  Linear discriminant analysis (LDA) has been a useful tool in pattern
recognition and data analysis research and practice. While linearity of class
boundaries cannot always be expected, nonlinear projections through pre-trained
deep neural networks have served to map complex data onto feature spaces in
which linear discrimination has served well. The solution to binary LDA is
obtained by eigenvalue analysis of within-class and between-class scatter
matrices. It is well known that the multiclass LDA is solved by an extension to
the binary LDA, a generalised eigenvalue problem, from which the largest
subspace that can be extracted is of dimension one lower than the number of
classes in the given problem. In this paper, we show that, apart from the first
of the discriminant directions, the generalised eigenanalysis solution to
multiclass LDA does neither yield orthogonal discriminant directions nor
maximise discrimination of projected data along them. Surprisingly, to the best
of our knowledge, this has not been noted in decades of literature on LDA. To
overcome this drawback, we present a derivation with a strict theoretical
support for sequentially obtaining discriminant directions that are orthogonal
to previously computed ones and maximise in each step the Fisher criterion. We
show distributions of projections along these axes and demonstrate that
discrimination of data projected onto these discriminant directions has optimal
separation, which is much higher than those from the generalised eigenvectors
of the multiclass LDA. Using a wide range of benchmark tasks, we present a
comprehensive empirical demonstration that on a number of pattern recognition
and classification problems, the optimal discriminant subspaces obtained by the
proposed method, referred to as GO-LDA (Generalised Optimal LDA), can offer
superior accuracy.
