Multiplicative Algorithms for Symmetric Nonnegative Tensor Factorizations and Its Applications

Abstract

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.