Globally Convergent Jacobi-Type Algorithms for Robust Approximate Diagonalization of Higher-Order Cumulant Tensors

Abstract

The sums of the absolute values of higher-order cumulants are classical contrast functions for independent component analysis (ICA) and lead naturally to robust approximate diagonalization of higher-order cumulant tensors using $\ell_1$-based diagonalization over the orthogonal or unitary group. We propose Jacobi-SubG, a Jacobi-type framework that selects index pairs using Riemannian subdifferential information and performs plane rotations with a refinement step that preserves nonzero diagonal contractions. For the general tensor model, we prove global convergence of the entire iterate sequence from any initialization with nonzero diagonal contractions, provided that the subproblem solvers satisfy a uniform sufficient-ascent estimate. For real and complex third- and fourth-order cumulant tensors, which are widely used in ICA, we develop specialized subproblem solvers satisfying these conditions. The resulting algorithms thus enjoy global convergence under the simple initialization condition that all diagonal contractions are nonzero. Numerical experiments on ICA and tensor diagonalization show that the proposed Jacobi-SubG variants achieve better source recovery and higher final objective values in several settings, particularly at the higher tested noise levels.