论文标题
最佳排名近似比的概率界限
Probabilistic bounds on best rank-one approximation ratio
论文作者
论文摘要
我们提供新的上限和下限，以（部分）对称张量的光谱和Frobenius规范的最小比率。在一般张量的特定情况下，我们的结果恢复了已知的上限。对于对称张量，我们的上限揭开范围，即规范的比率与微不足道的下限$ 1/\ sqrt {n^{d-1}} $相同，当时张紧$ d $的顺序是固定的，并且底层载体空间$ n $倾向于Infinity。但是，当$ n $固定并且$ d $倾向于无穷大时，我们的下限优于$ 1/\ sqrt {n^{d-1}} $。
We provide new upper and lower bounds on the minimum possible ratio of the spectral and Frobenius norms of a (partially) symmetric tensor. In the particular case of general tensors our result recovers a known upper bound. For symmetric tensors our upper bound unveils that the ratio of norms has the same order of magnitude as the trivial lower bound $1/\sqrt{n^{d-1}}$, when the order of a tensor $d$ is fixed and the dimension of the underlying vector space $n$ tends to infinity. However, when $n$ is fixed and $d$ tends to infinity, our lower bound is better than $1/\sqrt{n^{d-1}}$.
