Convergence Analysis of Two Alternating Iterative Schemes for Tucker Decomposition
For researchers in tensor decomposition, this fills gaps in convergence theory for complex tensors, but the result is incremental as it extends existing analysis.
The paper provides convergence analysis for two Tucker decomposition methods (HOOI and ASI), proving global convergence to stationary points under mild conditions for complex tensors, with monotonic increase of the objective function.
The higher-order orthogonal iteration (HOOI) and the alternating subspace iteration (ASI) are two popular numerical methods for computing the Tucker decomposition of a multiple-mode tensor. Xu [Linear and Multilinear Algebra, 66(11):2247--2265, 2018] proposed a variation of HOOI, called the greedy HOOI, which has an extra alignment action between consecutive approximations. Kroonenberg and De Leeuw [Psychometrika, 45(1):69--97, 1980] analyzed the convergence of ASI but their analysis has gaps. These analysis were for a real tensor only. In this paper, we present detailed convergence analysis of the two methods that is applicable to a complex tensor with a real tensor being a special case, and it is shown both methods are globally convergent to stationary points under mild conditions while the objective function monotonically increases. Numerical examples are presented to demonstrate the convergence behavior of the methods.