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Robust Multi-Dimensional Scaling via Accelerated Alternating Projections

Tong Deng, Tianming Wang

TL;DR

This work tackles robust multi-dimensional scaling (RMDS) where pairwise distances are corrupted by outliers. It introduces RMDS-AAP, an accelerated alternating projections algorithm that iteratively refines a sparse outlier matrix S^k and a low-rank Gram matrix L^k, using a tangent-space projection to speed up updates and a decaying threshold to isolate outliers. Under standard RPCA-like assumptions—incoherence of L^* and sparsity of S^*—the authors prove linear convergence of the reconstructed centered coordinates X_c up to a rotation, with explicit bounds depending on problem parameters. Empirically, RMDS-AAP achieves state-of-the-art performance in both noiseless and noisy-outlier scenarios on synthetic data, demonstrating robustness and computational efficiency for RMDS in practical applications.

Abstract

We consider the robust multi-dimensional scaling (RMDS) problem in this paper. The goal is to localize point locations from pairwise distances that may be corrupted by outliers. Inspired by classic MDS theories, and nonconvex works for the robust principal component analysis (RPCA) problem, we propose an alternating projection based algorithm that is further accelerated by the tangent space projection technique. For the proposed algorithm, if the outliers are sparse enough, we can establish linear convergence of the reconstructed points to the original points after centering and rotation alignment. Numerical experiments verify the state-of-the-art performances of the proposed algorithm.

Robust Multi-Dimensional Scaling via Accelerated Alternating Projections

TL;DR

This work tackles robust multi-dimensional scaling (RMDS) where pairwise distances are corrupted by outliers. It introduces RMDS-AAP, an accelerated alternating projections algorithm that iteratively refines a sparse outlier matrix S^k and a low-rank Gram matrix L^k, using a tangent-space projection to speed up updates and a decaying threshold to isolate outliers. Under standard RPCA-like assumptions—incoherence of L^* and sparsity of S^*—the authors prove linear convergence of the reconstructed centered coordinates X_c up to a rotation, with explicit bounds depending on problem parameters. Empirically, RMDS-AAP achieves state-of-the-art performance in both noiseless and noisy-outlier scenarios on synthetic data, demonstrating robustness and computational efficiency for RMDS in practical applications.

Abstract

We consider the robust multi-dimensional scaling (RMDS) problem in this paper. The goal is to localize point locations from pairwise distances that may be corrupted by outliers. Inspired by classic MDS theories, and nonconvex works for the robust principal component analysis (RPCA) problem, we propose an alternating projection based algorithm that is further accelerated by the tangent space projection technique. For the proposed algorithm, if the outliers are sparse enough, we can establish linear convergence of the reconstructed points to the original points after centering and rotation alignment. Numerical experiments verify the state-of-the-art performances of the proposed algorithm.
Paper Structure (16 sections, 11 theorems, 78 equations, 2 figures, 1 algorithm)

This paper contains 16 sections, 11 theorems, 78 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation and Assumptions
  3. Assumption 1.
  4. Assumption 2.
  5. Algorithm
  6. Numerical Experiments
  7. Proofs
  8. Useful Lemmas
  9. Proof for the Initialization Phase
  10. Proof for the Iteration Phase
  11. Proof for Proposition \ref{['prop']}
  12. Implementation Details of the Computation of $L^{k+1}$
  13. Proofs for Some Useful Lemmas
  14. Proof for Lemma \ref{['lem:A']}
  15. Proof for Lemma \ref{['lem:S_infinity']}
  16. ...and 1 more sections

Key Result

Theorem 1

Suppose that RMDS-AAP is provided with $\xi^0$ that satisfies $\|D^{\star}\|_{\infty} \leq \xi^0 \leq 3\|D^{\star}\|_{\infty}$, and $\gamma\in[\frac{1}{3},1)$. Denote $\kappa:=\frac{\lambda_1^{\star}}{\lambda_r^{\star}}$. If then for $\forall k\geq 0$, $\text{supp}(S^k)\subseteq \text{supp}(S^{\star})$, and

Figures (2)

  • Figure 1: Performances of RMDS-AAP for the plus sign (101 points), where the distances are only corrupted by outliers.
  • Figure 2: Performance comparisons for the plus sign (25 points) with 4 anchor points, where the noisy distances are further corrupted by outliers. The error bars show the standard deviation values of the two methods.

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • Lemma 1: Ding2020
  • Lemma 2: Ding2020
  • Lemma 3: Ma2019
  • Lemma 4: netrapalli2014non
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8: accaltprj2019
  • ...and 1 more