Subspace Least Squares Multidimensional Scaling

TL;DR

This paper introduces a spectral domain approach to the least squares variant of Multidimensional Scaling (LS-MDS), revealing a multiresolution property that accelerates optimization and improves embedding quality.

Contribution

It presents a novel spectral domain formulation of LS-MDS, enabling faster computations and potentially better embeddings compared to traditional methods.

Findings

Spectral domain formulation speeds up LS-MDS optimization.

Multiresolution property improves embedding quality.

Comparable or better embeddings achieved with faster convergence.

Abstract

Multidimensional Scaling (MDS) is one of the most popular methods for dimensionality reduction and visualization of high dimensional data. Apart from these tasks, it also found applications in the field of geometry processing for the analysis and reconstruction of non-rigid shapes. In this regard, MDS can be thought of as a \textit{shape from metric} algorithm, consisting of finding a configuration of points in the Euclidean space that realize, as isometrically as possible, some given distance structure. In the present work we cast the least squares variant of MDS (LS-MDS) in the spectral domain. This uncovers a multiresolution property of distance scaling which speeds up the optimization by a significant amount, while producing comparable, and sometimes even better, embeddings.