The non-abelian tensor product of groups which has its origins in algebraic K-theory as well as inhomotopy theory, was introduced by Brown and Loday in 1987. Group theoretical aspects of non-abelian tensor products have been studied extensively. In particular, some studies focused on the relationship between the exponent of a group and exponent of its tensor square. On the other hand, com...

we consider the semigroup $s$ of highest weights appearing in tensor powers $v^{otimes k}$ of a finite dimensional representation $v$ of a connected reductive group. we describe the cone generated by $s$ as the cone over the weight polytope of $v$ intersected with the positive weyl chamber. from this we get a description for the asymptotic of the number of highest weights appearing in $v^{otime...

Tensor decomposition is an important tool for big data analysis. In this paper,we resolve many of the key algorithmic questions regarding robustness, memoryefficiency, and differential privacy of tensor decomposition. We propose simplevariants of the tensor power method which enjoy these strong properties. We presentthe first guarantees for online tensor power method which has a...

The recent BICEP2 detection of degree scale CMB B-mode polarization, coupled with a deficit of observed power in large angle temperature anisotropy, suggest that the slow-roll parameter H , the fractional variation in the Hubble rate per efold, is both relatively large and may evolve from an even larger value on scales greater than the horizon at recombination. The relatively large tensor contr...

We present a novel analysis of the dynamics of tensor power iterations in the overcomplete regime where the tensor CP rank is larger than the input dimension. Finding the CP decomposition of an overcomplete tensor is NP-hard in general. We consider the case where the tensor components are randomly drawn, and show that the simple power iteration recovers the components with bounded error under m...

Tensor CANDECOMP/PARAFAC (CP) decomposition has wide applications in statistical learning of latent variable models and in data mining. In this paper, we propose fast and randomized tensor CP decomposition algorithms based on sketching. We build on the idea of count sketches, but introduce many novel ideas which are unique to tensors. We develop novel methods for randomized computation of tenso...

We introduce Banach algebras arising from tensor norms. By these Banach algebras we make Arensregular Banach algebras such that tensor product becomes irregular, where is tensor norm. Weillustrate injective tensor product, does not preserve bounded approximate identity and it is notalgebra norm.

We study the tensor rank of the tensor corresponding to the algebra of n-variate complex polynomials modulo the dth power of each variable. As a result we find a sequence of tensors with a large gap between rank and border rank, and thus a counterexample to a conjecture of Rhodes. At the same time we obtain a new lower bound on the tensor rank of tensor powers of the generalised W-state tensor....