A Unified Algorithmic Framework for Multi-Dimensional Scaling

TL;DR

This paper introduces a flexible, convergent algorithmic framework for various multi-dimensional scaling problems, capable of handling different cost functions and target spaces, including spherical embeddings, with improved accuracy and efficiency.

Contribution

A modular, unified iterative algorithmic framework for multi-dimensional scaling that guarantees convergence and extends to spherical embeddings, outperforming existing methods.

Findings

Converges to higher quality solutions than existing methods.

Flexible framework adaptable to various MDS variants.

Extends Johnson-Lindenstrauss Lemma to spherical settings.

Abstract

In this paper, we propose a unified algorithmic framework for solving many known variants of \mds. Our algorithm is a simple iterative scheme with guaranteed convergence, and is \emph{modular}; by changing the internals of a single subroutine in the algorithm, we can switch cost functions and target spaces easily. In addition to the formal guarantees of convergence, our algorithms are accurate; in most cases, they converge to better quality solutions than existing methods, in comparable time. We expect that this framework will be useful for a number of \mds variants that have not yet been studied. Our framework extends to embedding high-dimensional points lying on a sphere to points on a lower dimensional sphere, preserving geodesic distances. As a compliment to this result, we also extend the Johnson-Lindenstrauss Lemma to this spherical setting, where projecting to a random $O((1/\eps^2) \log n)$-dimensional sphere causes $\eps$-distortion.