High Dimensional Mode Hunting Using Pettiest Components Analysis

• Published in: IEEE transactions on pattern analysis and machine intelligence Vol. 45; no. 4; pp. 4637 - 4649
• Main Authors: Liu, Tianhao, Diaz-Pachon, Daniel Andres, Rao, J. Sunil, Dazard, Jean-Eudes
• Format: Journal Article
• Language: English
• Published: United States IEEE 01.04.2023; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Active information; Boxes; bump hunting; dimension reduction; Eigenvalues and eigenfunctions; Image reconstruction; Kernel; mode hunting; Partitioning algorithms; Principal component analysis; Principal components analysis; Time complexity; Tumors
• ISSN: 0162-8828; 1939-3539; 1939-3539; 2160-9292
• DOI: 10.1109/TPAMI.2022.3195462

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.