Linear discriminant analysis assumes Gaussian class-conditional densities that share a single covariance matrix across all classes, so the class densities may be translated but not reshaped. Setting the log-odds of two class posteriors to zero yields the decision boundary; the shared covariance causes the quadratic terms in the exponent to cancel, leaving an expression linear in the input, so the separating surface is a hyperplane and the model reduces to a set of linear discriminant functions. This places LDA in supervised statistical classification as the equal-covariance special case of the general Gaussian generative classifier, whose relaxation gives quadratic discriminant analysis, and as the class-aware analogue of principal component analysis when viewed as a supervised dimensionality-reduction criterion.
Linear discriminant analysis assumes Gaussian class-conditional densities that share a single covariance matrix across all classes, so the class densities may be translated but not reshaped. Setting …