Related papers: Inner Rank and Lower Bounds for Matrix Multiplicat…

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…

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite fields and we establish new asymptotical and not asymptotical upper bounds about it.

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of…

We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed…

Matrices of rank at most k are defined by the vanishing of polynomials of degree k + 1 in their entries (namely, their (k + 1)-times-(k + 1)-subdeterminants), regardless of the size of the matrix. We prove a qualitative analogue of this…

We show that lower bounds on the border rank of matrix multiplication can be used to non-trivially derandomize polynomial identity testing for small algebraic circuits. Letting $\underline{R}(n)$ denote the border rank of $n \times n \times…

In this paper, we give a survey of the known results concerning the tensor rank of the multiplication in finite extensions of finite fields, enriched with some not published recent results as well as analyzes enhancing the qualitative…

This educational article highlights the geometric and algebraic complexities that distinguish tensors from matrices, to supplement coverage in advanced courses on linear algebra, matrix analysis, and tensor decompositions. Using the case of…

The Rank Minimization Problem asks to find a matrix of lowest rank inside a linear variety of the space of n x n matrices. The Low Rank Matrix Completion problem asks to complete a partially filled matrix such that the resulting matrix has…

In this paper, we show that the low rank matrix completion problem can be reduced to the problem of finding the rank of a certain tensor.

In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two…

There are many notions of rank in multilinear algebra: tensor rank, partition rank, slice rank, and strength (or Schmidt rank) are a few examples. Typically the rank $\le r$ locus is not Zariski closed, and understanding the closure (the…

Motivated by recent works on statistics of matrices over sets of number theoretic interest, we study matrices with entries from arbitrary finite subsets $\mathcal A$ of finite rank multiplicative groups infields of characteristic zero. We…

In this article the notion of the nondecreasing (ND) rank of a matrix or tensor is introduced. A tensor has an ND rank of r if it can be represented as a sum of r outer products of vectors, with each vector satisfying a monotonicity…

Whereas matrix rank is additive under direct sum, in 1981 Sch\"onhage showed that one of its generalizations to the tensor setting, tensor border rank, can be strictly subadditive for tensors of order three. Whether border rank is additive…

Moment polytopes of tensors, the study of which is deeply rooted in invariant theory, representation theory and symplectic geometry, have found relevance in numerous places, from quantum information (entanglement polytopes) and algebraic…

We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high…

The determination of the maximal ranks of a set of a given type of tensors is a basic problem both in theory and application. In statistical applications, the maximal rank is related to the number of necessary parameters to be built in a…

We provide a nontrivial bound on the rank of any tensor $T$ over the quaternions $\mathbb{H}$ in the $n_1\times n_2\times n_3$ cases where $2\leq n_i\leq 3$. We describe a decomposition of $T$ into $3$ simple tensors in the $2\times 2\times…

Motivated by questions arising in signal processing, computational complexity, and other areas, we study the ranks and border ranks of symmetric tensors using geometric methods. We provide improved lower bounds for the rank of a symmetric…