论文标题
边界格式以外的张量的奇异元素的跨度
The span of singular tuples of a tensor beyond the boundary format
论文作者
论文摘要
张量$ t格式$（n_1，\ ldots，n_k）$的单数$ k $ - $ t $ t $ t $ t $ t $ t $的临界点是一个复杂的关键点，从$ t $限制为格式$（n_1，\ ldots，n_k，n_k）$排名的$ t t $。通用张量有限许多复杂的单数$ k $ tuplass，它们的数字仅取决于张量格式。此外，如果我们修复了第一个$ k-1 $ dimensions $ n_i $，那么在一个整数变量$ n_k $中，通用张量的奇异$ k $ - the the the $（n_1，n_1，\ ldots，n_k，n_k，n_k）$ sos a vacte a toces a docess a docess a docorment a docess a docorment a docess a dourage形式。 在本文中，我们研究了通用张量的单数$ k $的线性跨度。它的尺寸也仅取决于张量格式。特别是，我们专注于特殊订单的三个张量和订单 - 格式$（2，\ ldots，2，n）$的订单 - $ k $张量。结果，如果我们再次修复了第一个$ k-1 $ dimensions $ n_i $，并让$ n_k $增加，我们表明，在这些特殊格式中，线性跨度的尺寸也可以稳定，但在某些简洁的非核能格式中。我们猜想这种现象具有$ k> 3 $的任意格式。最后，我们为某些特殊的非式装订格式的通用顺序的三个张量$ t $ t $ t $ the的线性跨度提供了方程。从这些方程式中，我们得出结论，$ t $属于其单数三元组的线性跨度，并且我们猜想每种张量格式都是这种情况。
A singular $k$-tuple of a tensor $T$ of format $(n_1,\ldots,n_k)$ is essentially a complex critical point of the distance function from $T$ constrained to the cone of tensors of format $(n_1,\ldots,n_k)$ of rank at most one. A generic tensor has finitely many complex singular $k$-tuples, and their number depends only on the tensor format. Furthermore, if we fix the first $k-1$ dimensions $n_i$, then the number of singular $k$-tuples of a generic tensor becomes a monotone non-decreasing function in one integer variable $n_k$, that stabilizes when $(n_1,\ldots,n_k)$ reaches a boundary format. In this paper, we study the linear span of singular $k$-tuples of a generic tensor. Its dimension also depends only on the tensor format. In particular, we concentrate on special order three tensors and order-$k$ tensors of format $(2,\ldots,2,n)$. As a consequence, if again we fix the first $k-1$ dimensions $n_i$ and let $n_k$ increase, we show that in these special formats, the dimension of the linear span stabilizes as well, but at some concise non-sub-boundary format. We conjecture that this phenomenon holds for an arbitrary format with $k>3$. Finally, we provide equations for the linear span of singular triples of a generic order three tensor $T$ of some special non-sub-boundary format. From these equations, we conclude that $T$ belongs to the linear span of its singular triples, and we conjecture that this is the case for every tensor format.
