Robust conditional dimension reduction for dissimilarity data

Summary

The paper proposes rcMDS, a robust variant of conditional multidimensional scaling (cMDS, Bui 2024) that learns low-dimensional coordinates from dissimilarity data while conditioning on observed covariates. It replaces the outlier-sensitive squared conditional stress with the Fair loss ρ(r)=|r|−η·ln(1+|r|/η), solved by rcSMACOF: an outer loop reweights residuals as q_k=w_k/(|r_k|+η) and each inner step performs a fixed-weight conditional SMACOF update (Algorithm 1, equations 7-8). Theoretically, the authors prove NP-hardness even for p=1 with zero covariates and unit weights (Thm 1), a quadratic majorization legitimizing the reweighting (Thm 2), and monotone objective descent (Thm 3) — though the printed proof of Thm 3 has notational defects and no stationary-point guarantee; Prop 1 shows q_k>0 keeps V⊤HV≻0 so the B-update inverse stays valid. Experiments on synthetic car-brand data (n=30, 0-20% inflation contamination), Borg-Leutner rectangles (Table 1: width-height Procrustes 0.01 vs 0.50 over 100 replications), and Abelson-Sermat faces (0.17→0.26 vs 0.14→0.59 at 10% contamination) show rcSMACOF staying stable where cSMACOF degrades sharply. Limitations: comparison only against non-robust cMDS, inflation-only contamination, toy-scale data (n≤30), an η-setting contradiction between §4 (0.01) and §5.1 (0.03·s_Δ), no runtime measurements, and no code release. Major revision recommended.