Parallel Tensor Research Articles

Second order parallel tensors on some paracontact manifolds

The object of the present paper is to study the symmetric and skew-symmetric properties of a second order parallel tensor on paracontact metric (k, µ)-spaces and almost β-para-Kenmotsu (k, µ)-spaces. In this paper, we prove that if there exists a second order symmetric parallel tensor on a paracontact metric (k, µ)-space M, then either M is locally isometric to a product of a flat n + 1-dimensional manifold and an n-dimensional manifold of constant sectional curvature −4, or the second order parallel tensor is a constant multiple of the associated metric tensor g of M2n+1. If there is a second order parallel tensor on an almost β-para-Kenmotsu (k, µ)-space with k ≠ 0, then it is a constant multiple of the associated metric tensor g of M2n+1.

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.

Spatial and semantical label inference for social media

Exploring the spatial and semantical knowledge from messages in social media offers us an opportunity to get a deeper understanding about the mobility and activity of users, which can be leveraged to improve the service quality of online applications like recommender systems. In this paper, we investigate the problem of the spatial and semantical label inference, where the challenges come from three aspects: diverse heterogeneous information, uncertainty of individual mobility, and large-scale sparse data. We address the challenges by exploring two types of data fusion, the fusion of heterogeneous social networks and the fusion of heterogeneous features. We build a 4-dimensional tensor, called spatial---temporal semantical tensor (STST), to model the individual mobility and activity by fusing two heterogeneous social networks, a social media network and a location-based social network (LBSN). To address the challenge arising from diverse heterogeneous information and the uncertainty of individual mobility, we construct three types of heterogeneous features and fuse them with STST by exploring their interdependency relationships. Particularly, a spatial tendency feature is constructed to constrain the inference of individual mobility and reduce the uncertainty. To deal with large-scale sparse data, we propose a parallel contextual tensor factorization (PCTF) to concurrently factorize STST. Finally, we integrate these components into an inference framework, called spatial and semantical label inference SSLI. The results of extensive experiments conducted on real datasets and synthetic datasets verify the effectiveness and efficiency of SSLI.

A high performance data parallel tensor contraction framework: Application to coupled electro-mechanics

The paper presents aspects of implementation of a new high performance tensor contraction framework for the numerical analysis of coupled and multi-physics problems on streaming architectures. In addition to explicit SIMD instructions and smart expression templates, the framework introduces domain specific constructs for the tensor cross product and its associated algebra recently rediscovered by Bonet et al. (2015, 2016) in the context of solid mechanics. The two key ingredients of the presented expression template engine are as follows. First, the capability to mathematically transform complex chains of operations to simpler equivalent expressions, while potentially avoiding routes with higher levels of computational complexity and, second, to perform a compile time depth-first or breadth-first search to find the optimal contraction indices of a large tensor network in order to minimise the number of floating point operations. For optimisations of tensor contraction such as loop transformation, loop fusion and data locality optimisations, the framework relies heavily on compile time technologies rather than source-to-source translation or JIT techniques. Every aspect of the framework is examined through relevant performance benchmarks, including the impact of data parallelism on the performance of isomorphic and nonisomorphic tensor products, the FLOP and memory I/O optimality in the evaluation of tensor networks, the compilation cost and memory footprint of the framework and the performance of tensor cross product kernels. The framework is then applied to finite element analysis of coupled electro-mechanical problems to assess the speed-ups achieved in kernel-based numerical integration of complex electroelastic energy functionals. In this context, domain-aware expression templates combined with SIMD instructions are shown to provide a significant speed-up over the classical low-level style programming techniques.

Tensor and its tucker core: The invariance relationships

SummaryIn one study, Hillar and Lim famously demonstrated that “multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are nondeterministic polynomial‐time hard.” Despite many recent advancements, the state‐of‐the‐art methods for computing such “tensor analogues” still suffer severely from the curse of dimensionality. In this paper, we show that the Tucker core of a tensor, however, retains many properties of the original tensor, including the CANDECOMP/PARAFAC (CP) rank, the border rank, the tensor Schatten quasi norms, and the Z‐eigenvalues. When the core tensor is smaller than the original tensor, this property leads to considerable computational advantages as confirmed by our numerical experiments. In our analysis, we in fact work with a generalized Tucker‐like decomposition that can accommodate any full column‐rank factor matrices.

Parallel second-order tensors on Vaisman manifolds

The aim of this paper is to study the class of parallel tensor fields [Formula: see text] of [Formula: see text]-type in a Vaisman geometry [Formula: see text]. A sufficient condition for the reduction of such symmetric tensors [Formula: see text] to a constant multiple of [Formula: see text] is given by the skew-symmetry of [Formula: see text] with respect to the complex structure [Formula: see text]. As an application of the main result, we prove that certain vector fields on a [Formula: see text]-manifold turn out to be Killing. Also, we connect our main result with the Weyl connection of conformal geometry as well as with possible Ricci solitons in [Formula: see text] manifolds.

Torse-forming η-Ricci solitons in almost paracontact η-Einstein geometry

Torse-forming ?-Ricci solitons are studied in the framework of almost paracontact metric ?-Einstein manifolds. By adding a technical condition, called regularity and concerning with the scalars provided by the two ?-conditions, is obtained a reduction result for the parallel symmetric covariant tensor fields of order two.

Classification of Ricci semisymmetric contact metric manifolds

The object of the present paper is to study Ricci semisymmetric contact metric manifolds. As a consequence of the main result we deduce some important corollaries. Besides these we study contact metric manifolds satisfying the curvature condition Q,R = 0, where Q and R denote the Ricci operator and curvature tensor respectively. Also we study the symmetric properties of a second order parallel tensor in contact metric manifolds. Finally, we give an example to verify the main result.

Certain results on generalized $${\varvec{(k,\,\mu )}}$$ ( k , μ ) -contact metric manifolds

The object of the present paper is to study second order symmetric parallel tensors in generalized \((k,\,\mu )\)-contact metric manifolds and its applications to Ricci solitons. Next, we prove that a generalized \((k,\,\mu )\)-contact metric manifold M admits a Ricci soliton whose potential vector field is the Reeb vector field \(\xi \) if and only if M is a Sasaki–Einstein manifold. Finally, we give some examples of generalized \((k,\,\mu )\)-contact metric manifold.

On three dimensional quasi-Sasakian manifolds

Let M be a 3-dimensional quasi-Sasakian manifold. Olszak [6] proved that M is conformally flat with constant scalar curvature and hence its structure function $\beta$ is constant. We have shown that in such M, a second order symmetric parallel tensor is a constant multiple of the associated metric tensor. A necessary and sufficient condition for such a manifold to be minimal has been obtained. Finally if such M satisfies $R(X,Y).S =0$, then, S has two different non-zero eigen values.

Conformal regular spacelike hypersurfaces in a conformal space $$\mathbb {Q}^{n+1}_1$$ Q 1 n + 1

Let \(x: M \rightarrow \mathbb {Q}^{n+1}_1\) be an \(n (n\ge 4)\)-dimensional conformal regular spacelike hypersurface in the conformal space \(\mathbb {Q}^{n+1}_1\) with vanishing conformal form. If x is a conformal Blaschke isoparametric spacelike hypersurface in \(\mathbb {Q}^{n+1}_1\) with three distinct conformal principal curvatures, one of which is simple or x is of parallel conformal Blaschke tensor, we obtain some classification of such conformal regular spacelike hypersurface x.

Ricci solitons on CR submanifolds of maximal CR dimension in complex projective space

We study parallel and symmetric second order tensor fields on CR submanifolds of maximal CR dimension of the complex projective space. Under some natural conditions, these tensors are scalar multiples of the metric; an example of submanifold satisfying these assumptions is the geodesic hypersphere. The result is used for Ricci solitons on these CR submanifolds.

NON-ABELIAN TENSOR ANALOGUES OF 2-AUTO ENGEL GROUPS

The concept of tensor analogues of right 2-Engel elements in groups were defined and studied by Biddle and Kappe [1] and Moravec [9]. Using the automorphisms of a given group G, we introduce the notion of tensor analogue of 2-auto Engel elements in G and investigate their properties. Also the concept of <TEX>$2_{\otimes}$</TEX>-auto Engel groups is introduced and we prove that if G is a <TEX>$2_{\otimes}$</TEX>-auto Engel group, then <TEX>$G{\otimes}$</TEX> Aut(G) is abelian. Finally, we construct a non-abelian 2-auto-Engel group G so that its non-abelian tensor product by Aut(G) is abelian.

On some curvature conditions of pseudosymmetry type

It is known that the difference tensor \(R \cdot C - C \cdot R\) and the Tachibana tensor \(Q(S,C)\) of any semi-Riemannian Einstein manifold \((M,g)\) of dimension \(n \ge 4\) are linearly dependent at every point of \(M\). More precisely \(R \cdot C - C \cdot R = (1/(n-1))\, Q(S,C)\) holds on \(M\). In the paper we show that there are quasi-Einstein, as well as non-quasi-Einstein semi-Riemannian manifolds for which the above mentioned tensors are linearly dependent. For instance, we prove that every non-locally symmetric and non-conformally flat manifold with parallel Weyl tensor (essentially conformally symmetric manifold) satisfies \(R \cdot C = C \cdot R = Q(S,C) = 0\). Manifolds with parallel Weyl tensor having Ricci tensor of rank two form a subclass of the class of Roter type manifolds. Therefore we also investigate Roter type manifolds for which the tensors \(R \cdot C - C \cdot R\) and \(Q(S,C)\) are linearly dependent. We determine necessary and sufficient conditions for a Roter type manifold to be a manifold having that property.

An efficient tensor transpose algorithm for multicore CPU, Intel Xeon Phi, and NVidia Tesla GPU

An efficient parallel tensor transpose algorithm is suggested for shared-memory computing units, namely, multicore CPU, Intel Xeon Phi, and NVidia GPU. The algorithm operates on dense tensors (multidimensional arrays) and is based on the optimization of cache utilization on x86 CPU and the use of shared memory on NVidia GPU. From the applied side, the ultimate goal is to minimize the overhead encountered in the transformation of tensor contractions into matrix multiplications in computer implementations of advanced methods of quantum many-body theory (e.g., in electronic structure theory and nuclear physics). A particular accent is made on higher-dimensional tensors that typically appear in the so-called multireference correlated methods of electronic structure theory. Depending on tensor dimensionality, the presented optimized algorithms can achieve an order of magnitude speedup on x86 CPUs and 2–3 times speedup on NVidia Tesla K20X GPU with respect to the naïve scattering algorithm (no memory access optimization). The tensor transpose routines developed in this work have been incorporated into a general-purpose tensor algebra library (TAL-SH).

GPUTENSOR: Efficient tensor factorization for context-aware recommendations

Recommendation systems play an important role in many practical applications that help users manage information and provide personalized recommendations. The context in which a choice is made is an important factor for recommendation systems. Recently, researchers extended the classical matrix factorization to enable generic integration of contextual information by modeling the relevant data as a tensor. However, current tensor factorization methods have two important limitations: (1) computing costs can be very high in practice and (2) they can only be applied to static conditions. In this paper, we propose the GPUTENSOR, a parallel tensor factorization algorithm, to accelerate the tensor factorization of large datasets to support efficient context-aware recommendations. The basic idea of this algorithm is to partition a tensor into smaller blocks and then exploit the inherent parallelism and high memory bandwidth of Graphics Processing Units (GPU) to perform tensor related operations in parallel. Furthermore, based on the observation that contextual information changes dynamically and frequently, we present GPUTENSOR+, an effective incremental method that models the apparent changes of a tensor by adaptively updating its previously factorized components instead of recomputing them on the whole dataset every time data are changed. Experiments indicate that the proposed methods can achieve significant timesavings without a significant loss in accuracy. As an important data mining tool, these methods can be used a potential basis for many other interesting tensor applications, such as clustering, trend detection, social network analysis, and latent concept discovery.

Certain results on a type of contact metric manifold

Let \(M\) be a \(3\)-dimensional almost contact metric manifold satisfying \((*)\) condition. We denote such a manifold by \(M^{*}\). At first we study symmetric and skew-symmetric parallel tensor of type \((0,2)\) in \(M^{*}\). Next we prove that a non-cosymplectic manifold \(M^{*}\) is Ricci semisymmetric if and only if it is Einstein. Also we study locally \(\phi \)-symmetry and \(\eta \)-parallel Ricci tensor of \(M^{*}\). Finally, we prove that if a non-cosymplectic \(M^{*}\) is Einstein, then the manifold is Sasakian.

Clusters in dense-inertial granular flows: two new views of the conundrum

There is a conundrum in the rheological behavior of Dense-inertial granular flows. Dense inertial flows are dominated by clusters of particles which provide a path for internal force transmission through the elasticity of the interparticle contacts. Yet arguably the presence or absence of the clusters have no quantitative effect on the stresses. One possible explanation is that, even though the clusters exist they are mechanically inactive so the flow behaves as if it were a collisional flow. This paper addresses that possibility in two ways. First, computer simulations that assess both the actual contact stress tensor and a parallel collisional stress tensor show that the collisional stresses can be a small fraction of the total stresses until the flow approaches a fully collisional state, supporting the idea that clusters are indeed important. The second method was to compute force correlations between particles that inhabit the same cluster, as a way of assessing the efficiency of the cluster in force transmission. But once again, the results indicate that forces can be well correlated within a cluster and thus do provide an elastic path for force transmission. Both indicate that clusters are important in internal transport within a dense-inertial granular flow and thus shed no light on why clusters do not quantitatively affect the stresses.

Affine hypersurfaces with parallel difference tensor relative to affineα-connection

Li and Zhang (2014) studied affine hypersurfaces of R n + 1 with parallel difference tensor relative to the affine α -connection ∇ ( α ) , and characterized the generalized Cayley hypersurfaces by K n − 1 ≠ 0 and ∇ ( α ) K = 0 for some nonzero constant α , where the affine α -connection ∇ ( α ) of information geometry was introduced on affine hypersurface. In this paper, by a slightly different method we continue to study affine hypersurfaces with ∇ ( α ) K = 0 , if α = 0 we further assume that the Pick invariant vanishes and affine metric is of constant sectional curvature. It is proved that they are either hyperquadrics or improper affine hypersphere with flat indefinite affine metric, the latter can be locally given as a graph of a polynomial of at most degree n + 1 with constant Hessian determinant. In particular, if the affine metric is definite, Lorentzian, or its negative index is 2, we complete the classification of such hypersurfaces.

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.