Confidence Sets for Multidimensional Scaling
This provides a formal statistical framework for uncertainty quantification in CMDS, addressing a gap in multidimensional scaling applications.
The authors tackled the problem of constructing confidence sets for classical multidimensional scaling (CMDS) embeddings from noisy dissimilarity data, establishing distributional convergence results and proposing bootstrap procedures that improve finite-sample accuracy.
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.