### Introduction

In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. Random projection methods are powerful methods known for their simplicity and less erroneous output compared with other methods. According to experimental results, random projection preserve distances well, but empirical results are sparse.

Consider a problem as follows: We have a set of n points in a high-dimensional Euclidean space $\mathbf{R}^d$. We want to project the points onto a space of low dimension $\mathbf{R}^k$ in such a way that pairwise distances of the points are approximately the same as before.