On the convergence of Jacobi-type algorithms for Independent Component Analysis

TL;DR

This paper reviews recent results on the convergence of Jacobi-type algorithms used in independent component analysis, focusing on their weak and global convergence properties based on the Lojasiewicz gradient inequality.

Contribution

It provides an accessible review of new theoretical convergence results for Jacobi-type algorithms in ICA, emphasizing the role of the Lojasiewicz gradient inequality.

Findings

Established weak convergence of Jacobi algorithms in ICA

Proved global convergence under certain conditions

Connected convergence analysis to the Lojasiewicz gradient inequality

Abstract

Jacobi-type algorithms for simultaneous approximate diagonalization of real (or complex) symmetric tensors have been widely used in independent component analysis (ICA) because of their good performance. One natural way of choosing the index pairs in Jacobi-type algorithms is the classical cyclic ordering, while the other way is based on the Riemannian gradient in each iteration. In this paper, we mainly review in an accessible manner our recent results in a series of papers about weak and global convergence of these Jacobi-type algorithms. These results are mainly based on the Lojasiewicz gradient inequality.