This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These characterize large-deviation behavior the extreme eigenvalue random We extend Laplace transform method and Lieb's concavity theorem from matrices to tensors, apply these tools generalize classical associated with names Chernoff, Bennett, Bernstein scalar tensor setting. Tail norm a sum rectangula...

Summary In this article, we prove that for isotropic functions depend on $P$ vectors, $N$ symmetric tensors and $M$ non-symmetric (a) the minimal number of irreducible invariants a scalar-valued function is $3P+9M+6N-3,$ (b) vectors vector-valued $3$ (c) tensor-valued at most $9$. The numbers given in (a), are, general, much lower than obtained literature. This significant reduction has potenti...

It has been shown that a best rank-R approximation of an order-k tensor may not exist when R ≥ 2 and k ≥ 3. This poses a serious problem to data analysts using tensor decompositions. It has been observed numerically that, generally, this issue cannot be solved by consecutively computing and subtracting best rank-1 approximations. The reason for this is that subtracting a best rank-1 approximati...

In this paper we study multivariate polynomial functions in complex variables and the corresponding associated symmetric tensor representations. The focus is on finding conditions under which such complex polynomials/tensors always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, and present characteristic conditions for such complex polynomial...

We show that the irreducible variety of 4 × 4 × 4 complex valued tensors of border rank at most 4 is the zero set of polynomial equations of degree 5 (the Strassen commutative conditions), of degree 6 (the Landsberg-Manivel polynomials), and of degree 9 (the symmetrization conditions).

Uniqueness and regularity results for local vector-valued generalized minimizers and for local stress tensors associated to variational problems with linear growth conditions are established. Assuming that the energy density f has the structure f(Z) = h(|Z|), only very weak ellipticity assumptions are required. For the proof we combine arguments from measure theory and convex analysis with the ...

Tensor factorization methods provide a useful way to extract latent factors from complex multirelational data, and also for predicting missing data. Developing tensor factorization methods for massive tensors, especially when the data are binaryor count-valued (which is true of most real-world tensors), however, remains a challenge. We develop a scalable probabilistic tensor factorization frame...

This study of gauge field theories on κ-deformed Minkowski spacetime extends previous work on field theories on this example of a noncommutative spacetime. We construct deformed gauge theories for arbitrary compact Lie groups using the concept of enveloping algebra-valued gauge transformations and the SeibergWitten formalism. Derivative-valued gauge fields lead to field strength tensors as the ...