  Linear discriminant analysis (LDA) based classifiers tend to falter in many
practical settings where the training data size is smaller than, or comparable
to, the number of features. As a remedy, different regularized LDA (RLDA)
methods have been proposed. These methods may still perform poorly depending on
the size and quality of the available training data. In particular, the test
data deviation from the training data model, for example, due to noise
contamination, can cause severe performance degradation. Moreover, these
methods commit further to the Gaussian assumption (upon which LDA is
established) to tune their regularization parameters, which may compromise
accuracy when dealing with real data. To address these issues, we propose a
doubly regularized LDA classifier that we denote as R2LDA. In the proposed
R2LDA approach, the RLDA score function is converted into an inner product of
two vectors. By substituting the expressions of the regularized estimators of
these vectors, we obtain the R2LDA score function that involves two
regularization parameters. To set the values of these parameters, we adopt
three existing regularization techniques; the constrained perturbation
regularization approach (COPRA), the bounded perturbation regularization (BPR)
algorithm, and the generalized cross-validation (GCV) method. These methods are
used to tune the regularization parameters based on linear estimation models,
with the sample covariance matrix's square root being the linear operator.
Results obtained from both synthetic and real data demonstrate the consistency
and effectiveness of the proposed R2LDA approach, especially in scenarios
involving test data contaminated with noise that is not observed during the
training phase.
