Large distortion dimension reduction using random variable

TL;DR

This paper investigates the probability that a random matrix preserves distances within a distortion factor for multiple affine subspaces, establishing tight bounds on the matrix dimension needed for high-probability guarantees.

Contribution

It provides tight bounds on the dimension of random matrices needed to preserve distances across multiple subspaces with high probability, extending understanding of dimension reduction.

Findings

Dimension m is sufficient for high-probability distance preservation.

Results apply to various classes of random matrices, including Gaussian.

Bounds on m are tight with respect to parameters k, p, and D.

Abstract

Consider a random matrix $H:\mathbb{R}^n\longrightarrow\mathbb{R}^m$. Let $D\geq2$ and let $\{W_l\}_{l=1}^{p}$ be a set of $k$-dimensional affine subspaces of $\mathbb{R}^n$. We ask what is the probability that for all $1\leq l\leq p$ and $x,y\in W_l$, \[ \|x-y\|_2\leq\|Hx-Hy\|_2\leq D\|x-y\|_2. \] We show that for $m=O\big(k+\frac{\ln{p}}{\ln{D}}\big)$ and a variety of different classes of random matrices $H$, which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on $m$ is tight in terms of $k,p,D$.