# Bootstrapped Coordinate Search for Multidimensional Scaling

## TL;DR

This paper introduces a bootstrapped coordinate search method for multidimensional scaling that improves efficiency by focusing on promising directions, achieving faster convergence without sacrificing accuracy.

## Contribution

It proposes a novel bootstrapped coordinate search framework for gradient-free MDS that enhances speed and efficiency over existing methods.

## Key findings

- BS CSMDS reduces function evaluations significantly.
- It achieves faster convergence compared to other CSMDS methods.
- Performs consistently better on synthetic and real datasets.

## Abstract

In this work, a unified framework for gradient-free Multidimensional Scaling (MDS) based on Coordinate Search (CS) is proposed. This family of algorithms is an instance of General Pattern Search (GPS) methods which avoid the explicit computation of derivatives but instead evaluate the objective function while searching on coordinate steps of the embedding space. The backbone element of CSMDS framework is the corresponding probability matrix that correspond to how likely is each corresponding coordinate to be evaluated. We propose a Bootstrapped instance of CSMDS (BS CSMDS) which enhances the probability of the direction that decreases the most the objective function while also reducing the corresponding probability of all the other coordinates. BS CSMDS manages to avoid unnecessary function evaluations and result to significant speedup over other CSMDS alternatives while also obtaining the same error rate. Experiments on both synthetic and real data reveal that BS CSMDS performs consistently better than other CSMDS alternatives under various experimental setups.