  In this paper, we propose a novel linear discriminant analysis criterion via
the Bhattacharyya error bound estimation based on a novel L1-norm (L1BLDA) and
L2-norm (L2BLDA). Both L1BLDA and L2BLDA maximize the between-class scatters
which are measured by the weighted pairwise distances of class means and
meanwhile minimize the within-class scatters under the L1-norm and L2-norm,
respectively. The proposed models can avoid the small sample size (SSS) problem
and have no rank limit that may encounter in LDA. It is worth mentioning that,
the employment of L1-norm gives a robust performance of L1BLDA, and L1BLDA is
solved through an effective non-greedy alternating direction method of
multipliers (ADMM), where all the projection vectors can be obtained once for
all. In addition, the weighting constants of L1BLDA and L2BLDA between the
between-class and within-class terms are determined by the involved data set,
which makes our L1BLDA and L2BLDA adaptive. The experimental results on both
benchmark data sets as well as the handwritten digit databases demonstrate the
effectiveness of the proposed methods.
