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February 21, 2026Lobachevskii Journal of Mathematics0 citations

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

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CTChanon Thongprayoon

Key Points

  • The aim is to establish a robust formulation of multidimensional scaling influenced by Gaussian noise in distance measurements.
  • Developed a stress objective function from a probabilistic model.
  • Analyzed the effects of independent additive Gaussian noise on pairwise distances.
  • Proved the consistency of the estimator under noise reduction.
  • Derived finite-sample probability bounds on estimation error.
  • Established the estimator remains consistent as noise vanishes.
  • Provided theoretical bounds on estimation errors for finite samples.
  • Indicated improved accuracy of multidimensional scaling with better distance measurements.

Abstract

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.

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Cite This Study

Chanon Thongprayoon (2025) studied this question.

synapsesocial.com/papers/69994ad4873532290d01f3d2https://doi.org/10.1134/s1995080225611841
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