On Projections to Linear Subspaces

TL;DR

This paper explores the effects of projecting data onto linear subspaces, providing new bounds for distances and inner products, and relating these to intrinsic dimensionality estimation.

Contribution

It introduces a new family of bounds for Euclidean distances and inner products in subspace projections, advancing understanding of variance preservation and intrinsic dimensionality.

Findings

New bounds for Euclidean distances and inner products

Insights into variance preservation in subspace projections

Relation between projections and intrinsic dimensionality estimation

Abstract

The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched and the effect of random linear projections on measures such as intrinsic dimensionality still is an ongoing effort. In this paper, we investigate the less explored depths of linear projections onto explicit subspaces of varying dimensionality and the expectations of variance that ensue. The result is a new family of bounds for Euclidean distances and inner products. We showcase the quality of these bounds as well as investigate the intimate relation to intrinsic dimensionality estimation.