Confidence Sets for Multidimensional Scaling

TL;DR

This paper introduces a statistical framework for classical multidimensional scaling (CMDS) with noisy data, enabling the construction of valid confidence sets for the true configuration and improving finite-sample accuracy through bootstrap methods.

Contribution

It develops distributional convergence results for CMDS embeddings under noise, proposes bootstrap procedures for confidence sets, and provides theoretical guarantees for their validity.

Findings

Multiplier bootstrap adapts to heteroscedastic noise

Empirical bootstrap requires homoscedasticity

Bootstrap methods improve finite-sample accuracy

Abstract

We develop a formal statistical framework for classical multidimensional scaling (CMDS) applied to noisy dissimilarity data. We establish distributional convergence results for the embeddings produced by CMDS for various noise models, which enable the construction of \emph{bona~fide} uniform confidence sets for the latent configuration, up to rigid transformations. We further propose bootstrap procedures for constructing these confidence sets and provide theoretical guarantees for their validity. We find that the multiplier bootstrap adapts automatically to heteroscedastic noise such as multiplicative noise, while the empirical bootstrap seems to require homoscedasticity. Either form of bootstrap, when valid, is shown to substantially improve finite-sample accuracy. The empirical performance of the proposed methods is demonstrated through numerical experiments.