论文标题
实际张量的等级和对称等级的下限
Lower bounds on the rank and symmetric rank of real tensors
论文作者
论文摘要
我们使用Sylvester的等级不平等，通过线性组合的三个级别组合来降低张量的排名。以类似的方式，我们通过三个展开的对称等级的线性组合降低了对称等级。张量的等级和对称等级的下限对于查找Comon的猜想的反例很重要。 Comon猜想的真正反例是张量，其真正的等级和真实对称等级不同。以前，只有一个真正的反例。我们将构造分为三个步骤。第一步涉及二进制张量的线性空间。第二步考虑了较大的可分解张量的线性空间。第三步是根据感兴趣的张量来验证对称等级的猜想。我们使用构造来构建订单六个真正的张量，其实际等级和实际对称等级不同。
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 three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.
