Convergence Analysis of Two Alternating Iterative Schemes for Tucker Decomposition

TL;DR

This paper provides a detailed convergence analysis of two popular iterative methods, HOOI and ASI, for Tucker tensor decomposition, applicable to complex tensors and demonstrating their global convergence under mild conditions.

Contribution

It offers the first comprehensive convergence proof for both methods on complex tensors, filling gaps in previous analyses and establishing their global convergence.

Findings

Both methods are globally convergent to stationary points.

The objective function increases monotonically during iterations.

Numerical examples confirm the convergence behavior.

Abstract

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.