Affine Reduction of Dimensionality: An Origin-Centric Perspective

TL;DR

This paper introduces an origin-centric approach to multivariate dimensionality reduction that emphasizes the importance of the origin in affine transformations, enhancing interpretability and visualization of data.

Contribution

It presents a novel origin-centric perspective on affine-invariant dimensionality reduction methods, differing from traditional PCA by focusing on the fixed point at the origin.

Findings

Origin-centric visualizations improve interpretability.

Choice of origin affects scatter representation.

Method enhances graphical perception in data visualization.

Abstract

We consider statistical methods for reduction of multivariate dimensionality that have invariance and/or commutativity properties under the affine group of transformations (origin translations plus linear combinations of coordinates along initial axes). The methods discussed here differ from traditional principal component and coordinate approaches in that they are origin-centric. Because all Cartesian coordinates of the origin are zero, it is the unique fixed point for subsequent linear transformations of point scatters. Whenever visualizations allow shifting between and/or combining of Cartesian and polar coordinate representations, as in Biplots, the location of this origin is critical. Specifically, origin-centric visualizations enhance the psychology of graphical perception by yielding scatters that can be interpreted as Dyson swarms. The key factor is typically the analyst's choice of origin via an initial "centering" translation; this choice determines whether the recovered scatter will have either no points depicted as being near the origin or else one (or more) points exactly coincident with this origin.