Exact Cluster Recovery via Classical Multidimensional Scaling

Abstract: Classical multidimensional scaling is an important dimension reduction technique. Yet few theoretical results characterizing its statistical performance exist. This paper provides a theoretical framework for analyzing the quality of embedded samples produced by classical multidimensional scaling. This lays the foundation for various downstream statistical analyses, and we focus on clustering noisy data. Our results provide scaling conditions on the sample size, ambient dimensionality, between-class distance, and noise level under which classical multidimensional scaling followed by a distance-based clustering algorithm can recover the cluster labels of all samples with high probability. Numerical simulations confirm these scaling conditions are near-sharp. Applications to both human genomics data and natural language data lend strong support to the methodology and theory.