Extreme ratio between spectral and Frobenius norms of nonnegative tensors

Shengyu Cao, Simai He, Zhening Li, Zhen Wang

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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 cases for real tensors and complex tensors.
Original languageEnglish
Pages (from-to)919-944
JournalSIAM Journal on Matrix Analysis and Applications
Issue number2
Early online date12 Jun 2023
Publication statusPublished - 12 Jun 2023


  • extreme ratio
  • spectral norm
  • Frobenius norm
  • nonnegative tensors
  • symmetric tensors
  • nuclear norm
  • rank-one approximation
  • norm equivalence inequality

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