  Linear and Quadratic Discriminant analysis (LDA/QDA) are common tools for
classification problems. For these methods we assume observations are normally
distributed within group. We estimate a mean and covariance matrix for each
group and classify using Bayes theorem. With LDA, we estimate a single, pooled
covariance matrix, while for QDA we estimate a separate covariance matrix for
each group. Rarely do we believe in a homogeneous covariance structure between
groups, but often there is insufficient data to separately estimate covariance
matrices. We propose L1- PDA, a regularized model which adaptively pools
elements of the precision matrices. Adaptively pooling these matrices decreases
the variance of our estimates (as in LDA), without overly biasing them. In this
paper, we propose and discuss this method, give an efficient algorithm to fit
it for moderate sized problems, and show its efficacy on real and simulated
datasets.
