Dimension reduction for data of unknown cluster structure

TL;DR

This paper introduces a dimension reduction method for data from Gaussian mixtures that preserves clustering structure without prior knowledge of the clusters, using a data transformation followed by PCA.

Contribution

The paper proposes a novel transformation approach that aligns data variability with class separation directions, enabling PCA to approximate Fisher's subspace without prior cluster information.

Findings

Transformation aligns variability with class separation

PCA approximates Fisher's subspace after transformation

Preserves clustering structure in reduced dimensions

Abstract

For numerous reasons there raises a need for dimension reduction that preserves certain characteristics of data. In this work we focus on data coming from a mixture of Gaussian distributions and we propose a method that preserves distinctness of clustering structure, although the structure is assumed to be yet unknown. The rationale behind the method is the following: (i) had one known the clusters (classes) within the data, one could facilitate further analysis and reduce space dimension by projecting the data to the Fisher's linear subspace, which -- by definition -- preserves the structure of the given classes best (ii) under some reasonable assumptions, this can be done, albeit approximately, without the prior knowledge of the clusters (classes). In the paper, we show how this approach works. We present a method of preliminary data transformation that brings the directions of largest overall variability close to the directions of the best between-class separation. Hence, for the transformed data, simple PCA provides an approximation to the Fisher's subspace. We show that the transformation preserves distinctness of unknown structure in the data to a great extent.