相关论文: The slice rank of a direct sum

Cayley's first hyperdeterminant is a straightforward generalization of determinants for tensors. We prove that nonzero hyperdeterminants imply lower bounds on some types of tensor ranks. This result applies to the slice rank introduced by…

The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been…

Following the breakthrough of Croot, Lev, and Pach, Tao introduced a symmetrized version of their argument, which is now known as the slice rank method. In this paper, we introduce a more general version of the slice rank of a tensor, which…

We address the problem of the additivity of the tensor rank. That is for two independent tensors we study if the rank of their direct sum is equal to the sum of their individual ranks. A positive answer to this problem was previously known…

The rank of tensors is not additive with respect to the direct sum.

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a sum of symmetric outer product of vectors. A rank-1 order-k…

The article is concerned with the problem of the additivity of the tensor rank. That is for two independent tensors we study when the rank of their direct sum is equal to the sum of their individual ranks. The statement saying that…

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 study the semialgebraic structure of $D_r$, the set of nonnegative tensors of nonnegative rank not more than $r$, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for…

We prove a lower bound on the rank of tensors constructed from families of linear maps that `expand' the dimension of every subspace. Such families, called {\em dimension expanders} have been studied for many years with several known…

In the past few years, the slice-rank lemma of Tao has been applied successfully to many problems in extremal combinatorics. In this paper, first, we define a new notion of triangular tensors which generalizes that of triangular matrices…

We show that the sum of ranks of two matrix polynomials is the same as the sum of the rank of the matrix obtained by applying the greatest common divisor of the polynomials, with the rank of the matrix obtained by applying the lowest common…

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…

In this paper, we define the minimum (maximum) rank, term rank and the sign nonsingular of tensors. The sufficiency and necessity for the minimum rank of a real tensor to be $1$ is given. And we show that the maximum rank of a tensor is not…

A tensor is a multi-way array that can represent, in addition to a data set, the expression of a joint law or a multivariate function. As such it contains the description of the interactions between the variables corresponding to each of…

An important conjecture in additive combinatorics, number theory, and algebraic geometry posits that the partition rank and analytic rank of tensors are equal up to a constant, over any finite field. We prove the conjecture up to a…

The analytic rank of a tensor, first defined by Gowers and Wolf in the context of higher-order Fourier analysis, is defined to be the logarithm of the bias of the tensor. We prove that it is a subadditive measure of rank: that is, the…

Tensor type data are becoming important recently in various application fields. We determine a rank of a tensor T so that A+T is diagonalizable for a given 3-tensor A with 2 slices over the complex and real number field.

We study the subrank of real order-three tensors and give an upper bound to the subrank of a real tensor given its complex subrank. Using similar arguments to those used by Bernardi-Blekherman-Ottaviani, we show that all subranks between…

It has recently been shown that the tensor rank can be strictly submultiplicative under the tensor product, where the tensor product of two tensors is a tensor whose order is the sum of the orders of the two factors. The necessary upper…