In this paper we introduce a statistical model based on a permanental process
for supervised classification problems. Unlike many research work in the
literature, we assume only exchangeability instead of independence on
observations. Regardless of the number of classes or the dimension of the
feature variables, the model may require only 2-3 parameters for fitting the
covariance structure within clusters. It works well even if each class occupies
non-convex, disjoint regions, or regions overlapped with other classes in the
feature space. To calculate the weighted permanental ratio involved, we propose
analytic approximations based on its cyclic expansion, which require only
polynomial time up to order three. It works well for classification purpose. An
application to DNA microarray analysis indicates that the permanental model
with cyclic approximations is more capable of handling high-dimensional data.
It can employ more feature variables in an efficient way and reduce the
prediction error significantly. This is critical when the true classification
relies on non-reducible high-dimensional features.