Confidence Sets for Multidimensional Scaling

2025-10-25

AI Summary

Classical multidimensional scaling (CMDS) lacks a statistically rigorous framework for constructing reliable unified confidence sets—up to rigid transformations—for the underlying configuration when confronted with noisy, heterogeneous dissimilarity data. Method: This paper establishes, for the first time, the distributional convergence theory of CMDS embedding estimators and develops a multiplier bootstrap-based statistical inference framework. The method adaptively accommodates heteroscedastic noise and enables construction of unified confidence sets under the rigid transformation group with asymptotic validity. Contribution/Results: We prove the asymptotic validity of the proposed bootstrap procedure. Numerical experiments demonstrate substantial improvements in finite-sample coverage probability and estimation precision over existing approaches, especially under the multiplier bootstrap setting. This work provides the first uncertainty quantification tool for CMDS endowed with rigorous statistical guarantees.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Multiagent Systems: Multiagent Systems under Uncertainty
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• Develop statistical framework for noisy multidimensional scaling
• Establish convergence results for CMDS embeddings
• Construct confidence sets for latent configurations

Innovation

• Statistical framework for noisy multidimensional scaling
• Bootstrap procedures for valid confidence sets
• Adaptive multiplier bootstrap for heteroscedastic noise