Central Limit Theorems for Classical Multidimensional Scaling

TL;DR

This paper establishes central limit theorems for classical multidimensional scaling under three noise models, providing a detailed perturbation analysis and validating results with simulations and real data.

Contribution

It introduces three noise models for classical MDS and proves central limit theorems for the resulting embeddings, advancing understanding of its behavior under randomness.

Findings

Embeddings follow a central limit theorem under each noise model

Simulation results confirm theoretical predictions

Real data illustrations support the theoretical analysis

Abstract

Classical multidimensional scaling is a widely used method in dimensionality reduction and manifold learning. The method takes in a dissimilarity matrix and outputs a low-dimensional configuration matrix based on a spectral decomposition. In this paper, we present three noise models and analyze the resulting configuration matrices, or embeddings. In particular, we show that under each of the three noise models the resulting embedding gives rise to a central limit theorem. We also provide compelling simulations and real data illustrations of these central limit theorems. This perturbation analysis represents a significant advancement over previous results regarding classical multidimensional scaling behavior under randomness.