论文标题
非负张量的光谱和Frobenius规范之间的极高比率
Extreme ratio between spectral and Frobenius norms of nonnegative tensors
论文作者
论文摘要
近年来，多线性代数的基本问题之一是张量的频谱和frobenius规范之间的最小比率。虽然大多数值对于真实和复杂的张量未知，但已经建立了渐近的数量级和紧密的下限。但是，对非负张量知之甚少。在本文中，我们介绍了非负张量的比率几乎完整的图片。特别是，我们提供了一个紧密的下限，在简单且充分的条件下，一类广泛的非负张量可以实现，这有助于表征极端张量并获得诸如渐近级数量级之类的结果。我们表明，对称张量的比率仅比仅取决于张量的顺序乘以常数的比率，因此确定真实，复杂和非负对称量的渐近数量级。我们还发现，非负张量的Frobenius和核规范之间的最小比率通常不同，这与实际张量和复杂张量的情况形成了鲜明的对比。
One of the fundamental problems in multilinear algebra, the minimum ratio between the spectral and Frobenius norms of tensors, has received considerable attention in recent years. While most values are unknown for real and complex tensors, the asymptotic order of magnitude and tight lower bounds have been established. However, little is known about nonnegative tensors. In this paper, we present an almost complete picture of the ratio for nonnegative tensors. In particular, we provide a tight lower bound that can be achieved by a wide class of nonnegative tensors under a simple necessary and sufficient condition, which helps to characterize the extreme tensors and obtain results such as the asymptotic order of magnitude. We show that the ratio for symmetric tensors is no more than that for general tensors multiplied by a constant depending only on the order of tensors, hence determining the asymptotic order of magnitude for real, complex, and nonnegative symmetric tensors. We also find that the ratio is in general different to the minimum ratio between the Frobenius and nuclear norms for nonnegative tensors, a sharp contrast to the case for real tensors and complex tensors.
