# research paper on tensor flow

**Braided tensor categories**

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Categories enriched with tensor products, here called tensor categories, but also called monoidal categories, have been studied and used extensively in the literature [ML EK, ML SR, DM J. Large examples such as the categories of Abelian groups and of Banach Large-scale problems have always been a challenge for numerical computations. An example is the treatment of fully populated n n matrices when n2 is close to or beyond the computers memory capacity. Here the technique of hierarchical matrices can reduce the

**MR diffusion tensor spectroscopy and imaging**

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This paper describes a new NMR imaging modality--MR diffusion tensor imaging. It consists of estimating an effective diffusion tensor , Deff, within a voxel, and then displaying useful quantities derived from it. We show how the phenomenon of anisotropic diffusion of water (or

**Tensor subspace analysis**

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Previous work has demonstrated that the image variations of many objects (human faces in particular) under variable lighting can be effectively modeled by low dimensional linear spaces. The typical linear subspace learning algorithms include Principal Component

**Eigenvalues of a real supersymmetric tensor**

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In this paper, we define the symmetric hyperdeterminant, eigenvalues and E-eigenvalues of a real supersymmetric tensor . We show that eigenvalues are roots of a one-dimensional polynomial, and when the order of the tensor is even, E-eigenvalues are roots of anotherIn the previous chapter we discussed isotropic and anisotropic tensor functions and their general representations. Of particular interest in continuum mechanics are isotropic tensor - valued functions of one arbitrary (not necessarily symmetric) tensor . For example, the

**Finite tensor categories**

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We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (ie for fusion categories), obtained recently in our joint work with D. Nikshych. In particular, we generalize to the categorical setting the

**Tensor decompositions for learning latent variable models**

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This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation which exploits a certain tensor structure in

**The geometry of tensor calculus, I**

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The goal of this first paper is to formalise the use of certain diagrams for a wide variety of situations in pure and applied mathematics. The main examples are the Feynman diagrams describing patterns of particle interactions in space-time. Other examples are given by circuit

**Tensor products of operator spaces**

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In this paper we lay the foundations for a systematic study of tensor products of subspaces and subalgebras of C*-algebras. It is now widely acknowledged that in order to fully understand subspaces of C*-algebras it is necessary to take into account the matrix normedThrough several centuries there has been a lively interaction between mathematics and mechanics. On the one side, mechanics has used mathematics to formulate the basic laws and to apply them to a host of problems that call for the quantitative prediction of the The mechanical behavior of materials with oriented internal structures, produced by formation processes and manufacturing procedures (crystal arrangements, stratification, fibrosity, porosity, etc.) or induced by permanent deformation (anisotropic hardening

**Estimation of the effective self-diffusion tensor from the NMR spin echo**

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The diagonal and off-diagonal elements of the effective selfdiffusion tensor , D , are related to the echo intensity in an NMR spin-echo experiment. This relationship is used to design experiments from which Delf is estimated. This estimate is validated using isotropic and

**Tensor products of operator algebras**

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An early result of Murray and von Neumann states that if R is a factor on a Hilbert space ZY, Ris its commutant, and R@ Ris the algebraic tensor product, then the homomorphism that carries r@ Yonto rris an isomorphism. Since R@ R be realized as an algebra of

**Escaping from saddle points online stochastic gradient for tensor decomposition**

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We analyze stochastic gradient descent for optimizing non-convex functions. In many cases for non-convex functions the goal is to find a reasonable local minimum, and the main concern is that gradient updates are trapped in saddle points. In this paper we identify strict

**Tensor product variable binding and the representation of symbolic structures in connectionist systems**

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ABSTRACT A general method, the tensor product representation, is defined for the connectionist representation of value~ variable bindings. The technique is a formalization of the idea that a set of value~ variable pairs can be represented by accumulating activity in a

**Tensor structures arising from affine Lie algebras. I**

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Let g be a simple Lie algebra of type A, D, or E. There is a natural central extension of Lie algebras 0-+ Cl-+ i-+ C (( )) g-+ 0 (see 1.3) which has been extensively studied by physicists and mathematicians.(1 is a distinguished element of the centre.) A i-module is

**Self-interacting tensor multiplets in N= 2 superspace**

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Superspace formulations of extended theories are notoriously difficult to construct. In particular, as the number of anticommuting coordinates increases, the number of unphysical components in a superfield grows and the mass dimension of the measure increases while

**Tensor triangular geometry**

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Tensor triangular geometry is the study of tensor triangulated categories by algebro- geometric methods. We invite the reader to discover this relatively new subject. A great charm of this theory is the profusion of examples to be found throughout pure mathematics, be it in algebraic

**Tensor algebras and harmonic analysis**

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The material presented in this paper is a systematic exposition of the theory of tensor algebras and their applications and connections with harmonic analysis. We shall not attempt here in the introduction to describe or summarize the methods and results of this

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