A probabilistic framework for joint approximate diagonalization

Joint approximate diagonalization (JAD) is one of the well-known methods for solving independent component analysis and blind source separation. It estimates a separating matrix which diagonalizes many cumulant matrices of given observed signals as accurately as possible. It is derived by not a probabilistic model but a linear algebraic approach. Therefore, its validity is rigorously guaranteed only if the diagonalization succeeds completely. However, the condition is not satisfied in practical cases, where JAD lacks the theoretical foundation. In this paper, we propose a probabilistic framework for JAD. The framework uses a probabilistic model of the estimation errors of cumulants instead of source signals. By applying the central limit theorem to the errors, a likelihood function of cumulants is derived. It is shown that a lower bound of the likelihood function is maximized by JAD. Numerical experiments verify the validity of the proposed framework.