Efficient Inter-Geodesic Distance Computation and Fast Classical Scaling

TL;DR

This paper introduces an efficient method for classical scaling in multidimensional scaling, reducing computational complexity and enabling fast geodesic distance computation by extrapolating from a subset of data points.

Contribution

The paper presents a novel solver for classical scaling that reduces complexity from quadratic to quasi-linear and incorporates local and global data information for improved distance approximation.

Findings

Reduced computational complexity from quadratic to quasi-linear

Enables fast approximation of inter-geodesic distances

Provides a low-rank approximation of geodesic distances

Abstract

Multidimensional scaling (MDS) is a dimensionality reduction tool used for information analysis, data visualization and manifold learning. Most MDS procedures embed data points in low-dimensional Euclidean (flat) domains, such that distances between the points are as close as possible to given inter-point dissimilarities. We present an efficient solver for classical scaling, a specific MDS model, by extrapolating the information provided by distances measured from a subset of the points to the remainder. The computational and space complexities of the new MDS methods are thereby reduced from quadratic to quasi-linear in the number of data points. Incorporating both local and global information about the data allows us to construct a low-rank approximation of the inter-geodesic distances between the data points. As a by-product, the proposed method allows for efficient computation of geodesic distances.