Multidimensional scaling with noisy data

Yicheng Zeng, Archer Gong Zhang, Hengchao Chen, Qiang Sun

(The first two authors make equal contributions.)

Abstract: Multidimensional Scaling (MDS) is a technique that embeds data points into a lower-dimensional space while preserving pairwise distance patterns. This approach renders high-dimensional data relationships visually accessible, facilitating easier understanding and interpretation. Most existing literature focuses on deterministic and noise-free settings. This paper provides an overview of MDS methods for noisy data. First, we review the general formulation of MDS from an optimization perspective and categorize variants of Classical MDS (CMDS) based on this framework. Next, we explore the impact of noise levels and dimensionality on CMDS embeddings under signal-plus-noise models. We also address outliers as a form of noise and discuss three robust MDS methods. Finally, we close with several potential research directions.