I know that every class has the same covariance matrix $\Sigma$ in linear discriminant analysis (LDA), and in quadratic discriminant analysis (QDA) they are different. When using gaussian mixture model (GMM) in supervised classification, we fit a Gaussian with its own mean and variance to each class in data. So what is the difference between QDA and GMM?
Is my understanding of GMM wrong? Maybe I should fit more than one Gaussian to each class in order to model subgroups in it. But I am not sure if this is true or not.