Jacobi-type algorithms for simultaneous approximate diagonalization of symmetric real tensors (or partially symmetric complex tensors) have been widely used in independent component analysis (ICA) because of its high 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 our recent results in a series of papers about the weak convergence and global convergence of these Jacobi-type algorithms, under both of two pair selection rules. These results are mainly based on the Lojasiewicz gradient inequality.

independent component analysis; approximate tensor diagonalization; optimization on manifold; Jacobi-type algorithm; weak convergence; global convergence