Re-understanding of Gaussian Mixture Model Training Mechanism

There is no direct and rigorous mathematical proof for the convergence of the iterative algorithm of the expectation-maximum principle in the Gaussian mixture distribution model. Usually, after the introduction of hidden variables, the optimization objective function is transformed from the average sum of log-likelihood of each data into the joint entropy of a hidden variable and data variables, which traditional iterative algorithms maximize. Combined with the concept of membership degree of fuzzy mathematics, a fuzzy multi-Gaussian model is introduced as a copy of the Gaussian mixture model, which has the convergence property of strict mathematical proof. Combined with this copy, the working mechanism of the expected maximum algorithm of the Gaussian mixture model can be mathematically explained, and through random data experiments, it is confirmed that the data experiments of the fuzzy multi-Gaussian model are consistent with the theoretical analysis, and the effect of the fuzzy multi-Gaussian model and the Gaussian mixture model is consistent.