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.

翻译：本文针对含噪声的相异性数据，建立了经典多维标度分析（CMDS）的正式统计框架。我们针对多种噪声模型，证明了CMDS所生成嵌入向量的分布收敛性，从而能够为潜在构型（在刚性变换意义下）构建严格意义上的均匀置信集。进一步提出了构建此类置信集的bootstrap方法，并提供了其有效性的理论保证。研究发现乘数bootstrap能自动适应异方差噪声（如乘性噪声），而经验bootstrap似乎需要同方差性假设。当方法有效时，两种bootstrap形式均被证明能显著提升有限样本的准确性。通过数值实验验证了所提方法的实证性能。