Non-asymptotic Analysis of Compressive Fisher Discriminants in terms of the Effective Dimension

We provide a non-asymptotic analysis of the generalisation error of compressive Fisher linear discriminant (FLD) classification that is dimension free under mild assumptions. Our analysis includes the effects that random projection has on classification performance under covariance model misspecification, as well as various good and bad effects of random projections that contribute to the overall performance of compressive FLD. We also give an asymptotic bound as a corollary of our finite sample result. An important ingredient of our analysis is to develop new dimension-free bounds on the largest and smallest eigenvalue of the compressive covariance, which may be of independent interest.