Averaging orthogonal projectors

TL;DR

This paper introduces a generalized weighted distance for combining subspaces from different dimension reduction methods, enabling more flexible analysis of complex data structures.

Contribution

It proposes a novel weighted distance metric for aggregating subspaces of varying dimensions, extending the Crone and Crosby distance, with applications in combining multiple dimension reduction techniques.

Findings

Weighted distances effectively combine different dimension reduction methods.

Simulations demonstrate improved data structure capturing.

The approach offers a flexible framework for subspace averaging.

Abstract

Dimensionality is a major concern in analyzing large data sets. Some well known dimension reduction methods are for example principal component analysis (PCA), invariant coordinate selection (ICS), sliced inverse regression (SIR), sliced average variance estimate (SAVE), principal hessian directions (PHD) and inverse regression estimator (IRE). However, these methods are usually adequate of finding only certain types of structures or dependencies within the data. This calls the need to combine information coming from several different dimension reduction methods. We propose a generalization of the Crone and Crosby distance, a weighted distance that allows to combine subspaces of different dimensions. Some natural choices of weights are considered in detail. Based on the weighted distance metric we discuss the concept of averages of subspaces as well to combine various dimension reduction methods. The performance of the weighted distances and the combining approach is illustrated via simulations.