On the Consistency of Multidimensional Scaling under Gaussian Noise: A Maximum Likelihood Framework

This work establishes a formulation of the classical multidimensional scaling in which the pairwise distances are corrupted by independent additive Gaussian noise. This formulation yields a stress objective function, derived from an explicit probabilistic model, allowing for a rigorous statistical analysis. Up to rotations and translations, we also prove the consistency of the estimator when the noise vanishes and prove finite-sample probability bounds on the estimation error. Our results complement the recently developed theories by focusing on a fixed size of data with increasingly accurate distance measurements.