Both QDA and GMM are based on the premise that the underlying data is generated from two or more Gaussian distributions, but QDA is usually used in the supervised learning context, while GMM is used in the unsupervised context. Basically, QDA assumes the observations from each class are drawn from a Gaussian distribution, where each class has a separate covariance structure, while GMM assumes the underlying data is generated from an unknown number of Gaussian distributions.
What is the difference between QDA and Gaussian Mixture Models (GMM)?
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