什么时候减去一个秩一的近似会降低张量的秩？

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-01-23 DOI:10.1016/j.laa.2025.01.025

Emil Horobeţ , Ettore Teixeira Turatti

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引用次数: 0

摘要

从矩阵中减去临界秩1近似总是得到一个秩更低的矩阵。这对一般张量来说是不成立的。受此启发，我们提出了一个问题：那些张量集合的闭包是什么？对于这些张量集合，减去它的一些临界秩一近似值并重复这个过程，我们最终会得到零？在这篇文章中，我们展示了如何构造这种张量的变化，我们展示了它是如何连接到一阶张量变化的瓶颈点（一般来说是超行列式的奇异轨迹），以及这种变化如何等于，在某些情况下是如何大于（弱）正交可分解张量的。

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When does subtracting a rank-one approximation decrease tensor rank?

Subtracting a critical rank-one approximation from a matrix always results in a matrix with a lower rank. This is not true for tensors in general. Motivated by this, we ask the question: what is the closure of the set of those tensors for which subtracting some of its critical rank-one approximation from it and repeating the process we will eventually get to zero? In this article, we show how to construct this variety of tensors and we show how this is connected to the bottleneck points of the variety of rank-one tensors (and in general to the singular locus of the hyperdeterminant), and how this variety can be equal to and in some cases be more than (weakly) orthogonally decomposable tensors.

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.