Robust conditional dimension reduction for dissimilarity data
Authors: Xiao Ling, Anh Bui
Organizations: Department of Mathematics, Auburn University at Montgomery Montgomery, AL 36117, USA · Department of Mathematics and Statistics, Virginia Commonwealth University Richmond, VA 23284, USA
Conditional dimension reduction (cDR) learns low-dimensional latent coordinates while accounting for observed covariates that represent known sources of variation in the data. Conditional Multidimensional Scaling (cMDS) is a cDR technique that works directly with dissimilarity data. Its standard squared-stress formulation, however, is sensitive to contaminated dissimilarity, since outliers can dominate the objective and distort the learned configuration. We proposed Robust Conditional Multidimensional Scaling (rcMDS) by replacing the squared-stress criterion with a Fair M-estimation objective. We developed a reweighted conditional SMACOF algorithm to optimize this objective. The proposed algorithm admits computationally tractable updates, and its stabilized objective values decrease monotonically and converge to a finite limit. Experiments on synthetic and real data show that the pro
Figures & tables
Figure 1: Loss ( ρ ), influence ( ψ ), and weight ( w ) functions for least-squares, least-absolute, and Fair M-estimators.
rk[v]=δk−dk(U[v],B[v]).
Algorithm 1 rcSMACOF
Figure 2: Canonical correlations (cc) and Procrustes statistic (ss) between the estimated and true configurations under three different scenarios.
Figure 3: Scatterplots of the recovered two-dimensional configurations for the 14 kinship terms. The first row shows the results from cSMACOF, and the second row shows the results from rcSMACOF. The first column presents the configurations obtained after conditioning the gender feature, while the second column presents the configurations obtained after conditioning both the gender and kinship-degree features. The dotted arrows indicate the dominant direction of the remaining kinship structure in the recovered configurations.
Figure 4: Design configurations in gray and recovered two-dimensional configurations in black for the rectangle data under contaminated dissimilarities. The first row shows the width–height representation: (a) cSMACOF configuration and (b) rcSMACOF configuration. The second row shows the size–shape representation: (c) cSMACOF configuration and (d) rcSMACOF configuration.
Width and Height
Size and Shape
Metric
cSMACOF
rcSMACOF
cSMACOF
rcSMACOF
mean
0.50
0.01
0.47
0.02
std
0.11
0.00
0.20
0.00
Table 1: Procrustes statistic (SS) between the unknown feature space and the set of known variables.
Figure 5: Procrustes statistic (ss) between the estimated and true facial-expression configurations across contamination levels. Panels are ordered row-wise as follows: PU known, AR known, TS known, PU–AR known, PU–TS known, and AR–TS known. Error bars indicate standard deviations over ten replications.
Figure 6: Marker shapes distinguish the three known-feature scenarios. Solid curves report the average Procrustes statistic (ss, left axis), while dashed curves report the average number of iterations (right axis).
Figure 7: Procrustes statistic (ss) versus the number of iterations under 5% and 10% contaminated dissimilarities. Lower values indicate better recovery.