Conditional Multidimensional Scaling

• Conditional MDS is an extension of classical MDS that integrates auxiliary known features to recover low-dimensional structures from pairwise dissimilarities.
• The method employs a majorization–minimization strategy with iterative closed-form updates for configuration, transformation, and missing feature imputation, ensuring rigorous convergence.
• It is applicable to diverse fields such as consumer studies and sociolinguistics, delivering robust estimation even in high-noise, high-missingness scenarios while enhancing statistical efficiency.

Conditional Multidimensional Scaling (MDS) is an extension of classical multidimensional scaling designed to recover the low-dimensional structure of data from pairwise dissimilarities while systematically accounting for auxiliary information encoded in "known features." By integrating these known features into the embedding process, conditional MDS aims to improve estimation accuracy, facilitate knowledge discovery, and support the simultaneous imputation of missing features, all with rigorous convergence and computational guarantees. Recent advancements enable conditional MDS to operate robustly even in the presence of missing conditioning data, making it applicable to scenarios with incomplete measurements and limited resource environments (Bui, 20 Sep 2025).

1. Theoretical Framework and Objectives

Conditional MDS models the observed objects as points represented by (i) unknown feature coordinates, which are to be inferred, and (ii) known feature coordinates, which may be measured, partially observed, or controlled. The method seeks to find a low-dimensional configuration $U \in \mathbb{R}^{N \times p}$ for $N$ objects (target dimension $p$), a transformation matrix $B \in \mathbb{R}^{q \times q}$ for $q$ known features, and—where conditioning data are incomplete—imputations $V_2$ for the missing entries in the feature matrix $V = [V_1^\top ; V_2^\top]^\top$.

The optimization is performed by minimizing the conditional stress function:

$$o(U, B, V_2) = \sum_{i<j} W_{ij} \left( \delta_{ij} - d_{ij}(U, [B^\top V_1 ; V_2]) \right)^2$$

where:

• $\delta_{ij}$: observed pairwise dissimilarity,
• $W_{ij}$: nonnegative weights (can be arbitrary or all equal),
• $d_{ij}(U, V)$: Euclidean or similar distance in configuration and feature space, typically $d_{ij}(U,V) = \|u_i-u_j\| + \|v_i-v_j\|$.

Minimization is carried out jointly over $U$, $B$, and $V_2$, enabling simultaneous embedding and imputation.

2. Algorithmic Design and Update Procedures

A majorization–minimization strategy underpins the conditional MDS algorithm. Key steps include:

• Iterative Updates: Each parameter ($U$, $B$, $V_2$) is updated in closed form by minimizing (locally convex) quadratic majorizing functions, whose derivatives are explicitly derived (see Theorem 1 (Bui, 20 Sep 2025)).
• Configuration Update:

$U^{(l)} = H^+ C^{(l-1)}$

where $H^+$ is the Moore–Penrose inverse of the weight matrix $H$, and $C$ is an update matrix dependent on current estimates.

• Transformation Matrix Update: Similar closed-form expressions are provided for updating $B$, ensuring consistent alignment between the known and embedded spaces.
• Imputation of Missing Features: When $B$ is invertible, missing feature block $V_2$ is updated according to

$V_2 = V_2 B^{-1}$

or, more generally in the presence of missing data indicators $M$,

$$V_2 = V_2 \circ (1 - M) + (Y_2 - (V_2 \circ (1-M)) B) \circ M B^{-1}$$

where $\circ$ denotes the Hadamard (elementwise) product.

These updates guarantee a monotonic decrease in the conditional stress objective, and convergence to a stationary configuration is proven under mild conditions.

3. Handling Incomplete Conditioning Data

Traditional conditional MDS requires complete auxiliary feature information for all objects. The proposed extension admits arbitrary missingness patterns, using observed known features and partial dissimilarities to "borrow strength" both for the embedding and imputation of $V_2$. This capability enables practitioners to avoid data exclusion, thereby enhancing both statistical efficiency and practical feasibility. Imputed feature values are generated as a byproduct of the embedding, delivering substantive insight into ambiguous or partially measured objects.

4. Computational Implementation and Practical Features

Two algorithmic variants are implemented in the cml R package (available on CRAN):

• General Weights Algorithm: Utilizes arbitrary weighting schemes, as for nonlinear mappings (e.g., Sammon mapping). Key computational matrices are precomputed for efficiency.
• Equal Weights Algorithm: Assumes all weights $W_{ij}$ are equal, leading to algorithmic simplification: only a single $q \times q$ inversion is required, and memory and computational overhead are significantly reduced.

Initialization options include naive random starts and integration with complete-data conditional SMACOF runs. Convergence is monitored via the normalized conditional stress, enabling rigorous stopping criteria.

5. Applications and Empirical Results

Conditional MDS with incomplete conditioning data is suited for a wide range of applications:

• Consumer Perception Studies: In car-brand simulation experiments, practitioners generate dissimilarity matrices using weighted Euclidean distances over seven features. When up to 80% of the known features are missing, the proposed method achieves superior performance in metrics such as average canonical correlation (ACC), Procrustes statistic (PS), and mean squared errors for both $B$ (MSEB) and $V_2$ (MSEV), compared to methods that discard incomplete data.
• Sociolinguistic Analysis: Imputation of missing attributes—such as gender—of kinship terms produces results consistent with domain understanding, with distinct separation of kinship degree and generational differences observed in the embedded configuration.
• Knowledge Discovery: Simultaneous imputation enhances the interpretability of latent structure in the data.

Experimental evidence demonstrates robustness to both high missingness ratios and noise, with estimation quality improved by incorporating all available data.

6. Advantages, Limitations, and Implications

Advantages:

• Utilizes all available observations, increasing statistical power and reducing bias from exclusion.
• Simultaneously imputes auxiliary missing features, providing additional data-driven insights.
• Enables practitioners to reduce the cost and effort of data collection without sacrificing estimation quality.
• Demonstrates resilience in high-noise and high-missingness scenarios.

Limitations:

• The conditional stress minimization problem is globally nonconvex, although each subproblem is tractable.
• Feature interpretation (i.e., labeling embedded "unknown" dimensions) remains a substantive challenge.
• In large datasets ($N \gg 1000$), quadratic scaling of computational time may pose constraints, although efficient special-case algorithms reduce practical overhead in common scenarios.

A plausible implication is that similar majorization approaches could be extended to more general cases involving structured missingness, dynamic features, or nonlinear metric models for the conditioning data.

7. Future Directions

Future research directions include the development of stochastic optimization strategies to address computational scalability ($\mathcal{O}(N)$ time), further generalization to arbitrary weight structures and metric spaces, and exploration of domains such as network analysis or dynamic time series where missing conditioning data is the norm. The interplay between imputation accuracy and embedding quality opens avenues for joint methodological innovation in unsupervised learning and data augmentation.

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In conclusion, the conditional multidimensional scaling methodology with incomplete conditioning data substantially advances the utility of MDS in practical scientific applications, addressing limitations of conventional approaches and enabling richer modeling in environments where complete auxiliary information is unavailable (Bui, 20 Sep 2025).