High Dimensional Mode Hunting Using Pettiest Components Analysis

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.

Automated summary

Principal components analysis has long been used for dimensionality reduction. This paper demonstrates the greater importance of components with the smallest variance in mode detection. It proves that implementing petty component analysis leads to boxes of optimal volume in multivariate normal or Laplace distribution, with minimal volume compared to all possible boxes with the same dimensions and fixed probability. Experimental results show that petty components outperform their competitors in a simulation and in finding modal patterns of hand-written numbers using the MNIST database. In fact, the modes obtained with petty components produce better written digits for MNIST than those obtained with principal components.