We introduce a Bayesian extension of the tensor factorization problem to multiple coupled tensors. For a single tensor it reduces to standard PARAFAC-type Bayesian factorization, and for two tensors it is the first Bayesian Tensor Canonical Correlation Analysis method. It can also be seen to solve a tensorial extension of the recent Group Factor Analysis problem. The method decomposes the set o...

2. Projective Tensors. Theorems of the projective geometry of paths appear to have their statement in terms of what may be called projective tensors, i. e. tensors which are independent of the particular set of functions T\^ defining the paths. We shall show how a set of projective tensors may be derived by covariant differentiation from an w-uple of mutually independent vectors. Let h(a)i deno...

This paper deals with the class of Q-tensors, that is, a Q-tensor is a real tensor A such that the tensor complementarity problem (q,A): finding x ∈ R such that x ≥ 0,q+Axm−1 ≥ 0, and x⊤(q+Axm−1) = 0, has a solution for each vector q ∈ Rn. Several subclasses of Q-tensors are given: P-tensors, R-tensors, strictly semi-positive tensors and semi-positive R0-tensors. We prove that a nonnegative ten...

An important building block in all current asymptotically fast algorithms for matrix multiplication are tensors with low border rank, that is, tensors whose border rank is equal or very close to their size. To find new asymptotically fast algorithms for matrix multiplication, it seems to be important to understand those tensors whose border rank is as small as possible, so called tensors of min...

In spacetimes admitting Yano tensors the classical theory of the spinning particle possesses enhanced worldline supersymmetry. Quantum mechanically generators of extra supersymmetries correspond to operators that in the classical limit commute with the Dirac operator and generate conserved quantities. We show that the result is preserved in the full quantum theory, that is, Yano symmetries are ...

Several compositional distributional semantic methods use tensors to model multi-way interactions between vectors. Unfortunately, the size of the tensors can make their use impractical in large-scale implementations. In this paper, we investigate whether we can match the performance of full tensors with low-rank approximations that use a fraction of the original number of parameters. We investi...

Tensor extrapolation attempts to forecast relational time series data using multi-linear algebra. It proceeds as follows. Multi-way are arranged in the form of tensors, ie multi-dimensional arrays. decompositions then used retrieve periodic patterns data. Afterwards, these serve input for methods. However, previous approaches tensor limited preselected and binary To permit automatic forecasting...

In this paper we introduce three new classes of nonnegative forms (or equivalently, symmetric tensors) and their extensions. The newly identified nonnegative symmetric tensors constitute distinctive convex cones in the space of general symmetric tensors (order 6 or above). For the special case of quartic forms, they collapse into the set of convex quartic homogeneous polynomial functions. We di...

We begin by defining tensor products of vector spaces over a field and then we investigate some basic properties of these tensors, in particular the existence of bases and duality. After this, we investigate special kinds of tensors, namely, symmetric tensors and skew-symmetric tensors. Tensor products of modules over a commutative ring with identity will be discussed very briefly. They show up...