Related papers: Bounds on the tensor rank
Decompositions of higher-order tensors into sums of simple terms are ubiquitous. We show that in order to verify that two tensors are generated by the same (possibly scaled) terms it is not necessary to compute the individual…
In this paper we suggest a new algorithm for the computation of a best rank one approximation of tensors, called alternating singular value decomposition. This method is based on the computation of maximal singular values and the…
We introduce the Bhattacharya-Mesner rank of third order hypermatrices as a relaxation to the tensor rank and devise from it some bounds for the tensor rank. We use the Bhattacharya-Mesner rank to extend to third order hypermatrices the…
We give an upper-bound for the $X$-rank of points with respect to a non-degenerate irreducible variety $X$ in the case that sub-generic $X$-rank points generate a hypersurface. We give examples where this bound is sharp and it improves the…
In this paper, we present a partial survey of the tools borrowed from tensor algebra, which have been utilized recently in Statistics and Signal Processing. It is shown why the decompositions well known in linear algebra can hardly be…
Motivated by a problem in computational complexity, we consider the behavior of rank functions for tensors and polynomial maps under random coordinate restrictions. We show that, for a broad class of rank functions called natural rank…
Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…
We give a necessary and sufficient condition for a mixed quantum mechanical state to be separable. The criterion is formulated as a boundedness condition in terms of the greatest cross norm on the tensor product of trace class operators.
In this lecture note, we discuss a fundamental concept, referred to as the {\it characteristic rank}, which suggests a general framework for characterizing the basic properties of various low-dimensional models used in signal processing.…
Here, we give upper and lower bounds on the count of positive integers $n\le x$ dividing the $n$th term of a nondegenerate linearly recurrent sequence with simple roots.
In CANDECOMP/PARAFAC tensor decomposition, degeneracy often occurs in some difficult scenarios, e.g., when the rank exceeds the tensor dimension, or when the loading components are highly collinear in several or all modes, or when CPD does…
We present a new explicit formula for the determinant that contains superexponentially fewer terms than the usual Leibniz formula. As an immediate corollary of our formula, we show that the tensor rank of the $n \times n$ determinant tensor…
We introduce a new criterion which tests if a given decomposition of a given ternary form $T$ of even degree is unique. The criterion is based on the analysis of the Hilbert function of the projective set of points $Z$ associated to the…
We consider sets of fixed CP, multilinear, and TT rank tensors, and derive conditions for when (the smooth parts of) these sets are smooth homogeneous manifolds. For CP and TT ranks, the conditions are essentially that the rank is…
The Hadamard rank of a point with respect to a projective variety is, if it exists, the minimum number of points of the variety whose coordinate-wise product is the given point. We classify the projective varieties for which the Hadamard…
Tensor Network (TN) decompositions have emerged as an indispensable tool in Big Data analytics owing to their ability to provide compact low-rank representations, thus alleviating the ``Curse of Dimensionality'' inherent in handling…
The purpose of this note is to give a linear algebra algorithm to find out if a rank of a given tensor over a field $\F$ is at most $k$ over the algebraic closure of $\F$, where $k$ is a given positive integer. We estimate the arithmetic…
The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for…
A well studied problem in algebraic complexity theory is the determination of the complexity of problems relying on evaluations of bilinear maps. One measure of the complexity of a bilinear map (or 3-tensor) is the optimal number of…
In low-rank tensor completion tasks, due to the underlying multiple large-scale singular value decomposition (SVD) operations and rank selection problem of the traditional methods, they suffer from high computational cost and high…