论文标题
高级张量
High-rank subtensors of high-rank tensors
论文作者
论文摘要
令$ d \ ge 2 $为正整数。 We show that for a class of notions $R$ of rank for order-$d$ tensors, which includes in particular the tensor rank, the slice rank and the partition rank, there exist functions $F_{d,R}$ and $G_{d,R}$ such that if an order-$d$ tensor has $R$-rank at least $G_{d,R}(l)$ then we can restrict its entries to a product of集合$ x_1 \ times \ dots \ times x_d $，以使限制具有$ r $ -LANK至少$ l $，sets $ x_1，\ dots，x_d $ every最多有$ f_ {d，r}（d，r}（l）$。此外，我们的证明方法使我们可以证明，在非常自然的条件下，我们可以需要$ x_1，\ dots，x_d $是成对的脱节。
Let $d \ge 2$ be a positive integer. We show that for a class of notions $R$ of rank for order-$d$ tensors, which includes in particular the tensor rank, the slice rank and the partition rank, there exist functions $F_{d,R}$ and $G_{d,R}$ such that if an order-$d$ tensor has $R$-rank at least $G_{d,R}(l)$ then we can restrict its entries to a product of sets $X_1 \times \dots \times X_d$ such that the restriction has $R$-rank at least $l$ and the sets $X_1, \dots, X_d$ each have size at most $F_{d,R}(l)$. Furthermore, our proof methods allow us to show that under a very natural condition we can require the sets $X_1, \dots, X_d$ to be pairwise disjoint.
