In order to alleviate the problem of local convergence of the usual EM algorithm, a split-and-merge operation is introduced into the EM algorithm for Gaussian mixtures. The split-and-merge equations are 7rst presented theoretically. These equations show that the merge operation is a well-posed problem, whereas the split operation is an ill-posed problem because it is the inverse procedure of th...

The regularization of linear ill-posed problems is based on their conditional well-posedness when restricting the problem to certain classes of solutions. Given such class one may consider several related real-valued functions, which measure the well-posedness of the problem on such class. Among those functions the modulus of continuity is best studied. For solution classes which enjoy the addi...

The problem of topological classification is fundamental in the study dynamical systems. However, when we consider systems without well-posedness, it unclear how to generalize notion equivalence. For example, a system has trajectories distinguished only by parametrization, cannot apply usual definition equivalence based on phase space, which presupposes uniqueness trajectories. In this study, f...