Minimax Optimality of Classical Scaling Under General Noise Conditions

TL;DR

This paper proves that classical scaling is statistically optimal under broad noise conditions, requiring only finite fourth moments, and provides convergence rates and minimax lower bounds for its performance.

Contribution

It establishes the minimax optimality of classical scaling under general noise models with minimal assumptions, including finite fourth moments.

Findings

Classical scaling is consistent under broad noise conditions.

Convergence rates for classical scaling are derived.

Matching minimax lower bounds demonstrate optimality.

Abstract

We establish the consistency of classical scaling under a broad class of noise models, encompassing many commonly studied cases in literature. Our approach requires only finite fourth moments of the noise, significantly weakening standard assumptions. We derive convergence rates for classical scaling and establish matching minimax lower bounds, demonstrating that classical scaling achieves minimax optimality in recovering the true configuration even when the input dissimilarities are corrupted by noise.