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Minimax Optimality of Classical Scaling Under General Noise Conditions

Siddharth Vishwanath, Ery Arias-Castro

TL;DR

This work analyzes classical scaling under broad noise models where observed dissimilarities are $D=\Delta+\mathcal{E}$ with $\mathcal{E}=\Psi(\Delta,\Xi)$ and $q>4$ finite moments. The authors establish consistency and explicit convergence rates for the spectral embedding $\widehat{X}=\mathsf{CS}(D,p)$ in operator, Frobenius, and $L_{2\to\infty}$ norms, for both fixed and random configurations $X$, and provide matching minimax lower bounds, proving minimax optimality. Key results show $\min_{g\in\mathcal{G}(p)}\|\widehat{X}-g(X)\|_2 \lesssim \sigma\kappa$ and $\mathsf{L}_{rmse}(\widehat{X},X) \lesssim \sigma\kappa/\sqrt{n}$, along with $\mathsf{L}_{2\to\infty}(\widehat{X},X) \lesssim \sigma\sqrt{(\log n)/n}$ (up to constants depending on $\kappa,\varpi$); minimax lower bounds match these rates, including a $\sqrt{\log n / n}$ term for the uniform bound. Experiments corroborate the theory, highlighting the necessity of finite fourth moments and illustrating robustness across noise models. Overall, the paper positions classical scaling as minimax-optimal under broad, realistic noise settings and clarifies its statistical limits for latent-position recovery.

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.

Minimax Optimality of Classical Scaling Under General Noise Conditions

TL;DR

This work analyzes classical scaling under broad noise models where observed dissimilarities are with and finite moments. The authors establish consistency and explicit convergence rates for the spectral embedding in operator, Frobenius, and norms, for both fixed and random configurations , and provide matching minimax lower bounds, proving minimax optimality. Key results show and , along with (up to constants depending on ); minimax lower bounds match these rates, including a term for the uniform bound. Experiments corroborate the theory, highlighting the necessity of finite fourth moments and illustrating robustness across noise models. Overall, the paper positions classical scaling as minimax-optimal under broad, realistic noise settings and clarifies its statistical limits for latent-position recovery.

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.

Paper Structure

This paper contains 24 sections, 25 theorems, 226 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions
  4. Setting
  5. Performance Analysis of Classical Scaling
  6. Lower Bounds on the Reconstruction Error
  7. Numerical Experiments
  8. Discussion
  9. Proofs
  10. Proof of \ref{['lemma:iid']}
  11. Proof of \ref{['lemma:noise-models']}
  12. Proof of \ref{['lem:D-opnorm']}
  13. Proof of \ref{['thm:X-opnorm']}
  14. Proof of \ref{['cor:frobenius-norm']}
  15. Proof of \ref{['thm:X-ttinf']}
  16. ...and 9 more sections

Key Result

Corollary

For fixed $p$, suppose the rows of $\text{X} \in \mathds{R}^{n \times p}$ are sampled iid from a probability distribution $F$ with $\textup{diam}(\textup{supp}(F)) \le \varpi$ and $\frac{1}{\kappa^2}I_p \preccurlyeq \mathrm{Cov}(F) \preccurlyeq \kappa^2 I_p$ where $\kappa, \varpi > 0$ and let $\Delt where $\boldsymbol{c}(\kappa, \varpi)$ is a constant depending only on $\kappa, \varpi$.

Figures (4)

  • Figure 1: $\mathsf{L}_\textup{rmse}(\widehat{\text{X}}, \text{X})$ reconstruction error vs. $n$ for the setup in \ref{['exp:1']} on a $\log$-$\log$ scale.
  • Figure 2: $\mathsf{L}_{2\to\infty}(\widehat{\text{X}}, \text{X})$ reconstruction error vs. $n$ for the setup in \ref{['exp:1']} on a $\log$-$\log$ scale.
  • Figure 3: $\mathsf{L}_\textup{rmse}(\widehat{\text{X}}, \text{X})$ reconstruction error vs. $n$ for the setup in \ref{['exp:2']} on a $\log$-$\log$ scale.
  • Figure 4: $\mathsf{L}_{2\to\infty}(\widehat{\text{X}}, \text{X})$ reconstruction error vs. $n$ for the setup in \ref{['exp:2']} on a $\log$-$\log$ scale.

Theorems & Definitions (38)

  • Corollary
  • Definition 1: Realizable setting
  • Definition 2: Noisy realizable setting
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Corollary
  • Theorem 3
  • Remark 1
  • ...and 28 more