Algorithms and geometric analysis of data sets that are invariant under a group action

Abstract

We apply and develop pattern analysis techniques in the setting of data sets that are invariant under a group action. We apply Principal Component Analysis to data sets of images of a rotating object in Chapter 5 as a means of obtaining visual and low-dimensional representations of data. In Chapter 6, we propose an algorithm for finding distributions of points in a base space that are (locally) optimal in the sense that subspaces in the associated data bundle are distributed with locally maximal distance between neighbors. In Chapter 7, we define a distortion function that measures the quality of an approximation of a vector bundle by a set of points. We then use this function to compare the behavior of four standard distance metrics and one non-metric. Finally, in Chapter 8, we develop an algorithm to find the approximate intersection of two data sets.