AI 中文总结
本文利用张量切片导出矩阵的交换性性质，对张量的误差次数和退化阶给出指数级改进的多项式界，在特定假设下误差次数至多为1，得到边界秩r对应张量秩至多2r的去边界化结果。
AI 中文摘要
若张量T可表示为T=lim(ε→0)T(ε)，其中所有足够小ε对应的T(ε)秩均不超过r，则称T的边界秩至多为r。已知映射ε→T(ε)可假定为取值于张量的多项式，该映射的最小可能次数称为T的误差次数。误差次数及相关的退化阶是本文研究的两个核心量。研究动机之一来自去边界化：通过对映射ε→T(ε)进行多项式插值，可对T的张量秩给出上界。对于3阶张量，Lehmkuhl与Lickteig于近40年前（1989年）给出了误差次数和退化阶的指数上界，此后未获改进。本文给出适用于广泛类张量的界，较Lehmkuhl与Lickteig（1989）的结果实现指数级改进。本文结果对具有3个切片（格式为m×n×3）的张量最为通用，此时核心假设为矩阵切片的秩；还给出适用于任意矩形格式（m×n×p）张量的界，此时需对张量的一个切片附加1-正则性假设（矩阵的特征空间为1维时称为1-正则）。在上述假设下，本文证明误差次数至多为1，得到非平凡的去边界化结果：边界秩为r时，张量秩至多为2r。Lehmkuhl与Lickteig（1989）的结果依赖于边界秩至多为r的张量簇次数的上界，本文则转而利用该代数簇更具体的性质，特别是由张量切片导出的某些矩阵的交换性性质。
英文摘要
A tensor has border rank at most $r$ if it can be written as $T=\lim_{\varepsilon \rightarrow 0} T(\varepsilon)$ where $T(\varepsilon)$ has rank at most $r$ for all sufficiently small $\varepsilon$. It is known that the map $\varepsilon \mapsto T(\varepsilon)$ can be assumed to be a (tensor valued) polynomial in $\varepsilon$. The smallest possible degree of such a map is called the error degree of $T$. The error degree and the related notion of order of degeneration are the two key quantities that we study in this paper. One motivation comes from debordering: by polynomial interpolation on the map $\varepsilon \mapsto T(\varepsilon)$ we can upper bound the tensor rank of $T$. For order 3 tensors, exponential upper bounds on the error degree and degeneration order were given almost 40 years ago in (Lehmkuhl Lickteig, 1989) and were not improved ever since. In this paper we give bounds that apply to a wide class of tensors, exponentially improving on (Lehmkuhl Lickteig, 1989). Our results are most general for tensors with 3 slices (format $m \times n \times 3$). In this case, our main assumption is on the rank of the matrix slices. We also give bounds that apply to arbitrary rectangular formats ($m \times n \times p$). In this case, we need an additional 1-regularity assumption on one of the slices of the tensor (recall that a matrix is said to be 1-regular if its eigenspaces are 1-dimensional). Under these assumptions we show that the error degree is at most 1, which yields a nontrivial debordering result (tensor rank at most $2r$ for border rank $r$). The results in (Lehmkuhl Lickteig, 1989) rely on an upper bound on the degree of the variety of tensors of border rank at most $r$. We rely instead on more specific properties of this algebraic variety, and in particular on commutativity properties of certain matrices derived from the tensor slices.
Comments35 pages
