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.
翻译:多线性变代数的一个根本问题,即光谱和富比尼乌斯温度规范之间的最小比重,近年来受到相当重视。虽然大多数值对于真实的和复杂的发热器来说并不为人所知,但已经确定了无症状的强度和紧窄的下限。然而,对非阴性发热器所知甚少。在本文中,我们几乎完整地描绘了非阴性发热器的比例。特别是,我们提供了一个较窄的底线,可以在一个简单必要和足够的条件下,由一大批非阴性色的发热器达到,这有助于确定极端发热器的特点,并取得诸如程度的无症状等结果。我们表明,对一般发热器的比重不大于常值乘以仅取决于发热器的值,从而确定真实的、复杂的和非阴性对称性发热器的等量。我们还发现,对于真实的温度和复合体的温度比率一般不同于最起码的比重,对于非正反色体和复合体的反色体和核规范。