Nonnegative tensor factorization (NTF) and nonnegative Tucker decomposition (NTD) have been widely applied in high-dimensional nonnegative tensor data analysis. This paper focuses on symmetric NTF and symmetric NTD, which are the special cases of NTF and NTD, respectively. By minimizing the Euclidean distance and the generalized KL divergence, the multiplicative updating rules are proposed and the convergence under mild conditions is proved. We also show that if the solution converges based on the multiplicative updating rules, then the limit satisfies the Karush–Kuhn–Tucker optimality conditions. We illustrate the efficiency of these multiplicative updating rules via several numerical examples.

非负张量因子分解（NTF）和非负Tucker分解（NTD）已被广泛应用于高维非负张量数据分析中。本文重点介绍对称NTF和对称NTD，它们分别是NTF和NTD的特例。通过最小化欧氏距离和广义KL散度，提出了乘法更新规则，并证明了在温和条件下的收敛性。我们还表明，如果解决方案基于乘法更新规则收敛，则该限制满足Karush-Kuhn-Tucker最优性条件。我们通过几个数值示例来说明这些乘法更新规则的效率。