On random embeddings and their application to optimisation

TL;DR

This paper investigates the theoretical aspects of norm-preserving random embeddings and explores their application in reducing the complexity of high-dimensional optimisation problems.

Contribution

It provides new insights into the properties of random embeddings and demonstrates their utility in improving optimisation efficiency.

Findings

Random embeddings approximately preserve pairwise distances.

They enable dimensionality reduction in optimisation tasks.

Application results show improved computational efficiency.

Abstract

Random embeddings project high-dimensional spaces to low-dimensional ones; they are careful constructions which allow the approximate preservation of key properties, such as the pair-wise distances between points. Often in the field of optimisation, one needs to explore high-dimensional spaces representing the problem data or its parameters and thus the computational cost of solving an optimisation problem is connected to the size of the data/variables. This thesis studies the theoretical properties of norm-preserving random embeddings, and their application to several classes of optimisation problems.