  Principal components analysis has been used to reduce the dimensionality of
datasets for a long time. In this paper, we will demonstrate that in mode
detection the components of smallest variance, the pettiest components, are
more important. We prove that for a multivariate normal or Laplace
distribution, we obtain boxes of optimal volume by implementing "pettiest
component analysis", in the sense that their volume is minimal over all
possible boxes with the same number of dimensions and fixed probability. This
reduction in volume produces an information gain that is measured using active
information. We illustrate our results with a simulation and a search for modal
patterns of digitized images of hand-written numbers using the famous MNIST
database; in both cases pettiest components work better than their competitors.
In fact, we show that modes obtained with pettiest components generate better
written digits for MNIST than principal components.
