Low Dimensional Euclidean Volume Preserving Embeddings

TL;DR

This paper investigates volume-preserving linear embeddings of finite point sets in Euclidean space, establishing bounds on how much the volume of subsets can be distorted during low-dimensional projections.

Contribution

It introduces a linear mapping that approximately preserves volumes of small subsets of points with explicit bounds on distortion, advancing understanding of volume preservation in embeddings.

Findings

Existence of a linear map preserving subset volumes within a specific factor

Bound on volume distortion depends on the number of points and target dimension

Provides quantitative guarantees for volume preservation in low-dimensional embeddings

Abstract

Let $\mathcal{P}$ be an $n$-point subset of Euclidean space and $d\geq 3$ be an integer. In this paper we study the following question: What is the smallest (normalized) relative change of the volume of subsets of $\mathcal{P}$ when it is projected into $\RR^d$. We prove that there exists a linear mapping $f:\mathcal{P} \mapsto \RR^d$ that relatively preserves the volume of all subsets of size up to $\lfloor d/2\rfloor$ within at most a factor of $O(n^{2/d}\sqrt{\log n \log\log n})$.