Abstract:

Multidimensional Scaling (MDS) is a fundamental data analysis technique that represents similarity or dissimilarity among objects by mapping them into a lower-dimensional space while preserving relative distances between data points. The Individual Differences Scaling (INDSCAL) model extends MDS by jointly analyzing multiple symmetric data matrices, capturing structural relationships among different subjects while accounting for individual scale variations. Mathematically, INDSCAL can be formulated as a multivariate matrix optimization problem subject to column orthogonality and non-negative diagonal constraints. This paper presents an efficient numerical algorithm for solving the Orthogonal-INDSCAL (O-INDSCAL) model. The original problem is first reformulated as a matrix fixed-point iteration using the alternating least squares (ALS) method. To enhance convergence, we incorporate the $\varepsilon$-algorithm, a vector sequence acceleration technique, into a corresponding $\varepsilon$-accelerated fixed-point iteration algorithm. Numerical experiments demonstrate that the proposed $\varepsilon$-accelerated fixed-point iteration algorithm significantly improves convergence speed in solving the O-INDSCAL model. Moreover, compared to existing approaches such as projected gradient flow algorithms and various first- and second-order methods in the Manopt toolbox, our method exhibits superior iterative efficiency, highlighting its practical advantages in large-scale optimization.

Key words: multidimensional scaling, individual differences scaling, vector sequence acceleration, $\varepsilon$-algorithm