physics. In this paper we address this challenge by introducing approximation theory and high-performance computational algorithms designed for solving FDEs on tensor manifolds. Our approach involves approximating FDEs using high-dimensional partial differential equations(PDEs), and then solving such high-dimensional PDEs on a low-rank tensor manifold leveraging high-performance (parallel) tensor algorithms. The effectiveness of the proposed approach is demonstrated through its application to the Burgers-Hopf FDE, which governs the characteristic functional of the stochastic solution to the Burgers equationevolving from a random initial state.

Personalized Multimedia Recommendation Systems Using Higher-Order Tensor Singular-Value-Decomposition

Because of rapid growth of multimedia data over the Internet, the infobesity has been emerging in recent years. Many recommender systems (RSs) have been proposed using a variety of techniques, including artificial intelligence, machine learning, statistical analysis, and data mining. Nowadays, E-commerce companies such as Amazon, eBay, Netflix, etc. have applied RSs conveniently to provide better customer services than before by enabling customized multimedia-content recommendations. It is well known that classical RSs are almost all based on the user-item attribute matrices. However, this two-fold user-item relationship should be extended to the emerging many-fold relationship among an arbitrary number of attribute groups so that the corresponding new RSs may lead to better performances. Therefore, in this paper, we introduce a novel personalized multimedia recommender system (PMRS) according to our proposed new parallel higher-order tensor singular-value-decomposition (PHOTSVD) algorithm. This PHOTSVD algorithm is capable of accommodating an arbitrary number of entities such as social-media channels, users, queries, webpages, etc. in a robust RS. Furthermore, we also investigate the cold start and data bias problems critical to RSs. Finally, we compare the performances of our proposed new PMRS and three other existing tensor-based RSs over realworld data with respect to the well-adopted metric, namely normalized root-mean-square error (NRMSE). Our proposed new PMRS greatly outperforms the other three existing tensor-based RSs in terms of NRMSE.

Conformal recurrent Kӓhler spaces

In this paper we study pseudo-Riemannian spaces with recurrent tensor of conformal curvature, which admit a Kähler structure. It is proved that Kähler conformally recurrent spaces other than recurrent spaces do not exist, if their dimension is four. Recurrent Kähler spaces are divided into two types. For each type, the internal necessary characteristic is given. Some properties of four-dimensional Kähler conformally recurrent Kähler spaces are studied.

The topology of compact rank-one ECS manifolds

AbstractPseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric, also known as essentially conformally symmetric (ECS) manifolds, have a natural local invariant, the rank, which equals 1 or 2, and is the rank of a certain distinguished null parallel distribution $\mathcal{D}$. All known examples of compact ECS manifolds are of rank one and have dimensions greater than 4. We prove that a compact rank-one ECS manifold, if not locally homogeneous, replaced when necessary by a twofold isometric covering, must be a bundle over the circle with leaves of $\mathcal{D}^\perp$ serving as the fibres. The same conclusion holds in the locally homogeneous case if one assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show that in the pseudo-Riemannian universal covering space of any compact rank-one ECS manifold, the leaves of $\mathcal{D}^\perp$ are the factor manifolds of a global product decomposition.

On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds

We introduce new metric structures on a smooth manifold (called “weak” structures) that generalize the almost contact, Sasakian, cosymplectic, etc. metric structures (φ,ξ,η,g) and allow us to take a fresh look at the classical theory and find new applications. This assertion is illustrated by generalizing several well-known results. It is proved that any Sasakian structure is rigid, i.e., our weak Sasakian structure is homothetically equivalent to a Sasakian structure. It is shown that a weak almost contact structure with parallel tensor φ is a weak cosymplectic structure and an example of such a structure on the product of manifolds is given. Conditions are found under which a vector field is a weak contact vector field.

A 28-nm 8-bit Floating-Point Tensor Core-Based Programmable CNN Training Processor With Dynamic Structured Sparsity

Training deep/convolutional neural networks (DNNs/CNNs) requires a large amount of memory and iterative computation, which necessitates speedup and energy reduction, especially for edge devices with resource/energy constraints. In this work, we present an 8-bit floating-point (FP8) training the processor which implements: 1) highly parallel tensor cores (fused multiply–add trees) that maintain high utilization throughout forward propagation (FP), backward propagation (BP), and weight update (WU) phases of the training process; 2) hardware-efficient channel gating for dynamic output activation sparsity; 3) dynamic weight sparsity (WS) based on group Lasso; and 4) gradient skipping based on the FP prediction error. We develop a custom instruction set architecture (ISA) to flexibly support different CNN topologies and training parameters. The 28-nm prototype chip demonstrates large improvements in floating-point operations (FLOPs) reduction (7.3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times$</tex-math> </inline-formula> ), energy efficiency (16.4 TFLOPS/W), and overall training latency speedup (4.7 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times$</tex-math> </inline-formula> ), for both supervised and self-supervised training tasks.

A fast parallel tensor decomposition with optimal stochastic gradient descent: an application in structural damage identification

AbstractStructural health monitoring (SHM) provides an economic approach which aims to enhance understanding the behavior of structures by continuously collecting data through multiple networked sensors attached to the structure. These data are then utilized to gain insight into the health of a structure and make timely and economic decisions about its maintenance. The generated SHM sensing data are non-stationary and exist in a correlated multi-way form which makes the batch/off-line learning and standard two-way matrix analysis unable to capture all of these correlations and relationships. In this sense, the online tensor data analysis has become an essential tool for capturing underlying structures in higher-order datasets stored in a tensor $${\mathcal {X}} \in {\mathbb {R}} ^{I_1 \times \cdots \times I_N} $$ X ∈ R I 1 × ⋯ × I N . The CANDECOMP/PARAFAC (CP) decomposition has been extensively studied and applied to approximate X by N loading matrices $$A(1),\ldots ,A(N)$$ A ( 1 ) , … , A ( N ) where N represents the order of the tensor. We propose a novel algorithm, FP-CPD, to parallelize the CANDECOMP/PARAFAC (CP) decomposition of a tensor $${\mathcal {X}} \in {\mathbb {R}} ^{I_1 \times \cdots \times I_N} $$ X ∈ R I 1 × ⋯ × I N . Our approach is based on stochastic gradient descent (SGD) algorithm which allows us to parallelize the learning process, and it is very useful in online setting since it updates $${\mathcal {X}}^{t+1}$$ X t + 1 in one single step. Our SGD algorithm is augmented with Nesterov’s accelerated gradient and perturbation methods to accelerate and guarantee convergence. The experimental results using laboratory-based and real-life structural datasets indicate fast convergence and good scalability.

Real-Time Equation-of-Motion Coupled-Cluster Cumulant Green's Function Method: Heterogeneous Parallel Implementation Based on the Tensor Algebra for Many-Body Methods Infrastructure.

We report the implementation of the real-time equation-of-motion coupled-cluster (RT-EOM-CC) cumulant Green's function method [ J. Chem. Phys. 2020, 152, 174113] within the Tensor Algebra for Many-body Methods (TAMM) infrastructure. TAMM is a massively parallel heterogeneous tensor library designed for utilizing forthcoming exascale computing resources. The two-body electron repulsion matrix elements are Cholesky-decomposed, and we imposed spin-explicit forms of the various operators when evaluating the tensor contractions. Unlike our previous real algebra Tensor Contraction Engine (TCE) implementation, the TAMM implementation supports fully complex algebra. The RT-EOM-CC singles (S) and doubles (D) time-dependent amplitudes are propagated using a first-order Adams-Moulton method. This new implementation shows excellent scalability tested up to 500 GPUs using the Zn-porphyrin molecule with 655 basis functions, with parallel efficiencies above 90% up to 400 GPUs. The TAMM RT-EOM-CCSD was used to study core photoemission spectra in the formaldehyde and ethyl trifluoroacetate (ESCA) molecules. Simulations of the latter involve as many as 71 occupied and 649 virtual orbitals. The relative quasiparticle ionization energies and overall spectral functions agree well with available experimental results.

End-to-end data-dependent routing in multi-path neural networks

Neural networks are known to give better performance with increased depth due to their ability to learn more abstract features. Although the deepening of networks has been well established, there is still room for efficient feature extraction within a layer, which would reduce the need for mere parameter increment. The conventional widening of networks by having more filters in each layer introduces a quadratic increment of parameters. Having multiple parallel convolutional/dense operations in each layer solves this problem, but without any context-dependent allocation of input among these operations: The parallel computations tend to learn similar features making the widening process less effective. Therefore, we propose the use of multi-path neural networks with data-dependent resource allocation from parallel computations within layers, which also lets an input be routed end-to-end through these parallel paths. To do this, we first introduce a cross-prediction-based algorithm between parallel tensors of subsequent layers. Second, we further reduce the routing overhead by introducing feature-dependent cross-connections between parallel tensors of successive layers. Using image recognition tasks, we show that our multi-path networks show superior performance to existing widening and adaptive feature extraction, even ensembles and deeper networks at similar complexity.

Anisotropic pressure effects on nanoconfined water within narrow graphene slit pores.

There is an increasing interest toward disclosing and explaining confinement effects on liquids, such as water or aqueous solutions, in slit pore setups. Particularly puzzling are the changes of physical and chemical properties in the nanoconfinement regime where no bulk-like water phase exists between the two interfacial water layers such that the density profile across the slit pore becomes highly stratified, ultimately leading to bilayer and monolayer water. These changes must be quantified with respect to some meaningful reference state of water, the most natural one being bulk water at the same pressure and temperature conditions. However, bulk water is a homogeneous liquid with isotropic properties, whereas water confined in slit pores is inhomogeneous, implying anisotropic properties as described by the perpendicular and parallel components of the respective tensors. In the case of pressure, the inhomogeneous nature of the setup results in a well-defined difference between the perpendicular and parallel pressure tensor components that is uniquely determined by the interfacial tension being a thermodynamic property. For bilayer water constrained in graphene slit pores that are only about 1 nm wide, we demonstrate that there exists a thermodynamic point where the pressure tensor of the inhomogeneous fluid, nanoconfined water, is effectively isotopic and the pressure is thus scalar as in the homogeneous fluid, bulk water. This specific point of vanishing effective interfacial tension is proposed to serve as a well-defined reference state to compare the properties of nanoconfined liquids to those of the corresponding bulk liquid at the same (isotropic) pressure and temperature conditions. In future work, this idea could be applied to assess confinement effects on chemical reactivity in aqueous solutions as well as to other nanoconfined liquids in other pores such as layered minerals.

Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

We study the covariant derivatives of an eigenfunction for the Laplace–Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.

Vanishing results from Lichnerowicz Laplacian on complete Kähler manifolds and applications

In this paper, we show several rigidity results for harmonic (p,q)-forms in complete Kähler manifolds. We also give several applications to study non-compact Kähler manifolds with parallel Bochner tensor or quaternion Kähler manifolds. Our results are natural extensions of Petersen and Wink's results in [13,15] in the setting of complete, non-compact Kähler manifolds.

A study on K-paracontact and (k,μ)-paracontact manifold admitting vanishing Cotton tensor and Bach tensor

"The object of the present paper is to study K-paracontact manifold admitting parallel Cotton tensor, vanishing Cotton tensor and to study Bach flatness on K-paracontact manifold. In that we prove for a K-paracontact metric manifold M^{2n+1} has parallel Cotton tensor if and only if M^{2n+1} is an η-Einstein manifold and r=-2n(2n+1). Further we show that if g is an η-Einstein K-paracontact metric and if g is Bach flat then g is an Einstein. Also we study vanishing Cotton tensor on (k,μ)-paracontact manifold for both k&gt;-1 and k&lt;-1. Finally, we prove that if M^{2n+1} is a (k,μ)-paracontact manifold for k ≠ -1 and if M^{2n+1} has vanishing Cotton tensor for μ ≠ k, then M^{2n+1} is an η-Einstein manifold."

Perfect Fluid Spacetimes and Gradient Solitons

In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a perfect fluid spacetime admitting concircular vector field, we prove that the velocity vector field annihilates the conformal curvature tensor. In addition, in dimension 4, we show that a perfect fluid spacetime is a generalized Robertson–Walker spacetime with Einstein fibre. It is proved that if a perfect fluid spacetime furnished with concircular vector field admits a second order symmetric parallel tensor P, then either the equation of state of the perfect fluid spacetime is characterized by p=frac{3-n}{n-1} sigma , or the tensor P is a constant multiple of the metric tensor. Finally, The perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons, and gradient m -quasi Einstein solitons, are characterized.

On Submanifolds of the Unit Sphere with Vanishing Möbius Form and Parallel Para-Blaschke Tensor

On Submanifolds of the Unit Sphere with Vanishing Möbius Form and Parallel Para-Blaschke Tensor

A Tensor SVD-based Classification Algorithm Applied to fMRI Data

To analyze the abundance of multidimensional data, tensor-based frameworks have been developed. Traditionally, the matrix singular value decomposition (SVD) is used to extract the most dominant features from a matrix containing the vectorized data. While the SVD is highly useful for data that can be appropriately represented as a matrix, this step of vectorization causes us to lose the high-dimensional relationships intrinsic to the data. To facilitate efficient multidimensional feature extraction, we utilize a projection-based classification algorithm using the t-SVDM, a tensor analog of the matrix SVD. Our work extends the t-SVDM framework and the classification algorithm, both initially proposed for tensors of order 3, to any number of dimensions. We then apply this algorithm to a classification task using the StarPlus fMRI dataset. Our numerical experiments demonstrate that there exists a superior tensor-based approach to fMRI classification than the best possible equivalent matrix-based approach. Our results illustrate the advantages of our chosen tensor framework, provide insight into beneficial choices of parameters, and could be further developed for classification of more complex imaging data. We provide our Python implementation at https://github.com/elizabethnewman/tensor-fmri.

On pull-backs of the universal connection

AbstractNarasihman and Ramanan proved in [Amer. J. Math. 83(1961), 563–572] that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to relate geometric properties of the classifying map to geometric properties of the pulled-back connection. More specifically, we describe conditions on the classifying map under which the pulled-back connection: (1) is fat (in the sphere bundle), (2) has a parallel curvature tensor, and (3) induces a connection metric with nonnegative sectional curvature on the vector bundle (or positive sectional curvature on the sphere bundle).

Parallel tensor decomposition with distributed memory based on hierarchical singular value decomposition

AbstractAs an important tool of multiway/tensor data analysis tool, Tucker decomposition has been applied widely in various fields. But traditional sequential Tucker algorithms have been outdated because tensor data is growing rapidly in term of size. To address this problem, in this article, we focus on parallel Tucker decomposition of dense tensors on distributed‐memory systems. The proposed method uses hierarchical SVD to accelerate the SVD step in traditional sequential algorithms, which usually takes up most computation time. The data distribution strategy is designed to follow the implementation of hierarchical SVD. We also find that compared with the state‐of‐the‐art method, the proposed method has lower communication cost in large‐scale parallel cases under the assumption of the α–β model.